%comment; # Table of Reflective Lorentzian Lattices of Rank 3
%comment; # Daniel Allcock
%comment; #
%comment; # The arcane comment syntax here makes this file readable by both TeX and Perl
%comment; # Arcane header of header/footer pair for wrapping a tex file to make it into a perl script.
%comment; # The purpose of the unused perl hash variable "%comment" is to make all this invisible to TeX.
%comment; # Daniel Allcock 2010

%comment; # Open the very file perl is parsing (ie this file), and extract lines with certain prefixes
%comment; open(FILE, $0) or die("Could not open $0");
%comment; if ($#ARGV==0 && $ARGV[0]=~/explicit/) { while (<FILE>)  { if (s/^%EXPLICIT%//) { print ; } } exit; }
%comment; if ($#ARGV==0 && $ARGV[0]=~/implicit/) { while (<FILE>)  { if (s/^%IMPLICIT%//) { print ; } } exit; }
%comment; if ($#ARGV==0 && $ARGV[0]=~/duality/) { while (<FILE>)  { if (s/^%DUALITY%//) { print ; } } exit; }
%comment; if ($#ARGV==0 && $ARGV[0]=~/all/) { while (<FILE>)  { if (s/^%(IMPLICIT|EXPLICIT|DUALITY)%//) { print ; } } exit; }

%comment; # If none of the above applied, print a usage message
%comment; print "usage:\nperl $0 all\nperl $0 explicit\nperl $0 implicit\nperl $0 duality\n";
%comment; print "Print the reflective rank 3 Lorentzian lattices in PARI-readable format.\n";
%comment; print "(All of them, or those printed explicitly in the paper, or those printed\n";
%comment; print "implicitly in terms of filling and mainification operations, or those obtained\n";
%comment; print "from the explicity/implicitly listed lattices by duality operations.)\n";
%comment; print "The format of each output line is [IP,EDs,roots,W,L].  IP is the\n";
%comment; print "inner product matrix.  EDs are the elementary divisors [a,b,c]\n";
%comment; print "of the lattice inside its dual, with c|b|a, in the sense that the dual\n";
%comment; print "lattice is spanned by (1/a,0,0), (0,1/b,0), (0,0,1/c). The columns of\n";
%comment; print "\"roots\" are the simple roots, W is the number of the Weyl group,\n";
%comment; print "and L is the number of the lattice among all those with that Weyl group.\n";
%comment; print "In this numbering, the explictly printed lattices come first,\n";
%comment; print "in the order printed in the paper, then the implicitly\n";
%comment; print "printed lattices, in some unspecified order, and then the\n";
%comment; print "ones obtained by duality, again in some unspecified order.\n";
%comment; print "Probably you want to redirect all this data to a file, say foo,  by\n";
%comment; print "appending \"> foo\" at the end of one of the command lines above.\n";
%comment; exit;

%comment; # Make Perl ignore the TeX code,  using Perl's slightly obscure "here document" feature.
%comment; # This is needed despite the "exit" command above.
%comment; # The single quotes are essential to stop Perl from parsing the TeX
%comment; # and trying to substitute in values of variables. 
%comment; $variable_to_hold_and_ignore_whole_tex_file = << 'IGNORE_EVERYTHING_HERE';

\hyphenation{Lor-entz-ian}
\def\pdual{\mathop{\hbox{\rm $p$-dual}}\nolimits}

\documentclass[12pt]{amsart}
\usepackage{amssymb}

% for uniform layout of super- and sub-scripts in corner symbols and genus symbols:
% (#1=dimen to lower superscript, #2=pts to raise subscript, #3=superscript, #4=subscript
\def\above#1#2#3#4{\genfrac{}{}{0pt}{}{\lower#1\hbox{\smash{$\scriptstyle #3$}}}{\raise#2\hbox{\smash{$\scriptstyle #4$}}}}

% 2's, 3's and \infty's with vertical lines through them, for weyl groups with dihedral diagram automorphism groups:
\newdimen{\twowidth}\settowidth{\twowidth}{2}
\newdimen{\halftwo}\setlength{\halftwo}{0.5\twowidth}
\def\slashtwo{2\llap{\hbox to 0pt{\hss$|$\hss}\kern\halftwo}}
\newdimen{\threewidth}\settowidth{\threewidth}{3}
\newdimen{\halfthree}\setlength{\halfthree}{0.5\threewidth}
\def\slashthree{3\llap{\hbox to 0pt{\hss$|$\hss}\kern\halfthree}}
\newdimen{\inftywidth}\settowidth{\inftywidth}{$\infty$}
\newdimen{\halfinfty}\setlength{\halfinfty}{0.5\inftywidth}
\def\slashinfty{\infty\llap{\hbox to 0pt{\hss$|$\hss}\kern\halfinfty}}

% these interact delicately to enable the same tex code to lay out several different cases in the table.
% don't tweak carelessly.  Typesetting goals (last most preferable; 2nd matrix may be absent):
%
% [xxxxxxxxxx].......
% [xxxxxxxxxx][xxxxxx
% ...xxxxxxxxxxxxxxx]
% or
% [xxxxxxxxxx].......
% [xxxxxxxxxx].......
% ......[xxxxxxxxxxx]
% or
% [xxxxxxx][xxxx][xxx
% ..xxxxxxxxxxxxxxxx]
% or
% [xxxxx][xxx].......
% ......[xxxxxxxxxxx]
% or
% [xxxxx][xxx]...[xx]
%
\def\BreakWithGlue#1#2#3{\nolinebreak\hskip #1\hbox{}\penalty #2\hbox{}\nolinebreak\hskip #3}
\def\EasyButWeakLineBreak{\BreakWithGlue{0pt plus 1 fill}{500}{0pt plus 1 fill}}
\def\HardButStrongLineBreak{\BreakWithGlue{0pt plus 1 filll}{600}{0pt plus -1 filll}}
%
% #1 is header material put before the "tabular", #2--#4 are column entries
\def\OneColumnOfMatrix#1#2#3#4{\begingroup #1\begin{tabular}{@{}c@{}}#2\\#3\\#4\end{tabular}\endgroup}

% other macros for the table
\newcommand\spacer{\ }
\def\instructions#1{$\langle #1\rangle$}
\def\pdualwitharg#1{\mathop{\hbox{$#1$-dual}}\nolimits}
\def\pfillwitharg#1{\mathop{\hbox{$#1$-fill}}\nolimits}
\def\main{\mathop{\rm main}\nolimits}
\def\iso{\cong}

\long\def\GramMatrix#1{#1} % This line second ==> prints Gram Matrices
\long\def\GramMatrix#1{}   % This line second ==> doesn't.
% computer-computed values for use in the paper
\def\NumLattices{8595}
\def\NumPrintedExplicitly{704}
\def\NumPrintedImplicitly{715}
\def\NumGotByDuality{7176}
\def\NumMainLattices{122}
\def\NumSSFbutNotMain{38}
\def\NumTriangles{46}
\def\NumWeylGroups{374}
\def\LargestArea{$12\pi$}
\def\SmallestArea{$\pi/12$}
\def\WeylsWithLargestArea{$W_{329}$, $W_{365}$ and $W_{371}$}
\def\WeylsWithSmallestArea{$W_{3}$}
\def\MostLatticesInWeyl{176}
\def\LeastLatticesInWeyl{1}
\def\WeylsWithMostLattices{$W_{68}$}
\def\WeylsWithLeastLattices{$W_{286}$}
\def\MostEdges{28}
\def\WeylsWithMostEdges{$W_{365}$}
\def\LargestCyclic{3}
\def\WeylsWithLargestCyclic{$W_{174}$}
\def\WeylsWithDihedralEight{$W_{182}$, $W_{184}$, $W_{238}$, $W_{285}$, $W_{286}$, $W_{307}$, $W_{324}$, $W_{349}$, $W_{359}$ and $W_{365}$}
\def\WeylsWithDihedralTwelve{$W_{234}$, $W_{320}$ and $W_{350}$}
\def\NumRegularWeyl{2}
\def\RegularWeyls{$W_{234}$ and $W_{286}$}
\def\NumIdealWeyl{3}
\def\IdealWeyls{$W_{247}$, $W_{285}$ and $W_{286}$}
\def\NumRightAngledWeyl{163}
\def\RightAngledWeyls{$W_{14}$, $W_{17}$, $W_{25}$, $W_{35}$, $W_{38}$, $W_{42}$, $W_{43}$, $W_{45}$, $W_{47}$, $W_{50}$, $W_{56}$, $W_{57}$, $W_{60}$, $W_{66}$, $W_{68}$, $W_{69}$, $W_{70}$, $W_{72}$, $W_{75}$, $W_{76}$, $W_{79}$, $W_{80}$, $W_{81}$, $W_{82}$, $W_{84}$, $W_{87}$, $W_{88}$, $W_{91}$, $W_{95}$, $W_{97}$, $W_{98}$, $W_{100}$, $W_{102}$, $W_{103}$, $W_{104}$, $W_{105}$, $W_{107}$, $W_{110}$, $W_{111}$, $W_{115}$, $W_{118}$, $W_{121}$, $W_{122}$, $W_{124}$, $W_{128}$, $W_{129}$, $W_{130}$, $W_{131}$, $W_{132}$, $W_{135}$, $W_{136}$, $W_{137}$, $W_{138}$, $W_{139}$, $W_{151}$, $W_{165}$, $W_{166}$, $W_{172}$, $W_{175}$, $W_{177}$, $W_{193}$, $W_{196}$, $W_{199}$, $W_{204}$, $W_{205}$, $W_{207}$, $W_{208}$, $W_{210}$, $W_{211}$, $W_{212}$, $W_{213}$, $W_{214}$, $W_{219}$, $W_{223}$, $W_{224}$, $W_{225}$, $W_{229}$, $W_{230}$, $W_{232}$, $W_{234}$, $W_{235}$, $W_{236}$, $W_{238}$, $W_{241}$, $W_{242}$, $W_{243}$, $W_{244}$, $W_{245}$, $W_{249}$, $W_{252}$, $W_{254}$, $W_{256}$, $W_{257}$, $W_{260}$, $W_{261}$, $W_{262}$, $W_{264}$, $W_{269}$, $W_{270}$, $W_{271}$, $W_{272}$, $W_{273}$, $W_{276}$, $W_{277}$, $W_{281}$, $W_{282}$, $W_{289}$, $W_{290}$, $W_{292}$, $W_{293}$, $W_{295}$, $W_{296}$, $W_{297}$, $W_{301}$, $W_{303}$, $W_{304}$, $W_{305}$, $W_{307}$, $W_{309}$, $W_{310}$, $W_{311}$, $W_{312}$, $W_{313}$, $W_{314}$, $W_{315}$, $W_{316}$, $W_{317}$, $W_{318}$, $W_{319}$, $W_{320}$, $W_{321}$, $W_{322}$, $W_{323}$, $W_{324}$, $W_{325}$, $W_{327}$, $W_{328}$, $W_{333}$, $W_{336}$, $W_{337}$, $W_{338}$, $W_{339}$, $W_{340}$, $W_{345}$, $W_{346}$, $W_{347}$, $W_{349}$, $W_{350}$, $W_{351}$, $W_{352}$, $W_{353}$, $W_{354}$, $W_{357}$, $W_{359}$, $W_{360}$, $W_{363}$, $W_{365}$, $W_{367}$, $W_{368}$, $W_{369}$, $W_{370}$, $W_{372}$ and $W_{373}$}
\def\NumCompactChambers{269}
\def\NumTriangularWeyls{7}
\def\TriangularWeyls{$W_{1}$, $W_{2}$, $W_{3}$, $W_{7}$, $W_{140}$, $W_{145}$ and $W_{155}$}
\def\NumCommClasses{39}
\def\NumGenera{8488}
\def\MostReflectivesInGenus{2}
\def\NumGeneraWithMostLattices{107}
\def\GeneraWithMostLattices{558, 881, 918, 1282, 1388, 1390, 1420, 1918, 1954, 2068, 2074, 2219, 2374, 2717, 2721, 2887, 2888, 2892, 2932, 2935, 3040, 3043, 3330, 3610, 3665, 3771, 3802, 3948, 3967, 3970, 4001, 4002, 4069, 4274, 4279, 4441, 4578, 4581, 4586, 4761, 4762, 4763, 4764, 4766, 4812, 4814, 4880, 4899, 4924, 4926, 5112, 5207, 5483, 5536, 5784, 5847, 5863, 5970, 5971, 5986, 6102, 6226, 6345, 6348, 6349, 6449, 6478, 6479, 6625, 6713, 6714, 6726, 6727, 6942, 6979, 6999, 7080, 7091, 7111, 7169, 7210, 7213, 7233, 7248, 7274, 7305, 7306, 7382, 7407, 7463, 7512, 7529, 7594, 7618, 7678, 7710, 7754, 8021, 8074, 8230, 8253, 8254, 8301, 8308, 8316, 8342 and 8449}
\def\GenusWithTwoWeyls{{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{1}{1}256\above{1pt}{1pt}{1}{1}$}}
\def\GenusWithTwoWeylsFirstLattice{{L_{341.1}}}
\def\GenusWithTwoWeylsSecondLattice{{L_{342.1}}}
\def\GenusWithTwoWeylsFirstCornerSymbols{{${256}\above{1pt}{1pt}{*}{2}{32}\above{1pt}{1pt}{32,17}{\infty z}{8}\above{1pt}{1pt}{32,9}{\infty}{32}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
}
\def\GenusWithTwoWeylsSecondCornerSymbols{{${256}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{b}{2}{256}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{32,17}{\infty}{32}\above{1pt}{1pt}{32,25}{\infty z}{8}\above{1pt}{1pt}{}{2}$}%
}
\def\GenusWithTwoWeylsFirstWeyl{$W_{341}$}
\def\GenusWithTwoWeylsSecondWeyl{$W_{342}$}
\def\MostNegativeDet{$-29811600$}
\def\MostNegativeDetOfMinimalInDualityClass{$-108000$}
\def\LargestPrimeInDet{97}
\def\LargestRootNorm{14400}
\def\LargestRootNormofMinimalInDualityClass{7200}
\def\NumDualityClasses{1419}
\def\NumEssentialDualityClasses{704}
\def\ImplicitGeneraTable{%
\begingroup
\parindent=0pt
\parskip=0pt\small
\raggedbottom
%
% LATTICE L10.4
%
\leavevmode%
$\pfillwitharg{2}({L_{10.1}})$%
\ shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}%
% END OF LATTICE L10.4
%
\hbox{}\par\smallskip%
%
% LATTICE L248.5
%
\leavevmode%
$\main({L_{248.2}})$%
\ shares genus with {$ {L_{248.1}}$}%
\nopagebreak\par%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}128\above{1pt}{1pt}{1}{1}$}%
% END OF LATTICE L248.5
%
\hbox{}\par\smallskip%
%
% LATTICE L194.4
%
\leavevmode%
$\pfillwitharg{2}({L_{194.1}})$%
\ shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}%
% END OF LATTICE L194.4
%
\hbox{}\par\smallskip%
%
% LATTICE L195.4
%
\leavevmode%
$\pfillwitharg{2}({L_{195.1}})$%
\ shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}%
% END OF LATTICE L195.4
%
\hbox{}\par\smallskip%
%
% LATTICE L73.4
%
\leavevmode%
$\pfillwitharg{2}({L_{73.1}})$%
\ shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}%
% END OF LATTICE L73.4
%
\hbox{}\par\smallskip%
%
% LATTICE L366.2
%
\leavevmode%
$\pfillwitharg{2}({L_{366.1}})$%
\ shares genus with 13-dual\nopagebreak\par%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-}{}169\above{1pt}{1pt}{-}{}$}%
% END OF LATTICE L366.2
%
\hbox{}\par\smallskip%
%
% LATTICE L345.4
%
\leavevmode%
$\pfillwitharg{2}({L_{345.1}})$%
\ shares genus with 3-dual; isometric to own %
7-dual\nopagebreak\par%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{1}{}49\above{1pt}{1pt}{1}{}$}%
% END OF LATTICE L345.4
%
\hbox{}\par\smallskip%
\endgroup
}%

\raggedbottom
\overfullrule=10pt

\begin{document}

\title[Unabridged table of reflective lattices]{Unabridged table of reflective Lorentzian lattices of rank 3}
%
\author{Daniel Allcock}
\address{Department of Mathematics\\U.T. Austin}
\email{allcock@math.utexas.edu}
\urladdr{http://www.math.utexas.edu/\textasciitilde allcock}
%
\thanks{Partly supported by NSF grant DMS-0600112.}
\subjclass[2000]{11H56 (20F55, 22E40)}
\keywords{Lorentzian lattice, Weyl group, Coxeter group, Vinberg's algorithm}
\date{November 19, 2010}
%\date{November 14, 2010}

\begin{abstract}
This is the unabriged table of all \NumLattices{} rank three
reflective Lorentzian lattices, intended as a supplement to the
author's paper classifying them.  The abridged table in that paper is
complete too, but less explicit, so the main purpose of this document
is archival.  The TeX sourcecode is simultaneously a Perl script,
which when run prints out all the lattices in computer-readable
format.
\end{abstract}

\maketitle

\noindent
See the author's paper \cite{Allcock-rk3} for the meaning of this
table, proofs, and references.  The following comments describe
features of the table present here but not in the paper.

{\it Names and constructions of Lattices:\/} For a given Weyl group
$W_n$, we index the lattices having that Weyl group as in \cite{Allcock-rk3},
followed by those that are printed implicitly there, followed by those
that are got by $p$-duality operations.  For the lattices printed here but not there,
the method of construction is stated.  When $L$ is printed explicitly
or implicitly in the paper, and $M$ is got from it by duality, we have
printed all chains of $p$-dualities from $L$ to $M$.  (Except that we
omit chains involving a prime $p$ if $L\iso\pdual(L)$ or equivalently
$M\iso\pdual(M)$.)  This means that some lattices have more than one
construction; no one of those constructions is
distinguished, but the set of all of them is natural.

{\it Reference to Nikulin's lattices:\/}  When one of our lattices appears on
Nikulin's tables \cite{Nikulin-rk3}, we indicate where it does so.

{\it Gram matrices:\/} By default, the Gram matrices are not printed,
because they add more than 1000 pages to the already-long table.
However, you can print them if you really want to.  One reason to do
this would be to check that our Gram matrices really are the same as
Nikulin's (up to negation).  To print them, look in the TeX source file,
find the two lines\par
\noindent{\tt$\backslash$long$\backslash$def$\backslash$GramMatrix\#1\{\#1\}}\par
\noindent{\tt$\backslash$long$\backslash$def$\backslash$GramMatrix\#1\{\}}\par
\noindent
and exchange them.  Then run latex as usual and enjoy the enormous
file.  The Gram matrix follows the rest of the data
about the lattice.  When the lattice has no diagram isometry we have
printed the whole Gram matrix.  When it does have one then the Gram
matrix has block form
$$
\begin{pmatrix}A&B\\B&A\end{pmatrix}.
$$
We printed the top half of the matrix if it fit in the width of the
page; otherwise we printed the left half.  Even with these
conventions, they did not all fit within the margins.  However, none
of the matrices run off the edge of the (US letter size) page.

\begin{thebibliography}{9}

\bibitem{Allcock-rk3} 
Allcock, Daniel, The reflective Lorentzian lattices of rank 3,
preprint 2010, {\it arXiv:1010.0486}

\bibitem{Nikulin-rk3}
Nikulin, V. V., On the classification of hyperbolic root systems of
rank three. {\it Tr. Mat. Inst. Steklova} {\bf 230}  (2000), 256 pp.;
English translation in {\it Proc. Steklov Inst. Math.} {\bf 230} (2000)
1--241.

\end{thebibliography}

\vfill\eject
\begin{center}
\bf Unabridged Table of Rank 3 Reflective Lorentzian Lattices
\end{center}
\medskip
\begingroup
\parindent=0pt
\parskip=0pt\small
\raggedbottom
%------------------------------WEYL-GROUP-W1--------------------------------
%
\leavevmode\llap{}%
$W_{1\phantom{0}\phantom{0}}$%
\qquad\llap{11} lattices, $\chi=3$%
\hfill%
$\infty24$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L1.1
%
\leavevmode%
\llap{\spacer}%
${L_{1.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}$}\spacer%
\instructions{2\rightarrow N_{1}}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty b}{4}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[4, 0, 0; 0, 0, -1; 0, -1, -2],[4, 1, 1],[-1, 1, 0; -2, 0, 2; 0, 0, -1],0,1]
%
% END OF LATTICE L1.1
%
\hbox{}\par\smallskip%
%
% LATTICE L1.2
%
\leavevmode%
\llap{\spacer}%
${L_{1.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}$}\spacer%
\instructions{2\rightarrow N'_{1}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty a}{8}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\16\\32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-1\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-3\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-168, 16, 32; 16, -1, -3; 32, -3, -6],[8, 1, 1],[0, -1, 1; 2, -4, -2; -1, -4, 6],0,2]
%
% END OF LATTICE L1.2
%
\hbox{}\par\smallskip%
%
% LATTICE L1.3
%
\leavevmode%
\llap{\spacer}%
${L_{1.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty b}{8}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-328\\32\\56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-3\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\-6\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\15\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-24\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-328, 32, 56; 32, -3, -6; 56, -6, -7],[8, 1, 1],[2, -3, -1; 15, -24, -7; 3, -4, -2],0,3]
%
% END OF LATTICE L1.3
%
\hbox{}\par\smallskip%
%
% LATTICE L1.4
%
\leavevmode%
\llap{\spacer}%
${L_{1.4}}=\pfillwitharg{2}({L_{1.1}})$%
{${}={}$}Nikulin $1$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{2,1}{\infty}{1}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[0, -1, 0; -1, 0, 0; 0, 0, 1],[1, 1, 1],[0, 0, -1; -1, 0, 1; -1, 1, 0],0,4]
%
% END OF LATTICE L1.4
%
\hbox{}\par\smallskip%
%
% LATTICE L1.5
%
\leavevmode%
\llap{\spacer}%
${L_{1.5}}=\pfillwitharg{2}({L_{1.2}})$%
{${}={}$}Nikulin $1'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty b}{2}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[2, 0, -2; 0, 1, -1; -2, -1, 2],[2, 1, 1],[0, 1, 0; 0, 0, 1; 1, 0, 0],0,5]
%
% END OF LATTICE L1.5
%
\hbox{}\par\smallskip%
%
% LATTICE L1.6
%
\leavevmode%
\llap{\spacer}%
${L_{1.6}}=\main({L_{1.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{4,3}{\infty}{4}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-4, 4, 4; 4, -2, -3; 4, -3, -3],[4, 1, 1],[0, -1, 1; 1, -2, -1; -1, 0, 2],0,6]
%
% END OF LATTICE L1.6
%
\hbox{}\par\smallskip%
%
% LATTICE L1.7
%
\leavevmode%
\llap{\spacer}%
${L_{1.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{1.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-2\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-2\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[0, -2, -2; -2, -2, -2; -2, -2, -1],[2, 2, 1],[-1, 0, 1; 2, -1, -1; -2, 1, 0],0,7]
%
% END OF LATTICE L1.7
%
\hbox{}\par\smallskip%
%
% LATTICE L1.8
%
\leavevmode%
\llap{\spacer}%
${L_{1.8}}=\pdualwitharg{2}(\main({L_{1.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty}{4}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4, -4, 0; -4, 8, 4; 0, 4, 3],[4, 4, 1],[1, 0, 0; 0, 1, 1; 0, -2, 0],0,8]
%
% END OF LATTICE L1.8
%
\hbox{}\par\smallskip%
%
% LATTICE L1.9
%
\leavevmode%
\llap{\spacer}%
${L_{1.9}}=\pdualwitharg{2}({L_{1.1}})$%
\nopagebreak\par%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8, -4, 0; -4, 0, 0; 0, 0, 1],[4, 4, 1],[0, 0, 1; 1, 0, 0; -2, 1, 0],0,9]
%
% END OF LATTICE L1.9
%
\hbox{}\par\smallskip%
%
% LATTICE L1.10
%
\leavevmode%
\llap{\spacer}%
${L_{1.10}}=\pdualwitharg{2}({L_{1.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}$}\EasyButWeakLineBreak%
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0\\1\\-8
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-48, -56, -8; -56, -56, -8; -8, -8, -1],[8, 8, 1],[-1, 1, 0; 0, -1, 1; 8, 2, -8],0,10]
%
% END OF LATTICE L1.10
%
\hbox{}\par\smallskip%
%
% LATTICE L1.11
%
\leavevmode%
\llap{\spacer}%
${L_{1.11}}=\pdualwitharg{2}({L_{1.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,5}{\infty z}{4}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{4}$}%
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{$\left[\!\llap{\phantom{%
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-48\\-24\\8
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-24\\-8\\8
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\kern3pt%
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8\\8\\3
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-5\\8\\-8
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1\\-1\\2
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3\\-5\\4
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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16\\-8\\-8
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-48, -24, 8; -24, -8, 8; 8, 8, 3],[8, 8, 1],[-5, 1, 3; 8, -1, -5; -8, 2, 4],0,11]
%
% END OF LATTICE L1.11
%

\medskip%
%------------------------------WEYL-GROUP-W2--------------------------------
%
\leavevmode\llap{}%
$W_{2\phantom{0}\phantom{0}}$%
\qquad\llap{4} lattices, $\chi=2$%
\hfill%
$\infty23$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L2.1
%
\leavevmode%
\llap{\spacer}%
${L_{2.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}$}\spacer%
\instructions{2\rightarrow N_{2}}%
\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-24\\8\\16
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-2\\-5
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-5\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
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0\\2\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-24, 8, 16; 8, -2, -5; 16, -5, -10],[8, 1, 1],[0, -1, 1; 2, -4, -1; -1, 0, 2],1,1]
%
% END OF LATTICE L2.1
%
\hbox{}\par\smallskip%
%
% LATTICE L2.2
%
\leavevmode%
\llap{\spacer}%
${L_{2.2}}=\pfillwitharg{2}({L_{2.1}})$%
{${}={}$}Nikulin $2$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty a}{2}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{-}{3}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2\\0\\0
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\kern3pt%
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0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
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-1\\0\\-1
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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2\\-2\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[2, 0, 0; 0, 0, -1; 0, -1, 0],[2, 1, 1],[-1, 1, 0; 0, 0, -1; -1, 0, 1],1,2]
%
% END OF LATTICE L2.2
%
\hbox{}\par\smallskip%
%
% LATTICE L2.3
%
\leavevmode%
\llap{\spacer}%
${L_{2.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{2.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{+}{3}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-2\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-2\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-2
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-2\\1\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4, -2, -4; -2, 0, -2; -4, -2, -3],[2, 2, 1],[2, -1, -1; -1, 0, 2; -2, 1, 0],1,3]
%
% END OF LATTICE L2.3
%
\hbox{}\par\smallskip%
%
% LATTICE L2.4
%
\leavevmode%
\llap{\spacer}%
${L_{2.4}}=\pdualwitharg{2}({L_{2.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,3}{\infty z}{4}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{-}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\24\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-16\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-8
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16, 24, -8; 24, -16, 8; -8, 8, -3],[8, 8, 1],[2, -1, -1; -1, 0, 2; -8, 2, 8],1,4]
%
% END OF LATTICE L2.4
%

\medskip%
%------------------------------WEYL-GROUP-W3--------------------------------
%
\leavevmode\llap{}%
$W_{3\phantom{0}\phantom{0}}$%
\qquad\llap{6} lattices, $\chi=1$%
\hfill%
$426$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L3.1
%
\leavevmode%
\llap{\spacer}%
${L_{3.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{3}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-516\\36\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\-2\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-5\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-516, 36, 72; 36, -2, -5; 72, -5, -10],[12, 1, 1],[0, 1, -1; 2, -2, -3; -1, 8, -6],2,1]
%
% END OF LATTICE L3.1
%
\hbox{}\par\smallskip%
%
% LATTICE L3.2
%
\leavevmode%
\llap{\spacer}%
${L_{3.2}}=\pfillwitharg{2}({L_{3.1}})$%
{${}={}$}Nikulin $3$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3, 0, 0; 0, 1, 0; 0, 0, 1],[3, 1, 1],[0, 0, -1; -1, 1, 0; 1, 0, -3],2,2]
%
% END OF LATTICE L3.2
%
\hbox{}\par\smallskip%
%
% LATTICE L3.3
%
\leavevmode%
\llap{\spacer}%
${L_{3.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{3.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{4}{3}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-33\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-33\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-6\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-1\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30, -33, -6; -33, -33, -6; -6, -6, -1],[3, 3, 1],[-1, -2, 5; 2, -1, -1; -6, 12, -13],2,3]
%
% END OF LATTICE L3.3
%
\hbox{}\par\smallskip%
%
% LATTICE L3.4
%
\leavevmode%
\llap{\spacer}%
${L_{3.4}}=\pdualwitharg{3}({L_{3.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{*}{4}{12}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-6\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-9\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12, -12, -12; -12, -6, -9; -12, -9, -10],[12, 3, 1],[1, 1, -1; 2, -2, -1; -3, 0, 2],2,4]
%
% END OF LATTICE L3.4
%
\hbox{}\par\smallskip%
%
% LATTICE L3.5
%
\leavevmode%
\llap{\spacer}%
${L_{3.5}}=\pdualwitharg{2}({L_{3.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
312\\-156\\84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\88\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-40\\23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\3\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-4\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[312, -156, 84; -156, 88, -40; 84, -40, 23],[12, 4, 1],[-1, 2, 8; 0, 1, 3; 4, -6, -24],2,5]
%
% END OF LATTICE L3.5
%
\hbox{}\par\smallskip%
%
% LATTICE L3.6
%
\leavevmode%
\llap{\spacer}%
${L_{3.6}}=\pdualwitharg{2.3}({L_{3.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{4}{12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-84\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\696\\192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\192\\53
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24, -84, -24; -84, 696, 192; -24, 192, 53],[12, 12, 1],[1, 0, 0; 0, -5, -1; 0, 18, 4],2,6]
%
% END OF LATTICE L3.6
%

\medskip%
%------------------------------WEYL-GROUP-W4--------------------------------
%
\leavevmode\llap{}%
$W_{4\phantom{0}\phantom{0}}$%
\qquad\llap{44} lattices, $\chi=6$%
\hfill%
$\infty222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L4.1
%
\leavevmode%
\llap{\spacer}%
${L_{4.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{4},3,2}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{3,2}{\infty b}{12}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2124\\0\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\3\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\28\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[2124, 0, -108; 0, -6, 3; -108, 3, 4],[36, 3, 1],[-1, 3, 1, -1; -10, 28, 9, -10; -18, 60, 18, -20],3,1]
%
% END OF LATTICE L4.1
%
\hbox{}\par\smallskip%
%
% LATTICE L4.2
%
\leavevmode%
\llap{\spacer}%
${L_{4.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3m,3,2}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,5}{\infty a}{24}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3096\\-1440\\432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1440\\-669\\198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\198\\-49
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\35\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-40\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-72\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\12\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3096, -1440, 432; -1440, -669, 198; 432, 198, -49],[72, 3, 1],[-15, 17, 31, -5; 35, -40, -72, 12; 9, -12, -18, 4],3,2]
%
% END OF LATTICE L4.2
%
\hbox{}\par\smallskip%
%
% LATTICE L4.3
%
\leavevmode%
\llap{\spacer}%
${L_{4.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{32\rightarrow N'_{2},3,m}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,5}{\infty b}{24}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9144\\-1800\\-1080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-354\\-213
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\-213\\-127
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-8\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\15\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\16\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-9144, -1800, -1080; -1800, -354, -213; -1080, -213, -127],[72, 3, 1],[1, 3, -4, -5; -5, -8, 15, 16; 0, -12, 9, 16],3,3]
%
% END OF LATTICE L4.3
%
\hbox{}\par\smallskip%
%
% LATTICE L4.4
%
\leavevmode%
\llap{\spacer}%
${L_{4.4}}=\pfillwitharg{2.3}({L_{4.1}})$%
{${}={}$}Nikulin $4$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{2,1}{\infty}{3}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-3\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[3, 0, 0; 0, 0, -1; 0, -1, 0],[3, 1, 1],[-1, 1, 0, -1; 0, 0, -1, -1; -3, 0, 1, -1],3,4]
%
% END OF LATTICE L4.4
%
\hbox{}\par\smallskip%
%
% LATTICE L4.5
%
\leavevmode%
\llap{\spacer}%
${L_{4.5}}=\pfillwitharg{3.2}({L_{4.3}})$%
{${}={}$}Nikulin $2'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{2,1}{\infty a}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\18\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\8\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-5\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\9\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[42, 18, -12; 18, 8, -5; -12, -5, 3],[6, 1, 1],[-7, 1, 1, -3; 9, 0, -2, 2; -18, 6, 1, -12],3,5]
%
% END OF LATTICE L4.5
%
\hbox{}\par\smallskip%
%
% LATTICE L4.6
%
\leavevmode%
\llap{\spacer}%
${L_{4.6}}=\main(\pfillwitharg{3}({L_{4.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{4,1}{\infty}{12}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\12\\12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-3\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-6\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-60, 12, 12; 12, -2, -3; 12, -3, -1],[12, 1, 1],[-1, -1, 1, 1; -3, -6, 3, 4; -3, 0, 2, 0],3,6]
%
% END OF LATTICE L4.6
%
\hbox{}\par\smallskip%
%
% LATTICE L4.7
%
\leavevmode%
\llap{\spacer}%
${L_{4.7}}=\pfillwitharg{3}({L_{4.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{1,0}{\infty b}{12}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[12, 0, 0; 0, 0, 1; 0, 1, 2],[12, 1, 1],[1, -1, 0, 1; 6, 0, 0, 4; 0, 0, -1, -2],3,7]
%
% END OF LATTICE L4.7
%
\hbox{}\par\smallskip%
%
% LATTICE L4.8
%
\leavevmode%
\llap{\spacer}%
${L_{4.8}}=\pfillwitharg{3}({L_{4.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{4,1}{\infty b}{24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\48\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-1\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-3\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\18\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1560, 48, 72; 48, -1, -3; 72, -3, -2],[24, 1, 1],[1, 1, -1, -1; 18, 12, -18, -16; 9, 12, -10, -12],3,8]
%
% END OF LATTICE L4.8
%
\hbox{}\par\smallskip%
%
% LATTICE L4.9
%
\leavevmode%
\llap{\spacer}%
${L_{4.9}}=\pfillwitharg{3}({L_{4.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{4,1}{\infty a}{24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-191352\\-21120\\2592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21120\\-2331\\286
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2592\\286\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\81\\69
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-12\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-30\\-23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-8\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-191352, -21120, 2592; -21120, -2331, 286; 2592, 286, -35],[24, 1, 1],[-8, 1, 3, 1; 81, -12, -30, -8; 69, -24, -23, 8],3,9]
%
% END OF LATTICE L4.9
%
\hbox{}\par\smallskip%
%
% LATTICE L4.10
%
\leavevmode%
\llap{\spacer}%
${L_{4.10}}=\pfillwitharg{2}({L_{4.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{6,5}{\infty}{3}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-3\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\1\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-3\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[18, 0, -9; 0, -15, -3; -9, -3, 4],[9, 3, 1],[2, -3, -1, 2; -1, 1, 0, -1; 3, -6, 0, 4],3,10]
%
% END OF LATTICE L4.10
%
\hbox{}\par\smallskip%
%
% LATTICE L4.11
%
\leavevmode%
\llap{\spacer}%
${L_{4.11}}=\pfillwitharg{2}({L_{4.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{6,5}{\infty a}{6}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
954\\-18\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-3\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\3\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\10\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[954, -18, -90; -18, -3, 3; -90, 3, 8],[18, 3, 1],[-1, 3, 1, -1; -4, 10, 3, -4; -9, 30, 9, -10],3,11]
%
% END OF LATTICE L4.11
%
\hbox{}\par\smallskip%
%
% LATTICE L4.12
%
\leavevmode%
\llap{\spacer}%
${L_{4.12}}=\main({L_{4.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{12,5}{\infty}{12}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1548\\108\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108\\-6\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\15\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-14\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\22\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\27\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-6\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1548, 108, -180; 108, -6, 15; -180, 15, -17],[36, 3, 1],[-2, 3, 4, -1; -14, 22, 27, -8; 9, -12, -18, 4],3,12]
%
% END OF LATTICE L4.12
%
\hbox{}\par\smallskip%
%
% LATTICE L4.13
%
\leavevmode%
\llap{\spacer}%
${L_{4.13}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{4.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{2,1}{\infty}{1}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-9\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-12\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-9\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6, -9, -6; -9, -12, -9; -6, -9, -5],[3, 3, 1],[3, -1, -3, 4; -1, 0, 2, 0; -1, 1, 0, -3],3,13]
%
% END OF LATTICE L4.13
%
\hbox{}\par\smallskip%
%
% LATTICE L4.14
%
\leavevmode%
\llap{\spacer}%
${L_{4.14}}=\pdualwitharg{2}(\pfillwitharg{3.2}({L_{4.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{4,3}{\infty z}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-18\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\6\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-8\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\36\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-6\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\11\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[48, -18, 24; -18, 6, -8; 24, -8, 11],[6, 2, 1],[-13, 2, 1, -4; 36, -6, -3, 11; 54, -9, -4, 17],3,14]
%
% END OF LATTICE L4.14
%
\hbox{}\par\smallskip%
%
% LATTICE L4.15
%
\leavevmode%
\llap{\spacer}%
${L_{4.15}}=\pdualwitharg{3}(\pfillwitharg{3.2}({L_{4.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty a}{2}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
126\\24\\54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-6\\9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\9\\23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-5\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\2\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\1\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-10\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[126, 24, 54; 24, -6, 9; 54, 9, 23],[6, 3, 1],[-26, 9, 6, -49; -5, 2, 1, -10; 64, -22, -15, 120],3,15]
%
% END OF LATTICE L4.15
%
\hbox{}\par\smallskip%
%
% LATTICE L4.16
%
\leavevmode%
\llap{\spacer}%
${L_{4.16}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{4.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{6,1}{\infty}{3}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-3\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9, 0, 0; 0, -6, -3; 0, -3, -1],[9, 3, 1],[0, 1, 0, -1; 1, -1, -1, 0; -3, 0, 2, 0],3,16]
%
% END OF LATTICE L4.16
%
\hbox{}\par\smallskip%
%
% LATTICE L4.17
%
\leavevmode%
\llap{\spacer}%
${L_{4.17}}=\pdualwitharg{3}(\main(\pfillwitharg{3}({L_{4.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{4,1}{\infty}{4}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-24\\12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-18\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\15\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12, -24, 12; -24, -18, 15; 12, 15, -11],[12, 3, 1],[1, -1, -2, 1; 3, -2, -7, 0; 5, -4, -12, 0],3,17]
%
% END OF LATTICE L4.17
%
\hbox{}\par\smallskip%
%
% LATTICE L4.18
%
\leavevmode%
\llap{\spacer}%
${L_{4.18}}=\pdualwitharg{2.3}(\pfillwitharg{3.2}({L_{4.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\294\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
294\\4368\\1782
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\1782\\727
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-43\\106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\22\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-23\\57
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18, 294, 120; 294, 4368, 1782; 120, 1782, 727],[6, 6, 1],[-6, 1, 1, -6; -43, 2, 22, -23; 106, -5, -54, 57],3,18]
%
% END OF LATTICE L4.18
%
\hbox{}\par\smallskip%
%
% LATTICE L4.19
%
\leavevmode%
\llap{\spacer}%
${L_{4.19}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{4.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty b}{4}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\-12\\12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\9\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-8\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[36, -12, 12; -12, -24, 9; 12, 9, -2],[12, 3, 1],[1, -3, -1, 3; -2, 8, 2, -8; -4, 16, 3, -18],3,19]
%
% END OF LATTICE L4.19
%
\hbox{}\par\smallskip%
%
% LATTICE L4.20
%
\leavevmode%
\llap{\spacer}%
${L_{4.20}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{4.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{2,1}{\infty z}{3}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\4\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\3\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24, 12, -12; 12, 0, 4; -12, 4, -5],[12, 4, 1],[-2, 1, 1, -1; 3, 0, -2, 0; 6, -3, -4, 2],3,20]
%
% END OF LATTICE L4.20
%
\hbox{}\par\smallskip%
%
% LATTICE L4.21
%
\leavevmode%
\llap{\spacer}%
${L_{4.21}}=\pdualwitharg{2}(\main(\pfillwitharg{3}({L_{4.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{4,3}{\infty}{12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12, -12, 0; -12, 8, 0; 0, 0, 1],[12, 4, 1],[1, 2, 0, 0; 0, 3, 1, 0; 0, -6, 0, 1],3,21]
%
% END OF LATTICE L4.21
%
\hbox{}\par\smallskip%
%
% LATTICE L4.22
%
\leavevmode%
\llap{\spacer}%
${L_{4.22}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{4.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{6,1}{\infty a}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18, 0, 0; 0, -3, -3; 0, -3, -2],[18, 3, 1],[0, 1, 0, -1; 2, -2, -1, 0; -3, 0, 1, 0],3,22]
%
% END OF LATTICE L4.22
%
\hbox{}\par\smallskip%
%
% LATTICE L4.23
%
\leavevmode%
\llap{\spacer}%
${L_{4.23}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{4.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,1}{\infty b}{8}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3720\\-2064\\288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2064\\-1143\\159
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\159\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\6\\17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-14\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3720, -2064, 288; -2064, -1143, 159; 288, 159, -22],[24, 3, 1],[-2, 1, 5, 1; 6, -4, -14, 0; 17, -16, -36, 12],3,23]
%
% END OF LATTICE L4.23
%
\hbox{}\par\smallskip%
%
% LATTICE L4.24
%
\leavevmode%
\llap{\spacer}%
${L_{4.24}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{4.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,1}{\infty a}{8}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\96\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\-9\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-6\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\7\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-936, 96, 48; 96, -9, -6; 48, -6, -1],[24, 3, 1],[1, -1, -1, 1; 7, -8, -7, 8; 5, -4, -6, 0],3,24]
%
% END OF LATTICE L4.24
%
\hbox{}\par\smallskip%
%
% LATTICE L4.25
%
\leavevmode%
\llap{\spacer}%
${L_{4.25}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{4.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,7}{\infty z}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
198\\54\\90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\12\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\24\\41
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\1\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[198, 54, 90; 54, 12, 24; 90, 24, 41],[18, 6, 1],[0, -7, -1, 4; 1, 1, 0, 0; 0, 15, 2, -9],3,25]
%
% END OF LATTICE L4.25
%
\hbox{}\par\smallskip%
%
% LATTICE L4.26
%
\leavevmode%
\llap{\spacer}%
${L_{4.26}}=\pdualwitharg{3}(\main({L_{4.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{12,1}{\infty}{12}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-6\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\12\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[36, 0, 0; 0, -6, -3; 0, -3, -1],[36, 3, 1],[0, -1, 0, 1; 1, -2, -1, 0; -3, 0, 2, 0],3,26]
%
% END OF LATTICE L4.26
%
\hbox{}\par\smallskip%
%
% LATTICE L4.27
%
\leavevmode%
\llap{\spacer}%
${L_{4.27}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{4.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,11}{\infty z}{3}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6066\\-990\\2610
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-990\\156\\-426
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2610\\-426\\1123
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\1\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\1\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\0\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6066, -990, 2610; -990, 156, -426; 2610, -426, 1123],[18, 6, 1],[26, 4, -31, -3; 1, 1, 0, 0; -60, -9, 72, 7],3,27]
%
% END OF LATTICE L4.27
%
\hbox{}\par\smallskip%
%
% LATTICE L4.28
%
\leavevmode%
\llap{\spacer}%
${L_{4.28}}=\pdualwitharg{3}({L_{4.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{3,1}{\infty a}{12}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\-6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-16\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-180, 36, 0; 36, -6, -3; 0, -3, 8],[36, 3, 1],[-3, 1, 1, -1; -16, 4, 5, -6; -6, 0, 2, 0],3,28]
%
% END OF LATTICE L4.28
%
\hbox{}\par\smallskip%
%
% LATTICE L4.29
%
\leavevmode%
\llap{\spacer}%
${L_{4.29}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{4.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{2,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\804\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
804\\2976\\888
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\888\\265
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[216, 804, 240; 804, 2976, 888; 240, 888, 265],[12, 12, 1],[0, -1, -3, -1; -3, 0, 8, 2; 10, 1, -24, -6],3,29]
%
% END OF LATTICE L4.29
%
\hbox{}\par\smallskip%
%
% LATTICE L4.30
%
\leavevmode%
\llap{\spacer}%
${L_{4.30}}=\pdualwitharg{2.3}(\main(\pfillwitharg{3}({L_{4.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty}{4}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
276\\132\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\72\\36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\36\\19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-4\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[276, 132, 72; 132, 72, 36; 72, 36, 19],[12, 12, 1],[-1, 0, -2, -2; -2, 1, -3, -4; 8, -2, 12, 15],3,30]
%
% END OF LATTICE L4.30
%
\hbox{}\par\smallskip%
%
% LATTICE L4.31
%
\leavevmode%
\llap{\spacer}%
${L_{4.31}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{4.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{8,3}{\infty z}{12}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4200\\264\\384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
264\\-16\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
384\\-24\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4200, 264, 384; 264, -16, -24; 384, -24, -35],[24, 8, 1],[2, -1, -1, 0; -3, 9, 1, -3; 24, -18, -12, 2],3,31]
%
% END OF LATTICE L4.31
%
\hbox{}\par\smallskip%
%
% LATTICE L4.32
%
\leavevmode%
\llap{\spacer}%
${L_{4.32}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{4.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{8,7}{\infty z}{12}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\552\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
552\\-8400\\1096
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\1096\\-143
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\18\\138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24, 552, -72; 552, -8400, 1096; -72, 1096, -143],[24, 8, 1],[2, -1, -1, 0; -3, 18, 1, -3; -24, 138, 8, -23],3,32]
%
% END OF LATTICE L4.32
%
\hbox{}\par\smallskip%
%
% LATTICE L4.33
%
\leavevmode%
\llap{\spacer}%
${L_{4.33}}=\pdualwitharg{3}({L_{4.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,1}{\infty b}{24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-1152\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1152\\-525\\33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\33\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
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-6\\0\\4\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2520, -1152, 72; -1152, -525, 33; 72, 33, -2],[72, 3, 1],[-1, 1, 1, 1; 2, -4, -2, 0; -3, -36, 2, 36],3,33]
%
% END OF LATTICE L4.33
%
\hbox{}\par\smallskip%
%
% LATTICE L4.34
%
\leavevmode%
\llap{\spacer}%
${L_{4.34}}=\pdualwitharg{3}({L_{4.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,1}{\infty a}{24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-191736\\1296\\2592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1296\\-3\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2592\\-18\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\17\\213
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-71
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-72
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\0\\72
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-191736, 1296, 2592; 1296, -3, -18; 2592, -18, -35],[72, 3, 1],[3, -1, -1, 1; 17, -4, -6, 0; 213, -72, -71, 72],3,34]
%
% END OF LATTICE L4.34
%
\hbox{}\par\smallskip%
%
% LATTICE L4.35
%
\leavevmode%
\llap{\spacer}%
${L_{4.35}}=\pdualwitharg{2.3}({L_{4.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{6,1}{\infty z}{3}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1440\\828\\288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
828\\480\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\168\\59
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\13\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\16\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-15\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1440, 828, 288; 828, 480, 168; 288, 168, 59],[36, 12, 1],[-4, -5, 1, 5; 13, 16, -3, -15; -18, -21, 4, 18],3,35]
%
% END OF LATTICE L4.35
%
\hbox{}\par\smallskip%
%
% LATTICE L4.36
%
\leavevmode%
\llap{\spacer}%
${L_{4.36}}=\pdualwitharg{2}({L_{4.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{6,5}{\infty z}{3}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64224\\8244\\-8496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8244\\1056\\-1092
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8496\\-1092\\1123
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\52\\-78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\19\\-27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-93\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-9\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[64224, 8244, -8496; 8244, 1056, -1092; -8496, -1092, 1123],[36, 12, 1],[-17, -6, 31, 3; 52, 19, -93, -9; -78, -27, 144, 14],3,36]
%
% END OF LATTICE L4.36
%
\hbox{}\par\smallskip%
%
% LATTICE L4.37
%
\leavevmode%
\llap{\spacer}%
${L_{4.37}}=\pdualwitharg{2}(\main({L_{4.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,11}{\infty}{12}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
828\\-1152\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1152\\2028\\1704
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1008\\1704\\1441
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\34\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\47\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\51\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\12\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[828, -1152, -1008; -1152, 2028, 1704; -1008, 1704, 1441],[36, 12, 1],[-11, -15, -17, -4; 34, 47, 51, 12; -48, -66, -72, -17],3,37]
%
% END OF LATTICE L4.37
%
\hbox{}\par\smallskip%
%
% LATTICE L4.38
%
\leavevmode%
\llap{\spacer}%
${L_{4.38}}=\pdualwitharg{2.3}(\main({L_{4.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,7}{\infty}{12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252\\-72\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\12\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\24\\41
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-6\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[252, -72, -108; -72, 12, 24; -108, 24, 41],[36, 12, 1],[0, -1, 1, 2; 1, 5, -3, -6; 0, -6, 4, 9],3,38]
%
% END OF LATTICE L4.38
%
\hbox{}\par\smallskip%
%
% LATTICE L4.39
%
\leavevmode%
\llap{\spacer}%
${L_{4.39}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{4.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,3}{\infty z}{4}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\72\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-48, -24, 0; -24, 72, -24; 0, -24, 7],[24, 24, 1],[-1, 1, 1, -1; 2, -3, -3, 2; 8, -10, -12, 6],3,39]
%
% END OF LATTICE L4.39
%
\hbox{}\par\smallskip%
%
% LATTICE L4.40
%
\leavevmode%
\llap{\spacer}%
${L_{4.40}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{4.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,7}{\infty z}{4}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10032\\8616\\1032
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8616\\-6984\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1032\\-840\\-101
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\18\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-31\\278
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-19\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\15\\-135
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10032, 8616, 1032; 8616, -6984, -840; 1032, -840, -101],[24, 24, 1],[-1, 2, 1, -1; 18, -31, -19, 15; -160, 278, 168, -135],3,40]
%
% END OF LATTICE L4.40
%
\hbox{}\par\smallskip%
%
% LATTICE L4.41
%
\leavevmode%
\llap{\spacer}%
${L_{4.41}}=\pdualwitharg{2.3}({L_{4.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,19}{\infty z}{12}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-360\\288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-168\\144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\144\\-115
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-720, -360, 288; -360, -168, 144; 288, 144, -115],[72, 24, 1],[-1, 3, -1, -7; 2, -1, -1, 0; 0, 6, -4, -18],3,41]
%
% END OF LATTICE L4.41
%
\hbox{}\par\smallskip%
%
% LATTICE L4.42
%
\leavevmode%
\llap{\spacer}%
${L_{4.42}}=\pdualwitharg{2.3}({L_{4.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,7}{\infty z}{12}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30096\\1080\\-9360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1080\\-24\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9360\\336\\-2911
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\2\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-1\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-1\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30096, 1080, -9360; 1080, -24, 336; -9360, 336, -2911],[72, 24, 1],[15, -28, -5, 14; 2, -1, -1, 0; -48, 90, 16, -45],3,42]
%
% END OF LATTICE L4.42
%
\hbox{}\par\smallskip%
%
% LATTICE L4.43
%
\leavevmode%
\llap{\spacer}%
${L_{4.43}}=\pdualwitharg{2}({L_{4.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,11}{\infty z}{12}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\792\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
792\\-4200\\384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\384\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-144, 792, -72; 792, -4200, 384; -72, 384, -35],[72, 24, 1],[-1, 3, 1, -1; 2, -1, -3, 0; 24, -18, -36, 2],3,43]
%
% END OF LATTICE L4.43
%
\hbox{}\par\smallskip%
%
% LATTICE L4.44
%
\leavevmode%
\llap{\spacer}%
${L_{4.44}}=\pdualwitharg{2}({L_{4.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,23}{\infty z}{12}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720720\\5112\\10152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5112\\-24\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10152\\-72\\-143
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-1\\426
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-71
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-720720, 5112, 10152; 5112, -24, -72; 10152, -72, -143],[72, 24, 1],[-1, 6, 1, -1; 2, -1, -3, 0; -72, 426, 72, -71],3,44]
%
% END OF LATTICE L4.44
%

\medskip%
%------------------------------WEYL-GROUP-W5--------------------------------
%
\leavevmode\llap{}%
$W_{5\phantom{0}\phantom{0}}$%
\qquad\llap{22} lattices, $\chi=9$%
\hfill%
$\infty242$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L5.1
%
\leavevmode%
\llap{\spacer}%
${L_{5.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{5}}%
\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{1,0}{\infty b}{20}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15020\\360\\680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-8\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
680\\-17\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-50\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\200\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\32\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-46\\-42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-20\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\20\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\2\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-2\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-15020, 360, 680; 360, -8, -17; 680, -17, -30],[20, 1, 1],[-3, 13, 2, -3; -50, 200, 32, -46; -40, 180, 27, -42],4,1]
%
% END OF LATTICE L5.1
%
\hbox{}\par\smallskip%
%
% LATTICE L5.2
%
\leavevmode%
\llap{\spacer}%
${L_{5.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N'_{5}}%
\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{4,3}{\infty a}{40}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8360\\240\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-6\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-11\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-25\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\100\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-11\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-20\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\1\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-8360, 240, 360; 240, -6, -11; 360, -11, -15],[40, 1, 1],[-2, 9, 1, -1; -25, 100, 12, -11; -30, 140, 15, -16],4,2]
%
% END OF LATTICE L5.2
%
\hbox{}\par\smallskip%
%
% LATTICE L5.3
%
\leavevmode%
\llap{\spacer}%
${L_{5.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{4,3}{\infty b}{40}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{4}{2}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34760\\200\\1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\-1\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1560\\-9\\-70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-20\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-2\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-34760, 200, 1560; 200, -1, -9; 1560, -9, -70],[40, 1, 1],[-2, -1, 1, 0; 0, -20, 2, 6; -45, -20, 22, -1],4,3]
%
% END OF LATTICE L5.3
%
\hbox{}\par\smallskip%
%
% LATTICE L5.4
%
\leavevmode%
\llap{\spacer}%
${L_{5.4}}=\pfillwitharg{2}({L_{5.1}})$%
{${}={}$}Nikulin $5$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{2,1}{\infty}{5}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-5\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\5\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-1\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[5, 0, 0; 0, 0, -1; 0, -1, 0],[5, 1, 1],[-1, 1, 0, -1; 0, 0, -1, -1; -5, 0, 1, -2],4,4]
%
% END OF LATTICE L5.4
%
\hbox{}\par\smallskip%
%
% LATTICE L5.5
%
\leavevmode%
\llap{\spacer}%
${L_{5.5}}=\pfillwitharg{2}({L_{5.2}})$%
{${}={}$}Nikulin $5'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{2,1}{\infty b}{10}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
310\\180\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\105\\139
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\139\\186
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
104\\-80\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
141\\-110\\-100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\3\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-10\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\1\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[310, 180, 240; 180, 105, 139; 240, 139, 186],[10, 1, 1],[104, 141, -4, -2; -80, -110, 3, 2; -75, -100, 3, 1],4,5]
%
% END OF LATTICE L5.5
%
\hbox{}\par\smallskip%
%
% LATTICE L5.6
%
\leavevmode%
\llap{\spacer}%
${L_{5.6}}=\main({L_{5.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{4,3}{\infty}{20}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-20\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-2\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-9\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-10\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\20\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-1\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-180, -20, -80; -20, -2, -9; -80, -9, -35],[20, 1, 1],[2, 1, -1, 0; 0, -10, 1, 3; -5, 0, 2, -1],4,6]
%
% END OF LATTICE L5.6
%
\hbox{}\par\smallskip%
%
% LATTICE L5.7
%
\leavevmode%
\llap{\spacer}%
${L_{5.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{5.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{4,1}{\infty z}{5}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{4}{4}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8530\\350\\-4010
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
350\\-12\\164
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4010\\164\\-1885
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
174\\-105\\-380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\-35\\-95
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\12\\50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-11\\-52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-10\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\5\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\2\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-2\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8530, 350, -4010; 350, -12, 164; -4010, 164, -1885],[10, 2, 1],[174, 43, -23, 24; -105, -35, 12, -11; -380, -95, 50, -52],4,7]
%
% END OF LATTICE L5.7
%
\hbox{}\par\smallskip%
%
% LATTICE L5.8
%
\leavevmode%
\llap{\spacer}%
${L_{5.8}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{5.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{2,1}{\infty}{1}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{}{4}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-45\\20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-20\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\15\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\-27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\2\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-1\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-5\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-65, -45, 20; -45, -20, 15; 20, 15, -6],[5, 5, 1],[-1, -7, -6, 5; -1, 0, 2, -1; -8, -27, -15, 15],4,8]
%
% END OF LATTICE L5.8
%
\hbox{}\par\smallskip%
%
% LATTICE L5.9
%
\leavevmode%
\llap{\spacer}%
${L_{5.9}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{5.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty b}{2}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{4}{10}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12970\\-1010\\-5320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1010\\-75\\-415
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5320\\-415\\-2182
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\-22\\-137
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-10\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\15\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-16\\-75
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\5\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-5\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12970, -1010, -5320; -1010, -75, -415; -5320, -415, -2182],[10, 5, 1],[58, 37, -34, 32; -22, -10, 15, -16; -137, -88, 80, -75],4,9]
%
% END OF LATTICE L5.9
%
\hbox{}\par\smallskip%
%
% LATTICE L5.10
%
\leavevmode%
\llap{\spacer}%
${L_{5.10}}=\pdualwitharg{2}(\main({L_{5.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{4,1}{\infty}{20}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{4}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1880\\-100\\440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\20\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
440\\-24\\103
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\5\\100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\3\\52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-20\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\20\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\8\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-4\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1880, -100, 440; -100, 20, -24; 440, -24, 103],[20, 4, 1],[-23, -7, -1, -12; 5, 0, 0, 3; 100, 30, 4, 52],4,10]
%
% END OF LATTICE L5.10
%
\hbox{}\par\smallskip%
%
% LATTICE L5.11
%
\leavevmode%
\llap{\spacer}%
${L_{5.11}}=\pdualwitharg{2}({L_{5.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{2,1}{\infty z}{5}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23920\\-7500\\4300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7500\\2360\\-1348
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4300\\-1348\\773
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\5\\-175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\5\\-136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\1\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-10\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\5\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\8\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-4\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[23920, -7500, 4300; -7500, 2360, -1348; 4300, -1348, 773],[20, 4, 1],[9, 33, 26, 5; 0, 5, 5, 1; -50, -175, -136, -26],4,11]
%
% END OF LATTICE L5.11
%
\hbox{}\par\smallskip%
%
% LATTICE L5.12
%
\leavevmode%
\llap{\spacer}%
${L_{5.12}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{5.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{4}{20}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1860\\-17310\\-7720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17310\\162250\\72360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7720\\72360\\32271
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\976\\-2192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\663\\-1489
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-187\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-98\\220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-10\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1860, -17310, -7720; -17310, 162250, 72360; -7720, 72360, 32271],[10, 10, 1],[-15, -10, 3, 1; 976, 663, -187, -98; -2192, -1489, 420, 220],4,12]
%
% END OF LATTICE L5.12
%
\hbox{}\par\smallskip%
%
% LATTICE L5.13
%
\leavevmode%
\llap{\spacer}%
${L_{5.13}}=\pdualwitharg{5}({L_{5.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty b}{4}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{*}{4}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1020\\220\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
220\\-40\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-25\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-10\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1020, 220, 120; 220, -40, -25; 120, -25, -14],[20, 5, 1],[1, 1, -2, -3; -2, 0, 4, 2; 12, 8, -25, -30],4,13]
%
% END OF LATTICE L5.13
%
\hbox{}\par\smallskip%
%
% LATTICE L5.14
%
\leavevmode%
\llap{\spacer}%
${L_{5.14}}=\pdualwitharg{5}(\main({L_{5.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{4,3}{\infty}{4}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{}{4}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2100\\-840\\200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-335\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\80\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-5\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2100, -840, 200; -840, -335, 80; 200, 80, -19],[20, 5, 1],[0, 1, -1, -2; 1, -2, -1, 4; 4, 2, -15, -5],4,14]
%
% END OF LATTICE L5.14
%
\hbox{}\par\smallskip%
%
% LATTICE L5.15
%
\leavevmode%
\llap{\spacer}%
${L_{5.15}}=\pdualwitharg{5}({L_{5.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty a}{8}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{4}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\80\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\-15\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\5\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-5\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-200, 80, 40; 80, -30, -15; 40, -15, -7],[40, 5, 1],[0, 1, 0, -1; -1, 4, 2, -3; 2, -4, -5, 0],4,15]
%
% END OF LATTICE L5.15
%
\hbox{}\par\smallskip%
%
% LATTICE L5.16
%
\leavevmode%
\llap{\spacer}%
${L_{5.16}}=\pdualwitharg{5}({L_{5.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty b}{8}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{*}{4}{10}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33000\\440\\1120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
440\\-5\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1120\\-15\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\8\\55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\20\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-10\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-33000, 440, 1120; 440, -5, -15; 1120, -15, -38],[40, 5, 1],[0, -1, 1, 2; 2, -4, -2, 8; -1, -28, 30, 55],4,16]
%
% END OF LATTICE L5.16
%
\hbox{}\par\smallskip%
%
% LATTICE L5.17
%
\leavevmode%
\llap{\spacer}%
${L_{5.17}}=\pdualwitharg{2}({L_{5.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{8,1}{\infty z}{20}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{4}{16}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\280\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\-48\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-80\\-97
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-15\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\15\\50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\7\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-7\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-40\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\20\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-8\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1320, 280, 360; 280, -48, -80; 360, -80, -97],[40, 8, 1],[-14, 17, 7, -8; -15, 15, 7, -7; -40, 50, 20, -24],4,17]
%
% END OF LATTICE L5.17
%
\hbox{}\par\smallskip%
%
% LATTICE L5.18
%
\leavevmode%
\llap{\spacer}%
${L_{5.18}}=\pdualwitharg{2}({L_{5.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{8,5}{\infty z}{20}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{4}{16}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27480\\2000\\760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2000\\-120\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
760\\-56\\-21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-5\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\10\\370
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\3\\136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-5\\-168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-40\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\20\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-8\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-27480, 2000, 760; 2000, -120, -56; 760, -56, -21],[40, 8, 1],[-7, 11, 4, -5; -5, 10, 3, -5; -240, 370, 136, -168],4,18]
%
% END OF LATTICE L5.18
%
\hbox{}\par\smallskip%
%
% LATTICE L5.19
%
\leavevmode%
\llap{\spacer}%
${L_{5.19}}=\pdualwitharg{2.5}(\main({L_{5.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty}{4}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{}{4}{20}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\-160\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\1060\\200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\200\\39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\4\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\7\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\17\\-100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-20\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[100, -160, -40; -160, 1060, 200; -40, 200, 39],[20, 20, 1],[-3, -1, -5, -12; 4, 1, 7, 17; -24, -6, -40, -100],4,19]
%
% END OF LATTICE L5.19
%
\hbox{}\par\smallskip%
%
% LATTICE L5.20
%
\leavevmode%
\llap{\spacer}%
${L_{5.20}}=\pdualwitharg{2.5}({L_{5.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{2,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{4}{20}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\580\\100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
580\\3040\\520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\520\\89
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\6\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-20\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120, 580, 100; 580, 3040, 520; 100, 520, 89],[20, 20, 1],[0, 1, 5, 1; 1, 1, 6, 5; -6, -7, -40, -30],4,20]
%
% END OF LATTICE L5.20
%
\hbox{}\par\smallskip%
%
% LATTICE L5.21
%
\leavevmode%
\llap{\spacer}%
${L_{5.21}}=\pdualwitharg{2.5}({L_{5.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,1}{\infty z}{4}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{4}{80}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\440\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
440\\-680\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-120\\-21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-40\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-240, 440, 80; 440, -680, -120; 80, -120, -21],[40, 40, 1],[-1, 0, 2, 1; -2, -1, 5, 8; 8, 6, -20, -40],4,21]
%
% END OF LATTICE L5.21
%
\hbox{}\par\smallskip%
%
% LATTICE L5.22
%
\leavevmode%
\llap{\spacer}%
${L_{5.22}}=\pdualwitharg{2.5}({L_{5.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,5}{\infty z}{4}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{4}{80}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-4360\\280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4360\\-77880\\5000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\5000\\-321
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-46
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\8\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-40\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-240, -4360, 280; -4360, -77880, 5000; 280, 5000, -321],[40, 40, 1],[1, 1, -2, -5; 2, -3, -5, 8; 32, -46, -80, 120],4,22]
%
% END OF LATTICE L5.22
%

\medskip%
%------------------------------WEYL-GROUP-W6--------------------------------
%
\leavevmode\llap{}%
$W_{6\phantom{0}\phantom{0}}$%
\qquad\llap{6} lattices, $\chi=2$%
\hfill%
$2223$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L6.1
%
\leavevmode%
\llap{\spacer}%
${L_{6.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{6}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{+}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1660\\80\\100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-2\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\-5\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-4\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1660, 80, 100; 80, -2, -5; 100, -5, -6],[20, 1, 1],[-1, -2, 1, 1; -1, -5, 0, 2; -16, -30, 16, 15],5,1]
%
% END OF LATTICE L6.1
%
\hbox{}\par\smallskip%
%
% LATTICE L6.2
%
\leavevmode%
\llap{\spacer}%
${L_{6.2}}=\pfillwitharg{2}({L_{6.1}})$%
{${}={}$}Nikulin $6$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{-}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[15, 0, -5; 0, -1, 0; -5, 0, 2],[5, 1, 1],[0, 2, 0, -1; 0, 0, -1, -1; 1, 5, -1, -3],5,2]
%
% END OF LATTICE L6.2
%
\hbox{}\par\smallskip%
%
% LATTICE L6.3
%
\leavevmode%
\llap{\spacer}%
${L_{6.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{6.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{+}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-25\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-30\\20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\20\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-5\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10, -25, 15; -25, -30, 20; 15, 20, -13],[5, 5, 1],[-1, -1, 0, 2; -6, -2, 3, 5; -10, -4, 5, 10],5,3]
%
% END OF LATTICE L6.3
%
\hbox{}\par\smallskip%
%
% LATTICE L6.4
%
\leavevmode%
\llap{\spacer}%
${L_{6.4}}=\pdualwitharg{2}({L_{6.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{+}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8440\\-220\\1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\8\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1540\\-40\\281
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\2\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-4\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-20\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8440, -220, 1540; -220, 8, -40; 1540, -40, 281],[20, 4, 1],[3, 11, 2, 0; 2, 0, 0, 1; -16, -60, -11, 0],5,4]
%
% END OF LATTICE L6.4
%
\hbox{}\par\smallskip%
%
% LATTICE L6.5
%
\leavevmode%
\llap{\spacer}%
${L_{6.5}}=\pdualwitharg{5}({L_{6.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{-}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-20\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-5\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20, 0, 0; 0, -10, -5; 0, -5, -2],[20, 5, 1],[-1, 0, 1, 0; -1, -1, 0, 2; 0, 2, 0, -5],5,5]
%
% END OF LATTICE L6.5
%
\hbox{}\par\smallskip%
%
% LATTICE L6.6
%
\leavevmode%
\llap{\spacer}%
${L_{6.6}}=\pdualwitharg{2.5}({L_{6.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{-}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\100\\20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\280\\60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\60\\13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-20\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-20\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40, 100, 20; 100, 280, 60; 20, 60, 13],[20, 20, 1],[2, 0, 0, 1; -1, -1, -1, 0; 0, 4, 5, 0],5,6]
%
% END OF LATTICE L6.6
%

\medskip%
%------------------------------WEYL-GROUP-W7--------------------------------
%
\leavevmode\llap{}%
$W_{7\phantom{0}\phantom{0}}$%
\qquad\llap{16} lattices, $\chi=4$%
\hfill%
$\infty26$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L7.1
%
\leavevmode%
\llap{\spacer}%
${L_{7.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{7},3,2}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,7}{\infty a}{24}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1224\\360\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-102\\39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\39\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\20\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1224, 360, -144; 360, -102, 39; -144, 39, -14],[72, 3, 1],[-1, 3, -1; -7, 20, -3; -9, 24, 0],6,1]
%
% END OF LATTICE L7.1
%
\hbox{}\par\smallskip%
%
% LATTICE L7.2
%
\leavevmode%
\llap{\spacer}%
${L_{7.2}}=\pfillwitharg{2.3}({L_{7.1}})$%
{${}={}$}Nikulin $7$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{2,1}{\infty b}{6}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[6, 0, 0; 0, 0, -1; 0, -1, 0],[6, 1, 1],[-1, 1, 0; 0, 0, -1; -3, 0, 1],6,2]
%
% END OF LATTICE L7.2
%
\hbox{}\par\smallskip%
%
% LATTICE L7.3
%
\leavevmode%
\llap{\spacer}%
${L_{7.3}}=\pfillwitharg{3}({L_{7.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{4,3}{\infty b}{24}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-24\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-2\\39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\39\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\144\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-60\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-93\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-264, -24, 336; -24, -2, 39; 336, 39, -34],[24, 1, 1],[-17, 7, 11; 144, -60, -93; -3, 0, 2],6,3]
%
% END OF LATTICE L7.3
%
\hbox{}\par\smallskip%
%
% LATTICE L7.4
%
\leavevmode%
\llap{\spacer}%
${L_{7.4}}=\pfillwitharg{2}({L_{7.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{6,1}{\infty b}{6}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\36\\36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\-12\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\-15\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\2\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-2\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-90, 36, 36; 36, -12, -15; 36, -15, -14],[18, 3, 1],[2, -3, -1; 2, -2, -3; 3, -6, 0],6,4]
%
% END OF LATTICE L7.4
%
\hbox{}\par\smallskip%
%
% LATTICE L7.5
%
\leavevmode%
\llap{\spacer}%
${L_{7.5}}=\pdualwitharg{2}(\pfillwitharg{2.3}({L_{7.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{4,1}{\infty z}{3}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\18\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-4\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\6\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\6\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-60, 18, -18; 18, -4, 6; -18, 6, -5],[6, 2, 1],[5, -1, -2; 6, 0, -3; -12, 3, 4],6,5]
%
% END OF LATTICE L7.5
%
\hbox{}\par\smallskip%
%
% LATTICE L7.6
%
\leavevmode%
\llap{\spacer}%
${L_{7.6}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{7.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty b}{2}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-18\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-12\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-9\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\2\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6, -18, -6; -18, -12, -9; -6, -9, -4],[6, 3, 1],[-1, -1, 3; -1, 0, 2; 4, 2, -9],6,6]
%
% END OF LATTICE L7.6
%
\hbox{}\par\smallskip%
%
% LATTICE L7.7
%
\leavevmode%
\llap{\spacer}%
${L_{7.7}}=\pdualwitharg{2.3}(\pfillwitharg{2.3}({L_{7.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\90\\36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\744\\300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\300\\121
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12, 90, 36; 90, 744, 300; 36, 300, 121],[6, 6, 1],[-2, 0, 1; 1, 2, 0; -2, -5, 0],6,7]
%
% END OF LATTICE L7.7
%
\hbox{}\par\smallskip%
%
% LATTICE L7.8
%
\leavevmode%
\llap{\spacer}%
${L_{7.8}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{7.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{6,5}{\infty b}{6}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\18\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-12\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-9\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18, 18, 18; 18, -12, -9; 18, -9, -4],[18, 3, 1],[2, -1, -1; 5, -2, -3; -3, 0, 2],6,8]
%
% END OF LATTICE L7.8
%
\hbox{}\par\smallskip%
%
% LATTICE L7.9
%
\leavevmode%
\llap{\spacer}%
${L_{7.9}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{7.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty b}{8}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3672\\48\\192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
192\\-3\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\4\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3672, 48, 192; 48, 6, -3; 192, -3, -10],[24, 3, 1],[-1, 3, 0; -1, 4, -1; -19, 56, 0],6,9]
%
% END OF LATTICE L7.9
%
\hbox{}\par\smallskip%
%
% LATTICE L7.10
%
\leavevmode%
\llap{\spacer}%
${L_{7.10}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{7.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,1}{\infty z}{3}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\18\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\12\\12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\12\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[36, 18, 18; 18, 12, 12; 18, 12, 11],[18, 6, 1],[0, 1, -1; 1, 1, 0; 0, -3, 0],6,10]
%
% END OF LATTICE L7.10
%
\hbox{}\par\smallskip%
%
% LATTICE L7.11
%
\leavevmode%
\llap{\spacer}%
${L_{7.11}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{7.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,5}{\infty z}{3}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1368\\342\\648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
342\\84\\162
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
648\\162\\307
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\1\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\1\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1368, 342, 648; 342, 84, 162; 648, 162, 307],[18, 6, 1],[-3, 4, -1; 1, 1, 0; 6, -9, 2],6,11]
%
% END OF LATTICE L7.11
%
\hbox{}\par\smallskip%
%
% LATTICE L7.12
%
\leavevmode%
\llap{\spacer}%
${L_{7.12}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{7.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{8,1}{\infty z}{12}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\-1704\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1704\\2416\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-240\\23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-15\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\-27\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\13\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1200, -1704, 168; -1704, 2416, -240; 168, -240, 23],[24, 8, 1],[-25, -46, 22; -15, -27, 13; 24, 54, -24],6,12]
%
% END OF LATTICE L7.12
%
\hbox{}\par\smallskip%
%
% LATTICE L7.13
%
\leavevmode%
\llap{\spacer}%
${L_{7.13}}=\pdualwitharg{3}({L_{7.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,11}{\infty b}{24}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\15\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-72, 0, 72; 0, 6, 15; 72, 15, -34],[72, 3, 1],[-3, 1, 2; 8, -4, -5; -3, 0, 2],6,13]
%
% END OF LATTICE L7.13
%
\hbox{}\par\smallskip%
%
% LATTICE L7.14
%
\leavevmode%
\llap{\spacer}%
${L_{7.14}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{7.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,1}{\infty z}{4}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\2\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-48, 24, -24; 24, 144, 0; -24, 0, -11],[24, 24, 1],[4, 1, -11; -1, 0, 2; -8, -2, 24],6,14]
%
% END OF LATTICE L7.14
%
\hbox{}\par\smallskip%
%
% LATTICE L7.15
%
\leavevmode%
\llap{\spacer}%
${L_{7.15}}=\pdualwitharg{2}({L_{7.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,1}{\infty z}{12}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-72\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\48\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\48\\23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-54\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-720, -72, 72; -72, 48, 48; 72, 48, 23],[72, 24, 1],[0, -1, 13; 1, 4, -54; 0, -6, 72],6,15]
%
% END OF LATTICE L7.15
%
\hbox{}\par\smallskip%
%
% LATTICE L7.16
%
\leavevmode%
\llap{\spacer}%
${L_{7.16}}=\pdualwitharg{2.3}({L_{7.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,17}{\infty z}{12}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
363600\\44136\\-6120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44136\\5232\\-744
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6120\\-744\\103
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\10\\-1176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\19\\-2298
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-9\\1064
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[363600, 44136, -6120; 44136, 5232, -744; -6120, -744, 103],[72, 24, 1],[-21, -41, 19; 10, 19, -9; -1176, -2298, 1064],6,16]
%
% END OF LATTICE L7.16
%

\medskip%
%------------------------------WEYL-GROUP-W8--------------------------------
%
\leavevmode\llap{}%
$W_{8\phantom{0}\phantom{0}}$%
\qquad\llap{6} lattices, $\chi=6$%
\hfill%
$4242\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L8.1
%
\leavevmode%
\llap{\spacer}%
${L_{8.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{8}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\56\\28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\-10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-5\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-308, 56, 28; 56, -10, -5; 28, -5, -2],[28, 1, 1],[0, -1, 0, 1; 1, -6, -1, 6; -3, 0, 2, -2],7,1]
%
% END OF LATTICE L8.1
%
\hbox{}\par\smallskip%
%
% LATTICE L8.2
%
\leavevmode%
\llap{\spacer}%
${L_{8.2}}=\pfillwitharg{2}({L_{8.1}})$%
{${}={}$}Nikulin $8$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\7\\14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-2\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-7, 0, 0; 0, 1, 0; 0, 0, 1],[7, 1, 1],[0, 0, -1, -1; -1, 1, 0, -2; 1, 0, -3, -2],7,2]
%
% END OF LATTICE L8.2
%
\hbox{}\par\smallskip%
%
% LATTICE L8.3
%
\leavevmode%
\llap{\spacer}%
${L_{8.3}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{8.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-91, -49, -14; -49, -21, -7; -14, -7, -2],[7, 7, 1],[0, -2, -1, 1; 2, -1, -3, 0; -7, 14, 14, -7],7,3]
%
% END OF LATTICE L8.3
%
\hbox{}\par\smallskip%
%
% LATTICE L8.4
%
\leavevmode%
\llap{\spacer}%
${L_{8.4}}=\pdualwitharg{2}({L_{8.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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140\\24\\-36
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[952, 140, -252; 140, 24, -36; -252, -36, 67],[28, 4, 1],[-7, -3, 1, -3; 5, 2, 0, 3; -24, -10, 4, -10],7,4]
%
% END OF LATTICE L8.4
%
\hbox{}\par\smallskip%
%
% LATTICE L8.5
%
\leavevmode%
\llap{\spacer}%
${L_{8.5}}=\pdualwitharg{7}({L_{8.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-364, -84, -28; -84, -14, -7; -28, -7, -2],[28, 7, 1],[-1, -1, 1, 1; 2, -2, -3, 0; 7, 14, -7, -14],7,5]
%
% END OF LATTICE L8.5
%
\hbox{}\par\smallskip%
%
% LATTICE L8.6
%
\leavevmode%
\llap{\spacer}%
${L_{8.6}}=\pdualwitharg{2.7}({L_{8.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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-308\\3864\\-1064
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[56, -308, 84; -308, 3864, -1064; 84, -1064, 293],[28, 28, 1],[1, 1, 1, 1; 0, 27, 31, 4; 0, 98, 112, 14],7,6]
%
% END OF LATTICE L8.6
%

\medskip%
%------------------------------WEYL-GROUP-W9--------------------------------
%
\leavevmode\llap{}%
$W_{9\phantom{0}\phantom{0}}$%
\qquad\llap{22} lattices, $\chi=12$%
\hfill%
$\infty2222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L9.1
%
\leavevmode%
\llap{\spacer}%
${L_{9.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{9}}%
\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-21252\\532\\1204
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532\\-12\\-33
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1204\\-33\\-62
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-31\\-602\\-280
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\56\\28
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\98\\45
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\28\\7
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-172\\-82
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{$\left.\llap{\phantom{%
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28\\-28\\-14\\-42\\0
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-28\\28\\0\\-56\\-28
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\2\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-56\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-4\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-21252, 532, 1204; 532, -12, -33; 1204, -33, -62],[28, 1, 1],[-31, 3, 5, 1, -9; -602, 56, 98, 28, -172; -280, 28, 45, 7, -82],8,1]
%
% END OF LATTICE L9.1
%
\hbox{}\par\smallskip%
%
% LATTICE L9.2
%
\leavevmode%
\llap{\spacer}%
${L_{9.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{7}}%
\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{4,1}{\infty a}{56}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39928\\224\\1176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
224\\-1\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1176\\-7\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\126\\77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\28\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-42\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-35\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\48\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-28\\-14\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\56\\0\\-56\\-56
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-14\\0\\4\\0\\-8
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-21\\-56\\0\\7\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-8\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-39928, 224, 1176; 224, -1, -7; 1176, -7, -34],[56, 1, 1],[3, 1, -1, -1, 1; 126, 28, -42, -35, 48; 77, 28, -26, -28, 24],8,2]
%
% END OF LATTICE L9.2
%
\hbox{}\par\smallskip%
%
% LATTICE L9.3
%
\leavevmode%
\llap{\spacer}%
${L_{9.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{4,1}{\infty b}{56}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-655704\\94472\\11312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
94472\\-13611\\-1630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11312\\-1630\\-195
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\287\\385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-28\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-42\\-55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\14\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\172\\244
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-28\\-7\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\56\\0\\-112\\-56
\end{tabular}\endgroup%
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-7\\0\\1\\0\\-4
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-42\\-112\\0\\28\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-4\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-655704, 94472, 11312; 94472, -13611, -1630; 11312, -1630, -195],[56, 1, 1],[48, -5, -7, 3, 29; 287, -28, -42, 14, 172; 385, -56, -55, 56, 244],8,3]
%
% END OF LATTICE L9.3
%
\hbox{}\par\smallskip%
%
% LATTICE L9.4
%
\leavevmode%
\llap{\spacer}%
${L_{9.4}}=\pfillwitharg{2}({L_{9.1}})$%
{${}={}$}Nikulin $9$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{2,1}{\infty}{7}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-7\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-7\\-7\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\7\\0\\-28\\-7
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\2\\0\\-2
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-28\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-2\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[7, 0, 0; 0, 0, -1; 0, -1, 0],[7, 1, 1],[-1, 1, 0, -4, -1; 0, 0, -1, -7, -1; -7, 0, 1, -7, -3],8,4]
%
% END OF LATTICE L9.4
%
\hbox{}\par\smallskip%
%
% LATTICE L9.5
%
\leavevmode%
\llap{\spacer}%
${L_{9.5}}=\pfillwitharg{2}({L_{9.2}})$%
{${}={}$}Nikulin $7'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{2,1}{\infty a}{14}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4830\\3374\\3108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3374\\2357\\2171
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3108\\2171\\2000
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\-28\\-49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
74\\-35\\-77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-18\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-14\\-7\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\14\\0\\-28\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\1\\0\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-28\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-2\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[4830, 3374, 3108; 3374, 2357, 2171; 3108, 2171, 2000],[14, 1, 1],[51, -9, 2, 74, 37; -28, 0, -1, -35, -18; -49, 14, -2, -77, -38],8,5]
%
% END OF LATTICE L9.5
%
\hbox{}\par\smallskip%
%
% LATTICE L9.6
%
\leavevmode%
\llap{\spacer}%
${L_{9.6}}=\main({L_{9.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{4,1}{\infty}{28}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-364\\28\\112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-2\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\-7\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\63\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\14\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-21\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-35\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\24\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-14\\-7\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\28\\0\\-56\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\2\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-56\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-4\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-364, 28, 112; 28, -2, -7; 112, -7, -17],[28, 1, 1],[3, 1, -1, -2, 1; 63, 14, -21, -35, 24; -7, 0, 2, 0, -4],8,6]
%
% END OF LATTICE L9.6
%
\hbox{}\par\smallskip%
%
% LATTICE L9.7
%
\leavevmode%
\llap{\spacer}%
${L_{9.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{9.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{4,3}{\infty z}{7}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41146\\1526\\-19754
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1526\\-48\\730
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19754\\730\\-9483
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
466\\-301\\-994
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\14\\49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\49\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\14\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\-43\\-145
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-14\\-14\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\7\\0\\-28\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\2\\0\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-28\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-2\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-41146, 1526, -19754; 1526, -48, 730; -19754, 730, -9483],[14, 2, 1],[466, -23, -75, -13, 68; -301, 14, 49, 14, -43; -994, 49, 160, 28, -145],8,7]
%
% END OF LATTICE L9.7
%
\hbox{}\par\smallskip%
%
% LATTICE L9.8
%
\leavevmode%
\llap{\spacer}%
${L_{9.8}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{9.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{2,1}{\infty}{1}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-49\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-168\\-49
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-49\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-2\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-7\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-4\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\14\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-14\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14, -49, -14; -49, -168, -49; -14, -49, -13],[7, 7, 1],[4, -1, -7, 3, 12; -1, 0, 2, 0, -2; -1, 1, 0, -4, -7],8,8]
%
% END OF LATTICE L9.8
%
\hbox{}\par\smallskip%
%
% LATTICE L9.9
%
\leavevmode%
\llap{\spacer}%
${L_{9.9}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{9.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty a}{2}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89950\\-4676\\-39130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4676\\-217\\-2037
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39130\\-2037\\-17022
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
160\\-40\\-363
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\4\\34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-182\\45\\413
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\1\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
321\\-82\\-728
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-7\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-4\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\7\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-14\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-89950, -4676, -39130; -4676, -217, -2037; -39130, -2037, -17022],[14, 7, 1],[160, -15, -182, -7, 321; -40, 4, 45, 1, -82; -363, 34, 413, 16, -728],8,9]
%
% END OF LATTICE L9.9
%
\hbox{}\par\smallskip%
%
% LATTICE L9.10
%
\leavevmode%
\llap{\spacer}%
${L_{9.10}}=\pdualwitharg{2}({L_{9.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{2,1}{\infty z}{7}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60592\\-33796\\-19040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33796\\18856\\10620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19040\\10620\\5983
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
155\\-14\\518
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-1\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
438\\-35\\1456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\-9\\366
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-14\\-28\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\7\\0\\-56\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\8\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-56\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-8\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[60592, -33796, -19040; -33796, 18856, 10620; -19040, 10620, 5983],[28, 4, 1],[155, -11, 12, 438, 110; -14, 0, -1, -35, -9; 518, -35, 40, 1456, 366],8,10]
%
% END OF LATTICE L9.10
%
\hbox{}\par\smallskip%
%
% LATTICE L9.11
%
\leavevmode%
\llap{\spacer}%
${L_{9.11}}=\pdualwitharg{2}(\main({L_{9.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{4,3}{\infty}{28}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3640\\1820\\-952
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1820\\908\\-476
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-952\\-476\\249
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\7\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\14\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\2\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-28\\-28\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\28\\0\\-112\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\8\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-112\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-4\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3640, 1820, -952; 1820, 908, -476; -952, -476, 249],[28, 4, 1],[-18, 11, 1, -51, -7; 7, 0, 0, 14, 2; -56, 42, 4, -168, -23],8,11]
%
% END OF LATTICE L9.11
%
\hbox{}\par\smallskip%
%
% LATTICE L9.12
%
\leavevmode%
\llap{\spacer}%
${L_{9.12}}=\pdualwitharg{7}(\main({L_{9.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{4,1}{\infty}{4}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2324\\-336\\252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-42\\35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252\\35\\-27
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\16\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-7\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-8\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\14\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-28\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2324, -336, 252; -336, -42, 35; 252, 35, -27],[28, 7, 1],[1, -1, -2, 1, 7; 3, -2, -7, 1, 16; 13, -12, -28, 10, 84],8,12]
%
% END OF LATTICE L9.12
%
\hbox{}\par\smallskip%
%
% LATTICE L9.13
%
\leavevmode%
\llap{\spacer}%
${L_{9.13}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{9.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28000\\-482146\\-222768
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-482146\\8303414\\3836462
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-222768\\3836462\\1772577
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\664\\-1438
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-54\\117
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\181\\-392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\953\\-2064
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\1671\\-3619
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-14\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-4\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\14\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-4\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-14\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28000, -482146, -222768; -482146, 8303414, 3836462; -222768, 3836462, 1772577],[14, 14, 1],[-7, 1, -2, -11, -19; 664, -54, 181, 953, 1671; -1438, 117, -392, -2064, -3619],8,13]
%
% END OF LATTICE L9.13
%
\hbox{}\par\smallskip%
%
% LATTICE L9.14
%
\leavevmode%
\llap{\spacer}%
${L_{9.14}}=\pdualwitharg{7}({L_{9.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty b}{4}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\784\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
784\\1456\\-259
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-259\\46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\14\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-4\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-14\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-8\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\14\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-8\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-28\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[420, 784, -140; 784, 1456, -259; -140, -259, 46],[28, 7, 1],[1, -3, -5, -1, 3; -2, 8, 14, 4, -4; -8, 36, 63, 19, -14],8,14]
%
% END OF LATTICE L9.14
%
\hbox{}\par\smallskip%
%
% LATTICE L9.15
%
\leavevmode%
\llap{\spacer}%
${L_{9.15}}=\pdualwitharg{7}({L_{9.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,1}{\infty a}{8}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75208\\-38920\\2016
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38920\\-20139\\1043
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2016\\1043\\-54
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\6\\41
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-14\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\32\\280
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-14\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-8\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\28\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-56\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-75208, -38920, 2016; -38920, -20139, 1043; 2016, 1043, -54],[56, 7, 1],[-2, 1, 5, 0, -9; 6, -4, -14, 1, 32; 41, -40, -84, 19, 280],8,15]
%
% END OF LATTICE L9.15
%
\hbox{}\par\smallskip%
%
% LATTICE L9.16
%
\leavevmode%
\llap{\spacer}%
${L_{9.16}}=\pdualwitharg{7}({L_{9.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,1}{\infty b}{8}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11368\\504\\280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\-21\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\-14\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\15\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\108\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-7\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-16\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\7\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-16\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-28\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11368, 504, 280; 504, -21, -14; 280, -14, -5],[56, 7, 1],[1, -1, -1, 1, 7; 15, -16, -15, 16, 108; 13, -12, -14, 10, 84],8,16]
%
% END OF LATTICE L9.16
%
\hbox{}\par\smallskip%
%
% LATTICE L9.17
%
\leavevmode%
\llap{\spacer}%
${L_{9.17}}=\pdualwitharg{2}({L_{9.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{8,3}{\infty z}{28}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-142072\\1512\\5264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1512\\-16\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5264\\-56\\-195
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-7\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\49\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\16\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\7\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\-56\\-56\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\28\\0\\-112\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\8\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-112\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-8\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-142072, 1512, 5264; 1512, -16, -56; 5264, -56, -195],[56, 8, 1],[2, -1, -1, -3, 0; -7, 49, 16, 7, -7; 56, -42, -32, -84, 2],8,17]
%
% END OF LATTICE L9.17
%
\hbox{}\par\smallskip%
%
% LATTICE L9.18
%
\leavevmode%
\llap{\spacer}%
${L_{9.18}}=\pdualwitharg{2}({L_{9.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{8,7}{\infty z}{28}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\3080\\-392
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3080\\-48336\\6152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-392\\6152\\-783
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-7\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\98\\770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\30\\236
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\7\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\-56\\-56\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\28\\0\\-112\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\8\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-112\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-4\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-168, 3080, -392; 3080, -48336, 6152; -392, 6152, -783],[56, 8, 1],[2, -1, -1, -3, 0; -7, 98, 30, 7, -7; -56, 770, 236, 56, -55],8,18]
%
% END OF LATTICE L9.18
%
\hbox{}\par\smallskip%
%
% LATTICE L9.19
%
\leavevmode%
\llap{\spacer}%
${L_{9.19}}=\pdualwitharg{2.7}({L_{9.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{2,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\-1764\\476
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1764\\11088\\-2996
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
476\\-2996\\809
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\1\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-12\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-6\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-2\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-28\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-8\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\56\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-8\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-56\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[280, -1764, 476; -1764, 11088, -2996; 476, -2996, 809],[28, 28, 1],[-4, -1, 21, 17, 11; 1, -1, -12, -6, -2; 6, -3, -56, -32, -14],8,19]
%
% END OF LATTICE L9.19
%
\hbox{}\par\smallskip%
%
% LATTICE L9.20
%
\leavevmode%
\llap{\spacer}%
${L_{9.20}}=\pdualwitharg{2.7}(\main({L_{9.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty}{4}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
756\\-2324\\700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2324\\7336\\-2212
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
700\\-2212\\667
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-23\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-20\\-63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-28\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-16\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\56\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-16\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-28\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[756, -2324, 700; -2324, 7336, -2212; 700, -2212, 667],[28, 28, 1],[-1, 0, -2, -4, -3; -10, -3, -9, -23, -20; -32, -10, -28, -72, -63],8,20]
%
% END OF LATTICE L9.20
%
\hbox{}\par\smallskip%
%
% LATTICE L9.21
%
\leavevmode%
\llap{\spacer}%
${L_{9.21}}=\pdualwitharg{2.7}({L_{9.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,3}{\infty z}{4}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\168\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-15\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-56\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-16\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\56\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-16\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-56\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-112, 168, -56; 168, 56, 0; -56, 0, -5],[56, 56, 1],[-1, 1, 4, 1, -1; 2, -5, -15, -3, 4; 8, -18, -56, -12, 14],8,21]
%
% END OF LATTICE L9.21
%
\hbox{}\par\smallskip%
%
% LATTICE L9.22
%
\leavevmode%
\llap{\spacer}%
${L_{9.22}}=\pdualwitharg{2.7}({L_{9.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,7}{\infty z}{4}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100464\\-38136\\4984
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38136\\-14168\\1848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4984\\1848\\-241
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\34\\240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-63\\-442
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-195\\-1372
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-35\\-248
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\31\\217
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-56\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-16\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\56\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-16\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-28\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-100464, -38136, 4984; -38136, -14168, 1848; 4984, 1848, -241],[56, 56, 1],[-1, 2, 6, 1, -1; 34, -63, -195, -35, 31; 240, -442, -1372, -248, 217],8,22]
%
% END OF LATTICE L9.22
%

\medskip%
%------------------------------WEYL-GROUP-W10--------------------------------
%
\leavevmode\llap{}%
$W_{10\phantom{0}}$%
\qquad\llap{16} lattices, $\chi=6$%
\hfill%
$\infty222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L10.1
%
\leavevmode%
\llap{\spacer}%
${L_{10.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{25\rightarrow N_{10},5,2*}%
\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{20,9}{\infty a}{40}\above{1pt}{1pt}{b}{2}{50}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95800\\1200\\-10800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\-10\\165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10800\\165\\-1042
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-443\\75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\476\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\885\\-150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-96\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-20\\-25\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\0\\50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-95800, 1200, -10800; 1200, -10, 165; -10800, 165, -1042],[200, 5, 1],[-14, 15, 28, -3; -443, 476, 885, -96; 75, -80, -150, 16],9,1]
%
% END OF LATTICE L10.1
%
\hbox{}\par\smallskip%
%
% LATTICE L10.2
%
\leavevmode%
\llap{\spacer}%
${L_{10.2}}=\pfillwitharg{2.5}({L_{10.1}})$%
{${}={}$}Nikulin $10$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{2,1}{\infty a}{10}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-10\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[10, 0, 0; 0, 0, -1; 0, -1, 0],[10, 1, 1],[-1, 1, 0, -1; 0, 0, -1, -2; -5, 0, 1, -2],9,2]
%
% END OF LATTICE L10.2
%
\hbox{}\par\smallskip%
%
% LATTICE L10.3
%
\leavevmode%
\llap{\spacer}%
${L_{10.3}}=\pfillwitharg{5}({L_{10.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{4,1}{\infty a}{40}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-760\\40\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\-2\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-5\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-30\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\15\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-20\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-760, 40, 80; 40, -2, -5; 80, -5, -2],[40, 1, 1],[-2, -1, 1, 1; -30, -20, 15, 16; -5, 0, 2, 0],9,3]
%
% END OF LATTICE L10.3
%
\hbox{}\par\smallskip%
%
% LATTICE L10.4
%
\leavevmode%
\llap{\spacer}%
${L_{10.4}}=\pfillwitharg{2}({L_{10.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{10,9}{\infty a}{10}\above{1pt}{1pt}{r}{2}{50}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-950\\200\\400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\-40\\-85
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\-85\\-168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\4\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-4\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\2\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-10\\-25\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\0\\50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-950, 200, 400; 200, -40, -85; 400, -85, -168],[50, 5, 1],[3, -5, -1, 3; 4, -4, -5, 2; 5, -10, 0, 6],9,4]
%
% END OF LATTICE L10.4
%
\hbox{}\par\smallskip%
%
% LATTICE L10.5
%
\leavevmode%
\llap{\spacer}%
${L_{10.5}}=\pdualwitharg{2}(\pfillwitharg{2.5}({L_{10.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{4,3}{\infty z}{5}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-580\\50\\-230
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\-4\\20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-230\\20\\-91
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\10\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-10\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\5\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-580, 50, -230; 50, -4, 20; -230, 20, -91],[10, 2, 1],[9, -2, -2, 2; 10, 0, -5, -1; -20, 5, 4, -5],9,5]
%
% END OF LATTICE L10.5
%
\hbox{}\par\smallskip%
%
% LATTICE L10.6
%
\leavevmode%
\llap{\spacer}%
${L_{10.6}}=\pdualwitharg{5}(\pfillwitharg{2.5}({L_{10.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty a}{2}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-60\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-25\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\2\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10, -50, 0; -50, -60, -25; 0, -25, -2],[10, 5, 1],[-3, -1, 10, 9; -1, 0, 2, 0; 8, 2, -25, -20],9,6]
%
% END OF LATTICE L10.6
%
\hbox{}\par\smallskip%
%
% LATTICE L10.7
%
\leavevmode%
\llap{\spacer}%
${L_{10.7}}=\pdualwitharg{2.5}(\pfillwitharg{2.5}({L_{10.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\450\\200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
450\\10120\\4500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\4500\\2001
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-17\\38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-38\\85
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20, 450, 200; 450, 10120, 4500; 200, 4500, 2001],[10, 10, 1],[2, 0, 1, 5; -17, 4, 0, -38; 38, -9, 0, 85],9,7]
%
% END OF LATTICE L10.7
%
\hbox{}\par\smallskip%
%
% LATTICE L10.8
%
\leavevmode%
\llap{\spacer}%
${L_{10.8}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{10.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,1}{\infty a}{8}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\-80\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-30\\25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\25\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-11\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40, -80, 40; -80, -30, 25; 40, 25, -18],[40, 5, 1],[1, -1, -2, 1; 5, -4, -11, 0; 9, -8, -20, 0],9,8]
%
% END OF LATTICE L10.8
%
\hbox{}\par\smallskip%
%
% LATTICE L10.9
%
\leavevmode%
\llap{\spacer}%
${L_{10.9}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{10.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{10,1}{\infty a}{10}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{50}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-10\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\10\\0\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-50\\0\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[50, 0, 0; 0, -10, -5; 0, -5, -2],[50, 5, 1],[0, 1, 0, -1; 2, -2, -1, 0; -5, 0, 2, 0],9,9]
%
% END OF LATTICE L10.9
%
\hbox{}\par\smallskip%
%
% LATTICE L10.10
%
\leavevmode%
\llap{\spacer}%
${L_{10.10}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{10.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{8,7}{\infty z}{20}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\1560\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1560\\-24272\\3112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\3112\\-399
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\50\\390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-40\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\20\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-80, 1560, -200; 1560, -24272, 3112; -200, 3112, -399],[40, 8, 1],[2, -1, -1, 0; -5, 50, 2, -5; -40, 390, 16, -39],9,10]
%
% END OF LATTICE L10.10
%
\hbox{}\par\smallskip%
%
% LATTICE L10.11
%
\leavevmode%
\llap{\spacer}%
${L_{10.11}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{10.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{20,11}{\infty z}{5}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2600\\250\\1250
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
250\\20\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1250\\120\\601
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\2\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-10\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\5\\0\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-25\\0\\25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2600, 250, 1250; 250, 20, 120; 1250, 120, 601],[50, 10, 1],[0, -17, -1, 12; 1, 2, 0, 0; 0, 35, 2, -25],9,11]
%
% END OF LATTICE L10.11
%
\hbox{}\par\smallskip%
%
% LATTICE L10.12
%
\leavevmode%
\llap{\spacer}%
${L_{10.12}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{10.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{20,19}{\infty z}{5}\above{1pt}{1pt}{r}{2}{100}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\50\\50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\80\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\40\\29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\3\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-10\\-50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\5\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\0\\100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[100, 50, 50; 50, 80, 40; 50, 40, 29],[50, 10, 1],[4, 2, -1, 1; 3, 1, 0, 1; -10, -5, 0, -3],9,12]
%
% END OF LATTICE L10.12
%
\hbox{}\par\smallskip%
%
% LATTICE L10.13
%
\leavevmode%
\llap{\spacer}%
${L_{10.13}}=\pdualwitharg{5}({L_{10.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{20,1}{\infty a}{40}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{200}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-20\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\40\\0\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-200\\0\\200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[200, 0, 0; 0, -10, -5; 0, -5, -2],[200, 5, 1],[0, -1, 0, 1; 2, -4, -1, 0; -5, 0, 2, 0],9,13]
%
% END OF LATTICE L10.13
%
\hbox{}\par\smallskip%
%
% LATTICE L10.14
%
\leavevmode%
\llap{\spacer}%
${L_{10.14}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{10.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,7}{\infty z}{4}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39440\\-41320\\2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41320\\-42000\\2680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2640\\2680\\-171
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\26\\392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-47\\-706
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-53\\-800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\23\\345
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-39440, -41320, 2640; -41320, -42000, 2680; 2640, 2680, -171],[40, 40, 1],[-1, 2, 2, -1; 26, -47, -53, 23; 392, -706, -800, 345],9,14]
%
% END OF LATTICE L10.14
%
\hbox{}\par\smallskip%
%
% LATTICE L10.15
%
\leavevmode%
\llap{\spacer}%
${L_{10.15}}=\pdualwitharg{2.5}({L_{10.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{40,31}{\infty z}{20}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1131600\\11000\\-222200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11000\\-80\\2160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-222200\\2160\\-43631
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\2\\-280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-1\\550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-1\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\0\\-275
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-40\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\20\\0\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-50\\0\\25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1131600, 11000, -222200; 11000, -80, 2160; -222200, 2160, -43631],[200, 40, 1],[55, -108, -22, 54; 2, -1, -1, 0; -280, 550, 112, -275],9,15]
%
% END OF LATTICE L10.15
%
\hbox{}\par\smallskip%
%
% LATTICE L10.16
%
\leavevmode%
\llap{\spacer}%
${L_{10.16}}=\pdualwitharg{2}({L_{10.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{40,39}{\infty z}{20}\above{1pt}{1pt}{*}{2}{400}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15800400\\39800\\79400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39800\\-80\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79400\\-200\\-399
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-1\\1990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-199
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-40\\-200\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\20\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\0\\400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15800400, 39800, 79400; 39800, -80, -200; 79400, -200, -399],[200, 40, 1],[-1, 10, 2, -1; 2, -1, -5, 0; -200, 1990, 400, -199],9,16]
%
% END OF LATTICE L10.16
%

\medskip%
%------------------------------WEYL-GROUP-W11--------------------------------
%
\leavevmode\llap{}%
$W_{11\phantom{0}}$%
\qquad\llap{6} lattices, $\chi=5$%
\hfill%
$4223$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L11.1
%
\leavevmode%
\llap{\spacer}%
${L_{11.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{11}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{+}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33924\\748\\1012
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
748\\-14\\-23
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1012\\-23\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\16\\55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\22\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-33\\-110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-15\\-56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-11\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-33924, 748, 1012; 748, -14, -23; 1012, -23, -30],[44, 1, 1],[2, 3, -4, -2; 16, 22, -33, -15; 55, 84, -110, -56],10,1]
%
% END OF LATTICE L11.1
%
\hbox{}\par\smallskip%
%
% LATTICE L11.2
%
\leavevmode%
\llap{\spacer}%
${L_{11.2}}=\pfillwitharg{2}({L_{11.1}})$%
{${}={}$}Nikulin $11$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{-}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-11\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-11, 0, 0; 0, 1, 0; 0, 0, 1],[11, 1, 1],[0, 0, -3, -1; -1, 1, 0, -2; 1, 0, -11, -3],10,2]
%
% END OF LATTICE L11.2
%
\hbox{}\par\smallskip%
%
% LATTICE L11.3
%
\leavevmode%
\llap{\spacer}%
${L_{11.3}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{11.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{}{4}{11}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{+}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-154\\-253\\121
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-253\\-330\\176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
121\\176\\-89
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-12\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-6\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\11\\44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\-11\\-11\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\11\\0\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-22\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-154, -253, 121; -253, -330, 176; 121, 176, -89],[11, 11, 1],[-15, -14, 2, 16; -12, -6, 3, 11; -44, -33, 8, 44],10,3]
%
% END OF LATTICE L11.3
%
\hbox{}\par\smallskip%
%
% LATTICE L11.4
%
\leavevmode%
\llap{\spacer}%
${L_{11.4}}=\pdualwitharg{2}({L_{11.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{+}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
682264\\-19932\\-188672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19932\\584\\5512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-188672\\5512\\52175
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\7\\-196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\2\\-134
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73\\0\\-264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\2\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-4\\-44\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\0\\88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-8\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[682264, -19932, -188672; -19932, 584, 5512; -188672, 5512, 52175],[44, 4, 1],[-54, -37, -73, -11; 7, 2, 0, 2; -196, -134, -264, -40],10,4]
%
% END OF LATTICE L11.4
%
\hbox{}\par\smallskip%
%
% LATTICE L11.5
%
\leavevmode%
\llap{\spacer}%
${L_{11.5}}=\pdualwitharg{11}({L_{11.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{*}{4}{44}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{-}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1804\\528\\264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\-154\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
264\\-77\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\-22\\-11\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\44\\0\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-44\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1804, 528, 264; 528, -154, -77; 264, -77, -38],[44, 11, 1],[0, 3, 0, -2; 5, 10, -1, -7; -11, 0, 2, 0],10,5]
%
% END OF LATTICE L11.5
%
\hbox{}\par\smallskip%
%
% LATTICE L11.6
%
\leavevmode%
\llap{\spacer}%
${L_{11.6}}=\pdualwitharg{2.11}({L_{11.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${88}\above{1pt}{1pt}{*}{4}{44}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{88}\above{1pt}{1pt}{-}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1144\\1540\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1540\\2200\\-572
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-572\\149
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-39\\-132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-7\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-13\\-44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\-44\\-44\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\44\\0\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-88\\0\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1144, 1540, -396; 1540, 2200, -572; -396, -572, 149],[44, 44, 1],[7, 2, 0, 2; -39, -7, 1, -13; -132, -22, 4, -44],10,6]
%
% END OF LATTICE L11.6
%

\medskip%
%------------------------------WEYL-GROUP-W12--------------------------------
%
\leavevmode\llap{}%
$W_{12\phantom{0}}$%
\qquad\llap{22} lattices, $\chi=36$%
\hfill%
$\infty222\infty222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L12.1
%
\leavevmode%
\llap{\spacer}%
${L_{12.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{12}}%
\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{1,0}{\infty b}{44}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2348148\\9768\\70268
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9768\\-40\\-293
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70268\\-293\\-2102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42041\\1285284\\1225224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-154\\-4709\\-4488
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1281\\-39162\\-37333
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-1826\\-1848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-157\\-4796\\-4576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-500\\-465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-1040\\-958
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\-44\\-22\\-176\\-1012\\-924\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\44\\0\\-132\\-924\\-1012\\-44\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\2\\0\\-22\\-44\\-4\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-132\\0\\4\\0\\-44\\-8\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\-44\\-22\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\44\\0\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-132\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1012\\-924\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\-1012\\-44\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-44\\-4\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-44\\-8\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2348148, 9768, 70268; 9768, -40, -293; 70268, -293, -2102],[44, 1, 1],[-63, -157, -16, -33, -91, 3, 9, 5; -1826, -4796, -500, -1040, -2882, 88, 286, 184; -1848, -4576, -465, -958, -2640, 88, 261, 142],11,1]
%
% END OF LATTICE L12.1
%
\hbox{}\par\smallskip%
%
% LATTICE L12.2
%
\leavevmode%
\llap{\spacer}%
${L_{12.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N'_{8}}%
\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{4,1}{\infty b}{88}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-238040\\528\\4840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\-1\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4840\\-11\\-98
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10032\\-1672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-31\\5
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-192\\31
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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5\\330\\209
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\44\\44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-66\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-160\\-132
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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22\\-44\\-22\\-176\\-506\\-924\\-22\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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22\\-44\\-22\\-176
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\88\\0\\-264
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-264\\0\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-506\\-924\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\-2024\\-88\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-88\\-8\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-88\\-16\\-72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-238040, 528, 4840; 528, -1, -11; 4840, -11, -98],[88, 1, 1],[5, 1, -1, -3, -5, -1, 1, 3; 330, 44, -66, -160, -198, 220, 78, 208; 209, 44, -42, -132, -231, -88, 40, 124],11,2]
%
% END OF LATTICE L12.2
%
\hbox{}\par\smallskip%
%
% LATTICE L12.3
%
\leavevmode%
\llap{\spacer}%
${L_{12.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{4,1}{\infty a}{88}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-1848\\968
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-1826\\945
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
968\\945\\-483
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
527\\-1320\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
472\\-1181\\-1416
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-218\\545\\653
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-105\\209\\198
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\44\\44
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-22\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\-160\\-160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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22\\-44\\-11\\-176\\-506\\-924\\-11\\0
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\kern3pt%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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22\\-44\\-11\\-176
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\88\\0\\-264
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-11\\0\\1\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-264\\0\\8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-506\\-924\\-11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\-2024\\-44\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-44\\-2\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-88\\-8\\-72
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1848, -1848, 968; -1848, -1826, 945; 968, 945, -483],[88, 1, 1],[-105, -21, 11, 77, 149, 109, -9, -61; 209, 44, -22, -160, -319, -264, 17, 120; 198, 44, -21, -160, -330, -308, 15, 112],11,3]
%
% END OF LATTICE L12.3
%
\hbox{}\par\smallskip%
%
% LATTICE L12.4
%
\leavevmode%
\llap{\spacer}%
${L_{12.4}}=\pfillwitharg{2}({L_{12.1}})$%
{${}={}$}Nikulin $12$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{2,1}{\infty}{11}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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-23\\-44\\-66
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6\\11\\18
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4\\8\\11
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\0\\-11
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1\\0\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
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\HardButStrongLineBreak\kern3pt%
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-3\\-7\\-7
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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11\\-11\\-11\\-44
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\11\\0\\-33
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-11\\0\\2\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-33\\0\\1
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-253\\-231\\-11\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-253\\-22\\-11
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-22\\-4\\-4
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-4\\-9
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[11, 0, 0; 0, 0, -1; 0, -1, 0],[11, 1, 1],[-1, 1, 0, -3, -21, -23, -2, -1; 0, 0, -1, -7, -44, -44, -3, -1; -11, 0, 1, -7, -55, -66, -7, -5],11,4]
%
% END OF LATTICE L12.4
%
\hbox{}\par\smallskip%
%
% LATTICE L12.5
%
\leavevmode%
\llap{\spacer}%
${L_{12.5}}=\pfillwitharg{2}({L_{12.2}})$%
{${}={}$}Nikulin $8'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{2,1}{\infty a}{22}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38918\\28094\\12386
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28094\\20281\\8941
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12386\\8941\\3942
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82\\47\\151
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\GramMatrix{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[38918, 28094, 12386; 28094, 20281, 8941; 12386, 8941, 3942],[22, 1, 1],[-1558, -1753, -82, -103, -202, -7, 2, -217; 902, 1012, 47, 58, 110, 0, -1, 126; 2849, 3212, 151, 192, 385, 22, -4, 396],11,5]
%
% END OF LATTICE L12.5
%
\hbox{}\par\smallskip%
%
% LATTICE L12.6
%
\leavevmode%
\llap{\spacer}%
${L_{12.6}}=\main({L_{12.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{4,1}{\infty}{44}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\44\\264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\-2\\-11
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
264\\-11\\-49
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-1\\792\\-264
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0\\-31\\10
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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1\\39\\-4
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3\\104\\-8
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-253\\-462\\-11\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\-1012\\-44\\-44
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-924, 44, 264; 44, -2, -11; 264, -11, -49],[44, 1, 1],[-5, -1, 1, 3, 5, 1, -1, -3; -99, 110, 39, 104, 165, 22, -33, -80; -11, -44, -4, -8, -11, 0, 2, 0],11,6]
%
% END OF LATTICE L12.6
%
\hbox{}\par\smallskip%
%
% LATTICE L12.7
%
\leavevmode%
\llap{\spacer}%
${L_{12.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{12.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{4,3}{\infty z}{11}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1056594\\13090\\-504724
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13090\\-160\\6252
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504724\\6252\\-241101
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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979329\\-901659\\-2073478
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-13340\\12281\\28244
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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4458\\-4081\\-9438
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3216\\-2959\\-6809
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443\\-410\\-938
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383\\-356\\-811
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-1012\\-462\\-22\\0
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-22\\-22\\-4\\-4
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1056594, 13090, -504724; 13090, -160, 6252; -504724, 6252, -241101],[22, 2, 1],[4458, 3216, 443, 383, 1922, -26, -153, 197; -4081, -2959, -410, -356, -1793, 22, 143, -178; -9438, -6809, -938, -811, -4070, 55, 324, -417],11,7]
%
% END OF LATTICE L12.7
%
\hbox{}\par\smallskip%
%
% LATTICE L12.8
%
\leavevmode%
\llap{\spacer}%
${L_{12.8}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{12.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{2,1}{\infty}{1}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-121\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\-660\\-121
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-121\\-21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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39\\-4\\-22
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-4\\-11
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-33\\0\\11\\0\\-11\\-44\\-99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-11\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-33\\0\\11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-21\\-11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-23\\-22\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-22\\-44\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-44\\-99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-22, -121, -22; -121, -660, -121; -22, -121, -21],[11, 11, 1],[22, 29, 39, 30, 6, -1, -11, 26; -1, -2, -4, -4, -1, 0, 2, 0; -21, -23, -22, -11, -1, 1, 0, -33],11,8]
%
% END OF LATTICE L12.8
%
\hbox{}\par\smallskip%
%
% LATTICE L12.9
%
\leavevmode%
\llap{\spacer}%
${L_{12.9}}=\pdualwitharg{2}({L_{12.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{2,1}{\infty z}{11}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12163536\\-1404788\\-3361556
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1404788\\162248\\388232
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3361556\\388232\\929011
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2816881\\42592\\-10210464
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
311370\\-4709\\1128636
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
778425\\-11770\\2821589
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29273\\-440\\106106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16081\\-242\\58289
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2846\\-43\\10316
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1626\\-25\\5894
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\-22\\-44\\-176\\-1012\\-462\\-44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\11\\0\\-66\\-462\\-253\\-44\\-22
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\-22\\-44\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\11\\0\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-66\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1012\\-462\\-44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\-253\\-44\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-44\\-16\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-22\\-16\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12163536, -1404788, -3361556; -1404788, 162248, 388232; -3361556, 388232, 929011],[44, 4, 1],[29273, 16081, 2846, 1626, 2737, -76, 64, 4194; -440, -242, -43, -25, -44, 0, -1, -63; 106106, 58289, 10316, 5894, 9922, -275, 232, 15202],11,9]
%
% END OF LATTICE L12.9
%
\hbox{}\par\smallskip%
%
% LATTICE L12.10
%
\leavevmode%
\llap{\spacer}%
${L_{12.10}}=\pdualwitharg{2}(\main({L_{12.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{4,3}{\infty}{44}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8888\\4444\\-2288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4444\\2220\\-1144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2288\\-1144\\589
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
439\\-110\\1496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-31\\408
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\30\\-409
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-412\\99\\-1408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-457\\110\\-1562
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\10\\-140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\3\\-37
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\-44\\-44\\-88\\-1012\\-924\\-44\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\44\\0\\-66\\-924\\-1012\\-88\\-22
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\-44\\-44\\-88
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\44\\0\\-66
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-44\\0\\8\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-66\\0\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1012\\-924\\-44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\-1012\\-88\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-88\\-16\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-22\\-8\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8888, 4444, -2288; 4444, 2220, -1144; -2288, -1144, 589],[44, 4, 1],[-412, -457, -41, -11, -28, 17, 1, -29; 99, 110, 10, 3, 11, 0, 0, 7; -1408, -1562, -140, -37, -88, 66, 4, -99],11,10]
%
% END OF LATTICE L12.10
%
\hbox{}\par\smallskip%
%
% LATTICE L12.11
%
\leavevmode%
\llap{\spacer}%
${L_{12.11}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{12.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty a}{2}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{}{2}{22}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1655170\\-138578\\-755832
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138578\\-11561\\-63283
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-755832\\-63283\\-345150
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2818745\\222240\\6131972
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-223734\\17639\\486717
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1287612\\101520\\2801105
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2116\\-168\\-4603
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3067\\-242\\-6672
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2336\\-183\\-5082
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4055\\-316\\-8822
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-11\\-88\\-46\\-42\\-11\\0
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-2\\2\\0\\-66\\-42\\-46\\-22\\-22
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-11\\0\\11\\0\\-11\\-22\\-22\\-44
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-88\\-66\\0\\22\\0\\-22\\-44\\-198
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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2\\-2\\-11\\-88
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-66
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\11\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-66\\0\\22
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\-42\\-11\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-46\\-22\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-22\\-22\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-22\\-44\\-198
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1655170, -138578, -755832; -138578, -11561, -63283; -755832, -63283, -345150],[22, 11, 1],[2116, 3067, 2336, 4055, 928, -23, -814, 2033; -168, -242, -183, -316, -72, 2, 63, -164; -4603, -6672, -5082, -8822, -2019, 50, 1771, -4422],11,11]
%
% END OF LATTICE L12.11
%
\hbox{}\par\smallskip%
%
% LATTICE L12.12
%
\leavevmode%
\llap{\spacer}%
${L_{12.12}}=\pdualwitharg{11}(\main({L_{12.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{4,1}{\infty}{4}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12980\\-968\\748
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-968\\-66\\55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
748\\55\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
799\\1400\\15400
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\125\\1386
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-84\\-925
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\11\\133
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\30\\328
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\21\\198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\32\\264
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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1\\-2\\-11\\-88\\-23\\-42\\-11\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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1\\-2\\-11\\-88
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-132
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-132\\0\\44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-42\\-11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-92\\-44\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-44\\-44\\-88
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12980, -968, 748; -968, -66, 55; 748, 55, -43],[44, 11, 1],[7, 17, 10, 13, 1, -1, -2, 19; 11, 30, 21, 32, 3, -2, -7, 24; 133, 328, 198, 264, 21, -20, -44, 352],11,12]
%
% END OF LATTICE L12.12
%
\hbox{}\par\smallskip%
%
% LATTICE L12.13
%
\leavevmode%
\llap{\spacer}%
${L_{12.13}}=\pdualwitharg{2.11}(\pfillwitharg{2}({L_{12.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{}{2}{22}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
86724\\-4675726\\-2227126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4675726\\252108758\\120083590
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2227126\\120083590\\57197809
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17639\\-2227920\\4678080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-934185\\117986929\\-247743320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-444969\\56199282\\-118004569
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
259\\-32882\\69044
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
146\\-18496\\38837
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
151\\-19027\\39952
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\-11939\\25069
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-22\\-88\\-92\\-42\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-33\\-42\\-23\\-22\\-11
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-22\\-88
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-33
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\22\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-33\\0\\11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\-42\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-23\\-22\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-22\\-44\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-44\\-99
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[86724, -4675726, -2227126; -4675726, 252108758, 120083590; -2227126, 120083590, 57197809],[22, 22, 1],[259, 146, 151, 96, 35, 1, -4, 198; -32882, -18496, -19027, -11939, -4250, -70, 461, -25193; 69044, 38837, 39952, 25069, 8924, 147, -968, 52899],11,13]
%
% END OF LATTICE L12.13
%
\hbox{}\par\smallskip%
%
% LATTICE L12.14
%
\leavevmode%
\llap{\spacer}%
${L_{12.14}}=\pdualwitharg{11}({L_{12.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty b}{4}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8932\\11660\\-1188
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11660\\15224\\-1551
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1188\\-1551\\158
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1221\\-4324\\-33088
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1638\\-5797\\-44352
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-169\\598\\4575
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\94\\716
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\84\\648
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\20\\165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-22\\-176\\-92\\-84\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-132\\-84\\-92\\-44\\-44
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-132\\0\\44\\0\\-44\\-88\\-396
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-22\\-176
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-132
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\22\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-132\\0\\44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\-84\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-92\\-44\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-44\\-44\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-44\\-88\\-396
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8932, 11660, -1188; 11660, 15224, -1551; -1188, -1551, 158],[44, 11, 1],[-27, -23, -4, 3, 1, -3, -9, -55; 94, 84, 20, 0, -2, 8, 26, 184; 716, 648, 165, 22, -12, 56, 187, 1386],11,14]
%
% END OF LATTICE L12.14
%
\hbox{}\par\smallskip%
%
% LATTICE L12.15
%
\leavevmode%
\llap{\spacer}%
${L_{12.15}}=\pdualwitharg{2}({L_{12.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{8,3}{\infty z}{44}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-892936\\3784\\20768
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3784\\-16\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20768\\-88\\-483
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3719\\94809\\-178464
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-409\\768
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
86\\-2193\\4127
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\1133\\-2200
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\440\\-990
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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176\\-88\\-88\\-352
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-352\\-132\\0\\4
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-4048\\-1848\\-88\\0
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-892936, 3784, 20768; 3784, -16, -88; 20768, -88, -483],[88, 8, 1],[-46, -21, -1, 0, 2, -1, -1, -4; 1133, 440, 4, -11, -11, 121, 47, 113; -2200, -990, -44, 2, 88, -66, -52, -194],11,15]
%
% END OF LATTICE L12.15
%
\hbox{}\par\smallskip%
%
% LATTICE L12.16
%
\leavevmode%
\llap{\spacer}%
${L_{12.16}}=\pdualwitharg{2}({L_{12.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{8,7}{\infty z}{44}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\7656\\-968
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7656\\-121040\\15304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-968\\15304\\-1935
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\-3135\\-24816
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1376\\65359\\517376
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
174\\-8265\\-65425
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-46\\2101\\16632
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\803\\6358
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-87
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\44\\0\\-66\\-1848\\-1012\\-88\\-22
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
176\\-88\\-88\\-176
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\44\\0\\-66
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\0\\8\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-66\\0\\1
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4048\\-1848\\-88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-1012\\-88\\-22
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-88\\-16\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-22\\-8\\-9
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-440, 7656, -968; 7656, -121040, 15304; -968, 15304, -1935],[88, 8, 1],[-46, -21, -1, 0, 2, -1, -1, -2; 2101, 803, 4, -11, -11, 242, 91, 106; 16632, 6358, 32, -87, -88, 1914, 720, 839],11,16]
%
% END OF LATTICE L12.16
%
\hbox{}\par\smallskip%
%
% LATTICE L12.17
%
\leavevmode%
\llap{\spacer}%
${L_{12.17}}=\pdualwitharg{11}({L_{12.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,1}{\infty b}{8}\above{1pt}{1pt}{s}{2}{44}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-324104\\-165440\\5280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165440\\-84447\\2695
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5280\\2695\\-86
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-2185\\10192\\184184
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1119\\5221\\94369
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36\\-168\\-3037
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-13\\60\\1076
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\42\\638
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\64\\836
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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-4\\8\\0\\-264\\-84\\-184\\-88\\-88
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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2\\-4\\-22\\-176
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-264
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-176\\-264\\0\\88
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\-84\\-22\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-184\\-88\\-88
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-88\\-88\\-176
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-88\\-176\\-792
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-324104, -165440, 5280; -165440, -84447, 2695; 5280, 2695, -86],[88, 11, 1],[-4, -13, -11, -19, -2, 1, 5, -5; 22, 60, 42, 64, 6, -4, -14, 48; 441, 1076, 638, 836, 65, -64, -132, 1188],11,17]
%
% END OF LATTICE L12.17
%
\hbox{}\par\smallskip%
%
% LATTICE L12.18
%
\leavevmode%
\llap{\spacer}%
${L_{12.18}}=\pdualwitharg{11}({L_{12.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43560\\1232\\704
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1232\\-33\\-22
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
704\\-22\\-9
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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1087\\25840\\20944
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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5\\118\\99
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\304\\264
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}{}{\hbox{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\-84\\-11\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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0\\-88\\-88\\-792
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-43560, 1232, 704; 1232, -33, -22; 704, -22, -9],[88, 11, 1],[7, 17, 5, 13, 1, -1, -1, 19; 167, 404, 118, 304, 23, -24, -23, 456; 133, 328, 99, 264, 21, -20, -22, 352],11,18]
%
% END OF LATTICE L12.18
%
\hbox{}\par\smallskip%
%
% LATTICE L12.19
%
\leavevmode%
\llap{\spacer}%
${L_{12.19}}=\pdualwitharg{2.11}({L_{12.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{2,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\220\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
220\\-10208\\2992
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\kern3pt%
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-88\\2992\\-871
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-5797\\-6394\\-21252
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\kern3pt%
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50904\\56155\\186648
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13734\\-15151\\-50359
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{$\left.\llap{\phantom{%
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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260\\287\\954
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127\\141\\469
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
175\\198\\660
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39\\46\\154
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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4\\-2\\-44\\-176\\-92\\-42\\-44\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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4\\-2\\-44\\-176
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-2\\1\\0\\-66
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-66\\0\\44
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\-42\\-44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-23\\-44\\-22
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-44\\-176\\-176
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[88, 220, -88; 220, -10208, 2992; -88, 2992, -871],[44, 44, 1],[260, 127, 175, 39, -8, -1, 77, 465; 287, 141, 198, 46, -9, -2, 80, 510; 954, 469, 660, 154, -30, -7, 264, 1694],11,19]
%
% END OF LATTICE L12.19
%
\hbox{}\par\smallskip%
%
% LATTICE L12.20
%
\leavevmode%
\llap{\spacer}%
${L_{12.20}}=\pdualwitharg{2.11}(\main({L_{12.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty}{4}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1364\\-8228\\2288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8228\\52360\\-14564
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2288\\-14564\\4051
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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125\\3276\\11704
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1098\\-28549\\-101992
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306\\7956\\28423
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-17\\-430\\-1536
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-18\\-463\\-1654
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-431\\-1540
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-120\\-429
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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4\\-4\\-44\\-88\\-92\\-84\\-44\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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4\\-4\\-44\\-88
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-4\\4\\0\\-66
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-88\\-66\\0\\11
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-92\\-84\\-44\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-92\\-88\\-22
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-44\\-88\\-176\\-88
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0\\-22\\-88\\-99
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1364, -8228, 2288; -8228, 52360, -14564; 2288, -14564, 4051],[44, 44, 1],[-17, -18, -16, -4, -1, 0, -2, -14; -430, -463, -431, -120, -38, -5, -37, -348; -1536, -1654, -1540, -429, -136, -18, -132, -1243],11,20]
%
% END OF LATTICE L12.20
%
\hbox{}\par\smallskip%
%
% LATTICE L12.21
%
\leavevmode%
\llap{\spacer}%
${L_{12.21}}=\pdualwitharg{2.11}({L_{12.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\616\\-176
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
616\\-1496\\440
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\440\\-129
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-89\\368\\1408
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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23\\-94\\-360
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10\\-39\\-150
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4\\-11\\-44
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-1\\6\\22
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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16\\-8\\-88\\-352\\-368\\-168\\-88\\0
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{$\left[\!\llap{\phantom{%
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-8\\4\\0\\-132
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-352\\-132\\0\\44
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-368\\-168\\-88\\0
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-168\\-92\\-88\\-44
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-88\\-88\\-176\\-176
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0\\-44\\-176\\-396
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-176, 616, -176; 616, -1496, 440; -176, 440, -129],[88, 88, 1],[23, 10, 4, -1, -1, 1, 7, 23; -94, -39, -11, 6, 2, -7, -35, -98; -360, -150, -44, 22, 8, -26, -132, -374],11,21]
%
% END OF LATTICE L12.21
%
\hbox{}\par\smallskip%
%
% LATTICE L12.22
%
\leavevmode%
\llap{\spacer}%
${L_{12.22}}=\pdualwitharg{2.11}({L_{12.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,7}{\infty z}{4}\above{1pt}{1pt}{l}{2}{88}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-363440\\3642584\\-992728
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3642584\\-36506888\\9949368
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-992728\\9949368\\-2711541
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12135\\779984\\2857536
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121915\\-7835511\\-28706040
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33226\\2135444\\7823375
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\4922\\18032
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\2085\\7638
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\697\\2552
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-141\\-517
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-66\\0\\11
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-368\\-168\\-88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-92\\-88\\-22
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-88\\-176\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-22\\-88\\-99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-363440, 3642584, -992728; 3642584, -36506888, 9949368; -992728, 9949368, -2711541],[88, 88, 1],[77, 34, 15, -1, -3, 3, 22, 38; 4922, 2085, 697, -141, -166, 293, 1681, 2519; 18032, 7638, 2552, -517, -608, 1074, 6160, 9229],11,22]
%
% END OF LATTICE L12.22
%

\medskip%
%------------------------------WEYL-GROUP-W13--------------------------------
%
\leavevmode\llap{}%
$W_{13\phantom{0}}$%
\qquad\llap{22} lattices, $\chi=42$%
\hfill%
$\infty242\infty242\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L13.1
%
\leavevmode%
\llap{\spacer}%
${L_{13.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{13}}%
\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{1,0}{\infty b}{52}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14425996\\261508\\523016
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
261508\\-4740\\-9481
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
523016\\-9481\\-18962
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
896635\\125112\\24667916
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16383\\-2287\\-450723
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32508\\-4536\\-894349
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1159\\-130\\-31902
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2351\\-312\\-64688
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-258\\-38\\-7097
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-10\\-850
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\-52\\-26\\-52\\-728\\-1300\\-130\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\-52\\-26\\-52
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\52\\0\\-52
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\0\\2\\-2
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-52\\-2\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-728\\-1300\\-130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1300\\-2756\\-312\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-312\\-38\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-52\\-10\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-14425996, 261508, 523016; 261508, -4740, -9481; 523016, -9481, -18962],[52, 1, 1],[-1159, -2351, -258, -31, 41, 115, 0, -55; -130, -312, -38, -10, -26, 0, 2, -2; -31902, -64688, -7097, -850, 1144, 3172, -1, -1516],12,1]
%
% END OF LATTICE L13.1
%
\hbox{}\par\smallskip%
%
% LATTICE L13.2
%
\leavevmode%
\llap{\spacer}%
${L_{13.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N'_{10}}%
\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{4,3}{\infty b}{104}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{4}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5304\\104\\2080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
104\\-9\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2080\\-65\\-198
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11233\\-527280\\54912
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
468\\21969\\-2288
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2196\\103090\\-10737
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-412\\-19344\\2015
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\HardButStrongLineBreak\kern3pt%
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-1893\\-88868\\9256
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-225\\-10562\\1100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-1220\\127
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-26\\-52\\-2\\2
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-364\\-1300\\-130\\0
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-1300\\-5512\\-624\\-52
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-624\\-76\\-10
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-52\\-10\\-4
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[5304, 104, 2080; 104, -9, -65; 2080, -65, -198],[104, 1, 1],[-412, -1893, -225, -26, -56, 21, 9, -10; -19344, -88868, -10562, -1220, -2626, 988, 422, -470; 2015, 9256, 1100, 127, 273, -104, -44, 49],12,2]
%
% END OF LATTICE L13.2
%
\hbox{}\par\smallskip%
%
% LATTICE L13.3
%
\leavevmode%
\llap{\spacer}%
${L_{13.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{4,3}{\infty a}{104}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2600\\-2600\\1352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2600\\-2574\\1325
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1352\\1325\\-675
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17109\\33488\\31304
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15463\\30267\\28294
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7191\\-14076\\-13159
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-26\\-25
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\-6
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-52\\-13\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\104\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\1\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-52\\-1\\2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-364\\-1300\\-65\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1300\\-5512\\-312\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-312\\-19\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-52\\-5\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2600, -2600, 1352; -2600, -2574, 1325; 1352, 1325, -675],[104, 1, 1],[-150, -25, 13, 2, -461, -2419, -154, -49; 299, 52, -26, -5, 897, 4732, 302, 97; 286, 52, -25, -6, 832, 4420, 283, 92],12,3]
%
% END OF LATTICE L13.3
%
\hbox{}\par\smallskip%
%
% LATTICE L13.4
%
\leavevmode%
\llap{\spacer}%
${L_{13.4}}=\pfillwitharg{2}({L_{13.1}})$%
{${}={}$}Nikulin $13$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${13}\above{1pt}{1pt}{2,1}{\infty}{13}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\-78\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\26\\81
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\9\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-13\\-13\\-13\\-182\\-325\\-65\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\13\\0\\-13\\-325\\-689\\-156\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\2\\-1\\-65\\-156\\-38\\-5
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-13\\-1\\1\\0\\-13\\-5\\-2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-13\\-13\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\13\\0\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-13\\-1\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-182\\-325\\-65\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-325\\-689\\-156\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-156\\-38\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-13\\-5\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[13, 0, 0; 0, 0, -1; 0, -1, 0],[13, 1, 1],[-1, 1, 0, -1, -25, -53, -12, -1; 0, 0, -1, -2, -39, -78, -17, -1; -13, 0, 1, -3, -104, -234, -55, -6],12,4]
%
% END OF LATTICE L13.4
%
\hbox{}\par\smallskip%
%
% LATTICE L13.5
%
\leavevmode%
\llap{\spacer}%
${L_{13.5}}=\pfillwitharg{2}({L_{13.2}})$%
{${}={}$}Nikulin $10'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{2,1}{\infty b}{26}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
246506\\0\\15002
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15002\\-1\\914
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51427\\-851916\\-845208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
138\\-2287\\-2268
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2990\\-49530\\-49141
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
526\\-8710\\-8645
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1177\\-19500\\-19344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
138\\-2287\\-2268
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-498\\-493
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-26\\-13\\-26\\-364\\-650\\-65\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\26\\0\\-26\\-650\\-1378\\-156\\-26
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-26\\-1\\2\\0\\-26\\-5\\-4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-26\\-13\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\26\\0\\-26
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\1\\-1
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-26\\-1\\2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-364\\-650\\-65\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-650\\-1378\\-156\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-156\\-19\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-26\\-5\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[246506, 0, 15002; 0, 1, -1; 15002, -1, 914],[26, 1, 1],[526, 1177, 138, 30, 72, 19, 0, 16; -8710, -19500, -2287, -498, -1196, -312, 1, -264; -8645, -19344, -2268, -493, -1183, -312, 0, -263],12,5]
%
% END OF LATTICE L13.5
%
\hbox{}\par\smallskip%
%
% LATTICE L13.6
%
\leavevmode%
\llap{\spacer}%
${L_{13.6}}=\main({L_{13.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${13}\above{1pt}{1pt}{4,3}{\infty}{52}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-373204\\15860\\6188
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15860\\-674\\-263
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6188\\-263\\-99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-125425\\-2943980\\-20904
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5328\\125059\\888
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2196\\51545\\365
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-412\\-9672\\-65
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\HardButStrongLineBreak\kern3pt%
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-1893\\-44434\\-312
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-225\\-5281\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-610\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-26\\-13\\-13\\-182\\-650\\-65\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\52\\0\\-26\\-650\\-2756\\-312\\-26
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-26\\-1\\1\\0\\-26\\-5\\-2
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-26\\-13\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\52\\0\\-26
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-26\\-1\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-182\\-650\\-65\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-650\\-2756\\-312\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-312\\-38\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-26\\-5\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-373204, 15860, 6188; 15860, -674, -263; 6188, -263, -99],[52, 1, 1],[-412, -1893, -225, -26, -56, 21, 9, -10; -9672, -44434, -5281, -610, -1313, 494, 211, -235; -65, -312, -38, -5, -13, 0, 2, -1],12,6]
%
% END OF LATTICE L13.6
%
\hbox{}\par\smallskip%
%
% LATTICE L13.7
%
\leavevmode%
\llap{\spacer}%
${L_{13.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{13.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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-4032158\\276510\\-1877824
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\kern3pt%
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276510\\-18960\\128774
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{$\left.\llap{\phantom{%
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3485\\-19\\-7486
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4032158, 276510, -1877824; 276510, -18960, 128774; -1877824, 128774, -874525],[26, 2, 1],[15676, 15887, 3485, 416, -570, -781, 1, 746; -65, -78, -19, -5, -13, 0, 1, -1; -33670, -34125, -7486, -894, 1222, 1677, -2, -1602],12,7]
%
% END OF LATTICE L13.7
%
\hbox{}\par\smallskip%
%
% LATTICE L13.8
%
\leavevmode%
\llap{\spacer}%
${L_{13.8}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{13.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-481\\-325\\286
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-325\\-156\\195
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
286\\195\\-170
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-5101\\204\\-8398
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4050\\161\\-6669
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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19\\-1\\31
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-40\\2\\-65
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\-1\\65
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-25\\-65\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-481, -325, 286; -325, -156, 195; 286, 195, -170],[13, 13, 1],[19, -3, -40, 39, 131, 303, 940, 111; -1, 0, 2, -1, -5, -12, -38, -5; 31, -5, -65, 65, 216, 499, 1547, 182],12,8]
%
% END OF LATTICE L13.8
%
\hbox{}\par\smallskip%
%
% LATTICE L13.9
%
\leavevmode%
\llap{\spacer}%
${L_{13.9}}=\pdualwitharg{2}(\main({L_{13.3}}))$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
460824\\218244\\115648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
218244\\103396\\54772
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115648\\54772\\29023
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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1332941\\265512\\-5812560
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\kern3pt%
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627835\\125059\\-2737800
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334350\\66600\\-1458001
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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8537\\1703\\-37232
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19049\\3796\\-83070
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4459\\888\\-19444
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479\\95\\-2088
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-52\\-52\\-4\\4
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-728\\-1300\\-260\\0
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-1300\\-2756\\-624\\-52
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-260\\-624\\-152\\-20
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0\\-52\\-20\\-8
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[460824, 218244, 115648; 218244, 103396, 54772; 115648, 54772, 29023],[52, 4, 1],[8537, 19049, 4459, 479, 1122, 269, -1, 264; 1703, 3796, 888, 95, 221, 52, 0, 53; -37232, -83070, -19444, -2088, -4888, -1170, 4, -1152],12,9]
%
% END OF LATTICE L13.9
%
\hbox{}\par\smallskip%
%
% LATTICE L13.10
%
\leavevmode%
\llap{\spacer}%
${L_{13.10}}=\pdualwitharg{2}({L_{13.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{2,1}{\infty z}{13}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68544528\\52\\15594384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\8\\12
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15594384\\12\\3547837
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53308007\\5007483\\-234313404
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24336\\-2287\\106968
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12127440\\1139190\\-53305721
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-24336\\-2287\\106968
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[68544528, 52, 15594384; 52, 8, 12; 15594384, 12, 3547837],[52, 4, 1],[-46381, -51891, -24336, -2645, -6347, -837, 0, -1411; -4355, -4875, -2287, -249, -598, -78, 1, -132; 203866, 228085, 106968, 11626, 27898, 3679, 0, 6202],12,10]
%
% END OF LATTICE L13.10
%
\hbox{}\par\smallskip%
%
% LATTICE L13.11
%
\leavevmode%
\llap{\spacer}%
${L_{13.11}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{13.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty b}{2}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{4}{26}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1581658\\-441532\\-763958
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-441532\\-123253\\-213265
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-763958\\-213265\\-369000
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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1569439\\-176800\\-3147040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
436212\\-49141\\-874692
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1189\\-136\\-2383
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2423\\-274\\-4858
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3468\\-389\\-6955
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
433\\-44\\-871
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-50\\-106\\-156\\-26
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1581658, -441532, -763958; -441532, -123253, -213265; -763958, -213265, -369000],[26, 13, 1],[1189, 2423, 3468, 433, -35, -115, -6, 721; -136, -274, -389, -44, 6, 14, -1, -86; -2383, -4858, -6955, -871, 69, 230, 13, -1443],12,11]
%
% END OF LATTICE L13.11
%
\hbox{}\par\smallskip%
%
% LATTICE L13.12
%
\leavevmode%
\llap{\spacer}%
${L_{13.12}}=\pdualwitharg{13}(\main({L_{13.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{4,3}{\infty}{4}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{}{4}{13}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114452\\-52988\\4212
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52988\\-24531\\1950
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4212\\1950\\-155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-6801\\19720\\61880
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{$\left[\!\llap{\phantom{%
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-1\\-3\\-65
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-6\\12\\-13
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{$\left[\!\llap{\phantom{%
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-50\\-212\\-312\\-26
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-114452, -52988, 4212; -52988, -24531, 1950; 4212, 1950, -155],[52, 13, 1],[0, 1, -1, -6, -10, -41, -59, -4; 2, -2, -3, 12, 27, 118, 177, 17; 25, 2, -65, -13, 66, 362, 611, 104],12,12]
%
% END OF LATTICE L13.12
%
\hbox{}\par\smallskip%
%
% LATTICE L13.13
%
\leavevmode%
\llap{\spacer}%
${L_{13.13}}=\pdualwitharg{2.13}(\pfillwitharg{2}({L_{13.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{}{2}{26}\above{1pt}{1pt}{}{4}{52}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23764\\-779454\\-374738
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\kern3pt%
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-779454\\25652874\\12333100
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-374738\\12333100\\5929369
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-49141\\2171130\\-4519060
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-56\\-50\\-130\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-53\\-156\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-156\\-494\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-26\\-130\\-104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[23764, -779454, -374738; -779454, 25652874, 12333100; -374738, 12333100, 5929369],[26, 26, 1],[-665, -744, -2268, -493, -91, -12, 0, -263; 29374, 32874, 100231, 21810, 4028, 528, -25, 11592; -61140, -68425, -208624, -45396, -8384, -1099, 52, -24128],12,13]
%
% END OF LATTICE L13.13
%
\hbox{}\par\smallskip%
%
% LATTICE L13.14
%
\leavevmode%
\llap{\spacer}%
${L_{13.14}}=\pdualwitharg{13}({L_{13.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty b}{4}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{*}{4}{52}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105820\\-103324\\8268
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103324\\-100880\\8073
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8268\\8073\\-646
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52983\\-4296\\623636
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51874\\-4207\\610571
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4144\\336\\-48777
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-2\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\2\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\-2\\806
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-26\\-52\\-56\\-100\\-130\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-52\\-100\\-212\\-312\\-52
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\kern3pt%
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-26\\0\\26\\-26\\-130\\-312\\-494\\-130
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\-4\\-26\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\0\\26\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-52\\-26\\52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-100\\-130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-212\\-312\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-312\\-494\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-52\\-130\\-104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-105820, -103324, 8268; -103324, -100880, 8073; 8268, 8073, -646],[52, 13, 1],[9, -5, -5, 65, 139, 301, 449, 83; -2, 0, 2, -2, -10, -24, -38, -10; 90, -64, -39, 806, 1652, 3548, 5265, 936],12,14]
%
% END OF LATTICE L13.14
%
\hbox{}\par\smallskip%
%
% LATTICE L13.15
%
\leavevmode%
\llap{\spacer}%
${L_{13.15}}=\pdualwitharg{2}({L_{13.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{8,1}{\infty z}{52}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{4}{16}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-356720\\-13624\\624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13624\\-520\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\24\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\910\\14560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\33\\544
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\52\\416
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\169\\130
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\53\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\24\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\-104\\-104\\-208\\-2912\\-2600\\-520\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\-104\\-104\\-208
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\52\\0\\-104
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\0\\8\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-104\\-8\\16
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2912\\-2600\\-520\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2600\\-2756\\-624\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-624\\-152\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-104\\-40\\-32
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-356720, -13624, 624; -13624, -520, 24; 624, 24, -1],[104, 8, 1],[-1, -6, -2, -1, 1, 6, 2, 1; 52, 169, 53, 24, -26, -143, -47, -22; 416, 130, -12, -48, 0, 286, 108, 80],12,15]
%
% END OF LATTICE L13.15
%
\hbox{}\par\smallskip%
%
% LATTICE L13.16
%
\leavevmode%
\llap{\spacer}%
${L_{13.16}}=\pdualwitharg{2}({L_{13.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{8,5}{\infty z}{52}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{4}{16}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-486512\\19032\\18200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19032\\-728\\-704
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18200\\-704\\-677
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9593\\-248378\\515944
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\10251\\-21296
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-369\\9553\\-19845
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
261\\-6760\\14040
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-6227\\12922
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-1273\\2640
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-28\\56
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\-104\\-104\\-208\\-2912\\-2600\\-520\\0
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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208\\-104\\-104\\-208
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\52\\0\\-104
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-104\\0\\8\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-104\\-8\\16
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2912\\-2600\\-520\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2600\\-2756\\-624\\-104
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-624\\-152\\-40
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-104\\-40\\-32
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-486512, 19032, 18200; 19032, -728, -704; 18200, -704, -677],[104, 8, 1],[261, 240, 49, 1, -27, -6, 5, 17; -6760, -6227, -1273, -28, 702, 169, -125, -438; 14040, 12922, 2640, 56, -1456, -338, 264, 912],12,16]
%
% END OF LATTICE L13.16
%
\hbox{}\par\smallskip%
%
% LATTICE L13.17
%
\leavevmode%
\llap{\spacer}%
${L_{13.17}}=\pdualwitharg{13}({L_{13.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty b}{8}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{*}{4}{26}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1829672\\4992\\23816
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4992\\-13\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23816\\-65\\-310
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19199\\111360\\1447680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-349\\-4524
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-250\\-1450\\-18851
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-76
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\24\\455
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-26\\-26
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\0\\52\\-26
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-52\\-26\\26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-100\\-130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-424\\-624\\-52
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1829672, 4992, 23816; 4992, -13, -65; 23816, -65, -310],[104, 13, 1],[0, -1, 1, 6, 10, 41, 59, 4; 4, -4, -6, 24, 54, 236, 354, 34; -1, -76, 78, 455, 755, 3092, 4446, 299],12,17]
%
% END OF LATTICE L13.17
%
\hbox{}\par\smallskip%
%
% LATTICE L13.18
%
\leavevmode%
\llap{\spacer}%
${L_{13.18}}=\pdualwitharg{13}({L_{13.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty a}{8}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{}{4}{26}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-229320\\3224\\1664
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3224\\-39\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1664\\-26\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7643\\241080\\580944
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-3691\\-8892
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-1640\\-3953
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\379\\911
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\1672\\4028
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\1261\\3042
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\251\\611
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-13\\-26\\-28\\-100\\-65\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-52\\-100\\-424\\-312\\-52
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-13\\-26
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-52
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\13\\-13
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-52\\-13\\26
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-100\\-65\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-424\\-312\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-312\\-247\\-65
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-52\\-65\\-52
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-229320, 3224, 1664; 3224, -39, -26; 1664, -26, -11],[104, 13, 1],[12, 53, 40, 8, 1, -1, -1, 5; 379, 1672, 1261, 251, 31, -32, -31, 159; 911, 4028, 3042, 611, 77, -76, -78, 377],12,18]
%
% END OF LATTICE L13.18
%
\hbox{}\par\smallskip%
%
% LATTICE L13.19
%
\leavevmode%
\llap{\spacer}%
${L_{13.19}}=\pdualwitharg{2.13}(\main({L_{13.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty}{4}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{}{4}{52}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
884\\-5096\\-1196
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5096\\93028\\21372
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1196\\21372\\4911
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9076\\-49735\\218660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41003\\224664\\-987740
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9703\\53165\\-233741
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\131\\-576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\-22
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\71\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138\\757\\-3328
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-52\\-52\\-56\\-100\\-260\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-52\\-100\\-212\\-624\\-52
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-52\\-52
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-52
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\0\\104\\-52
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-52\\-52\\52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-100\\-260\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-212\\-624\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-260\\-624\\-1976\\-260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-52\\-260\\-104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[884, -5096, -1196; -5096, 93028, 21372; -1196, 21372, 4911],[52, 52, 1],[-24, -1, -13, -138, -289, -625, -1865, -175; 131, 5, 71, 757, 1584, 3425, 10219, 958; -576, -22, -312, -3328, -6964, -15058, -44928, -4212],12,19]
%
% END OF LATTICE L13.19
%
\hbox{}\par\smallskip%
%
% LATTICE L13.20
%
\leavevmode%
\llap{\spacer}%
${L_{13.20}}=\pdualwitharg{2.13}({L_{13.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{2,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{*}{4}{52}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
104\\988\\208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
988\\334464\\74152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\74152\\16441
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4207\\8520\\-38376
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-403075\\816499\\-3677700
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89027\\180340\\-812293
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-61\\123\\-554
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-9
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-287\\583\\-2626
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-52\\-52\\-56\\-50\\-260\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-26\\-50\\-53\\-312\\-26
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-52\\-26\\-52\\52\\0\\-26\\-260\\-104
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[104, 988, 208; 988, 334464, 74152; 208, 74152, 16441],[52, 52, 1],[-61, -1, 1, -287, -640, -700, -4207, -414; 123, 2, 0, 583, 1297, 1418, 8520, 837; -554, -9, 0, -2626, -5842, -6387, -38376, -3770],12,20]
%
% END OF LATTICE L13.20
%
\hbox{}\par\smallskip%
%
% LATTICE L13.21
%
\leavevmode%
\llap{\spacer}%
${L_{13.21}}=\pdualwitharg{2.13}({L_{13.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-624\\-20072\\-9776
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20072\\-614536\\-299312
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\kern3pt%
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-9776\\-299312\\-145781
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{$\left.\llap{\phantom{%
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0\\0\\0
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-85\\-5670\\11648
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-2316\\-156331\\321152
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{$\left[\!\llap{\phantom{%
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-200\\-212\\-624\\-104
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-624, -20072, -9776; -20072, -614536, -299312; -9776, -299312, -145781],[104, 104, 1],[1, 0, -4, -11, -13, -12, -32, -1; 74, -75, -633, -962, -884, -735, -1797, 152; -152, 154, 1300, 1976, 1816, 1510, 3692, -312],12,21]
%
% END OF LATTICE L13.21
%
\hbox{}\par\smallskip%
%
% LATTICE L13.22
%
\leavevmode%
\llap{\spacer}%
${L_{13.22}}=\pdualwitharg{2.13}({L_{13.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65624\\-72176\\-33488
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72176\\-79352\\-36816
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\kern3pt%
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-33488\\-36816\\-17081
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{$\left.\llap{\phantom{%
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0\\0\\0
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2141\\-17136\\32760
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2380\\-19041\\36400
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-224\\-200\\-520\\0
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-200\\-212\\-624\\-104
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-520\\-624\\-1976\\-520
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-65624, -72176, -33488; -72176, -79352, -36816; -33488, -36816, -17081],[104, 104, 1],[-3, 2, 21, 37, 37, 32, 81, -3; 25, -25, -212, -323, -297, -247, -604, 51; -48, 50, 416, 624, 568, 470, 1144, -104],12,22]
%
% END OF LATTICE L13.22
%

\medskip%
%------------------------------WEYL-GROUP-W14--------------------------------
%
\leavevmode\llap{}%
$W_{14\phantom{0}}$%
\qquad\llap{6} lattices, $\chi=12$%
\hfill%
$222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L14.1
%
\leavevmode%
\llap{\spacer}%
${L_{14.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{14}}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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-260\\-118\\25
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\kern3pt%
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52\\25\\6
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-572, -260, 52; -260, -118, 25; 52, 25, 6],[52, 1, 1],[17, 7, -6, -13, -6, 19; -36, -15, 13, 28, 13, -39; 0, 2, 0, -4, -3, -13],13,1]
%
% END OF LATTICE L14.1
%
\hbox{}\par\smallskip%
%
% LATTICE L14.2
%
\leavevmode%
\llap{\spacer}%
${L_{14.2}}=\pfillwitharg{2}({L_{14.1}})$%
{${}={}$}Nikulin $14$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\-13\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[78, -13, 0; -13, 2, 0; 0, 0, 1],[13, 1, 1],[0, 0, -10, -2, -1, -3; 0, 1, -65, -14, -8, -26; 1, 0, -26, -5, -2, 0],13,2]
%
% END OF LATTICE L14.2
%
\hbox{}\par\smallskip%
%
% LATTICE L14.3
%
\leavevmode%
\llap{\spacer}%
${L_{14.3}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{14.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13, -26, 0; -26, 39, 13; 0, 13, 2],[13, 13, 1],[3, 3, -1, -5, -4, 0; -2, -3, -1, 0, 2, 0; 13, 13, -1, -13, -13, 1],13,3]
%
% END OF LATTICE L14.3
%
\hbox{}\par\smallskip%
%
% LATTICE L14.4
%
\leavevmode%
\llap{\spacer}%
${L_{14.4}}=\pdualwitharg{2}({L_{14.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240760, -10660, 60528; -10660, 472, -2680; 60528, -2680, 15217],[52, 4, 1],[25, -1, -25, 3, 29, 389; -43, 0, 26, -6, -49, -663; -107, 4, 104, -13, -124, -1664],13,4]
%
% END OF LATTICE L14.4
%
\hbox{}\par\smallskip%
%
% LATTICE L14.5
%
\leavevmode%
\llap{\spacer}%
${L_{14.5}}=\pdualwitharg{13}({L_{14.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[52, 0, 0; 0, -26, 13; 0, 13, -6],[52, 13, 1],[-5, -1, 0, 1, 0, -1; -16, 1, 1, 0, -5, -5; -52, 0, 2, 0, -13, -15],13,5]
%
% END OF LATTICE L14.5
%
\hbox{}\par\smallskip%
%
% LATTICE L14.6
%
\leavevmode%
\llap{\spacer}%
${L_{14.6}}=\pdualwitharg{2.13}({L_{14.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\kern3pt%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-56\\208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\125\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\28\\-105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-11\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\0\\-52\\-65\\-52\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\104\\0\\-52\\-156\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\0\\8\\0\\-52\\-60
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\0\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-52\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-156\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-52\\-60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[312, -676, -156; -676, 1560, 364; -156, 364, 85],[52, 52, 1],[-5, -2, 0, 0, -3, -5; -11, -1, -1, -3, -13, -13; 39, 0, 4, 13, 52, 48],13,6]
%
% END OF LATTICE L14.6
%

\medskip%
%------------------------------WEYL-GROUP-W15--------------------------------
%
\leavevmode\llap{}%
$W_{15\phantom{0}}$%
\qquad\llap{8} lattices, $\chi=8$%
\hfill%
$\infty232$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L15.1
%
\leavevmode%
\llap{\spacer}%
${L_{15.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{15}}%
\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{4,3}{\infty a}{56}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{-}{3}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25256\\392\\672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
392\\-6\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
672\\-11\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-105\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\560\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\52\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-51\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-28\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\56\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-25256, 392, 672; 392, -6, -11; 672, -11, -14],[56, 1, 1],[-2, 11, 1, -1; -105, 560, 52, -51; -14, 84, 7, -8],14,1]
%
% END OF LATTICE L15.1
%
\hbox{}\par\smallskip%
%
% LATTICE L15.2
%
\leavevmode%
\llap{\spacer}%
${L_{15.2}}=\pfillwitharg{2}({L_{15.1}})$%
{${}={}$}Nikulin $15$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{2,1}{\infty b}{14}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{+}{3}{2}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-14\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\14\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[14, 0, 0; 0, 0, -1; 0, -1, 0],[14, 1, 1],[-1, 1, 0, -1; 0, 0, -1, -2; -7, 0, 1, -3],14,2]
%
% END OF LATTICE L15.2
%
\hbox{}\par\smallskip%
%
% LATTICE L15.3
%
\leavevmode%
\llap{\spacer}%
${L_{15.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{15.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{4,1}{\infty z}{7}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{-}{3}{4}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10472\\378\\-4956
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
378\\-12\\178
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4956\\178\\-2345
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\-49\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\63\\119
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\24\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-23\\-44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-14\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\7\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-2\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10472, 378, -4956; 378, -12, 178; -4956, 178, -2345],[14, 2, 1],[38, -54, -19, 20; -49, 63, 24, -23; -84, 119, 42, -44],14,3]
%
% END OF LATTICE L15.3
%
\hbox{}\par\smallskip%
%
% LATTICE L15.4
%
\leavevmode%
\llap{\spacer}%
${L_{15.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{15.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty b}{2}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{+}{3}{14}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17318\\-1428\\-7546
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1428\\-98\\-623
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7546\\-623\\-3288
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-2\\-57
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\6\\148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-86\\7\\196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\-8\\-189
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\14\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-7\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17318, -1428, -7546; -1428, -98, -623; -7546, -623, -3288],[14, 7, 1],[25, -65, -86, 83; -2, 6, 7, -8; -57, 148, 196, -189],14,4]
%
% END OF LATTICE L15.4
%
\hbox{}\par\smallskip%
%
% LATTICE L15.5
%
\leavevmode%
\llap{\spacer}%
${L_{15.5}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{15.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{-}{3}{28}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
532\\-25214\\-11648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25214\\1206184\\557214
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11648\\557214\\257413
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-24\\52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-127\\275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-291\\630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-194\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\28\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-14\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[532, -25214, -11648; -25214, 1206184, 557214; -11648, 557214, 257413],[14, 14, 1],[1, 2, 2, 1; -24, -127, -291, -194; 52, 275, 630, 420],14,5]
%
% END OF LATTICE L15.5
%
\hbox{}\par\smallskip%
%
% LATTICE L15.6
%
\leavevmode%
\llap{\spacer}%
${L_{15.6}}=\pdualwitharg{7}({L_{15.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty a}{8}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{-}{3}{14}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\112\\56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\-21\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\14\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-7\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-280, 112, 56; 112, -42, -21; 56, -21, -10],[56, 7, 1],[0, 1, 0, -1; -1, 4, 3, -3; 2, -4, -7, 0],14,6]
%
% END OF LATTICE L15.6
%
\hbox{}\par\smallskip%
%
% LATTICE L15.7
%
\leavevmode%
\llap{\spacer}%
${L_{15.7}}=\pdualwitharg{2}({L_{15.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{8,5}{\infty z}{28}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{+}{3}{16}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9968\\-1512\\224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1512\\-112\\32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
224\\32\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-7\\-448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\7\\406
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\3\\200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-3\\-152
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\-56\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\28\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\16\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-8\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9968, -1512, 224; -1512, -112, 32; 224, 32, -5],[56, 8, 1],[-9, 8, 4, -3; -7, 7, 3, -3; -448, 406, 200, -152],14,7]
%
% END OF LATTICE L15.7
%
\hbox{}\par\smallskip%
%
% LATTICE L15.8
%
\leavevmode%
\llap{\spacer}%
${L_{15.8}}=\pdualwitharg{2.7}({L_{15.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,5}{\infty z}{4}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{+}{3}{112}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\280\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\112\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-7\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\112\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-56\\112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-336, 280, -56; 280, 112, 0; -56, 0, -3],[56, 56, 1],[-1, 1, 3, -3; 2, -3, -7, 8; 16, -22, -56, 56],14,8]
%
% END OF LATTICE L15.8
%

\medskip%
%------------------------------WEYL-GROUP-W16--------------------------------
%
\leavevmode\llap{}%
$W_{16\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=4$%
\hfill%
$2262$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L16.1
%
\leavevmode%
\llap{\spacer}%
${L_{16.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{23\rightarrow N_{16},3,2}%
\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-324180\\-9360\\3600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9360\\-66\\69
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3600\\69\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-138\\-810
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\250\\1460
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\83\\486
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-111\\-649
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-324180, -9360, 3600; -9360, -66, 69; 3600, 69, -34],[180, 3, 1],[-5, 9, 3, -4; -138, 250, 83, -111; -810, 1460, 486, -649],15,1]
%
% END OF LATTICE L16.1
%
\hbox{}\par\smallskip%
%
% LATTICE L16.2
%
\leavevmode%
\llap{\spacer}%
${L_{16.2}}=\pfillwitharg{2.3}({L_{16.1}})$%
{${}={}$}Nikulin $16$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-30, 15, 0; 15, -7, 0; 0, 0, 1],[15, 1, 1],[0, 2, -1, -3; 0, 5, -3, -8; 1, 0, -3, 0],15,2]
%
% END OF LATTICE L16.2
%
\hbox{}\par\smallskip%
%
% LATTICE L16.3
%
\leavevmode%
\llap{\spacer}%
${L_{16.3}}=\pfillwitharg{3}({L_{16.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19380\\-7260\\900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7260\\-2694\\323
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
900\\323\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-38\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\210\\380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\69\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-93\\-169
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-19380, -7260, 900; -7260, -2694, 323; 900, 323, -34],[60, 1, 1],[11, -61, -20, 27; -38, 210, 69, -93; -70, 380, 126, -169],15,3]
%
% END OF LATTICE L16.3
%
\hbox{}\par\smallskip%
%
% LATTICE L16.4
%
\leavevmode%
\llap{\spacer}%
${L_{16.4}}=\pfillwitharg{2}({L_{16.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3555\\-180\\540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-3\\27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
540\\27\\-82
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-3\\-27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\2\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3555, -180, 540; -180, -3, 27; 540, 27, -82],[45, 3, 1],[-4, 3, 4, -3; -3, 0, 2, -1; -27, 20, 27, -20],15,4]
%
% END OF LATTICE L16.4
%
\hbox{}\par\smallskip%
%
% LATTICE L16.5
%
\leavevmode%
\llap{\spacer}%
${L_{16.5}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96285\\-12240\\33720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12240\\-1554\\4287
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33720\\4287\\-11809
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\14\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
74\\-45\\195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-16\\77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\39\\-180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-96285, -12240, 33720; -12240, -1554, 4287; 33720, 4287, -11809],[15, 3, 1],[-27, 74, 29, -68; 14, -45, -16, 39; -72, 195, 77, -180],15,5]
%
% END OF LATTICE L16.5
%
\hbox{}\par\smallskip%
%
% LATTICE L16.6
%
\leavevmode%
\llap{\spacer}%
${L_{16.6}}=\pdualwitharg{5}(\pfillwitharg{2.3}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21435\\-2970\\10065
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2970\\-410\\1395
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10065\\1395\\-4726
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-9\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\4\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\27\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\-20\\75
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-15\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21435, -2970, 10065; -2970, -410, 1395; 10065, 1395, -4726],[15, 5, 1],[13, -9, -46, 38; -9, 4, 27, -20; 25, -18, -90, 75],15,6]
%
% END OF LATTICE L16.6
%
\hbox{}\par\smallskip%
%
% LATTICE L16.7
%
\leavevmode%
\llap{\spacer}%
${L_{16.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{18}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9270\\-315\\495
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-3\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
495\\15\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-15\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-16\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\12\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\45\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-9\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9270, -315, 495; -315, -3, 15; 495, 15, -26],[45, 3, 1],[1, -2, -3, 2; 5, -15, -16, 12; 22, -45, -66, 45],15,7]
%
% END OF LATTICE L16.7
%
\hbox{}\par\smallskip%
%
% LATTICE L16.8
%
\leavevmode%
\llap{\spacer}%
${L_{16.8}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13860\\-60\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-18\\9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\9\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\52\\138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-21\\-55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\27
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13860, -60, -480; -60, -18, 9; -480, 9, 14],[60, 3, 1],[5, 1, -2, 1; 52, 10, -21, 10; 138, 30, -55, 27],15,8]
%
% END OF LATTICE L16.8
%
\hbox{}\par\smallskip%
%
% LATTICE L16.9
%
\leavevmode%
\llap{\spacer}%
${L_{16.9}}=\pdualwitharg{3.5}(\pfillwitharg{2.3}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2415\\-4590\\945
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4590\\8835\\-1815
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
945\\-1815\\373
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\20\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\9\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\18\\105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-15\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2415, -4590, 945; -4590, 8835, -1815; 945, -1815, 373],[15, 15, 1],[-9, -4, 0, -7; 20, 9, 1, 18; 120, 54, 5, 105],15,9]
%
% END OF LATTICE L16.9
%
\hbox{}\par\smallskip%
%
% LATTICE L16.10
%
\leavevmode%
\llap{\spacer}%
${L_{16.10}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20078520\\-449700\\-5223720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-449700\\10072\\116996
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5223720\\116996\\1359027
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\-49\\-138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-70\\-190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\12\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\-65\\-152
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-12\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20078520, -449700, -5223720; -449700, 10072, 116996; -5223720, 116996, 1359027],[60, 4, 1],[-37, -51, 19, -41; -49, -70, 12, -65; -138, -190, 72, -152],15,10]
%
% END OF LATTICE L16.10
%
\hbox{}\par\smallskip%
%
% LATTICE L16.11
%
\leavevmode%
\llap{\spacer}%
${L_{16.11}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55140\\16920\\-900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16920\\-5190\\275
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\275\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-32\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\18\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\57\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-36\\-65
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-15\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-55140, 16920, -900; 16920, -5190, 275; -900, 275, -14],[60, 5, 1],[-9, 5, 16, -10; -32, 18, 57, -36; -50, 32, 90, -65],15,11]
%
% END OF LATTICE L16.11
%
\hbox{}\par\smallskip%
%
% LATTICE L16.12
%
\leavevmode%
\llap{\spacer}%
${L_{16.12}}=\pdualwitharg{3}({L_{16.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{180}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{18}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16020\\360\\180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\42\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\-3\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\92
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-93
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\180\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-9\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16020, 360, 180; 360, 42, -3; 180, -3, -2],[180, 3, 1],[1, 1, -1, 1; -2, 0, 2, -3; 92, 90, -93, 90],15,12]
%
% END OF LATTICE L16.12
%
\hbox{}\par\smallskip%
%
% LATTICE L16.13
%
\leavevmode%
\llap{\spacer}%
${L_{16.13}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}{90}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14895\\2880\\855
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2880\\-555\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
855\\-165\\-49
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\-27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-45\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14895, 2880, 855; 2880, -555, -165; 855, -165, -49],[45, 15, 1],[0, -1, -1, 2; 3, 3, -5, -3; -10, -27, 0, 45],15,13]
%
% END OF LATTICE L16.13
%
\hbox{}\par\smallskip%
%
% LATTICE L16.14
%
\leavevmode%
\llap{\spacer}%
${L_{16.14}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-899865\\-4545\\26730
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4545\\-15\\135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26730\\135\\-794
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-3\\-135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\101
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\2\\135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-1\\-505
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-45\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-15\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-899865, -4545, 26730; -4545, -15, 135; 26730, 135, -794],[45, 15, 1],[-4, 3, 4, -15; -3, 0, 2, -1; -135, 101, 135, -505],15,14]
%
% END OF LATTICE L16.14
%
\hbox{}\par\smallskip%
%
% LATTICE L16.15
%
\leavevmode%
\llap{\spacer}%
${L_{16.15}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
924840\\-82020\\-249240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82020\\7272\\22104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-249240\\22104\\67169
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\1\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-137\\5\\-510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\1\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\1\\252
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-12\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[924840, -82020, -249240; -82020, 7272, 22104; -249240, 22104, 67169],[60, 12, 1],[-8, -137, -15, 68; 1, 5, 1, 1; -30, -510, -56, 252],15,15]
%
% END OF LATTICE L16.15
%
\hbox{}\par\smallskip%
%
% LATTICE L16.16
%
\leavevmode%
\llap{\spacer}%
${L_{16.16}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2580\\3180\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3180\\3690\\-735
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-735\\142
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\72\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-29\\-85
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\16\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-15\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2580, 3180, -660; 3180, 3690, -735; -660, -735, 142],[60, 15, 1],[-35, -1, 14, -8; 72, 2, -29, 16; 210, 6, -85, 45],15,16]
%
% END OF LATTICE L16.16
%
\hbox{}\par\smallskip%
%
% LATTICE L16.17
%
\leavevmode%
\llap{\spacer}%
${L_{16.17}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{}{6}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
285960\\60060\\31560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60060\\12680\\6580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31560\\6580\\3519
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-67\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\25\\34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-335\\933\\1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-218\\607\\820
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-60\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[285960, 60060, 31560; 60060, 12680, 6580; 31560, 6580, 3519],[60, 20, 1],[24, -9, -335, -218; -67, 25, 933, 607; -90, 34, 1260, 820],15,17]
%
% END OF LATTICE L16.17
%
\hbox{}\par\smallskip%
%
% LATTICE L16.18
%
\leavevmode%
\llap{\spacer}%
${L_{16.18}}=\pdualwitharg{2}({L_{16.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
944821080\\-11048220\\230823540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11048220\\129192\\-2699124
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
230823540\\-2699124\\56391107
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
409\\3\\-1674
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
193\\0\\-790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\-8\\192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
169\\-5\\-692
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-12\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[944821080, -11048220, 230823540; -11048220, 129192, -2699124; 230823540, -2699124, 56391107],[180, 12, 1],[409, 193, -47, 169; 3, 0, -8, -5; -1674, -790, 192, -692],15,18]
%
% END OF LATTICE L16.18
%
\hbox{}\par\smallskip%
%
% LATTICE L16.19
%
\leavevmode%
\llap{\spacer}%
${L_{16.19}}=\pdualwitharg{2.3}({L_{16.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{180}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{72}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11022120\\16020\\2691540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16020\\24\\3912
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2691540\\3912\\657259
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-86
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\15\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
211\\-12\\-864
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\180\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-36\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11022120, 16020, 2691540; 16020, 24, 3912; 2691540, 3912, 657259],[180, 12, 1],[21, -22, 0, 211; 0, 15, 1, -12; -86, 90, 0, -864],15,19]
%
% END OF LATTICE L16.19
%
\hbox{}\par\smallskip%
%
% LATTICE L16.20
%
\leavevmode%
\llap{\spacer}%
${L_{16.20}}=\pdualwitharg{3.5}({L_{16.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{90}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2340\\720\\-900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\210\\285
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\285\\-346
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-2\\100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\2\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-3\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-45\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2340, 720, -900; 720, 210, 285; -900, 285, -346],[180, 15, 1],[-39, -7, 41, -35; -2, 0, 2, -3; 100, 18, -105, 90],15,20]
%
% END OF LATTICE L16.20
%
\hbox{}\par\smallskip%
%
% LATTICE L16.21
%
\leavevmode%
\llap{\spacer}%
${L_{16.21}}=\pdualwitharg{5}({L_{16.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-147780\\-7560\\1440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7560\\-330\\75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1440\\75\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\12\\-450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\92
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-7\\270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\-185
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\0\\-90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-15\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-147780, -7560, 1440; -7560, -330, 75; 1440, 75, -14],[180, 15, 1],[-5, 1, 3, -2; 12, -2, -7, 4; -450, 92, 270, -185],15,21]
%
% END OF LATTICE L16.21
%
\hbox{}\par\smallskip%
%
% LATTICE L16.22
%
\leavevmode%
\llap{\spacer}%
${L_{16.22}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{16.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{}{6}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69240\\-336660\\87540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336660\\1636920\\-425640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
87540\\-425640\\110677
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-3\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\54\\180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-60\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[69240, -336660, 87540; -336660, 1636920, -425640; 87540, -425640, 110677],[60, 60, 1],[1, -7, -1, 35; 8, -3, 5, 54; 30, -6, 20, 180],15,22]
%
% END OF LATTICE L16.22
%
\hbox{}\par\smallskip%
%
% LATTICE L16.23
%
\leavevmode%
\llap{\spacer}%
${L_{16.23}}=\pdualwitharg{2.3.5}({L_{16.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{}{6}{360}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19587240\\-48060\\-4890060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48060\\120\\12000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4890060\\12000\\1220831
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\171\\-230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\-171\\234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-803\\2397\\-3240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-180\\360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19587240, -48060, -4890060; -48060, 120, 12000; -4890060, 12000, 1220831],[180, 60, 1],[-57, 58, 0, -803; 171, -171, 1, 2397; -230, 234, 0, -3240],15,23]
%
% END OF LATTICE L16.23
%
\hbox{}\par\smallskip%
%
% LATTICE L16.24
%
\leavevmode%
\llap{\spacer}%
${L_{16.24}}=\pdualwitharg{2.5}({L_{16.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{}{6}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21886920\\-183780\\-5479200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-183780\\1560\\46020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5479200\\46020\\1371679
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\-63\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\63\\-86
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-381\\1135\\-1560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-254\\757\\-1040
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\0\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-60\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[21886920, -183780, -5479200; -183780, 1560, 46020; -5479200, 46020, 1371679],[180, 60, 1],[22, -21, -381, -254; -63, 63, 1135, 757; 90, -86, -1560, -1040],15,24]
%
% END OF LATTICE L16.24
%

\medskip%
%------------------------------WEYL-GROUP-W17--------------------------------
%
\leavevmode\llap{}%
$W_{17\phantom{0}}$%
\qquad\llap{44} lattices, $\chi=6$%
\hfill%
$22222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L17.1
%
\leavevmode%
\llap{\spacer}%
${L_{17.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{17}}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22020\\360\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-4\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-5\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-28\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-30\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\80\\150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\28\\49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\30\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-6\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-60\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-22020, 360, 240; 360, -4, -5; 240, -5, -2],[60, 1, 1],[-1, -1, 3, 1, 1; -28, -30, 80, 28, 30; -50, -48, 150, 49, 45],16,1]
%
% END OF LATTICE L17.1
%
\hbox{}\par\smallskip%
%
% LATTICE L17.2
%
\leavevmode%
\llap{\spacer}%
${L_{17.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N'_{13}}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9960\\3240\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3240\\1054\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-39\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-28\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\40\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-90\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-40\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-3\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-120\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[9960, 3240, -120; 3240, 1054, -39; -120, -39, 1],[120, 1, 1],[9, -1, -13, 0, 29; -28, 3, 40, 0, -90; -8, 0, 0, -1, -30],16,2]
%
% END OF LATTICE L17.2
%
\hbox{}\par\smallskip%
%
% LATTICE L17.3
%
\leavevmode%
\llap{\spacer}%
${L_{17.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126840\\-3360\\600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3360\\-89\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
600\\16\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\112\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\75\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-260\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-112\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-75\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-40\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-6\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-6\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-60\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-126840, -3360, 600; -3360, -89, 16; 600, 16, -1],[120, 1, 1],[-3, -2, 7, 3, 2; 112, 75, -260, -112, -75; -8, -3, 20, 6, 0],16,3]
%
% END OF LATTICE L17.3
%
\hbox{}\par\smallskip%
%
% LATTICE L17.4
%
\leavevmode%
\llap{\spacer}%
${L_{17.4}}=\pfillwitharg{2}({L_{17.1}})$%
{${}={}$}Nikulin $17$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-10\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-30\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-5\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-3\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\5\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-30\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[30, -15, 0; -15, 7, 0; 0, 0, 1],[15, 1, 1],[0, 1, -4, -3, -11; 0, 3, -10, -8, -30; 1, 0, -5, -2, 0],16,4]
%
% END OF LATTICE L17.4
%
\hbox{}\par\smallskip%
%
% LATTICE L17.5
%
\leavevmode%
\llap{\spacer}%
${L_{17.5}}=\pfillwitharg{2}({L_{17.2}})$%
{${}={}$}Nikulin $13'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
510\\240\\330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\113\\155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
330\\155\\214
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\10\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\3\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\15\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-10\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-3\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-30\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[510, 240, 330; 240, 113, 155; 330, 155, 214],[30, 1, 1],[-1, -2, -11, -2, -7; 2, 0, 10, 3, 15; 0, 3, 10, 1, 0],16,5]
%
% END OF LATTICE L17.5
%
\hbox{}\par\smallskip%
%
% LATTICE L17.6
%
\leavevmode%
\llap{\spacer}%
${L_{17.6}}=\main({L_{17.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\-360\\60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-166\\29
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\29\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\32\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\15\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-40\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-17\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\15\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-3\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-3\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-60\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-780, -360, 60; -360, -166, 29; 60, 29, 7],[60, 1, 1],[-15, -7, 19, 8, -7; 32, 15, -40, -17, 15; -4, -3, 0, 2, 0],16,6]
%
% END OF LATTICE L17.6
%
\hbox{}\par\smallskip%
%
% LATTICE L17.7
%
\leavevmode%
\llap{\spacer}%
${L_{17.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{17.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
165\\15\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-3\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-6\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\1\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-5\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-2\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\-25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-15\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-3\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\15\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-3\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-30\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[165, 15, -75; 15, -3, -6; -75, -6, 34],[15, 3, 1],[4, 1, -34, -18, -12; 1, 0, -5, -2, 0; 9, 2, -75, -39, -25],16,7]
%
% END OF LATTICE L17.7
%
\hbox{}\par\smallskip%
%
% LATTICE L17.8
%
\leavevmode%
\llap{\spacer}%
${L_{17.8}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{17.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15270\\570\\-7530
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
570\\-16\\280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7530\\280\\-3713
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-7\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-15\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\20\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\14\\68
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\15\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-5\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-6\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\5\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-30\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15270, 570, -7530; 570, -16, 280; -7530, 280, -3713],[30, 2, 1],[17, 32, -51, -33, -29; -7, -15, 20, 14, 15; -35, -66, 105, 68, 60],16,8]
%
% END OF LATTICE L17.8
%
\hbox{}\par\smallskip%
%
% LATTICE L17.9
%
\leavevmode%
\llap{\spacer}%
${L_{17.9}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{17.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\-90\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\35\\35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\35\\19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-2\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-3\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\4\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\12\\-85
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\6\\-33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-5\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-15\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\1\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-15\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[105, -90, -45; -90, 35, 35; -45, 35, 19],[15, 5, 1],[7, 5, -11, -29, -11; -2, -3, 4, 12, 6; 20, 15, -32, -85, -33],16,9]
%
% END OF LATTICE L17.9
%
\hbox{}\par\smallskip%
%
% LATTICE L17.10
%
\leavevmode%
\llap{\spacer}%
${L_{17.10}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{17.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{30}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19830\\780\\-6510
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
780\\-24\\255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6510\\255\\-2137
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-14\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-5\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87\\40\\270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\14\\87
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\5\\25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-30\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-3\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-30\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19830, 780, -6510; 780, -24, 255; -6510, 255, -2137],[30, 3, 1],[29, 9, -87, -28, -8; -14, -5, 40, 14, 5; -90, -28, 270, 87, 25],16,10]
%
% END OF LATTICE L17.10
%
\hbox{}\par\smallskip%
%
% LATTICE L17.11
%
\leavevmode%
\llap{\spacer}%
${L_{17.11}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{17.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29130\\1200\\-5730
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\-40\\235
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5730\\235\\-1127
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-14\\-130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-15\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\8\\78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\14\\125
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\3\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-10\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-15\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-15\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-29130, 1200, -5730; 1200, -40, 235; -5730, 235, -1127],[30, 5, 1],[25, 23, -15, -24, -4; -14, -15, 8, 14, 3; -130, -120, 78, 125, 21],16,11]
%
% END OF LATTICE L17.11
%
\hbox{}\par\smallskip%
%
% LATTICE L17.12
%
\leavevmode%
\llap{\spacer}%
${L_{17.12}}=\pdualwitharg{3}(\main({L_{17.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2460\\-120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\6\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\3\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-60\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-3\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\60\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-3\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-60\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2460, -120, -60; -120, 6, 3; -60, 3, 1],[60, 3, 1],[-1, 0, 1, 0, -1; -16, 0, 20, 1, -15; -12, -1, 0, 0, -10],16,12]
%
% END OF LATTICE L17.12
%
\hbox{}\par\smallskip%
%
% LATTICE L17.13
%
\leavevmode%
\llap{\spacer}%
${L_{17.13}}=\pdualwitharg{3}({L_{17.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6060\\-180\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\6\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\3\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\90\\240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-25\\-70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-60\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-6\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\60\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-60\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6060, -180, 240; -180, 6, 3; 240, 3, -8],[60, 3, 1],[-1, 1, 7, 0, -2; -14, 12, 90, 1, -25; -36, 34, 240, 0, -70],16,13]
%
% END OF LATTICE L17.13
%
\hbox{}\par\smallskip%
%
% LATTICE L17.14
%
\leavevmode%
\llap{\spacer}%
${L_{17.14}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{17.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1790340\\-34770\\858870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34770\\678\\-16680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
858870\\-16680\\412021
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\1\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\0\\-148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
511\\5\\-1065
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
167\\3\\-348
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\5\\-350
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-15\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-6\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\15\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-30\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1790340, -34770, 858870; -34770, 678, -16680; 858870, -16680, 412021],[30, 6, 1],[36, 71, 511, 167, 168; 1, 0, 5, 3, 5; -75, -148, -1065, -348, -350],16,14]
%
% END OF LATTICE L17.14
%
\hbox{}\par\smallskip%
%
% LATTICE L17.15
%
\leavevmode%
\llap{\spacer}%
${L_{17.15}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{17.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\15\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\15\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-10\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-15\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\3\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-6\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30, -30, -15; -30, 15, 15; -15, 15, 8],[15, 15, 1],[0, -2, 2, 11, 3; 1, 1, -3, -10, -2; 0, -5, 6, 30, 8],16,15]
%
% END OF LATTICE L17.15
%
\hbox{}\par\smallskip%
%
% LATTICE L17.16
%
\leavevmode%
\llap{\spacer}%
${L_{17.16}}=\pdualwitharg{2}(\main({L_{17.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
424920\\-16020\\-105780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16020\\604\\3988
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105780\\3988\\26333
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-7\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\-57\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-50\\-125
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\30\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-5\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\5\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-12\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-60\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[424920, -16020, -105780; -16020, 604, 3988; -105780, 3988, 26333],[60, 4, 1],[-4, -38, -33, 1, 31; -7, -57, -50, 0, 30; -15, -144, -125, 4, 120],16,16]
%
% END OF LATTICE L17.16
%
\hbox{}\par\smallskip%
%
% LATTICE L17.17
%
\leavevmode%
\llap{\spacer}%
${L_{17.17}}=\pdualwitharg{2}({L_{17.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
526080\\-21780\\-141060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21780\\904\\5840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-141060\\5840\\37823
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\1\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\0\\-138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-158\\5\\-590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\3\\-172
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\15\\-420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-12\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-120\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[526080, -21780, -141060; -21780, 904, 5840; -141060, 5840, 37823],[60, 4, 1],[-8, -37, -158, -46, -112; 1, 0, 5, 3, 15; -30, -138, -590, -172, -420],16,17]
%
% END OF LATTICE L17.17
%
\hbox{}\par\smallskip%
%
% LATTICE L17.18
%
\leavevmode%
\llap{\spacer}%
${L_{17.18}}=\pdualwitharg{5}({L_{17.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2220\\600\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
600\\140\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-45\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-16\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-30\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-12\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2220, 600, -180; 600, 140, -45; -180, -45, 14],[60, 5, 1],[-3, -1, 1, 1, -1; -16, -6, 4, 4, -6; -90, -30, 26, 25, -33],16,18]
%
% END OF LATTICE L17.18
%
\hbox{}\par\smallskip%
%
% LATTICE L17.19
%
\leavevmode%
\llap{\spacer}%
${L_{17.19}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{17.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1579740\\-42150\\751830
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42150\\1130\\-20060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
751830\\-20060\\357811
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\1\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
257\\0\\-540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
129\\1\\-271
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
219\\3\\-460
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\3\\-294
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-5\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\1\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-30\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1579740, -42150, 751830; -42150, 1130, -20060; 751830, -20060, 357811],[30, 10, 1],[50, 257, 129, 219, 140; 1, 0, 1, 3, 3; -105, -540, -271, -460, -294],16,19]
%
% END OF LATTICE L17.19
%
\hbox{}\par\smallskip%
%
% LATTICE L17.20
%
\leavevmode%
\llap{\spacer}%
${L_{17.20}}=\pdualwitharg{5}(\main({L_{17.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-15\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-15\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-12\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[420, -120, 0; -120, 35, 0; 0, 0, -1],[60, 5, 1],[1, -1, -1, 0, 1; 4, -3, -4, -1, 3; 0, 0, -4, -5, -3],16,20]
%
% END OF LATTICE L17.20
%
\hbox{}\par\smallskip%
%
% LATTICE L17.21
%
\leavevmode%
\llap{\spacer}%
${L_{17.21}}=\pdualwitharg{3}({L_{17.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-217320\\1200\\2040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\-3\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2040\\-12\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\19\\95
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\140\\660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\-95
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-120\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-6\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\120\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-6\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-60\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-217320, 1200, 2040; 1200, -3, -12; 2040, -12, -19],[120, 3, 1],[1, 1, 7, -1, -1; 16, 19, 140, -18, -20; 96, 95, 660, -96, -95],16,21]
%
% END OF LATTICE L17.21
%
\hbox{}\par\smallskip%
%
% LATTICE L17.22
%
\leavevmode%
\llap{\spacer}%
${L_{17.22}}=\pdualwitharg{3}({L_{17.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-27\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-40\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\9\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\40\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-120\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-3\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\120\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-120\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120, 0, 120; 0, -6, -27; 120, -27, -1],[120, 3, 1],[1, -2, -9, 2, 9; 4, -9, -40, 9, 40; 0, 2, 0, -3, -10],16,22]
%
% END OF LATTICE L17.22
%
\hbox{}\par\smallskip%
%
% LATTICE L17.23
%
\leavevmode%
\llap{\spacer}%
${L_{17.23}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{17.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1564590\\51390\\-456360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51390\\1695\\-14985
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-456360\\-14985\\133114
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\152\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\128\\-205
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-193\\388\\-618
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-164\\331\\-525
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\71\\-112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\0\\-30\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-15\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-15\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-6\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1564590, 51390, -456360; 51390, 1695, -14985; -456360, -14985, 133114],[30, 15, 1],[-75, -64, -193, -164, -35; 152, 128, 388, 331, 71; -240, -205, -618, -525, -112],16,23]
%
% END OF LATTICE L17.23
%
\hbox{}\par\smallskip%
%
% LATTICE L17.24
%
\leavevmode%
\llap{\spacer}%
${L_{17.24}}=\pdualwitharg{5}({L_{17.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2040\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-40\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-15\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-15\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-24\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2040, 0, -120; 0, 5, 0; -120, 0, -7],[120, 5, 1],[1, -1, -1, 0, 1; 4, 3, 0, -1, 0; -20, 15, 16, 0, -18],16,24]
%
% END OF LATTICE L17.24
%
\hbox{}\par\smallskip%
%
% LATTICE L17.25
%
\leavevmode%
\llap{\spacer}%
${L_{17.25}}=\pdualwitharg{5}({L_{17.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8040\\2520\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2520\\790\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-75\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\16\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-40\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\8\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-30\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-12\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8040, 2520, -240; 2520, 790, -75; -240, -75, 7],[120, 5, 1],[-5, -1, 1, 1, -1; 16, 3, -4, -4, 3; 0, 0, -8, -10, -3],16,25]
%
% END OF LATTICE L17.25
%
\hbox{}\par\smallskip%
%
% LATTICE L17.26
%
\leavevmode%
\llap{\spacer}%
${L_{17.26}}=\pdualwitharg{2.3}(\main({L_{17.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\60\\420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\228\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\-12\\-113
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-1\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-1\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\2\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-15\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\15\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-60\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1560, 60, 420; 60, 228, -12; 420, -12, -113],[60, 12, 1],[12, 15, 4, -29, -11; -1, -1, 0, 2, 0; 45, 56, 15, -108, -40],16,26]
%
% END OF LATTICE L17.26
%
\hbox{}\par\smallskip%
%
% LATTICE L17.27
%
\leavevmode%
\llap{\spacer}%
${L_{17.27}}=\pdualwitharg{2.3}({L_{17.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
269160\\-22140\\-72360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22140\\1824\\5952
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72360\\5952\\19453
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-2\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-1\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\0\\-450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\2\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\0\\-160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-60\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\60\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-120\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[269160, -22140, -72360; -22140, 1824, 5952; -72360, 5952, 19453],[60, 12, 1],[-5, -6, -121, -45, -43; -2, -1, 0, 2, 0; -18, -22, -450, -168, -160],16,27]
%
% END OF LATTICE L17.27
%
\hbox{}\par\smallskip%
%
% LATTICE L17.28
%
\leavevmode%
\llap{\spacer}%
${L_{17.28}}=\pdualwitharg{3.5}(\main({L_{17.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\30\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\15\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\6\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\16\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-15\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-12\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[60, -60, -60; -60, 30, 15; -60, 15, -7],[60, 15, 1],[-1, 2, 5, 0, -1; -4, 6, 16, 1, -3; 0, -5, -12, 0, 2],16,28]
%
% END OF LATTICE L17.28
%
\hbox{}\par\smallskip%
%
% LATTICE L17.29
%
\leavevmode%
\llap{\spacer}%
${L_{17.29}}=\pdualwitharg{3.5}({L_{17.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2460\\1980\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1980\\-1560\\-285
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-285\\-52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-7\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-30\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-12\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2460, 1980, 360; 1980, -1560, -285; 360, -285, -52],[60, 15, 1],[3, -1, -1, 1, 1; -2, 6, 2, -7, -3; 30, -40, -18, 45, 23],16,29]
%
% END OF LATTICE L17.29
%
\hbox{}\par\smallskip%
%
% LATTICE L17.30
%
\leavevmode%
\llap{\spacer}%
${L_{17.30}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{17.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4383660\\-2611530\\-1073790
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2611530\\-1555770\\-639690
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1073790\\-639690\\-263023
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\573\\-1365
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\362\\-860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-341\\813
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-1109\\2640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-65\\154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-30\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\3\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-6\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4383660, -2611530, -1073790; -2611530, -1555770, -639690; -1073790, -639690, -263023],[30, 30, 1],[-7, -5, 4, 14, 1; 573, 362, -341, -1109, -65; -1365, -860, 813, 2640, 154],16,30]
%
% END OF LATTICE L17.30
%
\hbox{}\par\smallskip%
%
% LATTICE L17.31
%
\leavevmode%
\llap{\spacer}%
${L_{17.31}}=\pdualwitharg{2}({L_{17.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75480\\-3600\\1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3600\\-152\\56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\56\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\10\\155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\9\\152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-10\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-24\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\5\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-24\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-60\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-75480, -3600, 1200; -3600, -152, 56; 1200, 56, -19],[120, 8, 1],[-1, -1, 2, 2, 1; -5, -3, 10, 9, 0; -78, -72, 155, 152, 60],16,31]
%
% END OF LATTICE L17.31
%
\hbox{}\par\smallskip%
%
% LATTICE L17.32
%
\leavevmode%
\llap{\spacer}%
${L_{17.32}}=\pdualwitharg{2}({L_{17.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-40\\-79
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\45\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\19\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-10\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-24\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\20\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-24\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-120\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120, 0, 120; 0, -8, -40; 120, -40, -79],[120, 8, 1],[-1, -1, 9, 4, 1; -5, -3, 45, 19, 0; 1, 0, -10, -4, 0],16,32]
%
% END OF LATTICE L17.32
%
\hbox{}\par\smallskip%
%
% LATTICE L17.33
%
\leavevmode%
\llap{\spacer}%
${L_{17.33}}=\pdualwitharg{2.5}(\main({L_{17.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-60\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\140\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-120\\41
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-11\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-45\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\14\\19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\104\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\54\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-5\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\1\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-60\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-12\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-840, -60, -180; -60, 140, -120; -180, -120, 41],[60, 20, 1],[4, 16, -5, -37, -19; -11, -45, 14, 104, 54; -15, -60, 19, 140, 72],16,33]
%
% END OF LATTICE L17.33
%
\hbox{}\par\smallskip%
%
% LATTICE L17.34
%
\leavevmode%
\llap{\spacer}%
${L_{17.34}}=\pdualwitharg{2.5}({L_{17.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
152880\\8220\\34560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8220\\440\\1860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34560\\1860\\7811
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-23\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-132\\-150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-19\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\73\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\69\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\4\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-60\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[152880, 8220, 34560; 8220, 440, 1860; 34560, 1860, 7811],[60, 20, 1],[8, 41, 6, -22, -20; -23, -132, -19, 73, 69; -30, -150, -22, 80, 72],16,34]
%
% END OF LATTICE L17.34
%
\hbox{}\par\smallskip%
%
% LATTICE L17.35
%
\leavevmode%
\llap{\spacer}%
${L_{17.35}}=\pdualwitharg{3.5}({L_{17.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-60\\-119
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\19\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\44\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\-120\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-30\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\24\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-30\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-12\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120, 0, 120; 0, -15, -60; 120, -60, -119],[120, 15, 1],[1, 5, 11, -1, -1; 0, 19, 44, -2, -4; 0, -5, -12, 0, 1],16,35]
%
% END OF LATTICE L17.35
%
\hbox{}\par\smallskip%
%
% LATTICE L17.36
%
\leavevmode%
\llap{\spacer}%
${L_{17.36}}=\pdualwitharg{3.5}({L_{17.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6360\\3480\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3480\\1695\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-180\\19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-85
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\76
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\-120\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-15\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-15\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-24\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6360, 3480, -360; 3480, 1695, -180; -360, -180, 19],[120, 15, 1],[1, -1, -1, 1, 1; 4, -7, -8, 6, 6; 60, -85, -96, 75, 76],16,36]
%
% END OF LATTICE L17.36
%
\hbox{}\par\smallskip%
%
% LATTICE L17.37
%
\leavevmode%
\llap{\spacer}%
${L_{17.37}}=\pdualwitharg{2.3}({L_{17.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2760\\240\\1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-24\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\48\\451
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\51\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\115\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-15\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-50\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-30\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-24\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\60\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-120\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2760, 240, 1200; 240, -24, 48; 1200, 48, 451],[120, 24, 1],[1, 13, 29, -4, -13; 4, 51, 115, -15, -50; -3, -40, -90, 12, 40],16,37]
%
% END OF LATTICE L17.37
%
\hbox{}\par\smallskip%
%
% LATTICE L17.38
%
\leavevmode%
\llap{\spacer}%
${L_{17.38}}=\pdualwitharg{2.3}({L_{17.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216840\\-32520\\1920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32520\\-4848\\288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1920\\288\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-340
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-30\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-24\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\15\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-60\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-216840, -32520, 1920; -32520, -4848, 288; 1920, 288, -17],[120, 24, 1],[0, 2, 2, -1, -3; -1, -1, 0, 1, 0; -18, 208, 225, -96, -340],16,38]
%
% END OF LATTICE L17.38
%
\hbox{}\par\smallskip%
%
% LATTICE L17.39
%
\leavevmode%
\llap{\spacer}%
${L_{17.39}}=\pdualwitharg{2.3.5}(\main({L_{17.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4020\\7980\\-2040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7980\\15960\\-4080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2040\\-4080\\1043
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\12\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\21\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-60\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\3\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-12\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4020, 7980, -2040; 7980, 15960, -4080; -2040, -4080, 1043],[60, 60, 1],[-1, -1, 0, 2, 0; 12, 21, 10, -1, 1; 45, 80, 39, 0, 4],16,39]
%
% END OF LATTICE L17.39
%
\hbox{}\par\smallskip%
%
% LATTICE L17.40
%
\leavevmode%
\llap{\spacer}%
${L_{17.40}}=\pdualwitharg{2.3.5}({L_{17.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\10260\\-2760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10260\\62880\\-16920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2760\\-16920\\4553
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-27\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-17\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-13\\-44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-60\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\12\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-60\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1680, 10260, -2760; 10260, 62880, -16920; -2760, -16920, 4553],[60, 60, 1],[17, 2, 1, 5, 7; -27, -3, -5, -17, -13; -90, -10, -18, -60, -44],16,40]
%
% END OF LATTICE L17.40
%
\hbox{}\par\smallskip%
%
% LATTICE L17.41
%
\leavevmode%
\llap{\spacer}%
${L_{17.41}}=\pdualwitharg{2.5}({L_{17.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-4120\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\360\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\7\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-10\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\240\\0\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\1\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-120\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-12\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240, -120, 0; -120, -4120, 360; 0, 360, -31],[120, 40, 1],[-3, -1, 1, 4, -1; -6, 0, 2, 7, -3; -70, 0, 23, 80, -36],16,41]
%
% END OF LATTICE L17.41
%
\hbox{}\par\smallskip%
%
% LATTICE L17.42
%
\leavevmode%
\llap{\spacer}%
${L_{17.42}}=\pdualwitharg{2.5}({L_{17.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-520\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-10\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\240\\0\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\4\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-120\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240, -120, 0; -120, -520, 80; 0, 80, -11],[120, 40, 1],[-1, -1, 1, 2, -1; -2, 0, 2, 3, -3; -15, 0, 14, 20, -24],16,42]
%
% END OF LATTICE L17.42
%
\hbox{}\par\smallskip%
%
% LATTICE L17.43
%
\leavevmode%
\llap{\spacer}%
${L_{17.43}}=\pdualwitharg{2.3.5}({L_{17.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9960\\27600\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27600\\75720\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-1320\\23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\136
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-30\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-120\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\12\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9960, 27600, -480; 27600, 75720, -1320; -480, -1320, 23],[120, 120, 1],[1, -1, -1, 0, 1; 2, -1, -1, 1, 2; 135, -80, -78, 60, 136],16,43]
%
% END OF LATTICE L17.43
%
\hbox{}\par\smallskip%
%
% LATTICE L17.44
%
\leavevmode%
\llap{\spacer}%
${L_{17.44}}=\pdualwitharg{2.3.5}({L_{17.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\14280\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14280\\-814440\\20520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\20520\\-517
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\10\\390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\3\\116
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-30\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-120\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\3\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-12\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-240, 14280, -360; 14280, -814440, 20520; -360, 20520, -517],[120, 120, 1],[9, -1, -1, 1, 4; 10, 2, -1, -3, 3; 390, 80, -39, -120, 116],16,44]
%
% END OF LATTICE L17.44
%

\medskip%
%------------------------------WEYL-GROUP-W18--------------------------------
%
\leavevmode\llap{}%
$W_{18\phantom{0}}$%
\qquad\llap{88} lattices, $\chi=18$%
\hfill%
$\infty22222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L18.1
%
\leavevmode%
\llap{\spacer}%
${L_{18.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{18},3,2}%
\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{3,2}{\infty b}{60}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18654660\\443520\\38700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
443520\\-10542\\-921
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38700\\-921\\-80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\-1750\\-5490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\100\\300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\429\\1350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\65\\220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-331\\-1026
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-694\\-2168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-60\\-90\\-90\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\60\\0\\-60\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\18\\0\\-18\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-60\\0\\10\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-60\\-18\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-36\\-20\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-18654660, 443520, 38700; 443520, -10542, -921; 38700, -921, -80],[180, 3, 1],[-53, 3, 13, 2, -10, -21; -1750, 100, 429, 65, -331, -694; -5490, 300, 1350, 220, -1026, -2168],17,1]
%
% END OF LATTICE L18.1
%
\hbox{}\par\smallskip%
%
% LATTICE L18.2
%
\leavevmode%
\llap{\spacer}%
${L_{18.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3m,3,2}%
\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{12,5}{\infty b}{120}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116280\\-4680\\-5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4680\\-186\\-201
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5040\\-201\\-217
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-145\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-40\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\72\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\50\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-88\\104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-60\\-90\\-90\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\120\\0\\-120\\-60\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\36\\0\\-18\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-120\\0\\20\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\-18\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-72\\-40\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-116280, -4680, -5040; -4680, -186, -201; -5040, -201, -217],[360, 3, 1],[-2, -1, 1, 1, 0, -1; -145, -40, 72, 50, -10, -88; 180, 60, -90, -70, 9, 104],17,2]
%
% END OF LATTICE L18.2
%
\hbox{}\par\smallskip%
%
% LATTICE L18.3
%
\leavevmode%
\llap{\spacer}%
${L_{18.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{32\rightarrow N'_{12},3,m}%
\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{12,5}{\infty a}{120}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1572120\\-310680\\109800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-310680\\-61395\\21696
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
109800\\21696\\-7663
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-219\\1315\\585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\-260\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\-264\\-117
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\85\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-113\\680\\306
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\1172\\524
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-60\\-45\\-45\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\120\\0\\-60\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\9\\0\\-18\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-60\\0\\5\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-120\\-18\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-36\\-20\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1572120, -310680, 109800; -310680, -61395, 21696; 109800, 21696, -7663],[360, 3, 1],[-219, 43, 44, -14, -113, -195; 1315, -260, -264, 85, 680, 1172; 585, -120, -117, 40, 306, 524],17,3]
%
% END OF LATTICE L18.3
%
\hbox{}\par\smallskip%
%
% LATTICE L18.4
%
\leavevmode%
\llap{\spacer}%
${L_{18.4}}=\pfillwitharg{2.3}({L_{18.1}})$%
{${}={}$}Nikulin $18$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{2,1}{\infty}{15}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-15\\-15\\-45\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\15\\0\\-30\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\2\\0\\-6\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-30\\0\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-6\\-10\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[15, 0, 0; 0, 0, -1; 0, -1, 0],[15, 1, 1],[-1, 1, 0, -2, -2, -1; 0, 0, -1, -5, -3, -1; -15, 0, 1, -5, -9, -7],17,4]
%
% END OF LATTICE L18.4
%
\hbox{}\par\smallskip%
%
% LATTICE L18.5
%
\leavevmode%
\llap{\spacer}%
${L_{18.5}}=\pfillwitharg{3.2}({L_{18.3}})$%
{${}={}$}Nikulin $12'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{2,1}{\infty a}{30}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\0\\30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\-2003
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-2003\\23911
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
179\\-2145\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-12\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\-775\\-65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
74\\-894\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
97\\-1168\\-98
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-30\\-15\\-45\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\30\\0\\-30\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\1\\0\\-3\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-30\\0\\5\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-6\\-10\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[30, 0, 30; 0, 168, -2003; 30, -2003, 23911],[30, 1, 1],[179, 1, 1, 64, 74, 97; -2145, 0, -12, -775, -894, -1168; -180, 0, -1, -65, -75, -98],17,5]
%
% END OF LATTICE L18.5
%
\hbox{}\par\smallskip%
%
% LATTICE L18.6
%
\leavevmode%
\llap{\spacer}%
${L_{18.6}}=\main(\pfillwitharg{3}({L_{18.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{4,1}{\infty}{60}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1740\\60\\480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-2\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
480\\-15\\-97
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\315\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\30\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-45\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-35\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\99\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\232\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-30\\-15\\-45\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\60\\0\\-60\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\2\\0\\-6\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-60\\0\\10\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-12\\-20\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1740, 60, 480; 60, -2, -15; 480, -15, -97],[60, 1, 1],[7, 1, -1, -1, 2, 5; 315, 30, -45, -35, 99, 232; -15, 0, 2, 0, -6, -12],17,6]
%
% END OF LATTICE L18.6
%
\hbox{}\par\smallskip%
%
% LATTICE L18.7
%
\leavevmode%
\llap{\spacer}%
${L_{18.7}}=\pfillwitharg{3}({L_{18.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{1,0}{\infty b}{60}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-302340\\120180\\5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120180\\-47762\\-2003
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\-2003\\-84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\180\\-4350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-13\\310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\33\\-846
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\70\\-1728
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-60\\-30\\-90\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\60\\0\\-60\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\2\\0\\-6\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-60\\0\\10\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-60\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-12\\-20\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-302340, 120180, 5040; 120180, -47762, -2003; 5040, -2003, -84],[60, 1, 1],[-1, 1, 0, -1, -1, -1; 180, 0, -13, -5, 33, 70; -4350, 60, 310, 60, -846, -1728],17,7]
%
% END OF LATTICE L18.7
%
\hbox{}\par\smallskip%
%
% LATTICE L18.8
%
\leavevmode%
\llap{\spacer}%
${L_{18.8}}=\pfillwitharg{3}({L_{18.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{4,1}{\infty a}{120}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-817080\\960\\12600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
960\\-1\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12600\\-15\\-194
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\630\\405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\60\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-90\\-58
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-70\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\99\\57
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\464\\288
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-60\\-30\\-90\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\120\\0\\-120\\-60\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\4\\0\\-6\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-120\\0\\20\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\-6\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-24\\-40\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-817080, 960, 12600; 960, -1, -15; 12600, -15, -194],[120, 1, 1],[7, 1, -1, -1, 1, 5; 630, 60, -90, -70, 99, 464; 405, 60, -58, -60, 57, 288],17,8]
%
% END OF LATTICE L18.8
%
\hbox{}\par\smallskip%
%
% LATTICE L18.9
%
\leavevmode%
\llap{\spacer}%
${L_{18.9}}=\pfillwitharg{3}({L_{18.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{4,1}{\infty b}{120}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3480\\-3480\\1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3480\\-3450\\1769
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1800\\1769\\-899
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-203\\405\\390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\60\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-30\\-29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-35\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\108\\102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-143\\284\\272
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-60\\-15\\-45\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\120\\0\\-60\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\1\\0\\-6\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-60\\0\\5\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-120\\-6\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-12\\-20\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3480, -3480, 1800; -3480, -3450, 1769; 1800, 1769, -899],[120, 1, 1],[-203, -29, 15, 17, -55, -143; 405, 60, -30, -35, 108, 284; 390, 60, -29, -35, 102, 272],17,9]
%
% END OF LATTICE L18.9
%
\hbox{}\par\smallskip%
%
% LATTICE L18.10
%
\leavevmode%
\llap{\spacer}%
${L_{18.10}}=\pfillwitharg{2}({L_{18.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{6,5}{\infty}{15}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8460\\-1305\\-135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1305\\-201\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-21\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\35\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-12\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\10\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\22\\33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\17\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-15\\-45\\-45\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\15\\0\\-30\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\18\\0\\-18\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-30\\0\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\-18\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-18\\-10\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-8460, -1305, -135; -1305, -201, -21; -135, -21, -2],[45, 3, 1],[-6, 1, 2, -2, -4, -3; 35, -5, -12, 10, 22, 17; 30, -15, -9, 25, 33, 20],17,10]
%
% END OF LATTICE L18.10
%
\hbox{}\par\smallskip%
%
% LATTICE L18.11
%
\leavevmode%
\llap{\spacer}%
${L_{18.11}}=\pfillwitharg{2}({L_{18.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{6,5}{\infty a}{30}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-765630\\63630\\11070
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63630\\-5271\\-921
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11070\\-921\\-160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\-160\\-2745
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\10\\150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\39\\675
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-31\\-513
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-64\\-1084
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-30\\-45\\-45\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\30\\0\\-30\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\9\\0\\-9\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-30\\0\\5\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\-9\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-18\\-10\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-765630, 63630, 11070; 63630, -5271, -921; 11070, -921, -160],[90, 3, 1],[-53, 3, 13, 2, -10, -21; -160, 10, 39, 5, -31, -64; -2745, 150, 675, 110, -513, -1084],17,11]
%
% END OF LATTICE L18.11
%
\hbox{}\par\smallskip%
%
% LATTICE L18.12
%
\leavevmode%
\llap{\spacer}%
${L_{18.12}}=\main({L_{18.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{12,5}{\infty}{60}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58140\\-2340\\-2700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2340\\-93\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2700\\-108\\-125
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-55\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\27\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\15\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-36\\52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-30\\-45\\-45\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\60\\0\\-60\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\18\\0\\-18\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-60\\0\\10\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\-18\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-36\\-20\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-58140, -2340, -2700; -2340, -93, -108; -2700, -108, -125],[180, 3, 1],[-2, -1, 1, 1, 0, -1; -55, -10, 27, 15, -11, -36; 90, 30, -45, -35, 9, 52],17,12]
%
% END OF LATTICE L18.12
%
\hbox{}\par\smallskip%
%
% LATTICE L18.13
%
\leavevmode%
\llap{\spacer}%
${L_{18.13}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{2,1}{\infty}{5}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2955\\135\\1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
135\\-6\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1320\\-60\\-589
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\40\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-15\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\25\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\23\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\56\\-33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-5\\-15\\-45\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\5\\0\\-30\\-10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\6\\0\\-6\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-30\\0\\30\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-18\\-30\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2955, 135, 1320; 135, -6, -60; 1320, -60, -589],[15, 3, 1],[-7, 2, 2, -12, -6, -12; 40, -5, -15, 25, 23, 56; -20, 5, 6, -30, -16, -33],17,13]
%
% END OF LATTICE L18.13
%
\hbox{}\par\smallskip%
%
% LATTICE L18.14
%
\leavevmode%
\llap{\spacer}%
${L_{18.14}}=\pdualwitharg{2}(\pfillwitharg{3.2}({L_{18.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{4,3}{\infty z}{15}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-680340\\180270\\-332610
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180270\\-47762\\88132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-332610\\88132\\-162609
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2189\\180\\-4380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-155\\-13\\310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-5\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
437\\33\\-876
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
439\\35\\-879
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-30\\-30\\-90\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\15\\0\\-30\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\2\\0\\-6\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-30\\0\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-30\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-6\\-10\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-680340, 180270, -332610; 180270, -47762, 88132; -332610, 88132, -162609],[30, 2, 1],[2189, -22, -155, -16, 437, 439; 180, 0, -13, -5, 33, 35; -4380, 45, 310, 30, -876, -879],17,14]
%
% END OF LATTICE L18.14
%
\hbox{}\par\smallskip%
%
% LATTICE L18.15
%
\leavevmode%
\llap{\spacer}%
${L_{18.15}}=\pdualwitharg{5}(\pfillwitharg{2.3}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{2,1}{\infty}{3}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1365\\900\\570
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
900\\-560\\-375
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
570\\-375\\-238
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\-3\\117
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\2\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
128\\-6\\315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107\\-6\\265
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-3\\-15\\-9\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\0\\-6\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\10\\0\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-6\\0\\2\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-30\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1365, 900, 570; 900, -560, -375; 570, -375, -238],[15, 5, 1],[47, -5, -30, 8, 128, 107; -3, 0, 2, 0, -6, -6; 117, -12, -75, 19, 315, 265],17,15]
%
% END OF LATTICE L18.15
%
\hbox{}\par\smallskip%
%
% LATTICE L18.16
%
\leavevmode%
\llap{\spacer}%
${L_{18.16}}=\pdualwitharg{3}(\pfillwitharg{3.2}({L_{18.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{2,1}{\infty a}{10}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-806310\\240360\\-262050
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240360\\-71643\\78117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-262050\\78117\\-85166
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
728\\60\\-2185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-155\\-13\\465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-5\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\11\\-433
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
873\\70\\-2622
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-10\\-15\\-45\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\10\\0\\-30\\-10\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\3\\0\\-3\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-30\\0\\15\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-18\\-30\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-806310, 240360, -262050; 240360, -71643, 78117; -262050, 78117, -85166],[30, 3, 1],[728, -13, -155, -21, 144, 873; 60, 0, -13, -5, 11, 70; -2185, 40, 465, 60, -433, -2622],17,16]
%
% END OF LATTICE L18.16
%
\hbox{}\par\smallskip%
%
% LATTICE L18.17
%
\leavevmode%
\llap{\spacer}%
${L_{18.17}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{6,1}{\infty}{15}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-585\\-270\\540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\-93\\150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
540\\150\\-187
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-95\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71\\425\\135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\221\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-239\\1425\\450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\77\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-84\\-27
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-15\\-15\\-135\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\15\\0\\-90\\-30\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\2\\0\\-6\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-90\\0\\90\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-18\\-90\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-585, -270, 540; -270, -93, 150; 540, 150, -187],[45, 3, 1],[16, -71, -37, -239, -13, 14; -95, 425, 221, 1425, 77, -84; -30, 135, 70, 450, 24, -27],17,17]
%
% END OF LATTICE L18.17
%
\hbox{}\par\smallskip%
%
% LATTICE L18.18
%
\leavevmode%
\llap{\spacer}%
${L_{18.18}}=\pdualwitharg{5}(\pfillwitharg{3.2}({L_{18.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{2,1}{\infty a}{6}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1088490\\360540\\-211650
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360540\\-119405\\70105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-211650\\70105\\-41154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
436\\36\\-2181
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-155\\-13\\775
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-1\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
428\\33\\-2145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
869\\70\\-4350
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-15\\-9\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-6\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\5\\0\\-15\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-6\\0\\1\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\-15\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-30\\-10\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1088490, 360540, -211650; 360540, -119405, 70105; -211650, 70105, -41154],[30, 5, 1],[436, -7, -155, -5, 428, 869; 36, 0, -13, -1, 33, 70; -2181, 36, 775, 24, -2145, -4350],17,18]
%
% END OF LATTICE L18.18
%
\hbox{}\par\smallskip%
%
% LATTICE L18.19
%
\leavevmode%
\llap{\spacer}%
${L_{18.19}}=\pdualwitharg{3}(\main(\pfillwitharg{3}({L_{18.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{4,1}{\infty}{20}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19380\\360\\780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-6\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
780\\-15\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\30\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-35\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\52\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-10\\-15\\-45\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\20\\0\\-60\\-20\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\6\\0\\-6\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-60\\0\\30\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-20\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-36\\-60\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19380, 360, 780; 360, -6, -15; 780, -15, -31],[60, 3, 1],[2, 1, -1, -3, 0, 3; 30, 10, -15, -35, 3, 52; 35, 20, -18, -60, -2, 48],17,19]
%
% END OF LATTICE L18.19
%
\hbox{}\par\smallskip%
%
% LATTICE L18.20
%
\leavevmode%
\llap{\spacer}%
${L_{18.20}}=\pdualwitharg{2.3}(\pfillwitharg{3.2}({L_{18.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{4,3}{\infty z}{5}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89749770\\300450\\43079850
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
300450\\1008\\144216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43079850\\144216\\20678309
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5676\\-715\\11830
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\-12\\198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6103\\-775\\12720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2351\\-298\\4900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4619\\-584\\9627
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-10\\-30\\-90\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\5\\0\\-30\\-10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\6\\0\\-6\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-30\\0\\30\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-10\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-18\\-30\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[89749770, 300450, 43079850; 300450, 1008, 144216; 43079850, 144216, 20678309],[30, 6, 1],[-5676, -12, -95, -6103, -2351, -4619; -715, 0, -12, -775, -298, -584; 11830, 25, 198, 12720, 4900, 9627],17,20]
%
% END OF LATTICE L18.20
%
\hbox{}\par\smallskip%
%
% LATTICE L18.21
%
\leavevmode%
\llap{\spacer}%
${L_{18.21}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{1,0}{\infty b}{20}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4766220\\33900\\129420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33900\\-240\\-921
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
129420\\-921\\-3514
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\-770\\-1750
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\40\\100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\190\\429
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\100\\195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-142\\-331
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-908\\-2082
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-20\\-30\\-90\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\20\\0\\-60\\-20\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\6\\0\\-6\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-60\\0\\30\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-20\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-36\\-60\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4766220, 33900, 129420; 33900, -240, -921; 129420, -921, -3514],[60, 3, 1],[-53, 3, 13, 6, -10, -63; -770, 40, 190, 100, -142, -908; -1750, 100, 429, 195, -331, -2082],17,21]
%
% END OF LATTICE L18.21
%
\hbox{}\par\smallskip%
%
% LATTICE L18.22
%
\leavevmode%
\llap{\spacer}%
${L_{18.22}}=\pdualwitharg{3.5}(\pfillwitharg{2.3}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{2,1}{\infty}{1}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-225\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-225\\1905\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\1\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\2\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\4\\50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57\\8\\105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-15\\-9\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-6\\-10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\0\\-30\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-6\\0\\6\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\-30\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-90\\-30\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30, -225, 0; -225, 1905, -15; 0, -15, 1],[15, 15, 1],[7, 0, 1, 15, 29, 57; 1, 0, 0, 2, 4, 8; 14, 1, 0, 24, 50, 105],17,22]
%
% END OF LATTICE L18.22
%
\hbox{}\par\smallskip%
%
% LATTICE L18.23
%
\leavevmode%
\llap{\spacer}%
${L_{18.23}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{2,1}{\infty z}{15}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43032840\\120180\\-11475480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120180\\336\\-32048
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11475480\\-32048\\3060143
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1342\\-2145\\5010
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-24\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
959\\-1550\\3580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1109\\-1788\\4140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
727\\-1168\\2714
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-30\\-60\\-180\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\15\\0\\-60\\-60\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\8\\0\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-60\\0\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-24\\-40\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[43032840, 120180, -11475480; 120180, 336, -32048; -11475480, -32048, 3060143],[60, 4, 1],[1342, 4, 15, 959, 1109, 727; -2145, 0, -24, -1550, -1788, -1168; 5010, 15, 56, 3580, 4140, 2714],17,23]
%
% END OF LATTICE L18.23
%
\hbox{}\par\smallskip%
%
% LATTICE L18.24
%
\leavevmode%
\llap{\spacer}%
${L_{18.24}}=\pdualwitharg{2}(\main(\pfillwitharg{3}({L_{18.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{4,3}{\infty}{60}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16440\\8220\\-4200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8220\\4108\\-2100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4200\\-2100\\1073
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\15\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\0\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\10\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\12\\-192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\4\\-51
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-60\\-60\\-180\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\60\\0\\-120\\-120\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\8\\0\\-24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-120\\0\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-120\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-12\\-20\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16440, 8220, -4200; 8220, 4108, -2100; -4200, -2100, 1073],[60, 4, 1],[-38, 23, 1, -51, -55, -15; 15, 0, 0, 10, 12, 4; -120, 90, 4, -180, -192, -51],17,24]
%
% END OF LATTICE L18.24
%
\hbox{}\par\smallskip%
%
% LATTICE L18.25
%
\leavevmode%
\llap{\spacer}%
${L_{18.25}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{18.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{6,1}{\infty a}{30}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-186930\\-38340\\17370
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38340\\-7863\\3561
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17370\\3561\\-1610
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-190\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-147\\880\\360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\227\\92
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-244\\1455\\585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\77\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-174\\-72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-30\\-15\\-135\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\30\\0\\-90\\-30\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\1\\0\\-3\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-90\\0\\45\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-18\\-90\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-186930, -38340, 17370; -38340, -7863, 3561; 17370, 3561, -1610],[90, 3, 1],[32, -147, -38, -244, -13, 29; -190, 880, 227, 1455, 77, -174; -75, 360, 92, 585, 30, -72],17,25]
%
% END OF LATTICE L18.25
%
\hbox{}\par\smallskip%
%
% LATTICE L18.26
%
\leavevmode%
\llap{\spacer}%
${L_{18.26}}=\pdualwitharg{5}(\main(\pfillwitharg{3}({L_{18.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{4,1}{\infty}{12}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-15\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\15\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-6\\-15\\-9\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\12\\0\\-12\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\10\\0\\-30\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-12\\0\\2\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-60\\-20\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-780, -120, -120; -120, -10, -15; -120, -15, -17],[60, 5, 1],[-1, -1, 1, 1, 2, -1; -3, -6, 3, 5, 15, 4; 9, 12, -10, -12, -30, 0],17,26]
%
% END OF LATTICE L18.26
%
\hbox{}\par\smallskip%
%
% LATTICE L18.27
%
\leavevmode%
\llap{\spacer}%
${L_{18.27}}=\pdualwitharg{2.5}(\pfillwitharg{3.2}({L_{18.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{4,3}{\infty z}{3}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105502710\\420630\\50239350
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420630\\1680\\200300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50239350\\200300\\23923483
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4264\\-429\\8958
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119\\-12\\250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1527\\-155\\3208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8825\\-894\\18540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5781\\-584\\12145
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-30\\-18\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-6\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\0\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-6\\0\\2\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-30\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-30\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[105502710, 420630, 50239350; 420630, 1680, 200300; 50239350, 200300, 23923483],[30, 10, 1],[-4264, -10, -119, -1527, -8825, -5781; -429, 0, -12, -155, -894, -584; 8958, 21, 250, 3208, 18540, 12145],17,27]
%
% END OF LATTICE L18.27
%
\hbox{}\par\smallskip%
%
% LATTICE L18.28
%
\leavevmode%
\llap{\spacer}%
${L_{18.28}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{1,0}{\infty b}{12}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2580\\3840\\-7620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3840\\-5100\\9955
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7620\\9955\\-19382
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-199\\-1122\\-498
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\108\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
98\\552\\245
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\106\\47
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-161\\-912\\-405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-379\\-2140\\-950
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-30\\-18\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-12\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\0\\-30\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-12\\0\\2\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-60\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-60\\-20\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2580, 3840, -7620; 3840, -5100, 9955; -7620, 9955, -19382],[60, 5, 1],[-199, 19, 98, 19, -161, -379; -1122, 108, 552, 106, -912, -2140; -498, 48, 245, 47, -405, -950],17,28]
%
% END OF LATTICE L18.28
%
\hbox{}\par\smallskip%
%
% LATTICE L18.29
%
\leavevmode%
\llap{\spacer}%
${L_{18.29}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{18.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{4,1}{\infty a}{40}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38760\\360\\1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-3\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1560\\-15\\-62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\60\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-30\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-70\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\104\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-20\\-30\\-90\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\40\\0\\-120\\-20\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\12\\0\\-6\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-120\\0\\60\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-20\\-6\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-72\\-120\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38760, 360, 1560; 360, -3, -15; 1560, -15, -62],[120, 3, 1],[2, 1, -1, -3, 0, 3; 60, 20, -30, -70, 3, 104; 35, 20, -18, -60, -1, 48],17,29]
%
% END OF LATTICE L18.29
%
\hbox{}\par\smallskip%
%
% LATTICE L18.30
%
\leavevmode%
\llap{\spacer}%
${L_{18.30}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{4,1}{\infty b}{40}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-524040\\-84840\\18720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84840\\-13725\\3039
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18720\\3039\\-662
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
171\\-830\\1025
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\180\\-220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\165\\-204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\-185\\225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91\\-442\\544
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
463\\-2248\\2772
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-20\\-15\\-45\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\40\\0\\-60\\-40\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\3\\0\\-6\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-60\\0\\15\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-40\\-6\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-36\\-60\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-524040, -84840, 18720; -84840, -13725, 3039; 18720, 3039, -662],[120, 3, 1],[171, -37, -34, 38, 91, 463; -830, 180, 165, -185, -442, -2248; 1025, -220, -204, 225, 544, 2772],17,30]
%
% END OF LATTICE L18.30
%
\hbox{}\par\smallskip%
%
% LATTICE L18.31
%
\leavevmode%
\llap{\spacer}%
${L_{18.31}}=\pdualwitharg{3.5}(\pfillwitharg{3.2}({L_{18.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty a}{2}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89952690\\-475680\\-28407480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-475680\\2520\\150225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28407480\\150225\\8971217
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1063\\-2269\\3404
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-10\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89\\-190\\285
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1142\\-2439\\3657
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2200\\-4698\\7045
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8647\\-18462\\27690
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-15\\-9\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-6\\-10\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\15\\0\\-15\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-6\\0\\3\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\-15\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-90\\-30\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[89952690, -475680, -28407480; -475680, 2520, 150225; -28407480, 150225, 8971217],[30, 15, 1],[1063, 5, 89, 1142, 2200, 8647; -2269, -10, -190, -2439, -4698, -18462; 3404, 16, 285, 3657, 7045, 27690],17,31]
%
% END OF LATTICE L18.31
%
\hbox{}\par\smallskip%
%
% LATTICE L18.32
%
\leavevmode%
\llap{\spacer}%
${L_{18.32}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{18.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{12,7}{\infty z}{15}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{90}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6246810\\-562410\\3085380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-562410\\50640\\-277782
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3085380\\-277782\\1523909
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\5\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1437\\-250\\-2955
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
713\\-123\\-1466
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4903\\-840\\-10080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
397\\-67\\-816
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
92\\-15\\-189
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-30\\-30\\-270\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\15\\0\\-90\\-30\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\2\\0\\-6\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\-90\\0\\90\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-30\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-18\\-90\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6246810, -562410, 3085380; -562410, 50640, -277782; 3085380, -277782, 1523909],[90, 6, 1],[-44, 1437, 713, 4903, 397, 92; 5, -250, -123, -840, -67, -15; 90, -2955, -1466, -10080, -816, -189],17,32]
%
% END OF LATTICE L18.32
%
\hbox{}\par\smallskip%
%
% LATTICE L18.33
%
\leavevmode%
\llap{\spacer}%
${L_{18.33}}=\pdualwitharg{3}(\main({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{12,1}{\infty}{60}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-360\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-102\\105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\105\\-97
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\35\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\15\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\19\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\96\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-30\\-15\\-135\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\60\\0\\-180\\-60\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\2\\0\\-6\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-180\\0\\90\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-36\\-180\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1260, -360, 360; -360, -102, 105; 360, 105, -97],[180, 3, 1],[-14, 3, 2, -4, -7, -37; 35, -10, -5, 15, 19, 96; -15, 0, 2, 0, -6, -36],17,33]
%
% END OF LATTICE L18.33
%
\hbox{}\par\smallskip%
%
% LATTICE L18.34
%
\leavevmode%
\llap{\spacer}%
${L_{18.34}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{18.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{12,11}{\infty z}{15}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
182214810\\-731790\\89001630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-731790\\2940\\-357438
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89001630\\-357438\\43472263
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4954\\-295\\10140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\5\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-299\\-18\\612
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2179\\-135\\4460
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2307\\-142\\4722
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1431\\-87\\2929
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-30\\-90\\-90\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\15\\0\\-30\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\18\\0\\-18\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-30\\0\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-30\\-18\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-18\\-10\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[182214810, -731790, 89001630; -731790, 2940, -357438; 89001630, -357438, 43472263],[90, 6, 1],[-4954, 22, -299, -2179, -2307, -1431; -295, 5, -18, -135, -142, -87; 10140, -45, 612, 4460, 4722, 2929],17,34]
%
% END OF LATTICE L18.34
%
\hbox{}\par\smallskip%
%
% LATTICE L18.35
%
\leavevmode%
\llap{\spacer}%
${L_{18.35}}=\pdualwitharg{3}({L_{18.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{3,1}{\infty a}{60}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6660\\360\\-1080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-12\\69
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\69\\-160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\40\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-340\\240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-77\\55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74\\-435\\315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-11\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\78\\-54
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-60\\-30\\-270\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\60\\0\\-180\\-60\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\2\\0\\-6\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\-180\\0\\90\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-60\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-36\\-180\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6660, 360, -1080; 360, -12, 69; -1080, 69, -160],[180, 3, 1],[7, -57, -13, -74, -2, 13; 40, -340, -77, -435, -11, 78; -30, 240, 55, 315, 9, -54],17,35]
%
% END OF LATTICE L18.35
%
\hbox{}\par\smallskip%
%
% LATTICE L18.36
%
\leavevmode%
\llap{\spacer}%
${L_{18.36}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{18.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{4,1}{\infty a}{24}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10200\\240\\600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-5\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
600\\-15\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\18\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\32\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-30\\-18\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-24\\-60\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\20\\0\\-30\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-24\\0\\4\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\-30\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-120\\-40\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10200, 240, 600; 240, -5, -15; 600, -15, -34],[120, 5, 1],[1, 1, -1, -1, -1, 1; 18, 12, -18, -14, -9, 32; 9, 12, -10, -12, -15, 0],17,36]
%
% END OF LATTICE L18.36
%
\hbox{}\par\smallskip%
%
% LATTICE L18.37
%
\leavevmode%
\llap{\spacer}%
${L_{18.37}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{4,1}{\infty b}{24}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52188600\\-96360\\1354200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96360\\-175\\2500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1354200\\2500\\-35139
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-393\\-3111
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\120\\972
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\133\\1050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-38\\-311
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-233\\-1134\\-9060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-377\\-1844\\-14660
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-15\\-9\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-12\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\5\\0\\-30\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-12\\0\\1\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-120\\-30\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-60\\-20\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-52188600, -96360, 1354200; -96360, -175, 2500; 1354200, 2500, -35139],[120, 5, 1],[-80, 25, 27, -8, -233, -377; -393, 120, 133, -38, -1134, -1844; -3111, 972, 1050, -311, -9060, -14660],17,37]
%
% END OF LATTICE L18.37
%
\hbox{}\par\smallskip%
%
% LATTICE L18.38
%
\leavevmode%
\llap{\spacer}%
${L_{18.38}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{6,5}{\infty}{3}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\-45\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-3705\\765
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\765\\-157
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\5\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-8\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\14\\68
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\68\\330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\32\\155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-3\\-45\\-9\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\0\\-6\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\90\\0\\-90\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-6\\0\\2\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\-90\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-90\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[90, -45, 0; -45, -3705, 765; 0, 765, -157],[45, 15, 1],[3, -4, -1, 7, 35, 17; 5, -8, 0, 14, 68, 32; 24, -39, 0, 68, 330, 155],17,38]
%
% END OF LATTICE L18.38
%
\hbox{}\par\smallskip%
%
% LATTICE L18.39
%
\leavevmode%
\llap{\spacer}%
${L_{18.39}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{6,1}{\infty}{3}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2130\\-405
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-405\\-77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\13\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\27\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\17\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-3\\-15\\-27\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\0\\-18\\-30\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\10\\0\\-30\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-18\\0\\18\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-90\\-90\\0\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[45, 0, 0; 0, -2130, -405; 0, -405, -77],[45, 15, 1],[0, 1, 2, 2, 0, -1; -4, 1, 13, 27, 17, 0; 21, -6, -70, -144, -90, 0],17,39]
%
% END OF LATTICE L18.39
%
\hbox{}\par\smallskip%
%
% LATTICE L18.40
%
\leavevmode%
\llap{\spacer}%
${L_{18.40}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{2,1}{\infty z}{5}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20532480\\-2065740\\-5522520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2065740\\207864\\555612
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5522520\\555612\\1485365
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2593\\-25\\9650
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\-145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
316\\-3\\1176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7078\\-65\\26340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2484\\-23\\9244
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4574\\-43\\17022
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-10\\-60\\-180\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\5\\0\\-60\\-20\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\24\\0\\-24\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-60\\0\\120\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-20\\-24\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-72\\-120\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20532480, -2065740, -5522520; -2065740, 207864, 555612; -5522520, 555612, 1485365],[60, 12, 1],[2593, -39, 316, 7078, 2484, 4574; -25, 0, -3, -65, -23, -43; 9650, -145, 1176, 26340, 9244, 17022],17,40]
%
% END OF LATTICE L18.40
%
\hbox{}\par\smallskip%
%
% LATTICE L18.41
%
\leavevmode%
\llap{\spacer}%
${L_{18.41}}=\pdualwitharg{2.3}(\main(\pfillwitharg{3}({L_{18.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{4,3}{\infty}{20}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40920\\-7740\\-11100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7740\\1524\\2100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11100\\2100\\3011
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-151\\5\\-560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\0\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-291\\10\\-1080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111\\4\\-412
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114\\4\\-423
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-20\\-60\\-180\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\20\\0\\-120\\-40\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\24\\0\\-24\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-120\\0\\120\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-40\\-24\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-36\\-60\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40920, -7740, -11100; -7740, 1524, 2100; -11100, 2100, 3011],[60, 12, 1],[-151, -19, -13, -291, -111, -114; 5, 0, 0, 10, 4, 4; -560, -70, -48, -1080, -412, -423],17,41]
%
% END OF LATTICE L18.41
%
\hbox{}\par\smallskip%
%
% LATTICE L18.42
%
\leavevmode%
\llap{\spacer}%
${L_{18.42}}=\pdualwitharg{3.5}(\main(\pfillwitharg{3}({L_{18.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{4,1}{\infty}{4}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38100\\-1920\\1500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1920\\-90\\75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1500\\75\\-59
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\11\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\48\\540
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-15\\-9\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-12\\-20\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\0\\-30\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-12\\0\\6\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-20\\-30\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-180\\-60\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38100, -1920, 1500; -1920, -90, 75; 1500, 75, -59],[60, 15, 1],[1, -1, -2, 2, 5, 19; 3, -2, -7, 3, 11, 48; 29, -28, -60, 54, 140, 540],17,42]
%
% END OF LATTICE L18.42
%
\hbox{}\par\smallskip%
%
% LATTICE L18.43
%
\leavevmode%
\llap{\spacer}%
${L_{18.43}}=\pdualwitharg{2.3.5}(\pfillwitharg{3.2}({L_{18.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-181773330\\-102407970\\-42111690
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102407970\\-57694860\\-23724990
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42111690\\-23724990\\-9756071
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\5211\\-12724
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-44\\107
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-5553\\13560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-166\\408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\5133\\-12530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\15599\\-38085
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-30\\-18\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-6\\-10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\0\\-30\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-6\\0\\6\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-10\\-30\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-90\\-30\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-181773330, -102407970, -42111690; -102407970, -57694860, -23724990; -42111690, -23724990, -9756071],[30, 30, 1],[12, 0, -13, -1, 11, 35; 5211, -44, -5553, -166, 5133, 15599; -12724, 107, 13560, 408, -12530, -38085],17,43]
%
% END OF LATTICE L18.43
%
\hbox{}\par\smallskip%
%
% LATTICE L18.44
%
\leavevmode%
\llap{\spacer}%
${L_{18.44}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty b}{4}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45540\\54120\\-3900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54120\\64320\\-4635
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3900\\-4635\\334
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\76
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\38\\375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\20\\207
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\10\\115
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\4\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-30\\-18\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-12\\-20\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\0\\-30\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-12\\0\\6\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-20\\-30\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-180\\-60\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[45540, 54120, -3900; 54120, 64320, -4635; -3900, -4635, 334],[60, 15, 1],[1, -3, -13, -6, -2, 3; -2, 8, 38, 20, 10, 4; -16, 76, 375, 207, 115, 90],17,44]
%
% END OF LATTICE L18.44
%
\hbox{}\par\smallskip%
%
% LATTICE L18.45
%
\leavevmode%
\llap{\spacer}%
${L_{18.45}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{8,3}{\infty z}{60}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3129240\\7080\\53040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7080\\-16\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53040\\-120\\-899
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-15\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\225\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\94\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\165\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\-120\\-360\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\60\\0\\-120\\-120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\8\\0\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-120\\0\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-24\\-40\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3129240, 7080, 53040; 7080, -16, -120; 53040, -120, -899],[120, 8, 1],[2, -1, -1, -3, -1, 0; -15, 225, 94, 165, 6, -15; 120, -90, -72, -200, -60, 2],17,45]
%
% END OF LATTICE L18.45
%
\hbox{}\par\smallskip%
%
% LATTICE L18.46
%
\leavevmode%
\llap{\spacer}%
${L_{18.46}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{18.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{8,7}{\infty z}{60}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\14280\\-1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14280\\-226512\\28552
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\28552\\-3599
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-15\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\450\\3570
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\184\\1460
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\315\\2500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-119
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\-120\\-360\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\60\\0\\-120\\-120\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\8\\0\\-24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-120\\0\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-12\\-20\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-840, 14280, -1800; 14280, -226512, 28552; -1800, 28552, -3599],[120, 8, 1],[2, -1, -1, -3, -1, 0; -15, 450, 184, 315, 6, -15; -120, 3570, 1460, 2500, 48, -119],17,46]
%
% END OF LATTICE L18.46
%
\hbox{}\par\smallskip%
%
% LATTICE L18.47
%
\leavevmode%
\llap{\spacer}%
${L_{18.47}}=\pdualwitharg{3}({L_{18.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{12,1}{\infty a}{120}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{180}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6547320\\1971360\\35640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1971360\\-593565\\-10731
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35640\\-10731\\-194
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\70\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-20\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\30\\360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\19\\51
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\192\\216
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-60\\-30\\-270\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\120\\0\\-360\\-60\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\4\\0\\-6\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\-360\\0\\180\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\-6\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-72\\-360\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6547320, 1971360, 35640; 1971360, -593565, -10731; 35640, -10731, -194],[360, 3, 1],[21, -7, -3, 11, 6, 59; 70, -20, -10, 30, 19, 192; -15, -180, 2, 360, 51, 216],17,47]
%
% END OF LATTICE L18.47
%
\hbox{}\par\smallskip%
%
% LATTICE L18.48
%
\leavevmode%
\llap{\spacer}%
${L_{18.48}}=\pdualwitharg{3}({L_{18.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{12,1}{\infty b}{120}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117808920\\-23529240\\325440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23529240\\-4699347\\64998
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
325440\\64998\\-899
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74\\445\\5385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-20\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-30\\-359
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\15\\360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\188\\2370
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-181\\1092\\13428
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-60\\-15\\-135\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\120\\0\\-180\\-120\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\1\\0\\-6\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-180\\0\\45\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-120\\-6\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-36\\-180\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-117808920, -23529240, 325440; -23529240, -4699347, 64998; 325440, 64998, -899],[360, 3, 1],[-74, 3, 5, -2, -31, -181; 445, -20, -30, 15, 188, 1092; 5385, -360, -359, 360, 2370, 13428],17,48]
%
% END OF LATTICE L18.48
%
\hbox{}\par\smallskip%
%
% LATTICE L18.49
%
\leavevmode%
\llap{\spacer}%
${L_{18.49}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{2,1}{\infty z}{3}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13023840\\5914860\\-1048920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5914860\\2686280\\-476380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1048920\\-476380\\84483
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
487\\-1416\\-1938
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\6\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
206\\-599\\-820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
536\\-1559\\-2136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2590\\-7533\\-10320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1514\\-4403\\-6030
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-60\\-36\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-12\\-60\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\0\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-12\\0\\8\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-120\\-40\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13023840, 5914860, -1048920; 5914860, 2686280, -476380; -1048920, -476380, 84483],[60, 20, 1],[487, -2, 206, 536, 2590, 1514; -1416, 6, -599, -1559, -7533, -4403; -1938, 9, -820, -2136, -10320, -6030],17,49]
%
% END OF LATTICE L18.49
%
\hbox{}\par\smallskip%
%
% LATTICE L18.50
%
\leavevmode%
\llap{\spacer}%
${L_{18.50}}=\pdualwitharg{2.5}(\main(\pfillwitharg{3}({L_{18.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{4,3}{\infty}{12}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18300\\-2520\\6780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\380\\-960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6780\\-960\\2533
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\87\\108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\33\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\63\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\77\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109\\339\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\109\\135
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-60\\-36\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-24\\-120\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\0\\-120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-24\\0\\8\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-120\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-60\\-20\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18300, -2520, 6780; -2520, 380, -960; 6780, -960, 2533],[60, 20, 1],[-28, -11, -21, -25, -109, -35; 87, 33, 63, 77, 339, 109; 108, 42, 80, 96, 420, 135],17,50]
%
% END OF LATTICE L18.50
%
\hbox{}\par\smallskip%
%
% LATTICE L18.51
%
\leavevmode%
\llap{\spacer}%
${L_{18.51}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{18.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{6,5}{\infty a}{6}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
260730\\-108450\\-19980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108450\\45105\\8310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19980\\8310\\1531
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\7\\-51
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-16\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-60\\495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-7\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\2\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\16\\-100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-45\\-9\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-6\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\45\\0\\-45\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-6\\0\\1\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\-45\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-90\\-10\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[260730, -108450, -19980; -108450, 45105, 8310; -19980, 8310, 1531],[90, 15, 1],[-1, 3, 13, 2, 2, -1; 7, -16, -60, -7, 2, 16; -51, 126, 495, 64, 15, -100],17,51]
%
% END OF LATTICE L18.51
%
\hbox{}\par\smallskip%
%
% LATTICE L18.52
%
\leavevmode%
\llap{\spacer}%
${L_{18.52}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{18.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{6,1}{\infty a}{6}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{90}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1785\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-330\\-61
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\18\\-99
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-15\\-27\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-18\\-30\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\5\\0\\-15\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-18\\0\\9\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\-15\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-90\\-90\\0\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[90, 0, 0; 0, -1785, -330; 0, -330, -61],[90, 15, 1],[0, 1, 1, 1, 0, -1; -5, 2, 9, 18, 11, 0; 27, -12, -50, -99, -60, 0],17,52]
%
% END OF LATTICE L18.52
%
\hbox{}\par\smallskip%
%
% LATTICE L18.53
%
\leavevmode%
\llap{\spacer}%
${L_{18.53}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{18.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,1}{\infty a}{8}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-861000\\-436920\\10080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-436920\\-221715\\5115
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10080\\5115\\-118
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\6\\89
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-14\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\174
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\11\\220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\96\\1680
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-30\\-18\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-24\\-20\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\60\\0\\-30\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-24\\0\\12\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-20\\-30\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-360\\-120\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-861000, -436920, 10080; -436920, -221715, 5115; 10080, 5115, -118],[120, 15, 1],[-2, 1, 5, -1, -3, -29; 6, -4, -14, 6, 11, 96; 89, -88, -180, 174, 220, 1680],17,53]
%
% END OF LATTICE L18.53
%
\hbox{}\par\smallskip%
%
% LATTICE L18.54
%
\leavevmode%
\llap{\spacer}%
${L_{18.54}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,1}{\infty b}{8}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109800\\2280\\1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2280\\-45\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1320\\-30\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\31\\29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-32\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-31\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\32\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\158\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\596\\540
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-15\\-9\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-12\\-40\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\15\\0\\-30\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-12\\0\\3\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-40\\-30\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-180\\-60\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-109800, 2280, 1320; 2280, -45, -30; 1320, -30, -13],[120, 15, 1],[1, -1, -1, 1, 5, 19; 31, -32, -31, 32, 158, 596; 29, -28, -30, 27, 140, 540],17,54]
%
% END OF LATTICE L18.54
%
\hbox{}\par\smallskip%
%
% LATTICE L18.55
%
\leavevmode%
\llap{\spacer}%
${L_{18.55}}=\pdualwitharg{2.3}({L_{18.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{6,1}{\infty z}{15}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{360}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2852640\\138420\\705060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
138420\\6720\\34212
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
705060\\34212\\174263
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-5\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
196\\10\\-795
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
205\\13\\-832
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1463\\105\\-5940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127\\11\\-516
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\3\\-126
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-30\\-60\\-540\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\15\\0\\-180\\-60\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\8\\0\\-24\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\-180\\0\\360\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-72\\-360\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2852640, 138420, 705060; 138420, 6720, 34212; 705060, 34212, 174263],[180, 12, 1],[-22, 196, 205, 1463, 127, 31; -5, 10, 13, 105, 11, 3; 90, -795, -832, -5940, -516, -126],17,55]
%
% END OF LATTICE L18.55
%
\hbox{}\par\smallskip%
%
% LATTICE L18.56
%
\leavevmode%
\llap{\spacer}%
${L_{18.56}}=\pdualwitharg{2}({L_{18.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{6,5}{\infty z}{15}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
728859240\\-16073460\\178003260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16073460\\354480\\-3925488
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
178003260\\-3925488\\43472263
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8967\\-295\\36690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
121\\5\\-495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1091\\-36\\4464
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8119\\-270\\33220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8555\\-284\\35004
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5259\\-174\\21518
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-30\\-180\\-180\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\15\\0\\-60\\-60\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\72\\0\\-72\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-60\\0\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\-72\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-72\\-40\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[728859240, -16073460, 178003260; -16073460, 354480, -3925488; 178003260, -3925488, 43472263],[180, 12, 1],[-8967, 121, -1091, -8119, -8555, -5259; -295, 5, -36, -270, -284, -174; 36690, -495, 4464, 33220, 35004, 21518],17,56]
%
% END OF LATTICE L18.56
%
\hbox{}\par\smallskip%
%
% LATTICE L18.57
%
\leavevmode%
\llap{\spacer}%
${L_{18.57}}=\pdualwitharg{2}(\main({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{12,11}{\infty}{60}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1232460\\42480\\301860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42480\\1500\\10404
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
301860\\10404\\73933
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-574\\25\\2340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81\\5\\330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\3\\216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-363\\15\\1480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-415\\17\\1692
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-143\\6\\583
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-60\\-180\\-180\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\60\\0\\-120\\-120\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\72\\0\\-72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-120\\0\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-120\\-72\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-36\\-20\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1232460, 42480, 301860; 42480, 1500, 10404; 301860, 10404, 73933],[180, 12, 1],[-574, -81, -53, -363, -415, -143; 25, 5, 3, 15, 17, 6; 2340, 330, 216, 1480, 1692, 583],17,57]
%
% END OF LATTICE L18.57
%
\hbox{}\par\smallskip%
%
% LATTICE L18.58
%
\leavevmode%
\llap{\spacer}%
${L_{18.58}}=\pdualwitharg{2.3}(\main({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{12,7}{\infty}{60}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{360}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
158040\\-8820\\39780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8820\\492\\-2220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39780\\-2220\\10013
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-35\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-10\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\-30\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-16\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-21\\27
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-60\\-60\\-540\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\60\\0\\-360\\-120\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\8\\0\\-24\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\-360\\0\\360\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-120\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-36\\-180\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[158040, -8820, 39780; -8820, 492, -2220; 39780, -2220, 10013],[180, 12, 1],[-2, 7, -1, -47, -13, -8; -35, -10, 0, -30, -16, -21; 0, -30, 4, 180, 48, 27],17,58]
%
% END OF LATTICE L18.58
%
\hbox{}\par\smallskip%
%
% LATTICE L18.59
%
\leavevmode%
\llap{\spacer}%
${L_{18.59}}=\pdualwitharg{5}(\main({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{12,5}{\infty}{12}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72540\\-183240\\-31500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-183240\\-459105\\-78840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31500\\-78840\\-13537
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\-182\\939
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\178\\-918
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-107\\375\\-1935
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\-115\\593
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
254\\-887\\4575
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
325\\-1136\\5860
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-6\\-45\\-9\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\12\\0\\-12\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\90\\0\\-90\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-12\\0\\2\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\-90\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-180\\-20\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-72540, -183240, -31500; -183240, -459105, -78840; -31500, -78840, -13537],[180, 15, 1],[52, -51, -107, 33, 254, 325; -182, 178, 375, -115, -887, -1136; 939, -918, -1935, 593, 4575, 5860],17,59]
%
% END OF LATTICE L18.59
%
\hbox{}\par\smallskip%
%
% LATTICE L18.60
%
\leavevmode%
\llap{\spacer}%
${L_{18.60}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{18.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,7}{\infty z}{3}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
482130\\29700\\235710
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29700\\1830\\14520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
235710\\14520\\115237
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\1\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58\\13\\117
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-124\\29\\250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-143\\36\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\11\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\0\\-45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-30\\-54\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-18\\-30\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\0\\-30\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-18\\0\\18\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-30\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-90\\-90\\0\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[482130, 29700, 235710; 29700, 1830, 14520; 235710, 14520, 115237],[90, 30, 1],[-3, -58, -124, -143, -30, 22; 1, 13, 29, 36, 11, 0; 6, 117, 250, 288, 60, -45],17,60]
%
% END OF LATTICE L18.60
%
\hbox{}\par\smallskip%
%
% LATTICE L18.61
%
\leavevmode%
\llap{\spacer}%
${L_{18.61}}=\pdualwitharg{3.5}(\main({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{12,1}{\infty}{12}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{180}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17100\\5220\\-540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5220\\-1590\\165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\165\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\7\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\24\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-6\\-15\\-27\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\12\\0\\-36\\-60\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\10\\0\\-30\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-36\\0\\18\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-180\\-180\\0\\180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17100, 5220, -540; 5220, -1590, 165; -540, 165, -17],[180, 15, 1],[0, -1, 0, 2, 3, 7; 1, -2, -1, 3, 7, 24; 9, 12, -10, -36, -30, 0],17,61]
%
% END OF LATTICE L18.61
%
\hbox{}\par\smallskip%
%
% LATTICE L18.62
%
\leavevmode%
\llap{\spacer}%
${L_{18.62}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{18.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,11}{\infty z}{3}\above{1pt}{1pt}{}{2}{90}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3272490\\158400\\1598490
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
158400\\7590\\77370
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1598490\\77370\\780803
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
371\\25\\-762
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\1\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2629\\-180\\5400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-778\\-53\\1598
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2001\\-136\\4110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-426\\-29\\875
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-90\\-18\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-6\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\90\\0\\-90\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-6\\0\\2\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-30\\-90\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-90\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3272490, 158400, 1598490; 158400, 7590, 77370; 1598490, 77370, 780803],[90, 30, 1],[371, 19, -2629, -778, -2001, -426; 25, 1, -180, -53, -136, -29; -762, -39, 5400, 1598, 4110, 875],17,62]
%
% END OF LATTICE L18.62
%
\hbox{}\par\smallskip%
%
% LATTICE L18.63
%
\leavevmode%
\llap{\spacer}%
${L_{18.63}}=\pdualwitharg{5}({L_{18.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{3,2}{\infty b}{12}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2794860\\-383400\\-72720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-383400\\52590\\9975
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72720\\9975\\1892
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\28\\-186
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-76\\516
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-315\\2160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-45\\314
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-31\\240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\42\\-260
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-90\\-18\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-12\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\90\\0\\-90\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-12\\0\\2\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-60\\-90\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-180\\-20\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2794860, -383400, -72720; -383400, 52590, 9975; -72720, 9975, 1892],[180, 15, 1],[-1, 3, 13, 2, 2, -1; 28, -76, -315, -45, -31, 42; -186, 516, 2160, 314, 240, -260],17,63]
%
% END OF LATTICE L18.63
%
\hbox{}\par\smallskip%
%
% LATTICE L18.64
%
\leavevmode%
\llap{\spacer}%
${L_{18.64}}=\pdualwitharg{3.5}({L_{18.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{3,1}{\infty a}{12}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{180}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37620\\65160\\-10440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65160\\-112560\\18015
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10440\\18015\\-2882
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\50\\222
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-56\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-122\\-545
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-101\\-204\\-909
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-98\\-435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\60\\270
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-30\\-54\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-36\\-60\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\0\\-30\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-36\\0\\18\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-60\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-180\\-180\\0\\180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-37620, 65160, -10440; 65160, -112560, 18015; -10440, 18015, -2882],[180, 15, 1],[25, -27, -60, -101, -49, 29; 50, -56, -122, -204, -98, 60; 222, -252, -545, -909, -435, 270],17,64]
%
% END OF LATTICE L18.64
%
\hbox{}\par\smallskip%
%
% LATTICE L18.65
%
\leavevmode%
\llap{\spacer}%
${L_{18.65}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{8,3}{\infty z}{20}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85320\\83280\\-30120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83280\\-81288\\29400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30120\\29400\\-10633
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-5\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\25\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\34\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
148\\65\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\2\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-5\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-40\\-120\\-360\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\20\\0\\-120\\-40\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\24\\0\\-24\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-120\\0\\120\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\-24\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-72\\-120\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-85320, 83280, -30120; 83280, -81288, 29400; -30120, 29400, -10633],[120, 24, 1],[-19, 42, 67, 148, 9, -7; -5, 25, 34, 65, 2, -5; 40, -50, -96, -240, -20, 6],17,65]
%
% END OF LATTICE L18.65
%
\hbox{}\par\smallskip%
%
% LATTICE L18.66
%
\leavevmode%
\llap{\spacer}%
${L_{18.66}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{18.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{8,7}{\infty z}{20}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4213080\\-1126200\\106200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1126200\\-298800\\28392
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
106200\\28392\\-2677
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82\\-5\\3200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-791\\50\\-30850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1015\\64\\-39588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1829\\115\\-71340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\2\\-1288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79\\-5\\3081
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-40\\-120\\-360\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\20\\0\\-120\\-40\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\24\\0\\-24\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-120\\0\\120\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\-24\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-36\\-60\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4213080, -1126200, 106200; -1126200, -298800, 28392; 106200, 28392, -2677],[120, 24, 1],[82, -791, -1015, -1829, -33, 79; -5, 50, 64, 115, 2, -5; 3200, -30850, -39588, -71340, -1288, 3081],17,66]
%
% END OF LATTICE L18.66
%
\hbox{}\par\smallskip%
%
% LATTICE L18.67
%
\leavevmode%
\llap{\spacer}%
${L_{18.67}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{18.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{2,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\8220\\-2220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8220\\-138480\\37140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2220\\37140\\-9959
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-27\\-98
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
165\\364\\1320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
149\\334\\1212
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
129\\292\\1060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\190\\690
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-60\\-36\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-12\\-20\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\0\\-120\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-12\\0\\24\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-20\\-120\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-360\\-120\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-360, 8220, -2220; 8220, -138480, 37140; -2220, 37140, -9959],[60, 60, 1],[-12, -1, 165, 149, 129, 83; -27, -3, 364, 334, 292, 190; -98, -11, 1320, 1212, 1060, 690],17,67]
%
% END OF LATTICE L18.67
%
\hbox{}\par\smallskip%
%
% LATTICE L18.68
%
\leavevmode%
\llap{\spacer}%
${L_{18.68}}=\pdualwitharg{2.3.5}(\main(\pfillwitharg{3}({L_{18.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty}{4}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2100\\-19380\\5220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19380\\190440\\-51300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5220\\-51300\\13819
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-82\\-304
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-81\\-300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-217\\-804
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-367\\-1360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-344\\-1275
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-60\\-36\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-24\\-40\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\0\\-120\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-24\\0\\24\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-40\\-120\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-180\\-60\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2100, -19380, 5220; -19380, 190440, -51300; 5220, -51300, 13819],[60, 60, 1],[-1, 0, -2, -4, -6, -5; -82, -7, -81, -217, -367, -344; -304, -26, -300, -804, -1360, -1275],17,68]
%
% END OF LATTICE L18.68
%
\hbox{}\par\smallskip%
%
% LATTICE L18.69
%
\leavevmode%
\llap{\spacer}%
${L_{18.69}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{8,3}{\infty z}{12}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7800\\6960\\-3360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6960\\-6200\\3000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3360\\3000\\-1447
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-3\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\9\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\22\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\9\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\6\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-3\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-120\\-72\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-24\\-120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\40\\0\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-24\\0\\8\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-120\\-40\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7800, 6960, -3360; 6960, -6200, 3000; -3360, 3000, -1447],[120, 40, 1],[-13, 26, 71, 32, 31, -7; -3, 9, 22, 9, 6, -3; 24, -42, -120, -56, -60, 10],17,69]
%
% END OF LATTICE L18.69
%
\hbox{}\par\smallskip%
%
% LATTICE L18.70
%
\leavevmode%
\llap{\spacer}%
${L_{18.70}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{18.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{8,7}{\infty z}{12}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-948840\\329760\\39000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
329760\\-111080\\-13560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39000\\-13560\\-1603
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\-3\\-1848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
445\\18\\10674
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
993\\40\\23820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
374\\15\\8972
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
155\\6\\3720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74\\-3\\-1775
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-120\\-72\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-24\\-120\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\40\\0\\-120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-24\\0\\8\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-60\\-20\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-948840, 329760, 39000; 329760, -111080, -13560; 39000, -13560, -1603],[120, 40, 1],[-77, 445, 993, 374, 155, -74; -3, 18, 40, 15, 6, -3; -1848, 10674, 23820, 8972, 3720, -1775],17,70]
%
% END OF LATTICE L18.70
%
\hbox{}\par\smallskip%
%
% LATTICE L18.71
%
\leavevmode%
\llap{\spacer}%
${L_{18.71}}=\pdualwitharg{5}({L_{18.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,5}{\infty b}{24}\above{1pt}{1pt}{s}{2}{180}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2283480\\-2359440\\-394560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2359440\\-2436690\\-407445
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-394560\\-407445\\-68129
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\-581\\2994
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-83\\580\\-2988
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-167\\1170\\-6030
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\-384\\1978
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
207\\-1447\\7455
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
525\\-3672\\18920
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-90\\-18\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-24\\-60\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\180\\0\\-90\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-24\\0\\4\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\-90\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-360\\-40\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2283480, -2359440, -394560; -2359440, -2436690, -407445; -394560, -407445, -68129],[360, 15, 1],[83, -83, -167, 55, 207, 525; -581, 580, 1170, -384, -1447, -3672; 2994, -2988, -6030, 1978, 7455, 18920],17,71]
%
% END OF LATTICE L18.71
%
\hbox{}\par\smallskip%
%
% LATTICE L18.72
%
\leavevmode%
\llap{\spacer}%
${L_{18.72}}=\pdualwitharg{5}({L_{18.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,5}{\infty a}{24}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-447480\\-736920\\-118080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-736920\\-1213050\\-194355
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-118080\\-194355\\-31139
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-173\\894
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\160\\-828
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\183\\-945
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-50\\259
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
227\\-806\\4170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
297\\-1052\\5440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-45\\-9\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-12\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\45\\0\\-90\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-12\\0\\1\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-120\\-90\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-180\\-20\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-447480, -736920, -118080; -736920, -1213050, -194355; -118080, -194355, -31139],[360, 15, 1],[49, -45, -52, 14, 227, 297; -173, 160, 183, -50, -806, -1052; 894, -828, -945, 259, 4170, 5440],17,72]
%
% END OF LATTICE L18.72
%
\hbox{}\par\smallskip%
%
% LATTICE L18.73
%
\leavevmode%
\llap{\spacer}%
${L_{18.73}}=\pdualwitharg{3.5}({L_{18.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,1}{\infty a}{24}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{360}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34200\\-11880\\-1080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11880\\-4125\\-375
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\-375\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\48\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-30\\-54\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-72\\-60\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\20\\0\\-30\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-72\\0\\36\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\-30\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-360\\-360\\0\\360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-34200, -11880, -1080; -11880, -4125, -375; -1080, -375, -34],[360, 15, 1],[-1, 1, 1, -1, -2, -17; 2, -4, -2, 6, 7, 48; 9, 12, -10, -36, -15, 0],17,73]
%
% END OF LATTICE L18.73
%
\hbox{}\par\smallskip%
%
% LATTICE L18.74
%
\leavevmode%
\llap{\spacer}%
${L_{18.74}}=\pdualwitharg{3.5}({L_{18.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,1}{\infty b}{24}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{*}{2}{360}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1299960\\128520\\22680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
128520\\-12675\\-2235
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22680\\-2235\\-394
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\124\\-531
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-44\\192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-41\\175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\45\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\382\\-1650
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\1800\\-7740
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-15\\-27\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-36\\-120\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\5\\0\\-30\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-36\\0\\9\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-120\\-30\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-180\\-180\\0\\360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1299960, 128520, 22680; 128520, -12675, -2235; 22680, -2235, -394],[360, 15, 1],[3, -1, -1, 1, 9, 43; 124, -44, -41, 45, 382, 1800; -531, 192, 175, -198, -1650, -7740],17,74]
%
% END OF LATTICE L18.74
%
\hbox{}\par\smallskip%
%
% LATTICE L18.75
%
\leavevmode%
\llap{\spacer}%
${L_{18.75}}=\pdualwitharg{2.3}({L_{18.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{24,19}{\infty z}{60}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{360}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32815440\\-5913720\\-831960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5913720\\-1065720\\-149928
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-831960\\-149928\\-21091
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\200\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
344\\-2065\\1110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
145\\-871\\472
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
766\\-4605\\2520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-55\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73\\438\\-234
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\-120\\-1080\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\60\\0\\-360\\-120\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\8\\0\\-24\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\-360\\0\\360\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-72\\-360\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-32815440, -5913720, -831960; -5913720, -1065720, -149928; -831960, -149928, -21091],[360, 24, 1],[-33, 344, 145, 766, 9, -73; 200, -2065, -871, -4605, -55, 438; -120, 1110, 472, 2520, 36, -234],17,75]
%
% END OF LATTICE L18.75
%
\hbox{}\par\smallskip%
%
% LATTICE L18.76
%
\leavevmode%
\llap{\spacer}%
${L_{18.76}}=\pdualwitharg{2.3}({L_{18.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{24,7}{\infty z}{60}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{360}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87377040\\-17531640\\6124320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17531640\\-3517608\\1228800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6124320\\1228800\\-429247
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
167\\-1000\\-480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1641\\9845\\4770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-701\\4205\\2036
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3784\\22695\\10980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67\\401\\192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
164\\-984\\-477
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\-120\\-1080\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\60\\0\\-360\\-120\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\8\\0\\-24\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\-360\\0\\360\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-36\\-180\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-87377040, -17531640, 6124320; -17531640, -3517608, 1228800; 6124320, 1228800, -429247],[360, 24, 1],[167, -1641, -701, -3784, -67, 164; -1000, 9845, 4205, 22695, 401, -984; -480, 4770, 2036, 10980, 192, -477],17,76]
%
% END OF LATTICE L18.76
%
\hbox{}\par\smallskip%
%
% LATTICE L18.77
%
\leavevmode%
\llap{\spacer}%
${L_{18.77}}=\pdualwitharg{2}({L_{18.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{24,11}{\infty z}{60}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111381840\\18542520\\-316440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18542520\\-3086904\\52680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-316440\\52680\\-899
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-40\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\455\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
97\\579\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\345\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\17\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-30\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\-360\\-360\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\60\\0\\-120\\-120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\72\\0\\-72\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-120\\0\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-72\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-72\\-40\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-111381840, 18542520, -316440; 18542520, -3086904, 52680; -316440, 52680, -899],[360, 24, 1],[-7, 76, 97, 58, 3, -5; -40, 455, 579, 345, 17, -30; 120, -90, -216, -200, -60, 2],17,77]
%
% END OF LATTICE L18.77
%
\hbox{}\par\smallskip%
%
% LATTICE L18.78
%
\leavevmode%
\llap{\spacer}%
${L_{18.78}}=\pdualwitharg{2}({L_{18.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{24,23}{\infty z}{60}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-456117840\\-91351800\\1281240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-91351800\\-18296040\\256608
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1281240\\256608\\-3599
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-40\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-151\\905\\10770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\1119\\13212
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\645\\7540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\17\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-30\\-359
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\-360\\-360\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\60\\0\\-120\\-120\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\72\\0\\-72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-120\\0\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-72\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-36\\-20\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-456117840, -91351800, 1281240; -91351800, -18296040, 256608; 1281240, 256608, -3599],[360, 24, 1],[7, -151, -187, -108, -3, 5; -40, 905, 1119, 645, 17, -30; -360, 10770, 13212, 7540, 144, -359],17,78]
%
% END OF LATTICE L18.78
%
\hbox{}\par\smallskip%
%
% LATTICE L18.79
%
\leavevmode%
\llap{\spacer}%
${L_{18.79}}=\pdualwitharg{2.3.5}({L_{18.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{6,1}{\infty z}{3}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}{180}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20473200\\1976940\\-3489120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1976940\\190920\\-336900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3489120\\-336900\\594643
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\158\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-317\\1022\\-1281
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1356\\4371\\-5480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1630\\5253\\-6588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-460\\1481\\-1860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\69\\-90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-60\\-108\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-36\\-60\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\0\\-120\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-36\\0\\72\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-360\\-360\\0\\180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20473200, 1976940, -3489120; 1976940, 190920, -336900; -3489120, -336900, 594643],[180, 60, 1],[-49, -317, -1356, -1630, -460, -22; 158, 1022, 4371, 5253, 1481, 69; -198, -1281, -5480, -6588, -1860, -90],17,79]
%
% END OF LATTICE L18.79
%
\hbox{}\par\smallskip%
%
% LATTICE L18.80
%
\leavevmode%
\llap{\spacer}%
${L_{18.80}}=\pdualwitharg{2.5}({L_{18.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{6,5}{\infty z}{3}\above{1pt}{1pt}{r}{2}{360}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28461600\\-1753020\\-8238960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1753020\\107760\\507300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8238960\\507300\\2384867
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-452\\1375\\-1854
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\91\\-123
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6319\\-19227\\25920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1885\\-5735\\7732
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4871\\-14819\\19980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1041\\-3167\\4270
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-180\\-36\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-12\\-60\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\360\\0\\-360\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-12\\0\\8\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\-360\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-360\\-40\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28461600, -1753020, -8238960; -1753020, 107760, 507300; -8238960, 507300, 2384867],[180, 60, 1],[-452, -30, 6319, 1885, 4871, 1041; 1375, 91, -19227, -5735, -14819, -3167; -1854, -123, 25920, 7732, 19980, 4270],17,80]
%
% END OF LATTICE L18.80
%
\hbox{}\par\smallskip%
%
% LATTICE L18.81
%
\leavevmode%
\llap{\spacer}%
${L_{18.81}}=\pdualwitharg{2.5}(\main({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,11}{\infty}{12}\above{1pt}{1pt}{*}{2}{360}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280011960\\-30969900\\-90567900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30969900\\3425340\\10017000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90567900\\10017000\\29293553
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-583\\1607\\-2352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\136\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-581\\1590\\-2340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-517\\1420\\-2084
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2619\\7202\\-10560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-817\\2249\\-3295
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-180\\-36\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-24\\-120\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\360\\0\\-360\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-24\\0\\8\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-120\\-360\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-180\\-20\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[280011960, -30969900, -90567900; -30969900, 3425340, 10017000; -90567900, 10017000, 29293553],[180, 60, 1],[-583, -49, -581, -517, -2619, -817; 1607, 136, 1590, 1420, 7202, 2249; -2352, -198, -2340, -2084, -10560, -3295],17,81]
%
% END OF LATTICE L18.81
%
\hbox{}\par\smallskip%
%
% LATTICE L18.82
%
\leavevmode%
\llap{\spacer}%
${L_{18.82}}=\pdualwitharg{2.3.5}(\main({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,7}{\infty}{12}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
230220\\-3240\\-58860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3240\\300\\1020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58860\\1020\\15193
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\-307\\408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-77\\102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-151\\200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
247\\-759\\1008
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
397\\-1219\\1620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
386\\-1185\\1575
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-60\\-108\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-72\\-120\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\0\\-120\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-72\\0\\72\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-120\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-180\\-180\\0\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[230220, -3240, -58860; -3240, 300, 1020; -58860, 1020, 15193],[180, 60, 1],[100, 25, 49, 247, 397, 386; -307, -77, -151, -759, -1219, -1185; 408, 102, 200, 1008, 1620, 1575],17,82]
%
% END OF LATTICE L18.82
%
\hbox{}\par\smallskip%
%
% LATTICE L18.83
%
\leavevmode%
\llap{\spacer}%
${L_{18.83}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{18.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,3}{\infty z}{4}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\1320\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1320\\-6120\\1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\1680\\-461
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-9\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-63\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-25\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\8\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-120\\-72\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-24\\-40\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\120\\0\\-120\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-24\\0\\24\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\-120\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-360\\-120\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-240, 1320, -360; 1320, -6120, 1680; -360, 1680, -461],[120, 120, 1],[-1, 1, 10, 5, 2, -1; 2, -9, -63, -25, -5, 8; 8, -34, -240, -96, -20, 30],17,83]
%
% END OF LATTICE L18.83
%
\hbox{}\par\smallskip%
%
% LATTICE L18.84
%
\leavevmode%
\llap{\spacer}%
${L_{18.84}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{18.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,7}{\infty z}{4}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-891120\\12460440\\-3321840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12460440\\-174231000\\46448400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3321840\\46448400\\-12382721
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-286\\-1072
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\519\\1946
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\4273\\16020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\1911\\7164
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\587\\2200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-252\\-945
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-120\\-72\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-24\\-40\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\120\\0\\-120\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-24\\0\\24\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\-120\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-180\\-60\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-891120, 12460440, -3321840; 12460440, -174231000, 46448400; -3321840, 46448400, -12382721],[120, 120, 1],[-3, 3, 31, 16, 7, -1; -286, 519, 4273, 1911, 587, -252; -1072, 1946, 16020, 7164, 2200, -945],17,84]
%
% END OF LATTICE L18.84
%
\hbox{}\par\smallskip%
%
% LATTICE L18.85
%
\leavevmode%
\llap{\spacer}%
${L_{18.85}}=\pdualwitharg{2.3.5}({L_{18.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,19}{\infty z}{12}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{b}{2}{180}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3600\\-2520\\5400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\840\\-3000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5400\\-3000\\9553
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-250\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\203\\78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\729\\280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-139\\1125\\432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58\\469\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-234\\-90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-120\\-216\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-72\\-120\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\40\\0\\-120\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-72\\0\\72\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-360\\-360\\0\\180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3600, -2520, 5400; -2520, 840, -3000; 5400, -3000, 9553],[360, 120, 1],[31, -25, -90, -139, -58, 29; -250, 203, 729, 1125, 469, -234; -96, 78, 280, 432, 180, -90],17,85]
%
% END OF LATTICE L18.85
%
\hbox{}\par\smallskip%
%
% LATTICE L18.86
%
\leavevmode%
\llap{\spacer}%
${L_{18.86}}=\pdualwitharg{2.3.5}({L_{18.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,7}{\infty z}{12}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{b}{2}{72}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2673360\\-20480040\\-854280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20480040\\-156877800\\-6543840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-854280\\-6543840\\-272963
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\194\\-4848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-124\\-379\\9474
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-332\\-1017\\25420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-437\\-1341\\33516
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-389\\9720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62\\189\\-4725
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-120\\-216\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-72\\-120\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\40\\0\\-120\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-72\\0\\72\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-180\\-180\\0\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2673360, -20480040, -854280; -20480040, -156877800, -6543840; -854280, -6543840, -272963],[360, 120, 1],[63, -124, -332, -437, -126, 62; 194, -379, -1017, -1341, -389, 189; -4848, 9474, 25420, 33516, 9720, -4725],17,86]
%
% END OF LATTICE L18.86
%
\hbox{}\par\smallskip%
%
% LATTICE L18.87
%
\leavevmode%
\llap{\spacer}%
${L_{18.87}}=\pdualwitharg{2.5}({L_{18.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,11}{\infty z}{12}\above{1pt}{1pt}{l}{2}{360}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\2520\\-1080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2520\\-7800\\3360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\3360\\-1447
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-17\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\-147\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-23\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-25\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-360\\-72\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-24\\-120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\360\\0\\-360\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-24\\0\\8\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-360\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-360\\-40\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-720, 2520, -1080; 2520, -7800, 3360; -1080, 3360, -1447],[360, 120, 1],[-1, 3, 22, 3, 2, -1; 10, -17, -147, -23, -25, 4; 24, -42, -360, -56, -60, 10],17,87]
%
% END OF LATTICE L18.87
%
\hbox{}\par\smallskip%
%
% LATTICE L18.88
%
\leavevmode%
\llap{\spacer}%
${L_{18.88}}=\pdualwitharg{2.5}({L_{18.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,23}{\infty z}{12}\above{1pt}{1pt}{s}{2}{360}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3603600\\1857240\\76320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1857240\\-948840\\-39000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76320\\-39000\\-1603
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\74\\-1848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-427\\10674
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\-2859\\71460
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-359\\8972
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-149\\3720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\71\\-1775
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-360\\-72\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-24\\-120\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\360\\0\\-360\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-24\\0\\8\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-360\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-180\\-20\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3603600, 1857240, 76320; 1857240, -948840, -39000; 76320, -39000, -1603],[360, 120, 1],[-1, 6, 40, 5, 2, -1; 74, -427, -2859, -359, -149, 71; -1848, 10674, 71460, 8972, 3720, -1775],17,88]
%
% END OF LATTICE L18.88
%

\medskip%
%------------------------------WEYL-GROUP-W19--------------------------------
%
\leavevmode\llap{}%
$W_{19\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=3$%
\hfill%
$4222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L19.1
%
\leavevmode%
\llap{\spacer}%
${L_{19.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{19}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27060\\300\\540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
300\\-2\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
540\\-7\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\27\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-26\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-55\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\24\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-27060, 300, 540; 300, -2, -7; 540, -7, -10],[60, 1, 1],[1, -1, -2, 1; 27, -26, -55, 24; 35, -36, -70, 36],18,1]
%
% END OF LATTICE L19.1
%
\hbox{}\par\smallskip%
%
% LATTICE L19.2
%
\leavevmode%
\llap{\spacer}%
${L_{19.2}}=\pfillwitharg{2}({L_{19.1}})$%
{${}={}$}Nikulin $19$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-15, 0, 0; 0, 1, 0; 0, 0, 1],[15, 1, 1],[0, 0, -1, -1; -1, 1, 0, -3; 1, 0, -5, -3],18,2]
%
% END OF LATTICE L19.2
%
\hbox{}\par\smallskip%
%
% LATTICE L19.3
%
\leavevmode%
\llap{\spacer}%
${L_{19.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{19.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{4}{3}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3030\\255\\1065
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
255\\-21\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1065\\-90\\-374
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-10\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\9\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\20\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3030, 255, 1065; 255, -21, -90; 1065, -90, -374],[15, 3, 1],[-4, 5, 7, -2; -10, 9, 20, -3; -9, 12, 15, -5],18,3]
%
% END OF LATTICE L19.3
%
\hbox{}\par\smallskip%
%
% LATTICE L19.4
%
\leavevmode%
\llap{\spacer}%
${L_{19.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{19.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{4}{5}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4665\\-300\\1785
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\-10\\115
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1785\\115\\-683
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\2\\-65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-1\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-1\\26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\0\\-60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-5\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\5\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4665, -300, 1785; -300, -10, 115; 1785, 115, -683],[15, 5, 1],[-25, 23, 10, -23; 2, -1, -1, 0; -65, 60, 26, -60],18,4]
%
% END OF LATTICE L19.4
%
\hbox{}\par\smallskip%
%
% LATTICE L19.5
%
\leavevmode%
\llap{\spacer}%
${L_{19.5}}=\pdualwitharg{3}({L_{19.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{*}{4}{12}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11580\\360\\-3180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-6\\105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3180\\105\\-866
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-53\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\58\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\105\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-20\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11580, 360, -3180; 360, -6, 105; -3180, 105, -866],[60, 3, 1],[-14, 15, 28, -5; -53, 58, 105, -20; 45, -48, -90, 16],18,5]
%
% END OF LATTICE L19.5
%
\hbox{}\par\smallskip%
%
% LATTICE L19.6
%
\leavevmode%
\llap{\spacer}%
${L_{19.6}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{19.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{}{4}{15}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-150\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-150\\1005\\105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\105\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\2\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-15\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\15\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30, -150, -15; -150, 1005, 105; -15, 105, 11],[15, 15, 1],[1, 2, 0, 0; 0, 2, 1, 1; 0, -15, -9, -10],18,6]
%
% END OF LATTICE L19.6
%
\hbox{}\par\smallskip%
%
% LATTICE L19.7
%
\leavevmode%
\llap{\spacer}%
${L_{19.7}}=\pdualwitharg{2}({L_{19.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
422760\\-24300\\-113640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24300\\1400\\6532
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-113640\\6532\\30547
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\1\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\2\\-86
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\0\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\0\\-93
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-4\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[422760, -24300, -113640; -24300, 1400, 6532; -113640, 6532, 30547],[60, 4, 1],[-15, -23, -43, -25; 1, 2, 0, 0; -56, -86, -160, -93],18,7]
%
% END OF LATTICE L19.7
%
\hbox{}\par\smallskip%
%
% LATTICE L19.8
%
\leavevmode%
\llap{\spacer}%
${L_{19.8}}=\pdualwitharg{5}({L_{19.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{*}{4}{20}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-10\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\20\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[60, 0, 0; 0, -10, -5; 0, -5, -2],[60, 5, 1],[0, -1, 0, 1; 2, -2, -1, 0; -5, 0, 2, 0],18,8]
%
% END OF LATTICE L19.8
%
\hbox{}\par\smallskip%
%
% LATTICE L19.9
%
\leavevmode%
\llap{\spacer}%
${L_{19.9}}=\pdualwitharg{2.3}({L_{19.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{4}{12}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1515720\\159660\\-390180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
159660\\16824\\-41100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390180\\-41100\\100441
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\1\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\2\\-174
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-139\\0\\-540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\0\\-101
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-12\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1515720, 159660, -390180; 159660, 16824, -41100; -390180, -41100, 100441],[60, 12, 1],[-31, -45, -139, -26; 1, 2, 0, 0; -120, -174, -540, -101],18,9]
%
% END OF LATTICE L19.9
%
\hbox{}\par\smallskip%
%
% LATTICE L19.10
%
\leavevmode%
\llap{\spacer}%
${L_{19.10}}=\pdualwitharg{3.5}({L_{19.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{*}{4}{60}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6420\\-300\\300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
300\\-15\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\2\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-2\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-30\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\60\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6420, -300, 300; -300, -30, -15; 300, -15, 14],[60, 15, 1],[5, -3, -2, 1; 2, -2, -1, 0; -105, 60, 42, -20],18,10]
%
% END OF LATTICE L19.10
%
\hbox{}\par\smallskip%
%
% LATTICE L19.11
%
\leavevmode%
\llap{\spacer}%
${L_{19.11}}=\pdualwitharg{2.5}({L_{19.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{*}{4}{20}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
600\\-180\\300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
300\\-80\\143
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-7\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\12\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\-20\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\20\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[600, -180, 300; -180, 40, -80; 300, -80, 143],[60, 20, 1],[0, 3, -1, -4; 1, -7, 3, 12; 0, -10, 4, 15],18,11]
%
% END OF LATTICE L19.11
%
\hbox{}\par\smallskip%
%
% LATTICE L19.12
%
\leavevmode%
\llap{\spacer}%
${L_{19.12}}=\pdualwitharg{2.3.5}({L_{19.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{*}{4}{60}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\7500\\-1980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7500\\400920\\-105840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1980\\-105840\\27941
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-103\\-390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\19\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\33\\125
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-60\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\60\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120, 7500, -1980; 7500, 400920, -105840; -1980, -105840, 27941],[60, 60, 1],[1, 2, 0, 0; 0, -103, 19, 33; 0, -390, 72, 125],18,12]
%
% END OF LATTICE L19.12
%

\medskip%
%------------------------------WEYL-GROUP-W20--------------------------------
%
\leavevmode\llap{}%
$W_{20\phantom{0}}$%
\qquad\llap{22} lattices, $\chi=27$%
\hfill%
$\infty222242$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L20.1
%
\leavevmode%
\llap{\spacer}%
${L_{20.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{20}}%
\EasyButWeakLineBreak%
{${68}\above{1pt}{1pt}{1,0}{\infty b}{68}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{34}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53788\\1156\\2652
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1156\\-24\\-59
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2652\\-59\\-126
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-170\\-68
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75\\1768\\748
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\402\\169
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\3128\\1309
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\404\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\96\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-118\\-50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\-68\\-34\\-374\\-68\\-34\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\68\\0\\-204\\-68\\-68\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\0\\2\\0\\-4\\-8\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-374\\-204\\0\\34\\0\\-34\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-68\\-4\\0\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-68\\-8\\-34\\0\\2\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-68\\-14\\-102\\-12\\-2\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-53788, 1156, 2652; 1156, -24, -59; 2652, -59, -126],[68, 1, 1],[-7, 75, 17, 132, 17, 4, -5; -170, 1768, 402, 3128, 404, 96, -118; -68, 748, 169, 1309, 168, 39, -50],19,1]
%
% END OF LATTICE L20.1
%
\hbox{}\par\smallskip%
%
% LATTICE L20.2
%
\leavevmode%
\llap{\spacer}%
${L_{20.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{15}}%
\EasyButWeakLineBreak%
{${34}\above{1pt}{1pt}{4,3}{\infty b}{136}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{17}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2045576\\8160\\11424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8160\\-30\\-47
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11424\\-47\\-63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-289\\-510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\4216\\7548
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\501\\895
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\4012\\7157
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\1076\\1916
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\144\\255
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-143\\-256
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\-68\\-17\\-187\\-68\\-17\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\136\\0\\-204\\-136\\-68\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\1\\0\\-4\\-4\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\-204\\0\\17\\0\\-17\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-136\\-4\\0\\8\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-68\\-4\\-17\\0\\1\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-68\\-7\\-51\\-12\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2045576, 8160, 11424; 8160, -30, -47; 11424, -47, -63],[136, 1, 1],[-4, 59, 7, 56, 15, 2, -2; -289, 4216, 501, 4012, 1076, 144, -143; -510, 7548, 895, 7157, 1916, 255, -256],19,2]
%
% END OF LATTICE L20.2
%
\hbox{}\par\smallskip%
%
% LATTICE L20.3
%
\leavevmode%
\llap{\spacer}%
${L_{20.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${34}\above{1pt}{1pt}{4,3}{\infty a}{136}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{68}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{4}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62696\\-15640\\952
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15640\\-3901\\237
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
952\\237\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\170\\153
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-175\\748\\748
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\90\\94
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87\\374\\408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-30\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\-68\\-34\\-374\\-68\\-34\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\136\\0\\-408\\-136\\-136\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\0\\4\\0\\-8\\-16\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-374\\-408\\0\\68\\0\\-68\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-136\\-8\\0\\8\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-136\\-16\\-68\\0\\4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-68\\-14\\-102\\-12\\-2\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-62696, -15640, 952; -15640, -3901, 237; 952, 237, -14],[136, 1, 1],[-40, -175, -21, -87, 1, 7, -1; 170, 748, 90, 374, -4, -30, 4; 153, 748, 94, 408, 0, -32, -1],19,3]
%
% END OF LATTICE L20.3
%
\hbox{}\par\smallskip%
%
% LATTICE L20.4
%
\leavevmode%
\llap{\spacer}%
${L_{20.4}}=\pfillwitharg{2}({L_{20.1}})$%
{${}={}$}Nikulin $20$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${17}\above{1pt}{1pt}{2,1}{\infty}{17}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{34}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-17\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-17\\-17\\-187\\-17\\-17\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\17\\0\\-102\\-17\\-34\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\2\\0\\-2\\-8\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\-102\\0\\34\\0\\-34\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-17\\-2\\0\\1\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-34\\-8\\-34\\0\\2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-17\\-7\\-51\\-3\\-1\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[17, 0, 0; 0, 0, -1; 0, -1, 0],[17, 1, 1],[-1, 1, 0, -6, -1, -2, -1; 0, 0, -1, -17, -2, -3, -1; -17, 0, 1, -17, -4, -11, -8],19,4]
%
% END OF LATTICE L20.4
%
\hbox{}\par\smallskip%
%
% LATTICE L20.5
%
\leavevmode%
\llap{\spacer}%
${L_{20.5}}=\pfillwitharg{2}({L_{20.2}})$%
{${}={}$}Nikulin $15'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${34}\above{1pt}{1pt}{2,1}{\infty b}{34}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{17}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10846\\578\\9554
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
578\\31\\509
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9554\\509\\8416
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-155\\102\\170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\25\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-266\\221\\289
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\34\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\15\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\-34\\-17\\-187\\-34\\-17\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\34\\0\\-102\\-34\\-34\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\1\\0\\-2\\-4\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\-102\\0\\17\\0\\-17\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-34\\-2\\0\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-34\\-4\\-17\\0\\1\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-7\\-51\\-6\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[10846, 578, 9554; 578, 31, 509; 9554, 509, 8416],[34, 1, 1],[-15, -155, -33, -266, -37, -14, -1; 0, 102, 25, 221, 34, 15, 2; 17, 170, 36, 289, 40, 15, 1],19,5]
%
% END OF LATTICE L20.5
%
\hbox{}\par\smallskip%
%
% LATTICE L20.6
%
\leavevmode%
\llap{\spacer}%
${L_{20.6}}=\main({L_{20.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${17}\above{1pt}{1pt}{4,3}{\infty}{68}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{34}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31348\\-15164\\476
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15164\\-7335\\230
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
476\\230\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\85\\68
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-175\\374\\374
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\45\\49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87\\187\\221
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-15\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-34\\-17\\-187\\-34\\-17\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\68\\0\\-204\\-68\\-68\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\2\\0\\-4\\-8\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\-204\\0\\34\\0\\-34\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-68\\-4\\0\\4\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-68\\-8\\-34\\0\\2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-7\\-51\\-6\\-1\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-31348, -15164, 476; -15164, -7335, 230; 476, 230, -7],[68, 1, 1],[-40, -175, -21, -87, 1, 7, -1; 85, 374, 45, 187, -2, -15, 2; 68, 374, 49, 221, 2, -17, -3],19,6]
%
% END OF LATTICE L20.6
%
\hbox{}\par\smallskip%
%
% LATTICE L20.7
%
\leavevmode%
\llap{\spacer}%
${L_{20.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{20.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${68}\above{1pt}{1pt}{4,1}{\infty z}{17}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{34}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{4}{4}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-293930\\4522\\-143922
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4522\\-60\\2214
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-143922\\2214\\-70471
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
366\\-17\\-748
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1356\\51\\2771
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-643\\25\\1314
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5141\\204\\10506
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-344\\14\\703
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-183\\8\\374
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
184\\-7\\-376
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\-34\\-34\\-374\\-34\\-34\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\17\\0\\-102\\-17\\-34\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\0\\2\\0\\-2\\-8\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-374\\-102\\0\\34\\0\\-34\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-17\\-2\\0\\1\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-34\\-8\\-34\\0\\2\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-14\\-102\\-6\\-2\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-293930, 4522, -143922; 4522, -60, 2214; -143922, 2214, -70471],[34, 2, 1],[366, -1356, -643, -5141, -344, -183, 184; -17, 51, 25, 204, 14, 8, -7; -748, 2771, 1314, 10506, 703, 374, -376],19,7]
%
% END OF LATTICE L20.7
%
\hbox{}\par\smallskip%
%
% LATTICE L20.8
%
\leavevmode%
\llap{\spacer}%
${L_{20.8}}=\pdualwitharg{2}(\main({L_{20.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${68}\above{1pt}{1pt}{4,1}{\infty}{68}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{136}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{4}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1089972\\-21352\\266424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21352\\420\\-5220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
266424\\-5220\\65123
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
590\\-1275\\-2516
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
821\\-1785\\-3502
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
165\\-361\\-704
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
701\\-1547\\-2992
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-17\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-33\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91\\-196\\-388
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\-68\\-68\\-748\\-68\\-68\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\68\\0\\-408\\-68\\-136\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\0\\8\\0\\-8\\-32\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-748\\-408\\0\\136\\0\\-136\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-68\\-8\\0\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-136\\-32\\-136\\0\\8\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-68\\-28\\-204\\-12\\-4\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1089972, -21352, 266424; -21352, 420, -5220; 266424, -5220, 65123],[68, 4, 1],[590, 821, 165, 701, 7, 15, 91; -1275, -1785, -361, -1547, -17, -33, -196; -2516, -3502, -704, -2992, -30, -64, -388],19,8]
%
% END OF LATTICE L20.8
%
\hbox{}\par\smallskip%
%
% LATTICE L20.9
%
\leavevmode%
\llap{\spacer}%
${L_{20.9}}=\pdualwitharg{2}({L_{20.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${68}\above{1pt}{1pt}{2,1}{\infty z}{17}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{136}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
276896\\-64532\\62424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64532\\15048\\-14548
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62424\\-14548\\14073
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\0\\-102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
303\\34\\-1309
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\33\\-1208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2382\\289\\-10268
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\11\\-379
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
146\\19\\-628
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\1\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\-34\\-68\\-748\\-34\\-68\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\17\\0\\-204\\-17\\-68\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\0\\8\\0\\-4\\-32\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-748\\-204\\0\\136\\0\\-136\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-17\\-4\\0\\1\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-68\\-32\\-136\\0\\8\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-28\\-204\\-6\\-4\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[276896, -64532, 62424; -64532, 15048, -14548; 62424, -14548, 14073],[68, 4, 1],[23, 303, 280, 2382, 88, 146, 7; 0, 34, 33, 289, 11, 19, 1; -102, -1309, -1208, -10268, -379, -628, -30],19,9]
%
% END OF LATTICE L20.9
%
\hbox{}\par\smallskip%
%
% LATTICE L20.10
%
\leavevmode%
\llap{\spacer}%
${L_{20.10}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{20.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{2,1}{\infty}{1}\above{1pt}{1pt}{r}{2}{34}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{17}\above{1pt}{1pt}{r}{2}{34}\above{1pt}{1pt}{}{4}{17}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-561\\-289\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-289\\-136\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-17\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-8\\85
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\51
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-17\\-11\\-17\\-17\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-6\\-17\\-34\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\34\\0\\-34\\-136\\-119
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-6\\0\\2\\0\\-34\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-17\\-34\\0\\17\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-34\\-136\\-34\\0\\34\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-17\\-119\\-51\\-51\\-17\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-561, -289, -34; -289, -136, -17; -34, -17, -2],[17, 17, 1],[-1, -1, -2, 0, 1, 0, -3; -1, -2, -8, -2, 0, 2, -1; 22, 29, 85, 13, -17, -17, 51],19,10]
%
% END OF LATTICE L20.10
%
\hbox{}\par\smallskip%
%
% LATTICE L20.11
%
\leavevmode%
\llap{\spacer}%
${L_{20.11}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{20.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty b}{2}\above{1pt}{1pt}{}{2}{17}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{34}\above{1pt}{1pt}{}{2}{17}\above{1pt}{1pt}{}{4}{34}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1150526\\-35122\\-542946
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35122\\-1071\\-16575
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-542946\\-16575\\-256222
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-14\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-107\\104\\220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-438\\419\\901
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\197\\428
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-479\\448\\986
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\119\\272
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
124\\-120\\-255
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-17\\-11\\-34\\-17\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-6\\-34\\-34\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\17\\0\\-34\\-68\\-119
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-6\\0\\1\\0\\-17\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-34\\-34\\0\\34\\0\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-34\\-68\\-17\\0\\17\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-119\\-51\\-102\\-17\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1150526, -35122, -542946; -35122, -1071, -16575; -542946, -16575, -256222],[34, 17, 1],[16, -107, -438, -208, -479, -132, 124; -14, 104, 419, 197, 448, 119, -120; -33, 220, 901, 428, 986, 272, -255],19,11]
%
% END OF LATTICE L20.11
%
\hbox{}\par\smallskip%
%
% LATTICE L20.12
%
\leavevmode%
\llap{\spacer}%
${L_{20.12}}=\pdualwitharg{2}({L_{20.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${272}\above{1pt}{1pt}{8,1}{\infty z}{68}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{136}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{4}{16}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67320\\4216\\-5576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4216\\-240\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5576\\360\\-457
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\119\\-272
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-476\\1054
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-223\\496
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-437\\-1768\\3944
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-117\\262
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-60\\136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\65\\-144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
272\\-136\\-136\\-1496\\-136\\-136\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\68\\0\\-408\\-68\\-136\\-136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\0\\8\\0\\-8\\-32\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1496\\-408\\0\\136\\0\\-136\\-408
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\-68\\-8\\0\\4\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\-136\\-32\\-136\\0\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-136\\-56\\-408\\-24\\-8\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-67320, 4216, -5576; 4216, -240, 360; -5576, 360, -457],[136, 8, 1],[30, -117, -55, -437, -29, -15, 16; 119, -476, -223, -1768, -117, -60, 65; -272, 1054, 496, 3944, 262, 136, -144],19,12]
%
% END OF LATTICE L20.12
%
\hbox{}\par\smallskip%
%
% LATTICE L20.13
%
\leavevmode%
\llap{\spacer}%
${L_{20.13}}=\pdualwitharg{2}({L_{20.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${272}\above{1pt}{1pt}{8,5}{\infty z}{68}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{136}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{4}{16}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-273904\\-7208\\4760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7208\\-40\\96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4760\\96\\-77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-238\\-1224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\1071\\5474
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\495\\2532
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
244\\3893\\19924
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\255\\1306
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\127\\652
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-144\\-736
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
272\\-136\\-136\\-1496\\-136\\-136\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\68\\0\\-408\\-68\\-136\\-136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\0\\8\\0\\-8\\-32\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1496\\-408\\0\\136\\0\\-136\\-408
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\-68\\-8\\0\\4\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\-136\\-32\\-136\\0\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-136\\-56\\-408\\-24\\-8\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-273904, -7208, 4760; -7208, -40, 96; 4760, 96, -77],[136, 8, 1],[-15, 67, 31, 244, 16, 8, -9; -238, 1071, 495, 3893, 255, 127, -144; -1224, 5474, 2532, 19924, 1306, 652, -736],19,13]
%
% END OF LATTICE L20.13
%
\hbox{}\par\smallskip%
%
% LATTICE L20.14
%
\leavevmode%
\llap{\spacer}%
${L_{20.14}}=\pdualwitharg{17}(\main({L_{20.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{4,3}{\infty}{4}\above{1pt}{1pt}{b}{2}{34}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{68}\above{1pt}{1pt}{b}{2}{34}\above{1pt}{1pt}{}{4}{17}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2244\\544\\68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
544\\-119\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\-17\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\-51
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-17\\-11\\-34\\-17\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-12\\-68\\-68\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\34\\0\\-68\\-136\\-119
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-12\\0\\2\\0\\-34\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-68\\-68\\0\\68\\0\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-68\\-136\\-34\\0\\34\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-119\\-51\\-102\\-17\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2244, 544, 68; 544, -119, -17; 68, -17, -2],[68, 17, 1],[0, -1, -3, -1, -1, 1, 1; -1, -4, -8, -2, 0, 2, -1; 5, -10, -51, -21, -34, 17, 34],19,14]
%
% END OF LATTICE L20.14
%
\hbox{}\par\smallskip%
%
% LATTICE L20.15
%
\leavevmode%
\llap{\spacer}%
${L_{20.15}}=\pdualwitharg{2.17}(\pfillwitharg{2}({L_{20.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{}{2}{34}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{17}\above{1pt}{1pt}{}{2}{34}\above{1pt}{1pt}{}{4}{68}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
286144\\-1220770\\-592178
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1220770\\5216926\\2530654
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-592178\\2530654\\1227583
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-480\\990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-2579\\5319
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\-18777\\38726
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-8941\\18440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-10625\\21913
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-8111\\16728
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-610\\1258
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-34\\-22\\-34\\-34\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-6\\-17\\-34\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\0\\34\\0\\-34\\-136\\-238
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-6\\0\\2\\0\\-34\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-17\\-34\\0\\17\\0\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-34\\-136\\-34\\0\\34\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-238\\-102\\-102\\-34\\68
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[286144, -1220770, -592178; -1220770, 5216926, 2530654; -592178, 2530654, 1227583],[34, 34, 1],[1, 5, 36, 17, 20, 15, 1; -480, -2579, -18777, -8941, -10625, -8111, -610; 990, 5319, 38726, 18440, 21913, 16728, 1258],19,15]
%
% END OF LATTICE L20.15
%
\hbox{}\par\smallskip%
%
% LATTICE L20.16
%
\leavevmode%
\llap{\spacer}%
${L_{20.16}}=\pdualwitharg{17}({L_{20.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty b}{4}\above{1pt}{1pt}{r}{2}{34}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{68}\above{1pt}{1pt}{r}{2}{34}\above{1pt}{1pt}{*}{4}{68}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-284444\\10812\\8500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10812\\-408\\-323
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8500\\-323\\-254
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\10\\153
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\28\\-272
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\16\\-289
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\2\\-170
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-34\\-22\\-68\\-34\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-12\\-68\\-68\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\0\\34\\0\\-68\\-136\\-238
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-12\\0\\2\\0\\-34\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-68\\-68\\0\\68\\0\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-68\\-136\\-34\\0\\34\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-68\\-238\\-102\\-204\\-34\\68
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-284444, 10812, 8500; 10812, -408, -323; 8500, -323, -254],[68, 17, 1],[1, 3, 5, 0, -7, -8, -5; -2, 0, 10, 8, 28, 16, 2; 36, 100, 153, -11, -272, -289, -170],19,16]
%
% END OF LATTICE L20.16
%
\hbox{}\par\smallskip%
%
% LATTICE L20.17
%
\leavevmode%
\llap{\spacer}%
${L_{20.17}}=\pdualwitharg{17}({L_{20.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty a}{8}\above{1pt}{1pt}{s}{2}{68}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{136}\above{1pt}{1pt}{s}{2}{68}\above{1pt}{1pt}{*}{4}{34}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17000\\2040\\136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2040\\-238\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
136\\-17\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-20\\-102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-68
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\68
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-34\\-22\\-68\\-34\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-24\\-136\\-136\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\0\\68\\0\\-136\\-272\\-238
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-24\\0\\4\\0\\-68\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-136\\-136\\0\\136\\0\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-136\\-272\\-68\\0\\68\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-68\\-238\\-102\\-204\\-34\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17000, 2040, 136; 2040, -238, -17; 136, -17, -1],[136, 17, 1],[0, -1, -3, -1, -1, 1, 1; -1, -8, -20, -6, -4, 6, 3; 10, -20, -102, -42, -68, 34, 68],19,17]
%
% END OF LATTICE L20.17
%
\hbox{}\par\smallskip%
%
% LATTICE L20.18
%
\leavevmode%
\llap{\spacer}%
${L_{20.18}}=\pdualwitharg{17}({L_{20.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty b}{8}\above{1pt}{1pt}{l}{2}{17}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{136}\above{1pt}{1pt}{l}{2}{17}\above{1pt}{1pt}{}{4}{34}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\-272\\-136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-272\\-510\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\-255\\-127
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\5\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\4\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\28\\-68
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-17\\-11\\-68\\-17\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-12\\-136\\-68\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\17\\0\\-68\\-68\\-119
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-12\\0\\1\\0\\-17\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-136\\-68\\0\\136\\0\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-68\\-68\\-17\\0\\17\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-68\\-119\\-51\\-204\\-17\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-136, -272, -136; -272, -510, -255; -136, -255, -127],[136, 17, 1],[0, 3, 5, 2, 7, 0, -2; -1, 0, 5, 4, 28, 8, 1; 2, -4, -17, -11, -68, -17, 0],19,18]
%
% END OF LATTICE L20.18
%
\hbox{}\par\smallskip%
%
% LATTICE L20.19
%
\leavevmode%
\llap{\spacer}%
${L_{20.19}}=\pdualwitharg{2.17}(\main({L_{20.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty}{4}\above{1pt}{1pt}{*}{2}{136}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{68}\above{1pt}{1pt}{*}{2}{136}\above{1pt}{1pt}{}{4}{68}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
136\\-884\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-884\\20468\\4760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\4760\\1107
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\41\\-176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\62\\-266
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\222\\-952
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\56\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\95\\-408
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-68\\-44\\-68\\-68\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-24\\-68\\-136\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\0\\136\\0\\-136\\-544\\-476
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-24\\0\\8\\0\\-136\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-68\\-136\\0\\68\\0\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-136\\-544\\-136\\0\\136\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-68\\-476\\-204\\-204\\-68\\68
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[136, -884, -204; -884, 20468, 4760; -204, 4760, 1107],[68, 68, 1],[2, 3, 11, 3, 1, 1, 5; 41, 62, 222, 56, 8, 0, 95; -176, -266, -952, -240, -34, 0, -408],19,19]
%
% END OF LATTICE L20.19
%
\hbox{}\par\smallskip%
%
% LATTICE L20.20
%
\leavevmode%
\llap{\spacer}%
${L_{20.20}}=\pdualwitharg{2.17}({L_{20.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{2,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{136}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{17}\above{1pt}{1pt}{r}{2}{136}\above{1pt}{1pt}{*}{4}{68}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1224\\23596\\5508
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23596\\663136\\154768
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5508\\154768\\36121
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\171\\-733
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\2792\\-11968
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\1438\\-6164
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\932\\-3995
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\1634\\-7004
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\119\\-510
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-68\\-44\\-34\\-68\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-12\\-17\\-68\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\0\\136\\0\\-68\\-544\\-476
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-12\\0\\8\\0\\-136\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-17\\-68\\0\\17\\0\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-68\\-544\\-136\\0\\136\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-476\\-204\\-102\\-68\\68
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1224, 23596, 5508; 23596, 663136, 154768; 5508, 154768, 36121],[68, 68, 1],[0, 2, 33, 17, 11, 19, 1; 7, 171, 2792, 1438, 932, 1634, 119; -30, -733, -11968, -6164, -3995, -7004, -510],19,20]
%
% END OF LATTICE L20.20
%
\hbox{}\par\smallskip%
%
% LATTICE L20.21
%
\leavevmode%
\llap{\spacer}%
${L_{20.21}}=\pdualwitharg{2.17}({L_{20.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,1}{\infty z}{4}\above{1pt}{1pt}{l}{2}{136}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{68}\above{1pt}{1pt}{l}{2}{136}\above{1pt}{1pt}{}{4}{272}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4080\\39032\\-2312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39032\\-371960\\22032
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2312\\22032\\-1305
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\118
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\57\\952
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\27\\448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\31\\510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\17\\272
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-16\\-272
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-136\\-88\\-136\\-136\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-24\\-68\\-136\\-136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\0\\136\\0\\-136\\-544\\-952
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-24\\0\\8\\0\\-136\\-408
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\-68\\-136\\0\\68\\0\\-408
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\-136\\-544\\-136\\0\\136\\-136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-136\\-952\\-408\\-408\\-136\\272
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4080, 39032, -2312; 39032, -371960, 22032; -2312, 22032, -1305],[136, 136, 1],[-1, 0, 5, 4, 7, 8, 1; -2, 7, 57, 27, 31, 17, -16; -32, 118, 952, 448, 510, 272, -272],19,21]
%
% END OF LATTICE L20.21
%
\hbox{}\par\smallskip%
%
% LATTICE L20.22
%
\leavevmode%
\llap{\spacer}%
${L_{20.22}}=\pdualwitharg{2.17}({L_{20.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,5}{\infty z}{4}\above{1pt}{1pt}{s}{2}{136}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{68}\above{1pt}{1pt}{s}{2}{136}\above{1pt}{1pt}{*}{4}{272}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4080\\-25432\\-5848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25432\\-157352\\-36176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5848\\-36176\\-8317
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4080, -25432, -5848; -25432, -157352, -36176; -5848, -36176, -8317],[136, 136, 1],[1, 3, 15, 4, 0, -8, -9; -2, -23, -143, -53, -39, 17, 64; 8, 98, 612, 228, 170, -68, -272],19,22]
%
% END OF LATTICE L20.22
%

\medskip%
%------------------------------WEYL-GROUP-W21--------------------------------
%
\leavevmode\llap{}%
$W_{21\phantom{0}}$%
\qquad\llap{6} lattices, $\chi=18$%
\hfill%
$422422\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L21.1
%
\leavevmode%
\llap{\spacer}%
${L_{21.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{21}}%
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{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{b}{2}{38}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26372\\-18468\\1672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18468\\-12930\\1171
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\kern3pt%
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1672\\1171\\-106
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3191\\-1368\\35112
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2233\\-958\\24563
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-26372, -18468, 1672; -18468, -12930, 1171; 1672, 1171, -106],[76, 1, 1],[-4, 7, 139, 25, 7, -6; 2, -2, -57, -11, -4, 0; -41, 88, 1558, 272, 66, -95],20,1]
%
% END OF LATTICE L21.1
%
\hbox{}\par\smallskip%
%
% LATTICE L21.2
%
\leavevmode%
\llap{\spacer}%
${L_{21.2}}=\pfillwitharg{2}({L_{21.1}})$%
{${}={}$}Nikulin $21$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-19\\0\\0
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\kern3pt%
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0\\1\\0
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\kern3pt%
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0\\0\\1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-19, 0, 0; 0, 1, 0; 0, 0, 1],[19, 1, 1],[0, 0, -13, -3, -1, -6; -1, 1, 0, -2, -2, -19; 1, 0, -57, -13, -4, -19],20,2]
%
% END OF LATTICE L21.2
%
\hbox{}\par\smallskip%
%
% LATTICE L21.3
%
\leavevmode%
\llap{\spacer}%
${L_{21.3}}=\pdualwitharg{2}({L_{21.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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590824\\1900\\-159904
\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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-159904\\-516\\43279
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-84722\\-296989\\-316540
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-273\\-958\\-1020
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[590824, 1900, -159904; 1900, 8, -516; -159904, -516, 43279],[76, 4, 1],[0, -53, -1607, -273, -38, -122; 1, -184, -5624, -958, -135, -437; 0, -198, -6004, -1020, -142, -456],20,3]
%
% END OF LATTICE L21.3
%
\hbox{}\par\smallskip%
%
% LATTICE L21.4
%
\leavevmode%
\llap{\spacer}%
${L_{21.4}}=\pdualwitharg{19}(\pfillwitharg{2}({L_{21.1}}))$%
\nopagebreak\par%
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{${38}\above{1pt}{1pt}{}{4}{19}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-646\\-1007\\1216
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1007\\-1349\\1710
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1216\\1710\\-2133
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53821\\-73899\\-90459
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\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-646, -1007, 1216; -1007, -1349, 1710; 1216, 1710, -2133],[19, 19, 1],[-65, 66, -133, -715, -326, -127; -96, 95, -179, -975, -452, -178; -114, 114, -221, -1197, -551, -216],20,4]
%
% END OF LATTICE L21.4
%
\hbox{}\par\smallskip%
%
% LATTICE L21.5
%
\leavevmode%
\llap{\spacer}%
${L_{21.5}}=\pdualwitharg{19}({L_{21.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-3904652\\-464360\\51604
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\kern3pt%
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-464360\\-55214\\6137
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\kern3pt%
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53939\\-5220\\4033320
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3904652, -464360, 51604; -464360, -55214, 6137; 51604, 6137, -682],[76, 19, 1],[96, 5, -3, -3, 57, 34; -11, -4, 0, 2, -2, -3; 7163, 342, -227, -209, 4294, 2545],20,5]
%
% END OF LATTICE L21.5
%
\hbox{}\par\smallskip%
%
% LATTICE L21.6
%
\leavevmode%
\llap{\spacer}%
${L_{21.6}}=\pdualwitharg{2.19}({L_{21.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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-625\\-24789\\-93936
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[152, 14972, -3952; 14972, 34750696, -9170540; -3952, -9170540, 2420061],[76, 76, 1],[-625, -55, -1, 1, -153, -207; -24789, -2156, -38, 0, -6107, -8225; -93936, -8170, -144, 0, -23142, -31168],20,6]
%
% END OF LATTICE L21.6
%

\medskip%
%------------------------------WEYL-GROUP-W22--------------------------------
%
\leavevmode\llap{}%
$W_{22\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=4$%
\hfill%
$2622$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L22.1
%
\leavevmode%
\llap{\spacer}%
${L_{22.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{22}}%
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{$\left[\!\llap{\phantom{%
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588\\-10\\-7
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252\\-7\\-2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-25788, 588, 252; 588, -10, -7; 252, -7, -2],[84, 1, 1],[-1, -1, 1, 2; -20, -21, 20, 42; -56, -54, 55, 105],21,1]
%
% END OF LATTICE L22.1
%
\hbox{}\par\smallskip%
%
% LATTICE L22.2
%
\leavevmode%
\llap{\spacer}%
${L_{22.2}}=\pfillwitharg{2}({L_{22.1}})$%
{${}={}$}Nikulin $22$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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42\\21\\0
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0\\0\\1
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[42, 21, 0; 21, 10, 0; 0, 0, 1],[21, 1, 1],[0, -2, 3, 13; 0, 3, -5, -21; 1, 0, -2, 0],21,2]
%
% END OF LATTICE L22.2
%
\hbox{}\par\smallskip%
%
% LATTICE L22.3
%
\leavevmode%
\llap{\spacer}%
${L_{22.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{22.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35763\\1218\\-11739
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1218\\-39\\399
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11739\\399\\-3853
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
220\\-216\\-693
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-7\\-19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\40\\123
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
389\\-378\\-1225
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-3\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-3\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35763, 1218, -11739; 1218, -39, 399; -11739, 399, -3853],[21, 3, 1],[220, 6, -39, 389; -216, -7, 40, -378; -693, -19, 123, -1225],21,3]
%
% END OF LATTICE L22.3
%
\hbox{}\par\smallskip%
%
% LATTICE L22.4
%
\leavevmode%
\llap{\spacer}%
${L_{22.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{22.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{}{6}{14}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82110\\-1323\\-23856
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1323\\-14\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23856\\-385\\-6931
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
509\\-154\\-1743
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-12\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-94\\27\\322
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
382\\-117\\-1308
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\-21\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-21\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-82110, -1323, -23856; -1323, -14, -385; -23856, -385, -6931],[21, 7, 1],[509, 49, -94, 382; -154, -12, 27, -117; -1743, -168, 322, -1308],21,4]
%
% END OF LATTICE L22.4
%
\hbox{}\par\smallskip%
%
% LATTICE L22.5
%
\leavevmode%
\llap{\spacer}%
${L_{22.5}}=\pdualwitharg{3}({L_{22.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38388\\588\\924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
588\\-6\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
924\\-15\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-21\\-70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-3\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-3\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38388, 588, 924; 588, -6, -15; 924, -15, -22],[84, 3, 1],[1, 1, -1, -2; 8, 10, -9, -21; 36, 35, -36, -70],21,5]
%
% END OF LATTICE L22.5
%
\hbox{}\par\smallskip%
%
% LATTICE L22.6
%
\leavevmode%
\llap{\spacer}%
${L_{22.6}}=\pdualwitharg{2}({L_{22.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1586760\\-29988\\376320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29988\\568\\-7112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
376320\\-7112\\89249
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\1\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\0\\-156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\3\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\21\\-420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\-12\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-12\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1586760, -29988, 376320; -29988, 568, -7112; 376320, -7112, 89249],[84, 4, 1],[5, 37, 38, 100; 1, 0, 3, 21; -21, -156, -160, -420],21,6]
%
% END OF LATTICE L22.6
%
\hbox{}\par\smallskip%
%
% LATTICE L22.7
%
\leavevmode%
\llap{\spacer}%
${L_{22.7}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{22.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4326\\-95403\\-29379
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95403\\2113734\\650916
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29379\\650916\\200447
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\5321\\-17283
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-56\\182
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-569\\1848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\1455\\-4726
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\-21\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-21\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4326, -95403, -29379; -95403, 2113734, 650916; -29379, 650916, 200447],[21, 21, 1],[-28, 1, 2, -8; 5321, -56, -569, 1455; -17283, 182, 1848, -4726],21,7]
%
% END OF LATTICE L22.7
%
\hbox{}\par\smallskip%
%
% LATTICE L22.8
%
\leavevmode%
\llap{\spacer}%
${L_{22.8}}=\pdualwitharg{7}({L_{22.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{}{6}{14}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11172\\924\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
924\\-70\\105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\105\\-142
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\8\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\3\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-6\\49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\-21\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-21\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11172, 924, -1260; 924, -70, 105; -1260, 105, -142],[84, 7, 1],[7, 5, -6, -1; 8, 3, -6, 0; -56, -42, 49, 9],21,8]
%
% END OF LATTICE L22.8
%
\hbox{}\par\smallskip%
%
% LATTICE L22.9
%
\leavevmode%
\llap{\spacer}%
${L_{22.9}}=\pdualwitharg{2.3}({L_{22.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
634872\\-28812\\150192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28812\\1320\\-6816
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
150192\\-6816\\35531
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\0\\-93
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\1\\-152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\2\\-156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\0\\-224
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\-12\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[634872, -28812, 150192; -28812, 1320, -6816; 150192, -6816, 35531],[84, 12, 1],[22, 36, 37, 53; 0, 1, 2, 0; -93, -152, -156, -224],21,9]
%
% END OF LATTICE L22.9
%
\hbox{}\par\smallskip%
%
% LATTICE L22.10
%
\leavevmode%
\llap{\spacer}%
${L_{22.10}}=\pdualwitharg{3.7}({L_{22.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-588\\168\\84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-21\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\-21\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-21\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-588, 168, 84; 168, -42, -21; 84, -21, -10],[84, 21, 1],[1, 0, -1, 0; 4, 3, -5, -1; 0, -7, 0, 2],21,10]
%
% END OF LATTICE L22.10
%
\hbox{}\par\smallskip%
%
% LATTICE L22.11
%
\leavevmode%
\llap{\spacer}%
${L_{22.11}}=\pdualwitharg{2.7}({L_{22.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{}{6}{56}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
403368\\-12012\\-106344
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12012\\392\\3192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-106344\\3192\\28055
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\67\\-91
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\183\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\267\\-364
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58\\177\\-240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\-84\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-84\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[403368, -12012, -106344; -12012, 392, 3192; -106344, 3192, 28055],[84, 28, 1],[-22, -61, -88, -58; 67, 183, 267, 177; -91, -252, -364, -240],21,11]
%
% END OF LATTICE L22.11
%
\hbox{}\par\smallskip%
%
% LATTICE L22.12
%
\leavevmode%
\llap{\spacer}%
${L_{22.12}}=\pdualwitharg{2.3.7}({L_{22.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{}{6}{168}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1176\\2100\\504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2100\\3864\\924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\924\\221
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\13\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\39\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\-84\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1176, 2100, 504; 2100, 3864, 924; 504, 924, 221],[84, 84, 1],[0, 1, 2, 0; -5, 13, 39, -1; 21, -56, -168, 4],21,12]
%
% END OF LATTICE L22.12
%

\medskip%
%------------------------------WEYL-GROUP-W23--------------------------------
%
\leavevmode\llap{}%
$W_{23\phantom{0}}$%
\qquad\llap{88} lattices, $\chi=48$%
\hfill%
$\infty2222\infty2222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L23.1
%
\leavevmode%
\llap{\spacer}%
${L_{23.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{23},3,2}%
\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{3,2}{\infty b}{84}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16363116\\948024\\53676
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
948024\\-54906\\-3111
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53676\\-3111\\-176
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
505343\\4563888\\73432800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29696\\-268193\\-4315200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1632\\-14739\\-237151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-333\\-3010\\-48342
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-899\\-8120\\-130620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-403\\-3639\\-58572
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-259\\-2338\\-37656
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-442\\-7130
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-84\\-126\\-168\\-84\\-672\\-1092\\-378\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\84\\0\\-84\\-84\\-1092\\-2436\\-1008\\-588\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\0\\18\\0\\-18\\-378\\-1008\\-450\\-288\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-84\\0\\12\\0\\-168\\-588\\-288\\-204\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\-18\\0\\4\\0\\-84\\-54\\-48\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-84\\-126\\-168\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\84\\0\\-84\\-84
\end{tabular}\endgroup%
\kern3pt%
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-126\\0\\18\\0\\-18
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-168\\-84\\0\\12\\0
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\kern3pt%
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-84\\-84\\-18\\0\\4
\end{tabular}\endgroup%
\kern3pt%
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-672\\-1092\\-378\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1092\\-2436\\-1008\\-588\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-378\\-1008\\-450\\-288\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-588\\-288\\-204\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-54\\-48\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-16363116, 948024, 53676; 948024, -54906, -3111; 53676, -3111, -176],[252, 3, 1],[-333, -899, -403, -259, -49, -115, 3, 19, 3, -15; -3010, -8120, -3639, -2338, -442, -1036, 28, 171, 26, -136; -48342, -130620, -58572, -37656, -7130, -16758, 420, 2772, 456, -2170],22,1]
%
% END OF LATTICE L23.1
%
\hbox{}\par\smallskip%
%
% LATTICE L23.2
%
\leavevmode%
\llap{\spacer}%
${L_{23.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{3m,3,2}%
\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{12,11}{\infty b}{168}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-8168328\\-56448\\-59472
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56448\\-390\\-411
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59472\\-411\\-433
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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6299\\88200\\-945000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\601\\-6450
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\644\\-6901
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\168\\0\\-168\\-84
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\kern3pt%
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-126\\0\\36\\0\\-18
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-42\\-84\\-18\\0\\2
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\kern3pt%
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-336\\-1092\\-378\\-168\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1092\\-4872\\-2016\\-1176\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-378\\-2016\\-900\\-576\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-1176\\-576\\-408\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-54\\-48\\-8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-8168328, -56448, -59472; -56448, -390, -411; -59472, -411, -433],[504, 3, 1],[-4, -27, -13, -9, -1, -3, -1, 1, 1, 0; -133, -448, -156, -68, 1, 35, 56, -12, -44, -15; 672, 4116, 1926, 1296, 136, 378, 84, -126, -96, 14],22,2]
%
% END OF LATTICE L23.2
%
\hbox{}\par\smallskip%
%
% LATTICE L23.3
%
\leavevmode%
\llap{\spacer}%
${L_{23.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{32\rightarrow N'_{16},3,m}%
\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{12,11}{\infty a}{168}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22810536\\3272976\\-990864
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3272976\\-469623\\142176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-990864\\142176\\-43039
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-6504625\\-39054960\\20738592
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\kern3pt%
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932578\\5599369\\-2973324
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-283932\\-1704780\\905255
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-2617\\-15715\\8337
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-13491\\-81004\\43008
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-2972\\-17844\\9477
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-22810536, 3272976, -990864; 3272976, -469623, 142176; -990864, 142176, -43039],[504, 3, 1],[-2617, -13491, -2972, -3747, -338, -729, 107, 104, -77, -140; -15715, -81004, -17844, -22496, -2029, -4375, 644, 624, -464, -841; 8337, 43008, 9477, 11952, 1079, 2331, -336, -333, 240, 445],22,3]
%
% END OF LATTICE L23.3
%
\hbox{}\par\smallskip%
%
% LATTICE L23.4
%
\leavevmode%
\llap{\spacer}%
${L_{23.4}}=\pfillwitharg{2.3}({L_{23.1}})$%
{${}={}$}Nikulin $23$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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-1\\-2\\-5
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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-21\\21\\0\\-21\\-21
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-21\\0\\2\\0\\-3
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-168\\-273\\-63\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
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-273\\-609\\-168\\-147\\-21
\end{tabular}\endgroup%
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-63\\-168\\-50\\-48\\-9
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-42\\-147\\-48\\-51\\-12
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-9\\-12\\-4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[21, 0, 0; 0, 0, -1; 0, -1, 0],[21, 1, 1],[-1, 1, 0, -1, -1, -13, -29, -8, -7, -1; 0, 0, -1, -3, -2, -21, -42, -11, -9, -1; -21, 0, 1, -3, -5, -84, -210, -61, -57, -10],22,4]
%
% END OF LATTICE L23.4
%
\hbox{}\par\smallskip%
%
% LATTICE L23.5
%
\leavevmode%
\llap{\spacer}%
${L_{23.5}}=\pfillwitharg{3.2}({L_{23.3}})$%
{${}={}$}Nikulin $16'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{2,1}{\infty b}{42}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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42\\0\\42
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\360\\-6101
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\-6101\\103437
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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4591\\-78288\\-4620
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11480\\195719\\11550
\end{tabular}\endgroup%
\kern3pt%
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199096\\-3394344\\-200311
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1919\\-32739\\-1932
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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4591\\-78288\\-4620
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\HardButStrongLineBreak\kern3pt%
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655\\-11167\\-659
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1199\\-20436\\-1206
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
199\\-3389\\-200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\-42\\-21\\-84\\-42\\-336\\-546\\-63\\-84\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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-42\\42\\0\\-42\\-42
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-21\\0\\1\\0\\-3
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-84\\-42\\0\\6\\0
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-42\\-42\\-3\\0\\2
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-336\\-546\\-63\\-84\\0
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-546\\-1218\\-168\\-294\\-42
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-63\\-168\\-25\\-48\\-9
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\kern3pt%
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-84\\-294\\-48\\-102\\-24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-9\\-24\\-8
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[42, 0, 42; 0, 360, -6101; 42, -6101, 103437],[42, 1, 1],[1919, 4591, 655, 1199, 199, 377, 1, 1, 113, 129; -32739, -78288, -11167, -20436, -3389, -6405, 0, -17, -1932, -2203; -1932, -4620, -659, -1206, -200, -378, 0, -1, -114, -130],22,5]
%
% END OF LATTICE L23.5
%
\hbox{}\par\smallskip%
%
% LATTICE L23.6
%
\leavevmode%
\llap{\spacer}%
${L_{23.6}}=\main(\pfillwitharg{3}({L_{23.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{4,3}{\infty}{84}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-2\\-21
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
924\\-21\\-199
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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1399\\91560\\-3360
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-30\\-1963\\72
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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25\\1659\\-63
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
139\\9114\\-336
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\HardButStrongLineBreak\kern3pt%
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21\\1371\\-50
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41\\2664\\-96
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\257\\-9
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{$\left.\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-21\\-42\\-3\\0\\1
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-168\\-546\\-63\\-84\\0
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-546\\-2436\\-336\\-588\\-42
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0\\-42\\-9\\-24\\-4
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3444, 84, 924; 84, -2, -21; 924, -21, -199],[84, 1, 1],[25, 139, 21, 41, 4, 10, 1, -1, -1, 1; 1659, 9114, 1371, 2664, 257, 630, 42, -63, -48, 70; -63, -336, -50, -96, -9, -21, 0, 2, 0, -3],22,6]
%
% END OF LATTICE L23.6
%
\hbox{}\par\smallskip%
%
% LATTICE L23.7
%
\leavevmode%
\llap{\spacer}%
${L_{23.7}}=\pfillwitharg{3}({L_{23.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{1,0}{\infty b}{84}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1269996\\512484\\15120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
512484\\-206790\\-6101
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15120\\-6101\\-180
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-29\\2940\\-102060
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-4\\439\\-15212
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-7\\846\\-29256
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\160\\-5506
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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84\\-84\\-42\\-168\\-84\\-672\\-1092\\-126\\-168\\0
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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-672\\-1092\\-126\\-168\\0
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\kern3pt%
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-1092\\-2436\\-336\\-588\\-84
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1269996, 512484, 15120; 512484, -206790, -6101; 15120, -6101, -180],[84, 1, 1],[-13, -29, -4, -7, -1, -1, 1, 0, -1, -1; 1092, 2940, 439, 846, 160, 378, 0, -19, -6, 50; -38094, -102060, -15212, -29256, -5506, -12894, 84, 644, 120, -1778],22,7]
%
% END OF LATTICE L23.7
%
\hbox{}\par\smallskip%
%
% LATTICE L23.8
%
\leavevmode%
\llap{\spacer}%
${L_{23.8}}=\pfillwitharg{3}({L_{23.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{4,3}{\infty b}{168}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-3125640\\1848\\35280
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\kern3pt%
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1848\\-1\\-21
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\kern3pt%
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35280\\-21\\-398
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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21139\\2765112\\1725024
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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139\\18228\\11340
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21\\2742\\1714
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41\\5328\\3348
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4\\514\\327
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-336\\-1092\\-126\\-168\\0
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-1092\\-4872\\-672\\-1176\\-84
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3125640, 1848, 35280; 1848, -1, -21; 35280, -21, -398],[168, 1, 1],[25, 139, 21, 41, 4, 10, 1, -1, -1, 1; 3318, 18228, 2742, 5328, 514, 1260, 84, -126, -96, 140; 2037, 11340, 1714, 3348, 327, 819, 84, -82, -84, 81],22,8]
%
% END OF LATTICE L23.8
%
\hbox{}\par\smallskip%
%
% LATTICE L23.9
%
\leavevmode%
\llap{\spacer}%
${L_{23.9}}=\pfillwitharg{3}({L_{23.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{4,3}{\infty a}{168}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-6888\\-6888\\3528
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-6888\\-6846\\3485
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-5615\\11172\\10836
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-6888, -6888, 3528; -6888, -6846, 3485; 3528, 3485, -1763],[168, 1, 1],[-1004, -5615, -425, -1663, -163, -410, -41, 21, 47, -39; 1995, 11172, 846, 3312, 325, 819, 84, -42, -96, 77; 1932, 10836, 821, 3216, 316, 798, 84, -41, -96, 74],22,9]
%
% END OF LATTICE L23.9
%
\hbox{}\par\smallskip%
%
% LATTICE L23.10
%
\leavevmode%
\llap{\spacer}%
${L_{23.10}}=\pfillwitharg{2}({L_{23.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{6,5}{\infty}{21}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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3723\\22440\\-7039
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{$\left[\!\llap{\phantom{%
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-433\\-2611\\819
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-1044\\-6293\\1974
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-273\\-1645\\516
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-45\\-271\\85
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-21\\-21\\-9\\0\\1\\0\\-21\\-27\\-12\\-4
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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21\\-21\\-63\\-42\\-21
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-21\\21\\0\\-21\\-21
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-42\\-21\\0\\3\\0
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-168\\-273\\-189\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-273\\-609\\-504\\-147\\-21
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\kern3pt%
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-189\\-504\\-450\\-144\\-27
\end{tabular}\endgroup%
\kern3pt%
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-42\\-147\\-144\\-51\\-12
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[1638, -252, 63; -252, 21, -66; 63, -66, -176],[63, 3, 1],[-433, -1044, -895, -273, -45, -78, 22, 19, -19, -28; -2611, -6293, -5394, -1645, -271, -469, 133, 114, -115, -169; 819, 1974, 1692, 516, 85, 147, -42, -36, 36, 53],22,10]
%
% END OF LATTICE L23.10
%
\hbox{}\par\smallskip%
%
% LATTICE L23.11
%
\leavevmode%
\llap{\spacer}%
${L_{23.11}}=\pfillwitharg{2}({L_{23.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{6,5}{\infty b}{42}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8181558\\474012\\53676
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
474012\\-27453\\-3111
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53676\\-3111\\-352
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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505343\\4563888\\36716400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29696\\-268193\\-2157600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3264\\-29478\\-237151
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-333\\-3010\\-24171
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-899\\-8120\\-65310
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-403\\-3639\\-29286
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-259\\-2338\\-18828
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-442\\-3565
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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42\\-42\\-63\\-84\\-42\\-336\\-546\\-189\\-84\\0
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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-63\\0\\9\\0\\-9
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-84\\-42\\0\\6\\0
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-42\\-42\\-9\\0\\2
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-336\\-546\\-189\\-84\\0
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-546\\-1218\\-504\\-294\\-42
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\kern3pt%
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-189\\-504\\-225\\-144\\-27
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\kern3pt%
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-84\\-294\\-144\\-102\\-24
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-27\\-24\\-8
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-8181558, 474012, 53676; 474012, -27453, -3111; 53676, -3111, -352],[126, 3, 1],[-333, -899, -403, -259, -49, -115, 3, 19, 3, -15; -3010, -8120, -3639, -2338, -442, -1036, 28, 171, 26, -136; -24171, -65310, -29286, -18828, -3565, -8379, 210, 1386, 228, -1085],22,11]
%
% END OF LATTICE L23.11
%
\hbox{}\par\smallskip%
%
% LATTICE L23.12
%
\leavevmode%
\llap{\spacer}%
${L_{23.12}}=\main({L_{23.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{12,11}{\infty}{84}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-718452\\-11844\\-13104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11844\\-195\\-216
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13104\\-216\\-239
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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2687\\61824\\-201600
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43\\988\\-3225
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49\\1127\\-3676
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-4\\-133\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-27\\-658\\2058
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-13\\-285\\963
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-9\\-176\\648
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\68
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-21\\-42\\-9\\0\\1\\0\\-42\\-27\\-24\\-4
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-63\\0\\18\\0\\-9
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-84\\-84\\0\\12\\0
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-21\\-42\\-9\\0\\1
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-168\\-546\\-189\\-84\\0
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-546\\-2436\\-1008\\-588\\-42
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-189\\-1008\\-450\\-288\\-27
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-84\\-588\\-288\\-204\\-24
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0\\-42\\-27\\-24\\-4
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-718452, -11844, -13104; -11844, -195, -216; -13104, -216, -239],[252, 3, 1],[-4, -27, -13, -9, -1, -3, -1, 1, 1, 0; -133, -658, -285, -176, -15, -28, 14, 9, -8, -8; 336, 2058, 963, 648, 68, 189, 42, -63, -48, 7],22,12]
%
% END OF LATTICE L23.12
%
\hbox{}\par\smallskip%
%
% LATTICE L23.13
%
\leavevmode%
\llap{\spacer}%
${L_{23.13}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{23.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{2,1}{\infty}{7}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16401\\315\\-4368
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
315\\-6\\84
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4368\\84\\-1163
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7559\\50085\\-24570
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2032\\13462\\-6605
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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47\\294\\-154
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\HardButStrongLineBreak\kern3pt%
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114\\749\\-371
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98\\657\\-318
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\HardButStrongLineBreak\kern3pt%
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30\\206\\-97
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15\\110\\-48
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16401, 315, -4368; 315, -6, 84; -4368, 84, -1163],[21, 3, 1],[47, 114, 98, 30, 15, 9, -2, -2, 2, 9; 294, 749, 657, 206, 110, 77, -7, -21, 6, 49; -154, -371, -318, -97, -48, -28, 7, 6, -7, -30],22,13]
%
% END OF LATTICE L23.13
%
\hbox{}\par\smallskip%
%
% LATTICE L23.14
%
\leavevmode%
\llap{\spacer}%
${L_{23.14}}=\pdualwitharg{2}(\pfillwitharg{3.2}({L_{23.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2857596, 768726, -1406118; 768726, -206790, 378262; -1406118, 378262, -691899],[42, 2, 1],[19307, 25805, 7686, 7384, 2773, 6467, -31, -322, -20, 909; 1092, 1470, 439, 423, 160, 378, 0, -19, -3, 50; -38640, -51639, -15380, -14775, -5548, -12936, 63, 644, 39, -1820],22,14]
%
% END OF LATTICE L23.14
%
\hbox{}\par\smallskip%
%
% LATTICE L23.15
%
\leavevmode%
\llap{\spacer}%
${L_{23.15}}=\pdualwitharg{3}(\pfillwitharg{3.2}({L_{23.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{2,1}{\infty b}{14}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3386754, 1024968, 1149078; 1024968, -310185, -347757; 1149078, -347757, -389866],[42, 3, 1],[-6414, -17155, -7666, -4911, -2768, -2154, 19, 322, 15, -904; 364, 980, 439, 282, 160, 126, 0, -19, -2, 50; -19229, -51436, -22986, -14726, -8301, -6461, 56, 966, 46, -2709],22,15]
%
% END OF LATTICE L23.15
%
\hbox{}\par\smallskip%
%
% LATTICE L23.16
%
\leavevmode%
\llap{\spacer}%
${L_{23.16}}=\pdualwitharg{7}(\pfillwitharg{2.3}({L_{23.1}}))$%
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{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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-213\\-9\\749
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4389, 2646, -1218; 2646, -1540, 735; -1218, 735, -338],[21, 7, 1],[-239, -613, -1258, -1187, -213, -65, 5, 42, -29, -91; -9, -24, -50, -48, -9, -3, 0, 2, 0, -3; 843, 2160, 4431, 4179, 749, 228, -18, -147, 105, 322],22,16]
%
% END OF LATTICE L23.16
%
\hbox{}\par\smallskip%
%
% LATTICE L23.17
%
\leavevmode%
\llap{\spacer}%
${L_{23.17}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{23.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{6,1}{\infty}{21}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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4\\25\\-66
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4\\26\\-67
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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-15\\-98\\252
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\HardButStrongLineBreak\kern3pt%
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-20\\-133\\336
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-4\\-27\\67
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-2\\-14\\33
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1\\6\\-18
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1197, -189, 0; -189, 21, -3; 0, -3, -1],[63, 3, 1],[-15, -20, -4, -2, 1, 1, -8, -4, -6, -7; -98, -133, -27, -14, 6, 7, -49, -25, -38, -45; 252, 336, 67, 33, -18, -21, 126, 65, 99, 117],22,17]
%
% END OF LATTICE L23.17
%
\hbox{}\par\smallskip%
%
% LATTICE L23.18
%
\leavevmode%
\llap{\spacer}%
${L_{23.18}}=\pdualwitharg{3}(\main(\pfillwitharg{3}({L_{23.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{4,3}{\infty}{28}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\112\\203
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\686\\1400
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\HardButStrongLineBreak\kern3pt%
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13\\321\\678
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\216\\472
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\68\\159
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-21\\0\\6\\0\\-9
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-56\\-182\\-63\\-28\\0
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-182\\-812\\-336\\-196\\-42
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-239484, 1260, 3948; 1260, -6, -21; 3948, -21, -65],[84, 3, 1],[4, 27, 13, 9, 3, 3, 1, -1, -1, 0; 112, 686, 321, 216, 68, 63, 14, -21, -16, 7; 203, 1400, 678, 472, 159, 161, 56, -54, -56, -3],22,18]
%
% END OF LATTICE L23.18
%
\hbox{}\par\smallskip%
%
% LATTICE L23.19
%
\leavevmode%
\llap{\spacer}%
${L_{23.19}}=\pdualwitharg{2.3}(\pfillwitharg{3.2}({L_{23.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{4,1}{\infty z}{7}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30418542\\256242\\13036464
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\kern3pt%
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256242\\2160\\109818
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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32308457\\23916984\\-75856872
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264390\\195719\\-620760
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13843964\\10248272\\-32504177
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-15085\\-11167\\35418
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30418542, 256242, 13036464; 256242, 2160, 109818; 13036464, 109818, 5587033],[42, 6, 1],[-14734, -17623, -15085, -4602, -4582, -2892, -3, -23, -434, -2972; -10913, -13048, -11167, -3406, -3389, -2135, 0, -17, -322, -2203; 34594, 41377, 35418, 10805, 10758, 6790, 7, 54, 1019, 6978],22,19]
%
% END OF LATTICE L23.19
%
\hbox{}\par\smallskip%
%
% LATTICE L23.20
%
\leavevmode%
\llap{\spacer}%
${L_{23.20}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{23.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2862468, 38892, 228900; 38892, -528, -3111; 228900, -3111, -18302],[84, 3, 1],[-333, -899, -403, -259, -147, -115, 3, 19, 3, -45; -6790, -18368, -8240, -5300, -3014, -2366, 56, 392, 68, -910; -3010, -8120, -3639, -2338, -1326, -1036, 28, 171, 26, -408],22,20]
%
% END OF LATTICE L23.20
%
\hbox{}\par\smallskip%
%
% LATTICE L23.21
%
\leavevmode%
\llap{\spacer}%
${L_{23.21}}=\pdualwitharg{7}(\pfillwitharg{3.2}({L_{23.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{2,1}{\infty b}{6}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-52024854, -362082, -8308734; -362082, -2520, -57827; -8308734, -57827, -1326963],[42, 7, 1],[-143, -391, -411, -797, -153, -53, -1, 19, 13, -43; -2721, -7290, -7606, -14628, -2753, -921, 6, 322, 60, -889; 1014, 2766, 2905, 5628, 1078, 372, 6, -133, -84, 308],22,21]
%
% END OF LATTICE L23.21
%
\hbox{}\par\smallskip%
%
% LATTICE L23.22
%
\leavevmode%
\llap{\spacer}%
${L_{23.22}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{23.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{2,1}{\infty z}{21}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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79326589\\142405872\\-334012056
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\kern3pt%
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109025\\195719\\-459060
\end{tabular}\endgroup%
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18886245\\33904296\\-79522309
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-18224\\-32739\\76734
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-21800\\-39144\\91791
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-12441\\-22334\\52384
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-11387\\-20436\\47946
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1890\\-3389\\7958
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-84\\0\\8\\0\\-12
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-84\\-42\\-12\\0\\4
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-672\\-546\\-252\\-168\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-546\\-609\\-336\\-294\\-42
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\kern3pt%
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-252\\-336\\-200\\-192\\-36
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-168\\-294\\-192\\-204\\-48
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0\\-42\\-36\\-48\\-16
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[364911960, 512484, 86883720; 512484, 720, 122020; 86883720, 122020, 20686581],[84, 4, 1],[-18224, -21800, -12441, -11387, -1890, -3581, -5, -19, -1073, -1225; -32739, -39144, -22334, -20436, -3389, -6405, 0, -34, -1932, -2203; 76734, 91791, 52384, 47946, 7958, 15078, 21, 80, 4518, 5158],22,22]
%
% END OF LATTICE L23.22
%
\hbox{}\par\smallskip%
%
% LATTICE L23.23
%
\leavevmode%
\llap{\spacer}%
${L_{23.23}}=\pdualwitharg{2}(\main(\pfillwitharg{3}({L_{23.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{4,1}{\infty}{84}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32088\\16044\\7896
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16044\\8020\\3948
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7896\\3948\\1943
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13217\\-4032\\61824
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6431\\-1963\\30084
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3245\\-990\\15179
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
382\\105\\-1764
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
857\\252\\-3990
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\HardButStrongLineBreak\kern3pt%
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237\\72\\-1108
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
104\\33\\-489
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\11\\-144
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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84\\-84\\-84\\-84\\-84\\-672\\-1092\\-252\\-84\\0
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-84\\-84\\-12\\0\\4\\0\\-84\\-36\\-24\\-16
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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-84\\84\\0\\-42\\-84
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-84\\0\\8\\0\\-12
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-84\\-42\\0\\3\\0
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-84\\-84\\-12\\0\\4
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-672\\-1092\\-252\\-84\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1092\\-2436\\-672\\-294\\-84
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-672\\-200\\-96\\-36
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-294\\-96\\-51\\-24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-36\\-24\\-16
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[32088, 16044, 7896; 16044, 8020, 3948; 7896, 3948, 1943],[84, 4, 1],[382, 857, 237, 104, 30, 31, -31, -1, 14, 29; 105, 252, 72, 33, 11, 21, 0, 0, 3, 7; -1764, -3990, -1108, -489, -144, -168, 126, 4, -63, -132],22,23]
%
% END OF LATTICE L23.23
%
\hbox{}\par\smallskip%
%
% LATTICE L23.24
%
\leavevmode%
\llap{\spacer}%
${L_{23.24}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{23.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{6,1}{\infty b}{42}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-633402\\-291564\\24570
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-291564\\-134211\\11310
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24570\\11310\\-953
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-38305\\76608\\-78624
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\kern3pt%
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-17632\\35263\\-36192
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1482\\-2964\\3041
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-291\\581\\-609
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\HardButStrongLineBreak\kern3pt%
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-395\\784\\-882
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-40\\79\\-94
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\HardButStrongLineBreak\kern3pt%
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-41\\80\\-108
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-39\\27
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-126\\-126\\-9\\0\\18\\0\\-126\\-27\\-72\\-72
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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-42\\42\\0\\-42\\-126
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\1\\0\\-9
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-42\\0\\6\\0
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\kern3pt%
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-126\\-126\\-9\\0\\18
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-336\\-546\\-63\\-84\\0
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\kern3pt%
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-546\\-1218\\-168\\-294\\-126
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-63\\-168\\-25\\-48\\-27
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\kern3pt%
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-84\\-294\\-48\\-102\\-72
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-126\\-27\\-72\\-72
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-633402, -291564, 24570; -291564, -134211, 11310; 24570, 11310, -953],[126, 3, 1],[-291, -395, -40, -41, 19, 25, -137, -36, -111, -133; 581, 784, 79, 80, -39, -49, 280, 73, 224, 267; -609, -882, -94, -108, 27, 63, -210, -62, -204, -261],22,24]
%
% END OF LATTICE L23.24
%
\hbox{}\par\smallskip%
%
% LATTICE L23.25
%
\leavevmode%
\llap{\spacer}%
${L_{23.25}}=\pdualwitharg{3.7}(\pfillwitharg{2.3}({L_{23.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{2,1}{\infty}{1}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\5733\\1764
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5733\\834141\\256641
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[42, 5733, 1764; 5733, 834141, 256641; 1764, 256641, 78961],[21, 21, 1],[51, 122, 731, 223, 111, 10, 0, 1, 21, 72; 303, 736, 4440, 1366, 697, 67, 4, 0, 114, 413; -986, -2395, -14448, -4445, -2268, -218, -13, 0, -371, -1344],22,25]
%
% END OF LATTICE L23.25
%
\hbox{}\par\smallskip%
%
% LATTICE L23.26
%
\leavevmode%
\llap{\spacer}%
${L_{23.26}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{23.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{4,3}{\infty b}{56}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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13\\642\\1770
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2722776, 3024, 18816; 3024, -3, -21; 18816, -21, -130],[168, 3, 1],[4, 27, 13, 9, 3, 3, 1, -1, -1, 0; 224, 1372, 642, 432, 136, 126, 28, -42, -32, 14; 539, 3668, 1770, 1228, 411, 413, 140, -138, -140, -3],22,26]
%
% END OF LATTICE L23.26
%
\hbox{}\par\smallskip%
%
% LATTICE L23.27
%
\leavevmode%
\llap{\spacer}%
${L_{23.27}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{23.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{4,3}{\infty a}{56}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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-6267\\-44772\\-37660
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-1380\\-9859\\-8292
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1968120, 232680, 50904; 232680, -27501, -6027; 50904, -6027, -1306],[168, 3, 1],[-1217, -6267, -1380, -1739, -470, -337, 51, 48, -37, -196; -8694, -44772, -9859, -12424, -3358, -2408, 364, 343, -264, -1400; -7315, -37660, -8292, -10448, -2823, -2023, 308, 288, -224, -1179],22,27]
%
% END OF LATTICE L23.27
%
\hbox{}\par\smallskip%
%
% LATTICE L23.28
%
\leavevmode%
\llap{\spacer}%
${L_{23.28}}=\pdualwitharg{7}(\main(\pfillwitharg{3}({L_{23.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{4,3}{\infty}{12}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-112812, 33852, -1680; 33852, -10157, 504; -1680, 504, -25],[84, 7, 1],[2, 5, 3, 1, -1, -1, -1, 1, 7, 2; 18, 66, 57, 84, 2, -3, -6, 3, 36, 13; 225, 978, 931, 1596, 105, 6, -54, -7, 252, 126],22,28]
%
% END OF LATTICE L23.28
%
\hbox{}\par\smallskip%
%
% LATTICE L23.29
%
\leavevmode%
\llap{\spacer}%
${L_{23.29}}=\pdualwitharg{2.7}(\pfillwitharg{3.2}({L_{23.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3683568, 2303826, 1773576; 2303826, 1448118, 1109248; 1773576, 1109248, 853947],[42, 14, 1],[176173, 210667, 420719, 384995, 127721, 34507, 13, 641, 36365, 82959; -276, -330, -659, -603, -200, -54, 0, -1, -57, -130; -365538, -437109, -872942, -798819, -265006, -71598, -27, -1330, -75453, -172130],22,29]
%
% END OF LATTICE L23.29
%
\hbox{}\par\smallskip%
%
% LATTICE L23.30
%
\leavevmode%
\llap{\spacer}%
${L_{23.30}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{23.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{1,0}{\infty b}{12}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1128372, -35112, 276444; -35112, -1092, 8603; 276444, 8603, -67726],[84, 7, 1],[-223, -611, -642, -1243, -237, -79, 7, 38, 35, -65; 1074, 2916, 3054, 5892, 1114, 366, -36, -174, -132, 326; -774, -2124, -2233, -4326, -826, -276, 24, 133, 126, -224],22,30]
%
% END OF LATTICE L23.30
%
\hbox{}\par\smallskip%
%
% LATTICE L23.31
%
\leavevmode%
\llap{\spacer}%
${L_{23.31}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{23.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{12,1}{\infty z}{21}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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-3276\\2514\\-2028
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3200\\-1083\\-6512
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115\\-39\\-234
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[85302, -3276, 42462; -3276, 2514, -2028; 42462, -2028, 21203],[126, 6, 1],[18967, 14313, 3200, 2086, 115, 227, 4881, 2284, 3398, 8111; -6419, -4844, -1083, -706, -39, -77, -1652, -773, -1150, -2745; -38598, -29127, -6512, -4245, -234, -462, -9933, -4648, -6915, -16506],22,31]
%
% END OF LATTICE L23.31
%
\hbox{}\par\smallskip%
%
% LATTICE L23.32
%
\leavevmode%
\llap{\spacer}%
${L_{23.32}}=\pdualwitharg{3}(\main({L_{23.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{12,7}{\infty}{84}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-2772\\1008\\756
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\-366\\-273
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
756\\-273\\-199
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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2351\\7896\\-2016
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-868\\-2915\\744
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-658\\-2209\\563
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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20\\70\\-21
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-14\\0
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\HardButStrongLineBreak\kern3pt%
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-2\\-7\\2
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\HardButStrongLineBreak\kern3pt%
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3\\8\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\42\\-9
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2772, 1008, 756; 1008, -366, -273; 756, -273, -199],[252, 3, 1],[20, -5, -2, 3, 13, 78, 397, 58, 109, 29; 70, -14, -7, 8, 42, 259, 1330, 195, 368, 99; -21, 0, 2, 0, -9, -63, -336, -50, -96, -27],22,32]
%
% END OF LATTICE L23.32
%
\hbox{}\par\smallskip%
%
% LATTICE L23.33
%
\leavevmode%
\llap{\spacer}%
${L_{23.33}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{23.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{12,5}{\infty z}{21}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1570535694\\-9491202\\772451820
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-9491202\\57360\\-4668150
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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1092347885\\44062368\\-2220401484
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537259118\\21671584\\-1092079693
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-136020\\-5495\\276486
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-161135\\-6503\\327537
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-137363\\-5541\\279216
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-41698\\-1681\\84759
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13644\\-549\\27734
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-84\\-21\\0\\3\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1570535694, -9491202, 772451820; -9491202, 57360, -4668150; 772451820, -4668150, 379922479],[126, 6, 1],[-136020, -161135, -137363, -41698, -13644, -25022, 93, -673, -4314, -9362; -5495, -6503, -5541, -1681, -549, -1001, 7, -27, -175, -379; 276486, 327537, 279216, 84759, 27734, 50862, -189, 1368, 8769, 19030],22,33]
%
% END OF LATTICE L23.33
%
\hbox{}\par\smallskip%
%
% LATTICE L23.34
%
\leavevmode%
\llap{\spacer}%
${L_{23.34}}=\pdualwitharg{3}({L_{23.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-140364, -21168, -7056; -21168, -3192, -1065; -7056, -1065, -352],[252, 3, 1],[19, -131, -33, -99, -115, -243, -317, -31, -29, 19; -112, 784, 197, 590, 684, 1442, 1876, 183, 170, -114; -42, 252, 65, 198, 234, 504, 672, 67, 66, -36],22,34]
%
% END OF LATTICE L23.34
%
\hbox{}\par\smallskip%
%
% LATTICE L23.35
%
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${L_{23.35}}=\pdualwitharg{3.7}(\pfillwitharg{3.2}({L_{23.3}}))$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1481550, 1784580, -730674; 1784580, 2172177, -895923; -730674, -895923, 371408],[42, 21, 1],[47731, 114153, 341959, 208615, 103811, 9349, 7, 521, 19705, 67429; -95554, -228526, -684577, -417632, -207822, -18716, -14, -1043, -39448, -134988; -136597, -326684, -978621, -597016, -297087, -26755, -20, -1491, -56392, -192969],22,35]
%
% END OF LATTICE L23.35
%
\hbox{}\par\smallskip%
%
% LATTICE L23.36
%
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${L_{23.36}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{23.1}}))$%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2603328, -1533084, 595056; -1533084, 902952, -350424; 595056, -350424, 136015],[84, 12, 1],[-10715, -12692, -21638, -6568, -3223, -1969, 8, -106, -680, -2213; -301, -357, -609, -185, -91, -56, 0, -3, -19, -62; 46102, 54607, 93096, 28258, 13866, 8470, -35, 456, 2926, 9522],22,36]
%
% END OF LATTICE L23.36
%
\hbox{}\par\smallskip%
%
% LATTICE L23.37
%
\leavevmode%
\llap{\spacer}%
${L_{23.37}}=\pdualwitharg{2.3}(\main(\pfillwitharg{3}({L_{23.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{4,1}{\infty}{28}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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8275\\84\\-34874
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7127\\72\\-30036
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1119\\11\\-4716
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[905352, -50652, 214704; -50652, 2892, -12012; 214704, -12012, 50917],[84, 12, 1],[3415, 8275, 7127, 1096, 1119, 764, 83, 37, 98, 672; 35, 84, 72, 11, 11, 7, 0, 0, 1, 7; -14392, -34874, -30036, -4619, -4716, -3220, -350, -156, -413, -2832],22,37]
%
% END OF LATTICE L23.37
%
\hbox{}\par\smallskip%
%
% LATTICE L23.38
%
\leavevmode%
\llap{\spacer}%
${L_{23.38}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{23.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{4,3}{\infty b}{24}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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1\\18\\153
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1\\12\\156
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1300152, 3360, 8064; 3360, -7, -21; 8064, -21, -50],[168, 7, 1],[1, 1, -1, -7, -2, -2, -5, -3, -1, 1; 18, 12, -18, -96, -22, -12, 12, 42, 144, 28; 153, 156, -154, -1092, -315, -321, -828, -518, -252, 147],22,38]
%
% END OF LATTICE L23.38
%
\hbox{}\par\smallskip%
%
% LATTICE L23.39
%
\leavevmode%
\llap{\spacer}%
${L_{23.39}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{23.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{4,3}{\infty a}{24}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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-42\\-84\\-21\\0\\14
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-48\\-156\\-63\\-168\\0
\end{tabular}\endgroup%
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-156\\-696\\-336\\-1176\\-84
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-63\\-336\\-175\\-672\\-63
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-717528, 13272, 214872; 13272, -245, -3976; 214872, -3976, -64341],[168, 7, 1],[44, -13, -15, 61, 73, 182, 917, 467, 1747, 153; 387, -120, -131, 576, 659, 1629, 8184, 4163, 15552, 1357; 123, -36, -42, 168, 203, 507, 2556, 1302, 4872, 427],22,39]
%
% END OF LATTICE L23.39
%
\hbox{}\par\smallskip%
%
% LATTICE L23.40
%
\leavevmode%
\llap{\spacer}%
${L_{23.40}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{23.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{6,5}{\infty}{3}\above{1pt}{1pt}{r}{2}{126}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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-63\\-11193\\-1575
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-7399\\-14195\\101472
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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102\\196\\-1401
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589\\1128\\-8064
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171\\326\\-2331
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24\\45\\-322
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[126, -63, 0; -63, -11193, -1575; 0, -1575, -221],[63, 21, 1],[46, 102, 589, 171, 24, 3, -4, -1, 25, 25; 89, 196, 1128, 326, 45, 5, -8, 0, 50, 49; -636, -1401, -8064, -2331, -322, -36, 57, 0, -357, -350],22,40]
%
% END OF LATTICE L23.40
%
\hbox{}\par\smallskip%
%
% LATTICE L23.41
%
\leavevmode%
\llap{\spacer}%
${L_{23.41}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{23.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{6,1}{\infty}{3}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{63}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
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1\\26\\174
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1\\43\\288
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1\\70\\469
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0\\47\\315
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[63, 0, 0; 0, -4956, 735; 0, 735, -109],[63, 21, 1],[1, 1, 1, 0, -1, 0, 1, 3, 4, 4; 26, 43, 70, 47, 0, -4, 1, 18, 41, 66; 174, 288, 469, 315, 0, -27, 6, 119, 273, 441],22,41]
%
% END OF LATTICE L23.41
%
\hbox{}\par\smallskip%
%
% LATTICE L23.42
%
\leavevmode%
\llap{\spacer}%
${L_{23.42}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{23.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{8,1}{\infty z}{84}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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-2688\\-2184\\-504\\-336\\0
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-2184\\-2436\\-672\\-588\\-168
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2680944, -63672, 1680; -63672, -1512, 40; 1680, 40, -1],[168, 8, 1],[13, 4, 0, -1, -1, 1, 10, 4, 5, 3; -504, -147, 3, 42, 40, -42, -399, -159, -198, -118; 1512, 714, 92, -18, -80, 0, 798, 340, 450, 296],22,42]
%
% END OF LATTICE L23.42
%
\hbox{}\par\smallskip%
%
% LATTICE L23.43
%
\leavevmode%
\llap{\spacer}%
${L_{23.43}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{23.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{8,5}{\infty z}{84}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-3245424\\77784\\75768
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\kern3pt%
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77784\\-1848\\-1808
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\kern3pt%
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75768\\-1808\\-1765
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{$\left.\llap{\phantom{%
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-532\\22229\\-45600
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0\\0\\0
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49\\-2055\\4208
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17\\-714\\1461
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1\\-44\\88
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-2184\\-2436\\-672\\-294\\-168
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3245424, 77784, 75768; 77784, -1848, -1808; 75768, -1808, -1765],[168, 8, 1],[239, 206, 49, 17, 1, -43, -10, 7, 11, 27; -9996, -8631, -2055, -714, -44, 1806, 441, -285, -456, -1126; 20496, 17682, 4208, 1461, 88, -3696, -882, 592, 939, 2312],22,43]
%
% END OF LATTICE L23.43
%
\hbox{}\par\smallskip%
%
% LATTICE L23.44
%
\leavevmode%
\llap{\spacer}%
${L_{23.44}}=\pdualwitharg{3}({L_{23.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{12,7}{\infty a}{168}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-24518088\\8177904\\98784
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8177904\\-2727705\\-32949
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
98784\\-32949\\-398
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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385307\\1097544\\4763808
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-128535\\-366131\\-1589160
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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50\\140\\819
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-9\\-28\\84
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-5\\-14\\-82
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5\\16\\-84
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29\\84\\243
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24518088, 8177904, 98784; 8177904, -2727705, -32949; 98784, -32949, -398],[504, 3, 1],[50, -9, -5, 5, 29, 181, 933, 137, 259, 70; 140, -28, -14, 16, 84, 518, 2660, 390, 736, 198; 819, 84, -82, -84, 243, 2037, 11340, 1714, 3348, 981],22,44]
%
% END OF LATTICE L23.44
%
\hbox{}\par\smallskip%
%
% LATTICE L23.45
%
\leavevmode%
\llap{\spacer}%
${L_{23.45}}=\pdualwitharg{3}({L_{23.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{12,7}{\infty b}{168}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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146\\875\\10563
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3\\16\\360
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67\\399\\5031
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-451391976, 64548288, 892080; 64548288, -9230295, -127566; 892080, -127566, -1763],[504, 3, 1],[146, -5, -7, 3, 67, 456, 2413, 179, 685, 191; 875, -28, -42, 16, 399, 2723, 14420, 1070, 4096, 1143; 10563, -504, -503, 360, 5031, 33705, 177576, 13151, 50232, 13941],22,45]
%
% END OF LATTICE L23.45
%
\hbox{}\par\smallskip%
%
% LATTICE L23.46
%
\leavevmode%
\llap{\spacer}%
${L_{23.46}}=\pdualwitharg{3.7}(\main(\pfillwitharg{3}({L_{23.2}})))$%
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{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-116340, -3948, 3108; -3948, -126, 105; 3108, 105, -83],[84, 21, 1],[1, -1, -2, 5, 10, 7, 33, 98, 59, 14; 3, -2, -7, 8, 21, 16, 78, 235, 144, 36; 41, -40, -84, 196, 399, 281, 1328, 3948, 2380, 567],22,46]
%
% END OF LATTICE L23.46
%
\hbox{}\par\smallskip%
%
% LATTICE L23.47
%
\leavevmode%
\llap{\spacer}%
${L_{23.47}}=\pdualwitharg{2.3.7}(\pfillwitharg{3.2}({L_{23.3}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16375028124, 39767806806, 19326557838; 39767806806, -96578671266, -46935786156; 19326557838, -46935786156, -22810088327],[42, 42, 1],[-907, -1215, -7606, -2438, -2753, -307, 1, 322, 10, -889; 1448, 1518, 8523, 2381, 1724, -2, -72, 153, 511, 2614; -3748, -4153, -23982, -6965, -5880, -256, 149, -42, -1043, -6132],22,47]
%
% END OF LATTICE L23.47
%
\hbox{}\par\smallskip%
%
% LATTICE L23.48
%
\leavevmode%
\llap{\spacer}%
${L_{23.48}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{23.1}}))$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[421092, 18816, 17388; 18816, 840, 777; 17388, 777, 718],[84, 21, 1],[1, -3, -19, -21, -27, -9, -13, -29, -11, 3; -2, 8, 56, 68, 98, 38, 64, 160, 76, 10; -22, 64, 399, 434, 546, 176, 244, 525, 182, -84],22,48]
%
% END OF LATTICE L23.48
%
\hbox{}\par\smallskip%
%
% LATTICE L23.49
%
\leavevmode%
\llap{\spacer}%
${L_{23.49}}=\pdualwitharg{2.7}(\pfillwitharg{3}({L_{23.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4293408, 1290156, -140700; 1290156, 387688, -42280; -140700, -42280, 4611],[84, 28, 1],[197, 227, 892, 800, 125, 29, -1, 12, 104, 101; -702, -810, -3185, -2859, -448, -105, 3, -43, -369, -359; -426, -501, -1988, -1806, -294, -78, -3, -28, -210, -210],22,49]
%
% END OF LATTICE L23.49
%
\hbox{}\par\smallskip%
%
% LATTICE L23.50
%
\leavevmode%
\llap{\spacer}%
${L_{23.50}}=\pdualwitharg{2.7}(\main(\pfillwitharg{3}({L_{23.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{4,1}{\infty}{12}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[408072, 14532, -86436; 14532, 700, -2940; -86436, -2940, 18413],[84, 28, 1],[1447, 3437, 6853, 3134, 1043, 298, 53, 127, 356, 702; -4587, -10896, -21726, -9936, -3307, -945, -168, -402, -1128, -2225; 6060, 14394, 28700, 13125, 4368, 1248, 222, 532, 1491, 2940],22,50]
%
% END OF LATTICE L23.50
%
\hbox{}\par\smallskip%
%
% LATTICE L23.51
%
\leavevmode%
\llap{\spacer}%
${L_{23.51}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{23.2}}))$%
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{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[877590, 348264, -52920; 348264, 138201, -21000; -52920, -21000, 3191],[126, 21, 1],[-1, 3, 19, 21, 9, 9, 13, 29, 11, -1; 13, -34, -201, -206, -79, -65, -70, -111, -2, 27; 69, -174, -1008, -1008, -371, -279, -246, -252, 168, 161],22,51]
%
% END OF LATTICE L23.51
%
\hbox{}\par\smallskip%
%
% LATTICE L23.52
%
\leavevmode%
\llap{\spacer}%
${L_{23.52}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{23.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-630, -2394, -378; -2394, -7203, -1176; -378, -1176, -191],[126, 21, 1],[-4, 5, 15, 57, 82, 30, 47, 37, 47, -4; -7, 10, 28, 104, 147, 53, 82, 64, 80, -9; 51, -72, -203, -756, -1071, -387, -600, -469, -588, 63],22,52]
%
% END OF LATTICE L23.52
%
\hbox{}\par\smallskip%
%
% LATTICE L23.53
%
\leavevmode%
\llap{\spacer}%
${L_{23.53}}=\pdualwitharg{2.3}({L_{23.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8020656, -380268, -2015748; -380268, 18048, 95568; -2015748, 95568, 506597],[252, 12, 1],[32, 1048, 995, 1497, 1810, 4287, 3271, 1473, 971, 41; 7, 175, 165, 247, 297, 700, 532, 239, 157, 6; 126, 4137, 3928, 5910, 7146, 16926, 12915, 5816, 3834, 162],22,53]
%
% END OF LATTICE L23.53
%
\hbox{}\par\smallskip%
%
% LATTICE L23.54
%
\leavevmode%
\llap{\spacer}%
${L_{23.54}}=\pdualwitharg{2}({L_{23.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{6,5}{\infty z}{21}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6282142776, -18982404, -1596167748; -18982404, 57360, 4823052; -1596167748, 4823052, 405554533],[252, 12, 1],[70233, 83201, 141853, 43061, 7045, 12920, -48, 695, 4455, 4834; -5495, -6503, -11082, -3362, -549, -1001, 7, -54, -350, -379; 276486, 327537, 558432, 169518, 27734, 50862, -189, 2736, 17538, 19030],22,54]
%
% END OF LATTICE L23.54
%
\hbox{}\par\smallskip%
%
% LATTICE L23.55
%
\leavevmode%
\llap{\spacer}%
${L_{23.55}}=\pdualwitharg{2}(\main({L_{23.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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30421\\595\\119910
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4351284, 19152, -1104012; 19152, 228, -4860; -1104012, -4860, 280111],[252, 12, 1],[12552, 30421, 26203, 4030, 1372, 2813, 309, 137, 360, 823; 245, 595, 513, 79, 27, 56, 7, 3, 7, 16; 49476, 119910, 103284, 15885, 5408, 11088, 1218, 540, 1419, 3244],22,55]
%
% END OF LATTICE L23.55
%
\hbox{}\par\smallskip%
%
% LATTICE L23.56
%
\leavevmode%
\llap{\spacer}%
${L_{23.56}}=\pdualwitharg{2.3}(\main({L_{23.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{12,1}{\infty}{84}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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855288\\28476\\-214452
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28476\\948\\-7140
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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1\\0\\4
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[855288, 28476, -214452; 28476, 948, -7140; -214452, -7140, 53771],[252, 12, 1],[-3, -11, 1, 5, 25, 94, 193, 51, 21, 14; 91, 14, 0, 8, 63, 343, 854, 248, 116, 123; 0, -42, 4, 21, 108, 420, 882, 236, 99, 72],22,56]
%
% END OF LATTICE L23.56
%
\hbox{}\par\smallskip%
%
% LATTICE L23.57
%
\leavevmode%
\llap{\spacer}%
${L_{23.57}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{23.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-232680, -120288, 6216; -120288, -62181, 3213; 6216, 3213, -166],[168, 21, 1],[-2, 1, 5, -3, -11, -9, -45, -137, -85, -22; 6, -4, -14, 16, 42, 32, 156, 470, 288, 72; 41, -40, -84, 196, 399, 281, 1328, 3948, 2380, 567],22,57]
%
% END OF LATTICE L23.57
%
\hbox{}\par\smallskip%
%
% LATTICE L23.58
%
\leavevmode%
\llap{\spacer}%
${L_{23.58}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{23.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty a}{8}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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5\\240\\616
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-936264, 8232, 4368; 8232, -63, -42; 4368, -42, -19],[168, 21, 1],[1, -1, -1, 5, 10, 7, 33, 49, 59, 14; 47, -48, -47, 240, 477, 333, 1568, 2327, 2800, 663; 125, -124, -126, 616, 1239, 869, 4100, 6090, 7336, 1743],22,58]
%
% END OF LATTICE L23.58
%
\hbox{}\par\smallskip%
%
% LATTICE L23.59
%
\leavevmode%
\llap{\spacer}%
${L_{23.59}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{23.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{8,1}{\infty z}{28}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-3\\-91\\322
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-1\\-8\\38
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-1\\12\\-24
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-78288, -18648, -6048; -18648, -4440, -1440; -6048, -1440, -467],[168, 24, 1],[-1, -3, -3, -1, -1, 1, 3, 3, 1, 1; -140, -91, -53, -8, 12, 14, -35, -55, -28, -66; 448, 322, 204, 38, -24, -56, 70, 132, 74, 192],22,59]
%
% END OF LATTICE L23.59
%
\hbox{}\par\smallskip%
%
% LATTICE L23.60
%
\leavevmode%
\llap{\spacer}%
${L_{23.60}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{23.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{8,5}{\infty z}{28}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2844240, -347592, -154056; -347592, -42408, -18792; -154056, -18792, -8327],[168, 24, 1],[85, 73, 52, 6, 1, -15, -3, 8, 4, 29; -6776, -5803, -4127, -475, -72, 1190, 217, -661, -323, -2322; 13720, 11746, 8352, 961, 144, -2408, -434, 1344, 655, 4704],22,60]
%
% END OF LATTICE L23.60
%
\hbox{}\par\smallskip%
%
% LATTICE L23.61
%
\leavevmode%
\llap{\spacer}%
${L_{23.61}}=\pdualwitharg{7}(\main({L_{23.3}}))$%
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{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-260316, 629496, -102816; 629496, -1515129, 247590; -102816, 247590, -40457],[252, 21, 1],[76, -75, -155, 369, 249, 525, 2479, 7367, 4439, 352; -266, 262, 543, -1288, -870, -1835, -8666, -25755, -15520, -1231; -1821, 1794, 3717, -8820, -5957, -12564, -59334, -176337, -106260, -8428],22,61]
%
% END OF LATTICE L23.61
%
\hbox{}\par\smallskip%
%
% LATTICE L23.62
%
\leavevmode%
\llap{\spacer}%
${L_{23.62}}=\pdualwitharg{2.3.7}(\pfillwitharg{2}({L_{23.2}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4777794, 189504, 2356074; 189504, 7518, 93450; 2356074, 93450, 1161851],[126, 42, 1],[-3, -82, -299, -490, -1313, -497, -418, -701, -510, -187; 1, 16, 54, 83, 207, 73, 58, 94, 65, 15; 6, 165, 602, 987, 2646, 1002, 843, 1414, 1029, 378],22,62]
%
% END OF LATTICE L23.62
%
\hbox{}\par\smallskip%
%
% LATTICE L23.63
%
\leavevmode%
\llap{\spacer}%
${L_{23.63}}=\pdualwitharg{3.7}(\main({L_{23.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
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27\\80\\1596
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\18\\315
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-40572, -6804, 1008; -6804, -1113, 168; 1008, 168, -25],[252, 21, 1],[4, 17, 16, 27, 5, 0, -1, 0, 5, 7; 10, 46, 45, 80, 18, 1, -2, -1, 8, 15; 225, 978, 931, 1596, 315, 6, -54, -7, 252, 378],22,63]
%
% END OF LATTICE L23.63
%
\hbox{}\par\smallskip%
%
% LATTICE L23.64
%
\leavevmode%
\llap{\spacer}%
${L_{23.64}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{23.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,5}{\infty z}{3}\above{1pt}{1pt}{}{2}{126}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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12636666\\-760032\\6220620
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-760032\\45570\\-374136
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\kern3pt%
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6220620\\-374136\\3062209
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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3537125\\-155940\\-7204428
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-271284\\11959\\552552
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1742478\\-76820\\-3549085
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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707\\-31\\-1440
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28\\-1\\-57
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-7114\\315\\14490
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-8097\\357\\16492
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12636666, -760032, 6220620; -760032, 45570, -374136; 6220620, -374136, 3062209],[126, 42, 1],[707, 28, -7114, -6547, -8097, -11141, -10462, -55490, -14321, -2337; -31, -1, 315, 289, 357, 491, 461, 2445, 631, 103; -1440, -57, 14490, 13335, 16492, 22692, 21309, 113022, 29169, 4760],22,64]
%
% END OF LATTICE L23.64
%
\hbox{}\par\smallskip%
%
% LATTICE L23.65
%
\leavevmode%
\llap{\spacer}%
${L_{23.65}}=\pdualwitharg{7}({L_{23.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{3,2}{\infty b}{12}\above{1pt}{1pt}{r}{2}{126}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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1755180\\563472\\-83664
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\kern3pt%
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563472\\180894\\-26859
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\kern3pt%
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-83664\\-26859\\3988
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-1\\4\\6
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3\\-4\\36
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19\\-3\\378
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21\\22\\588
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\24\\350
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-156\\-348\\-1008\\-588\\-84
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-378\\-1008\\-3150\\-2016\\-378
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-168\\-588\\-2016\\-1428\\-336
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1755180, 563472, -83664; 563472, 180894, -26859; -83664, -26859, 3988],[252, 21, 1],[-1, 3, 19, 21, 9, 9, 13, 29, 11, -1; 4, -4, -3, 22, 24, 46, 104, 303, 178, 26; 6, 36, 378, 588, 350, 498, 972, 2646, 1428, 154],22,65]
%
% END OF LATTICE L23.65
%
\hbox{}\par\smallskip%
%
% LATTICE L23.66
%
\leavevmode%
\llap{\spacer}%
${L_{23.66}}=\pdualwitharg{3.7}({L_{23.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{3,1}{\infty a}{12}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{252}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105588\\-180936\\-30996
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180936\\-308742\\-52941
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30996\\-52941\\-9076
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{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-105588, -180936, -30996; -180936, -308742, -52941; -30996, -52941, -9076],[252, 21, 1],[-259, -405, -318, -401, 41, 37, -39, -126, -487, -707; -520, -812, -637, -802, 84, 74, -80, -255, -982, -1422; 3918, 6120, 4802, 6048, -630, -558, 600, 1918, 7392, 10710],22,66]
%
% END OF LATTICE L23.66
%
\hbox{}\par\smallskip%
%
% LATTICE L23.67
%
\leavevmode%
\llap{\spacer}%
${L_{23.67}}=\pdualwitharg{2.3.7}(\pfillwitharg{3}({L_{23.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1512, -18228, -4368; -18228, 447216, 107268; -4368, 107268, 25729],[84, 84, 1],[1, 0, -21, -19, -35, -16, -15, -159, -41, -10; -129, -6, 2578, 2380, 4421, 2029, 1906, 20222, 5220, 1279; 538, 25, -10752, -9926, -18438, -8462, -7949, -84336, -21770, -5334],22,67]
%
% END OF LATTICE L23.67
%
\hbox{}\par\smallskip%
%
% LATTICE L23.68
%
\leavevmode%
\llap{\spacer}%
${L_{23.68}}=\pdualwitharg{2.3.7}(\main(\pfillwitharg{3}({L_{23.2}})))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3444, -49308, -11676; -49308, 753480, 178416; -11676, 178416, 42247],[84, 84, 1],[-1, 0, -2, -2, -11, -7, -16, -94, -14, -13; 160, 9, 159, 217, 1312, 887, 2085, 12405, 1877, 1829; -676, -38, -672, -917, -5544, -3748, -8810, -52416, -7931, -7728],22,68]
%
% END OF LATTICE L23.68
%
\hbox{}\par\smallskip%
%
% LATTICE L23.69
%
\leavevmode%
\llap{\spacer}%
${L_{23.69}}=\pdualwitharg{2.7}(\pfillwitharg{3}({L_{23.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-162960, -38808, -16464; -38808, -9240, -3920; -16464, -3920, -1663],[168, 56, 1],[1, 1, 1, -1, -3, -5, -5, -9, -7, -1; 6, -27, -87, -120, -82, -60, -27, -21, 12, 28; -24, 54, 196, 294, 224, 192, 114, 140, 42, -56],22,69]
%
% END OF LATTICE L23.69
%
\hbox{}\par\smallskip%
%
% LATTICE L23.70
%
\leavevmode%
\llap{\spacer}%
${L_{23.70}}=\pdualwitharg{2.7}(\pfillwitharg{3}({L_{23.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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10\\-773\\1568
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33\\-2434\\4928
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-670992, -242760, -115416; -242760, -87528, -41608; -115416, -41608, -19779],[168, 56, 1],[-7, -1, 10, 14, 33, 41, 35, 58, 20, 1; 510, 57, -773, -1047, -2434, -3000, -2547, -4207, -1443, -56; -1032, -114, 1568, 2121, 4928, 6072, 5154, 8512, 2919, 112],22,70]
%
% END OF LATTICE L23.70
%
\hbox{}\par\smallskip%
%
% LATTICE L23.71
%
\leavevmode%
\llap{\spacer}%
${L_{23.71}}=\pdualwitharg{7}({L_{23.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,11}{\infty b}{24}\above{1pt}{1pt}{s}{2}{252}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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35\\-245\\-1674
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116\\-809\\-5530
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-520632, 573048, -94752; 573048, -627690, 103845; -94752, 103845, -17179],[504, 21, 1],[35, -35, -71, 173, 116, 244, 1151, 3419, 2059, 163; -245, 244, 498, -1204, -809, -1703, -8036, -23874, -14380, -1139; -1674, 1668, 3402, -8232, -5530, -11640, -54924, -163170, -98280, -7784],22,71]
%
% END OF LATTICE L23.71
%
\hbox{}\par\smallskip%
%
% LATTICE L23.72
%
\leavevmode%
\llap{\spacer}%
${L_{23.72}}=\pdualwitharg{7}({L_{23.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,11}{\infty a}{24}\above{1pt}{1pt}{l}{2}{63}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3647448, 5775336, -997416; 5775336, -9095898, 1572291; -997416, 1572291, -271741],[504, 21, 1],[813, -805, -820, 3997, 2682, 5645, 26637, 39568, 47667, 3776; -2561, 2536, 2583, -12592, -8449, -17783, -83912, -124647, -150160, -11895; -17802, 17628, 17955, -87528, -58730, -123612, -583284, -866439, -1043784, -82684],22,72]
%
% END OF LATTICE L23.72
%
\hbox{}\par\smallskip%
%
% LATTICE L23.73
%
\leavevmode%
\llap{\spacer}%
${L_{23.73}}=\pdualwitharg{3.7}({L_{23.2}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,7}{\infty a}{24}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{b}{2}{126}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-81144, -68544, 2016; -68544, -57855, 1701; 2016, 1701, -50],[504, 21, 1],[-1, 1, 1, -3, -8, -6, -29, -29, -53, -13; 2, -4, -2, 16, 30, 20, 92, 90, 160, 36; 27, -96, -28, 420, 693, 435, 1944, 1876, 3276, 693],22,73]
%
% END OF LATTICE L23.73
%
\hbox{}\par\smallskip%
%
% LATTICE L23.74
%
\leavevmode%
\llap{\spacer}%
${L_{23.74}}=\pdualwitharg{3.7}({L_{23.3}})$%
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{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,7}{\infty b}{24}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{126}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4393368, -312480, 49392; -312480, -22197, 3507; 49392, 3507, -554],[504, 21, 1],[3, -1, -1, 5, 16, 13, 65, 33, 123, 32; -164, 52, 55, -256, -852, -698, -3500, -1779, -6640, -1734; -771, 240, 259, -1176, -3969, -3261, -16368, -8323, -31080, -8127],22,74]
%
% END OF LATTICE L23.74
%
\hbox{}\par\smallskip%
%
% LATTICE L23.75
%
\leavevmode%
\llap{\spacer}%
${L_{23.75}}=\pdualwitharg{2.3}({L_{23.2}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{24,1}{\infty z}{84}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-234864, 10584, 504; 10584, -456, -24; 504, -24, -1],[504, 24, 1],[-1, -3, -1, -1, -1, 1, 3, 1, 1, 1; -28, -49, -15, -14, -12, 14, 35, 11, 10, 6; 0, -462, -172, -186, -216, 168, 630, 220, 234, 288],22,75]
%
% END OF LATTICE L23.75
%
\hbox{}\par\smallskip%
%
% LATTICE L23.76
%
\leavevmode%
\llap{\spacer}%
${L_{23.76}}=\pdualwitharg{2.3}({L_{23.3}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-32060952, -4050144, 237888; -4050144, -511512, 30048; 237888, 30048, -1765],[504, 24, 1],[253, 213, 50, 17, 1, -43, -3, 10, 13, 89; 1757, 1484, 349, 119, 9, -301, -28, 67, 89, 615; 64008, 53970, 12680, 4317, 288, -10920, -882, 2488, 3267, 22464],22,76]
%
% END OF LATTICE L23.76
%
\hbox{}\par\smallskip%
%
% LATTICE L23.77
%
\leavevmode%
\llap{\spacer}%
${L_{23.77}}=\pdualwitharg{2}({L_{23.2}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-306899208, 47940480, 9269568; 47940480, -7488744, -1447992; 9269568, -1447992, -279977],[504, 24, 1],[2701, 1971, 1265, 243, -41, -321, 409, 775, 437, 381; 16219, 11830, 7590, 1457, -247, -1925, 2464, 4662, 2627, 2289; 5544, 4074, 2628, 510, -80, -672, 798, 1548, 882, 776],22,77]
%
% END OF LATTICE L23.77
%
\hbox{}\par\smallskip%
%
% LATTICE L23.78
%
\leavevmode%
\llap{\spacer}%
${L_{23.78}}=\pdualwitharg{2}({L_{23.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1730736, -45864, 4032; -45864, -1176, 120; 4032, 120, -5],[504, 24, 1],[71, 66, 49, 6, 1, -15, -10, -1, 2, 7; -2114, -1967, -1461, -179, -30, 448, 301, 33, -59, -208; 6384, 5922, 4392, 537, 88, -1344, -882, -72, 183, 632],22,78]
%
% END OF LATTICE L23.78
%
\hbox{}\par\smallskip%
%
% LATTICE L23.79
%
\leavevmode%
\llap{\spacer}%
${L_{23.79}}=\pdualwitharg{2.3.7}({L_{23.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{6,1}{\infty z}{3}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{252}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[532252224, -20786220, 150518088; -20786220, 811776, -5878236; 150518088, -5878236, 42565715],[252, 84, 1],[2298, 1696, 5239, 3353, 32, 71, 673, 4237, 6123, 7075; -7241, -5342, -16497, -10553, -93, -224, -2123, -13363, -19307, -22302; -9126, -6735, -20804, -13314, -126, -282, -2673, -16828, -24318, -28098],22,79]
%
% END OF LATTICE L23.79
%
\hbox{}\par\smallskip%
%
% LATTICE L23.80
%
\leavevmode%
\llap{\spacer}%
${L_{23.80}}=\pdualwitharg{2.7}({L_{23.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2178288, -30996, -530964; -30996, 2352, -9744; -530964, -9744, -127709],[252, 84, 1],[338, 13, -6809, -6263, -3872, -5327, -5002, -53059, -13693, -1117; -1039, -40, 20931, 19253, 11903, 16376, 15377, 163113, 42095, 3434; -1326, -51, 26712, 24570, 15190, 20898, 19623, 208152, 53718, 4382],22,80]
%
% END OF LATTICE L23.80
%
\hbox{}\par\smallskip%
%
% LATTICE L23.81
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9585451512, 355372668, 2158799580; 355372668, 13175148, 80035704; 2158799580, 80035704, 486196777],[252, 84, 1],[1271, 71, 1267, 1727, 3479, 7054, 16579, 98633, 14923, 4846; -3593, -202, -3558, -4867, -9813, -19907, -46798, -278442, -42133, -13687; -5052, -282, -5040, -6867, -13832, -28044, -65910, -392112, -59325, -19264],22,81]
%
% END OF LATTICE L23.81
%
\hbox{}\par\smallskip%
%
% LATTICE L23.82
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${L_{23.82}}=\pdualwitharg{2.3.7}(\main({L_{23.3}}))$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L23.82
%
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%
% LATTICE L23.83
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-504\\-672\\-4200\\-1344\\-1512
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1008, -52584, -25872; -52584, -2619624, -1288896; -25872, -1288896, -634157],[168, 168, 1],[-7, -6, -30, -7, -1, 1, 0, -6, -5, -17; -976, -731, -3347, -668, 248, 122, -123, -1777, -1040, -2810; 1984, 1486, 6804, 1358, -504, -248, 250, 3612, 2114, 5712],22,83]
%
% END OF LATTICE L23.83
%
\hbox{}\par\smallskip%
%
% LATTICE L23.84
%
\leavevmode%
\llap{\spacer}%
${L_{23.84}}=\pdualwitharg{2.3.7}(\pfillwitharg{3}({L_{23.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,5}{\infty z}{4}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{336}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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-301392\\-319704\\-152544
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{$\left.\llap{\phantom{%
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85\\-1122\\2184
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9\\-112\\217
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-284088, -301392, -143808; -301392, -319704, -152544; -143808, -152544, -72785],[168, 168, 1],[23, 18, 85, 9, -3, -3, 2, 35, 11, 63; -327, -245, -1122, -112, 83, 41, -41, -594, -174, -941; 640, 478, 2184, 217, -168, -80, 82, 1176, 343, 1848],22,84]
%
% END OF LATTICE L23.84
%
\hbox{}\par\smallskip%
%
% LATTICE L23.85
%
\leavevmode%
\llap{\spacer}%
${L_{23.85}}=\pdualwitharg{2.3.7}({L_{23.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,1}{\infty z}{12}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{1008}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3881808, -584136, -82656; -584136, -87864, -12432; -82656, -12432, -1759],[504, 168, 1],[-7, -6, -10, -7, -1, 1, 0, -2, -5, -17; 298, 265, 451, 326, 78, -44, -11, 57, 182, 684; -1776, -1590, -2716, -1974, -504, 264, 78, -308, -1050, -4032],22,85]
%
% END OF LATTICE L23.85
%
\hbox{}\par\smallskip%
%
% LATTICE L23.86
%
\leavevmode%
\llap{\spacer}%
${L_{23.86}}=\pdualwitharg{2.3.7}({L_{23.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1315944, -9828000, 182952; -9828000, -73398696, 1366344; 182952, 1366344, -25435],[504, 168, 1],[55, 42, 65, 20, -11, -7, 6, 31, 28, 155; -107, -77, -114, -32, 39, 13, -17, -74, -62, -321; -5352, -3834, -5656, -1575, 2016, 648, -870, -3752, -3129, -16128],22,86]
%
% END OF LATTICE L23.86
%
\hbox{}\par\smallskip%
%
% LATTICE L23.87
%
\leavevmode%
\llap{\spacer}%
${L_{23.87}}=\pdualwitharg{2.7}({L_{23.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,17}{\infty z}{12}\above{1pt}{1pt}{s}{2}{504}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7477848, -1104264, -3719016; -1104264, -162960, -549192; -3719016, -549192, -1849607],[504, 168, 1],[346, -27, -2131, -1650, -1782, -2100, -1727, -8393, -1858, 28; 1, 1, 3, -1, -3, -5, -5, -27, -7, -1; -696, 54, 4284, 3318, 3584, 4224, 3474, 16884, 3738, -56],22,87]
%
% END OF LATTICE L23.87
%
\hbox{}\par\smallskip%
%
% LATTICE L23.88
%
\leavevmode%
\llap{\spacer}%
${L_{23.88}}=\pdualwitharg{2.7}({L_{23.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,5}{\infty z}{12}\above{1pt}{1pt}{l}{2}{504}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-809928, -72576, -29736; -72576, -6216, -2520; -29736, -2520, -1019],[504, 168, 1],[-3, -1, 8, 5, 13, 17, 15, 76, 9, 1; 181, 58, -501, -305, -787, -1025, -902, -4563, -539, -57; -360, -114, 1008, 609, 1568, 2040, 1794, 9072, 1071, 112],22,88]
%
% END OF LATTICE L23.88
%

\medskip%
%------------------------------WEYL-GROUP-W24--------------------------------
%
\leavevmode\llap{}%
$W_{24\phantom{0}}$%
\qquad\llap{44} lattices, $\chi=9$%
\hfill%
$22224$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L24.1
%
\leavevmode%
\llap{\spacer}%
${L_{24.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{24}}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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2436\\-20\\-27
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2856\\-27\\-26
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-3\\-198\\-124
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-3\\-196\\-126
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4\\264\\165
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13\\854\\539
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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4\\0\\-6\\-14\\-2
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0\\28\\0\\-56\\-28
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-6\\0\\6\\0\\-6
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-14\\-56\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-28\\-6\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-278796, 2436, 2856; 2436, -20, -27; 2856, -27, -26],[84, 1, 1],[-3, -3, 4, 13, 3; -198, -196, 264, 854, 196; -124, -126, 165, 539, 125],23,1]
%
% END OF LATTICE L24.1
%
\hbox{}\par\smallskip%
%
% LATTICE L24.2
%
\leavevmode%
\llap{\spacer}%
${L_{24.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{19}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1129128\\-560616\\11088
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560616\\-278347\\5504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11088\\5504\\-107
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
74\\-151\\-99
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
151\\-308\\-196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-197\\402\\264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-645\\1316\\854
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\153\\98
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-6\\-14\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-112\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\12\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-112\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-28\\-6\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1129128, -560616, 11088; -560616, -278347, 5504; 11088, 5504, -107],[168, 1, 1],[74, 151, -197, -645, -75; -151, -308, 402, 1316, 153; -99, -196, 264, 854, 98],23,2]
%
% END OF LATTICE L24.2
%
\hbox{}\par\smallskip%
%
% LATTICE L24.3
%
\leavevmode%
\llap{\spacer}%
${L_{24.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-514248\\3024\\1848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3024\\-17\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1848\\-12\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\119\\83
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\1792\\1232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\120\\81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-119\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-120\\-82
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-3\\-7\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\3\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-56\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-56\\-6\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-514248, 3024, 1848; 3024, -17, -12; 1848, -12, -5],[168, 1, 1],[1, 15, 1, -1, -1; 119, 1792, 120, -119, -120; 83, 1232, 81, -84, -82],23,3]
%
% END OF LATTICE L24.3
%
\hbox{}\par\smallskip%
%
% LATTICE L24.4
%
\leavevmode%
\llap{\spacer}%
${L_{24.4}}=\pfillwitharg{2}({L_{24.1}})$%
{${}={}$}Nikulin $24$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-49\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-23\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-7\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-28\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-28\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\-6\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-84, -21, 0; -21, -5, 0; 0, 0, 1],[21, 1, 1],[0, -2, 1, 15, 7; 0, 7, -3, -49, -23; 1, 0, -3, -7, -1],23,4]
%
% END OF LATTICE L24.4
%
\hbox{}\par\smallskip%
%
% LATTICE L24.5
%
\leavevmode%
\llap{\spacer}%
${L_{24.5}}=\pfillwitharg{2}({L_{24.2}})$%
{${}={}$}Nikulin $19'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8778\\6888\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6888\\5405\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-99\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
571\\-742\\-756
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
187\\-243\\-249
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
167\\-217\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\13\\13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-3\\-7\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-28\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\3\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-28\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\-3\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[8778, 6888, -126; 6888, 5405, -99; -126, -99, 2],[42, 1, 1],[0, 571, 187, 167, -10; 0, -742, -243, -217, 13; 1, -756, -249, -224, 13],23,5]
%
% END OF LATTICE L24.5
%
\hbox{}\par\smallskip%
%
% LATTICE L24.6
%
\leavevmode%
\llap{\spacer}%
${L_{24.6}}=\main({L_{24.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-151284\\1260\\1932
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1260\\-10\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1932\\-17\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\66\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\980\\532
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\129\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-133\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-65\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-7\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-56\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-56\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-28\\-6\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-151284, 1260, 1932; 1260, -10, -17; 1932, -17, -23],[84, 1, 1],[1, 15, 2, -2, -1; 66, 980, 129, -133, -65; 35, 532, 72, -70, -36],23,6]
%
% END OF LATTICE L24.6
%
\hbox{}\par\smallskip%
%
% LATTICE L24.7
%
\leavevmode%
\llap{\spacer}%
${L_{24.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{24.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76272\\1617\\-24654
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1617\\-30\\522
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24654\\522\\-7969
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\24\\147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
328\\-203\\-1029
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-69\\-376
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
550\\-301\\-1722
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-23\\-150
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-21\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-84\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-84\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-42\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-76272, 1617, -24654; 1617, -30, 522; -24654, 522, -7969],[21, 3, 1],[-47, 328, 120, 550, 48; 24, -203, -69, -301, -23; 147, -1029, -376, -1722, -150],23,7]
%
% END OF LATTICE L24.7
%
\hbox{}\par\smallskip%
%
% LATTICE L24.8
%
\leavevmode%
\llap{\spacer}%
${L_{24.8}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{24.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-170310\\3738\\-84000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3738\\-80\\1842
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84000\\1842\\-41429
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\-99\\-122
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
810\\-1309\\-1701
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
443\\-708\\-930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
267\\-413\\-560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59\\98\\124
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-14\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-28\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-28\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-14\\-6\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-170310, 3738, -84000; 3738, -80, 1842; -84000, 1842, -41429],[42, 2, 1],[58, 810, 443, 267, -59; -99, -1309, -708, -413, 98; -122, -1701, -930, -560, 124],23,8]
%
% END OF LATTICE L24.8
%
\hbox{}\par\smallskip%
%
% LATTICE L24.9
%
\leavevmode%
\llap{\spacer}%
${L_{24.9}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{24.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{l}{2}{42}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208866\\5040\\70686
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\-120\\-1707
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70686\\-1707\\-23921
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\-99\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1215\\-2618\\-3402
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111\\-236\\-311
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-202\\-413\\-567
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\98\\123
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-3\\-21\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-84\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-84\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-42\\-3\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-208866, 5040, 70686; 5040, -120, -1707; 70686, -1707, -23921],[42, 3, 1],[-43, -1215, -111, -202, 44; -99, -2618, -236, -413, 98; -120, -3402, -311, -567, 123],23,9]
%
% END OF LATTICE L24.9
%
\hbox{}\par\smallskip%
%
% LATTICE L24.10
%
\leavevmode%
\llap{\spacer}%
${L_{24.10}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{24.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128667\\-4620\\-37212
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-161\\-1337
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37212\\-1337\\-10762
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62\\35\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\-34\\-248
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
526\\-264\\-1785
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\-58\\-373
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-36\\-203
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-21\\-7\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\42\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-4\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-14\\-42\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-128667, -4620, -37212; -4620, -161, -1337; -37212, -1337, -10762],[21, 7, 1],[-62, 73, 526, 110, 60; 35, -34, -264, -58, -36; 210, -248, -1785, -373, -203],23,10]
%
% END OF LATTICE L24.10
%
\hbox{}\par\smallskip%
%
% LATTICE L24.11
%
\leavevmode%
\llap{\spacer}%
${L_{24.11}}=\pdualwitharg{3}(\main({L_{24.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14028\\672\\2772
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
672\\-30\\-141
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2772\\-141\\-517
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\80\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127\\1260\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\59\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-161\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-89\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-21\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-168\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-168\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-84\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14028, 672, 2772; 672, -30, -141; 2772, -141, -517],[84, 3, 1],[8, 127, 6, -16, -9; 80, 1260, 59, -161, -89; 21, 336, 16, -42, -24],23,11]
%
% END OF LATTICE L24.11
%
\hbox{}\par\smallskip%
%
% LATTICE L24.12
%
\leavevmode%
\llap{\spacer}%
${L_{24.12}}=\pdualwitharg{3}({L_{24.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41412\\1680\\8820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\-60\\-387
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8820\\-387\\-1774
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\-430\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-448\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\190\\53
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
188\\1876\\525
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\438\\123
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-42\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-168\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-168\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-84\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-41412, 1680, 8820; 1680, -60, -387; 8820, -387, -1774],[84, 3, 1],[-43, -45, 19, 188, 44; -430, -448, 190, 1876, 438; -120, -126, 53, 525, 123],23,12]
%
% END OF LATTICE L24.12
%
\hbox{}\par\smallskip%
%
% LATTICE L24.13
%
\leavevmode%
\llap{\spacer}%
${L_{24.13}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{24.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
190092\\-78498\\79086
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78498\\32430\\-32658
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79086\\-32658\\32903
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7029\\-371\\16527
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1537\\-81\\3614
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4126\\-217\\9702
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
245\\13\\-576
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-42\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-84\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-84\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-42\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[190092, -78498, 79086; -78498, 32430, -32658; 79086, -32658, 32903],[42, 6, 1],[5, -7029, -1537, -4126, 245; 0, -371, -81, -217, 13; -12, 16527, 3614, 9702, -576],23,13]
%
% END OF LATTICE L24.13
%
\hbox{}\par\smallskip%
%
% LATTICE L24.14
%
\leavevmode%
\llap{\spacer}%
${L_{24.14}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{24.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-416094\\-7308\\-60984
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7308\\-91\\-1071
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60984\\-1071\\-8938
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\2\\-819
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
455\\0\\-3104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
862\\-3\\-5880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-1\\-491
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119\\-1\\812
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-21\\-7\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-4\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\21\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-4\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-14\\-21\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-416094, -7308, -60984; -7308, -91, -1071; -60984, -1071, -8938],[42, 7, 1],[120, 455, 862, 72, -119; 2, 0, -3, -1, -1; -819, -3104, -5880, -491, 812],23,14]
%
% END OF LATTICE L24.14
%
\hbox{}\par\smallskip%
%
% LATTICE L24.15
%
\leavevmode%
\llap{\spacer}%
${L_{24.15}}=\pdualwitharg{2}({L_{24.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6138384\\-515172\\1453704
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-515172\\43240\\-122004
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1453704\\-122004\\344269
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\7\\-462
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
194\\9\\-816
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1044\\35\\-4396
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
434\\13\\-1828
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-28\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-112\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-112\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-56\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6138384, -515172, 1453704; -515172, 43240, -122004; 1453704, -122004, 344269],[84, 4, 1],[9, 110, 194, 1044, 434; 0, 7, 9, 35, 13; -38, -462, -816, -4396, -1828],23,15]
%
% END OF LATTICE L24.15
%
\hbox{}\par\smallskip%
%
% LATTICE L24.16
%
\leavevmode%
\llap{\spacer}%
${L_{24.16}}=\pdualwitharg{2}(\main({L_{24.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1236984\\-122052\\292656
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-122052\\12052\\-28876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
292656\\-28876\\69239
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115\\5\\-484
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
374\\14\\-1575
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
205\\6\\-864
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\0\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\2\\-164
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-28\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-56\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-56\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-28\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1236984, -122052, 292656; -122052, 12052, -28876; 292656, -28876, 69239],[84, 4, 1],[115, 374, 205, 53, 39; 5, 14, 6, 0, 2; -484, -1575, -864, -224, -164],23,16]
%
% END OF LATTICE L24.16
%
\hbox{}\par\smallskip%
%
% LATTICE L24.17
%
\leavevmode%
\llap{\spacer}%
${L_{24.17}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{24.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18690\\-204351\\-62958
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204351\\2237844\\689451
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62958\\689451\\212411
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-653\\2121
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\4052\\-13161
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\6862\\-22288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\3072\\-9978
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-194\\630
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-21\\-21\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-12\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\14\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-12\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-42\\-42\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18690, -204351, -62958; -204351, 2237844, 689451; -62958, 689451, 212411],[21, 21, 1],[5, -30, -51, -23, 1; -653, 4052, 6862, 3072, -194; 2121, -13161, -22288, -9978, 630],23,17]
%
% END OF LATTICE L24.17
%
\hbox{}\par\smallskip%
%
% LATTICE L24.18
%
\leavevmode%
\llap{\spacer}%
${L_{24.18}}=\pdualwitharg{3}({L_{24.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216888\\-105504\\16464
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105504\\-51321\\8001
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16464\\8001\\-1178
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
197\\-412\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
375\\-784\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-175\\366\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1667\\3486\\378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\391\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-6\\-42\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-336\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-336\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-84\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-216888, -105504, 16464; -105504, -51321, 8001; 16464, 8001, -1178],[168, 3, 1],[197, 375, -175, -1667, -187; -412, -784, 366, 3486, 391; -45, -84, 40, 378, 42],23,18]
%
% END OF LATTICE L24.18
%
\hbox{}\par\smallskip%
%
% LATTICE L24.19
%
\leavevmode%
\llap{\spacer}%
${L_{24.19}}=\pdualwitharg{3}({L_{24.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-338520\\4200\\-39144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4200\\-51\\504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39144\\504\\-4223
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-1341\\81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-377\\-19432\\1176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-412\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\1393\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\1288\\-78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-3\\-21\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-168\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-168\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-168\\-6\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-338520, 4200, -39144; 4200, -51, 504; -39144, 504, -4223],[168, 3, 1],[-26, -377, -8, 27, 25; -1341, -19432, -412, 1393, 1288; 81, 1176, 25, -84, -78],23,19]
%
% END OF LATTICE L24.19
%
\hbox{}\par\smallskip%
%
% LATTICE L24.20
%
\leavevmode%
\llap{\spacer}%
${L_{24.20}}=\pdualwitharg{7}(\main({L_{24.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1596\\336\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-70\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-35\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\9\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-21\\-7\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\42\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-8\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-28\\-42\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1596, 336, 168; 336, -70, -35; 168, -35, -17],[84, 7, 1],[0, 1, 2, 0, -1; 3, 8, 9, -1, -5; -7, -8, 0, 2, 0],23,20]
%
% END OF LATTICE L24.20
%
\hbox{}\par\smallskip%
%
% LATTICE L24.21
%
\leavevmode%
\llap{\spacer}%
${L_{24.21}}=\pdualwitharg{7}({L_{24.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-421428\\8904\\8904
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8904\\-182\\-189
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8904\\-189\\-188
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-16\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-18\\-124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\21\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\41\\290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\125\\868
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-42\\-14\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\42\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-8\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-28\\-42\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-421428, 8904, 8904; 8904, -182, -189; 8904, -189, -188],[84, 7, 1],[-3, -3, 4, 7, 21; -16, -18, 21, 41, 125; -126, -124, 168, 290, 868],23,21]
%
% END OF LATTICE L24.21
%
\hbox{}\par\smallskip%
%
% LATTICE L24.22
%
\leavevmode%
\llap{\spacer}%
${L_{24.22}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{24.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77959014\\44478\\37544010
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44478\\28\\21420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37544010\\21420\\18080689
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8854\\-54\\-18385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40595\\-249\\-84294
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5179\\-32\\-10754
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2171\\13\\4508
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-42\\-14\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\42\\0\\-42
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-4\\0\\2\\0
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-14\\-14\\-42\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[77959014, 44478, 37544010; 44478, 28, 21420; 37544010, 21420, 18080689],[42, 14, 1],[0, 8854, 40595, 5179, -2171; 1, -54, -249, -32, 13; 0, -18385, -84294, -10754, 4508],23,22]
%
% END OF LATTICE L24.22
%
\hbox{}\par\smallskip%
%
% LATTICE L24.23
%
\leavevmode%
\llap{\spacer}%
${L_{24.23}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{24.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64220898\\-49434\\22503726
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49434\\42\\-17325
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22503726\\-17325\\7885561
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14323\\-28754\\-40938
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10945\\-21973\\-31283
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4189\\-8410\\-11973
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1756\\3525\\5019
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-21\\-21\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-12\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\7\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-12\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-42\\-21\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[64220898, -49434, 22503726; -49434, 42, -17325; 22503726, -17325, 7885561],[42, 21, 1],[0, 14323, 10945, 4189, -1756; 1, -28754, -21973, -8410, 3525; 0, -40938, -31283, -11973, 5019],23,23]
%
% END OF LATTICE L24.23
%
\hbox{}\par\smallskip%
%
% LATTICE L24.24
%
\leavevmode%
\llap{\spacer}%
${L_{24.24}}=\pdualwitharg{2.3}({L_{24.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
442848\\523236\\122892
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
523236\\618216\\145200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
122892\\145200\\34103
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\7\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\3\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\35\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\13\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-84\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-336\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-336\\0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-168\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[442848, 523236, 122892; 523236, 618216, 145200; 122892, 145200, 34103],[84, 12, 1],[-5, 50, 12, -18, -32; 0, 7, 3, 35, 13; 18, -210, -56, -84, 60],23,24]
%
% END OF LATTICE L24.24
%
\hbox{}\par\smallskip%
%
% LATTICE L24.25
%
\leavevmode%
\llap{\spacer}%
${L_{24.25}}=\pdualwitharg{2.3}(\main({L_{24.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55608\\58548\\15204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58548\\61644\\16008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15204\\16008\\4157
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\5\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\14\\-189
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\0\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\2\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-84\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-168\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-168\\0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-84\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[55608, 58548, 15204; 58548, 61644, 16008; 15204, 16008, 4157],[84, 12, 1],[21, 37, -1, -23, 11; 5, 14, 2, 0, 2; -96, -189, -4, 84, -48],23,25]
%
% END OF LATTICE L24.25
%
\hbox{}\par\smallskip%
%
% LATTICE L24.26
%
\leavevmode%
\llap{\spacer}%
${L_{24.26}}=\pdualwitharg{7}({L_{24.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-182616\\33600\\4032
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33600\\-6181\\-742
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4032\\-742\\-89
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\12\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\10\\52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\12\\35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-42\\-14\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-16\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\84\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-16\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-28\\-42\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-182616, 33600, 4032; 33600, -6181, -742; 4032, -742, -89],[168, 7, 1],[-2, -1, 5, 3, 3; -5, -4, 12, 10, 12; -49, -12, 126, 52, 35],23,26]
%
% END OF LATTICE L24.26
%
\hbox{}\par\smallskip%
%
% LATTICE L24.27
%
\leavevmode%
\llap{\spacer}%
${L_{24.27}}=\pdualwitharg{7}({L_{24.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4998840\\14448\\28896
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14448\\-35\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28896\\-84\\-167
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\83\\1169
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\104\\1504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-167
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-21\\-7\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-8\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\21\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-8\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-56\\-42\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4998840, 14448, 28896; 14448, -35, -84; 28896, -84, -167],[168, 7, 1],[7, 9, 1, -1, -1; 83, 104, 9, -12, -10; 1169, 1504, 168, -167, -168],23,27]
%
% END OF LATTICE L24.27
%
\hbox{}\par\smallskip%
%
% LATTICE L24.28
%
\leavevmode%
\llap{\spacer}%
${L_{24.28}}=\pdualwitharg{2}({L_{24.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54768\\-6888\\2352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6888\\-856\\296
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2352\\296\\-101
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-7\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-1\\136
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-24\\-56\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-112\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-112\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-56\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-54768, -6888, 2352; -6888, -856, 296; 2352, 296, -101],[168, 8, 1],[-3, -3, 4, 19, 6; 2, 0, -3, -7, -1; -64, -70, 84, 420, 136],23,28]
%
% END OF LATTICE L24.28
%
\hbox{}\par\smallskip%
%
% LATTICE L24.29
%
\leavevmode%
\llap{\spacer}%
${L_{24.29}}=\pdualwitharg{2}({L_{24.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3889200\\20664\\40656
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20664\\-104\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40656\\-216\\-425
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\-288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-287
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\384
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-7\\2968
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-1\\1148
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-24\\-56\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-56\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-56\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-28\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3889200, 20664, 40656; 20664, -104, -216; 40656, -216, -425],[168, 8, 1],[-3, -3, 4, 31, 12; 2, 0, -3, -7, -1; -288, -287, 384, 2968, 1148],23,29]
%
% END OF LATTICE L24.29
%
\hbox{}\par\smallskip%
%
% LATTICE L24.30
%
\leavevmode%
\llap{\spacer}%
${L_{24.30}}=\pdualwitharg{3.7}(\main({L_{24.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145572\\107940\\4872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107940\\-80031\\-3612
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4872\\-3612\\-163
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\17\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\24\\-204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\51
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-21\\-21\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\14\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-24\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-84\\-42\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-145572, 107940, 4872; 107940, -80031, -3612; 4872, -3612, -163],[84, 21, 1],[7, 11, 2, -2, -3; 17, 24, 3, -5, -5; -168, -204, -7, 51, 21],23,30]
%
% END OF LATTICE L24.30
%
\hbox{}\par\smallskip%
%
% LATTICE L24.31
%
\leavevmode%
\llap{\spacer}%
${L_{24.31}}=\pdualwitharg{3.7}({L_{24.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132636\\8652\\41328
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8652\\-546\\-2709
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41328\\-2709\\-12868
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-88\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\-86\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\39\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\201\\282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
301\\601\\840
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-42\\-42\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\14\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-24\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-84\\-42\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-132636, 8652, 41328; 8652, -546, -2709; 41328, -2709, -12868],[84, 21, 1],[-45, -43, 20, 101, 301; -88, -86, 39, 201, 601; -126, -120, 56, 282, 840],23,31]
%
% END OF LATTICE L24.31
%
\hbox{}\par\smallskip%
%
% LATTICE L24.32
%
\leavevmode%
\llap{\spacer}%
${L_{24.32}}=\pdualwitharg{2.3.7}(\pfillwitharg{2}({L_{24.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-124770702\\302327718\\146928936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
302327718\\-732558876\\-356017956
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
146928936\\-356017956\\-173021977
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\40865\\-84084
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\77525\\-159519
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\97928\\-201502
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\24549\\-50514
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-40538\\83412
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-42\\-42\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-12\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\14\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-12\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-42\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-124770702, 302327718, 146928936; 302327718, -732558876, -356017956; 146928936, -356017956, -173021977],[42, 42, 1],[2, 0, -1, -1, -1; 40865, 77525, 97928, 24549, -40538; -84084, -159519, -201502, -50514, 83412],23,32]
%
% END OF LATTICE L24.32
%
\hbox{}\par\smallskip%
%
% LATTICE L24.33
%
\leavevmode%
\llap{\spacer}%
${L_{24.33}}=\pdualwitharg{2.7}({L_{24.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8476944\\385812\\-1733424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
385812\\17584\\-78876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1733424\\-78876\\354475
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\31\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\27\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-617\\1857\\-2604
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-601\\1811\\-2536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1805\\5441\\-7616
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-84\\-28\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-16\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\168\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-16\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-56\\-168\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8476944, 385812, -1733424; 385812, 17584, -78876; -1733424, -78876, 354475],[84, 28, 1],[-10, -9, -617, -601, -1805; 31, 27, 1857, 1811, 5441; -42, -38, -2604, -2536, -7616],23,33]
%
% END OF LATTICE L24.33
%
\hbox{}\par\smallskip%
%
% LATTICE L24.34
%
\leavevmode%
\llap{\spacer}%
${L_{24.34}}=\pdualwitharg{2.7}(\main({L_{24.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
924\\-168\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\364\\308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\308\\305
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\83\\-112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\29\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\63\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\41\\-56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-84\\-28\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-8\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\168\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-8\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-28\\-168\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[924, -168, -336; -168, 364, 308; -336, 308, 305],[84, 28, 1],[-26, -9, -19, 1, -13; 83, 29, 63, -3, 41; -112, -39, -84, 4, -56],23,34]
%
% END OF LATTICE L24.34
%
\hbox{}\par\smallskip%
%
% LATTICE L24.35
%
\leavevmode%
\llap{\spacer}%
${L_{24.35}}=\pdualwitharg{3.7}({L_{24.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-488712\\-96096\\14112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96096\\-18879\\2772
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14112\\2772\\-407
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-39\\-231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\28\\156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\38\\224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-20\\-102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-83\\-462
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-42\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-48\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\28\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-48\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-84\\-42\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-488712, -96096, 14112; -96096, -18879, 2772; 14112, 2772, -407],[168, 21, 1],[1, -1, -1, 1, 3; -39, 28, 38, -20, -83; -231, 156, 224, -102, -462],23,35]
%
% END OF LATTICE L24.35
%
\hbox{}\par\smallskip%
%
% LATTICE L24.36
%
\leavevmode%
\llap{\spacer}%
${L_{24.36}}=\pdualwitharg{3.7}({L_{24.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1627752\\14280\\533064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14280\\-105\\-4704
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
533064\\-4704\\-174533
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
392\\1567\\1155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
505\\2016\\1488
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\75\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-224\\-165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-226\\-168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-21\\-21\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-24\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\7\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-24\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-168\\-42\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1627752, 14280, 533064; 14280, -105, -4704; 533064, -4704, -174533],[168, 21, 1],[392, 505, 19, -56, -57; 1567, 2016, 75, -224, -226; 1155, 1488, 56, -165, -168],23,36]
%
% END OF LATTICE L24.36
%
\hbox{}\par\smallskip%
%
% LATTICE L24.37
%
\leavevmode%
\llap{\spacer}%
${L_{24.37}}=\pdualwitharg{2.3}({L_{24.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45248784\\-342888\\-791784
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-342888\\-2568\\-6000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-791784\\-6000\\-13855
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\2\\-3144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61\\0\\-3486
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-1\\1372
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-363\\-7\\20748
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-118\\-1\\6744
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-24\\-168\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-336\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-336\\0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-168\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-45248784, -342888, -791784; -342888, -2568, -6000; -791784, -6000, -13855],[168, 24, 1],[55, 61, -24, -363, -118; 2, 0, -1, -7, -1; -3144, -3486, 1372, 20748, 6744],23,37]
%
% END OF LATTICE L24.37
%
\hbox{}\par\smallskip%
%
% LATTICE L24.38
%
\leavevmode%
\llap{\spacer}%
${L_{24.38}}=\pdualwitharg{2.3}({L_{24.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2005584\\25704\\-651672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25704\\-312\\8352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-651672\\8352\\-211747
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
117\\2\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
116\\0\\-357
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-1\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1201\\-7\\3696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-464\\-1\\1428
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-24\\-168\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-168\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-84\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2005584, 25704, -651672; 25704, -312, 8352; -651672, 8352, -211747],[168, 24, 1],[117, 116, -52, -1201, -464; 2, 0, -1, -7, -1; -360, -357, 160, 3696, 1428],23,38]
%
% END OF LATTICE L24.38
%
\hbox{}\par\smallskip%
%
% LATTICE L24.39
%
\leavevmode%
\llap{\spacer}%
${L_{24.39}}=\pdualwitharg{2.3.7}({L_{24.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1243536\\753396\\209076
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
753396\\456456\\126672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
209076\\126672\\35153
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-81\\280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-203\\684
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-579\\1932
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-84\\-84\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-48\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\56\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-48\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-168\\-168\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1243536, 753396, 209076; 753396, 456456, 126672; 209076, 126672, 35153],[84, 84, 1],[1, 0, 2, 8, 26; 10, -5, -81, -203, -579; -42, 18, 280, 684, 1932],23,39]
%
% END OF LATTICE L24.39
%
\hbox{}\par\smallskip%
%
% LATTICE L24.40
%
\leavevmode%
\llap{\spacer}%
${L_{24.40}}=\pdualwitharg{2.3.7}(\main({L_{24.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8148\\91392\\21924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91392\\1048740\\251496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21924\\251496\\60311
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89\\516\\-2184
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\236\\-999
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\291\\-1232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\17\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\139\\-588
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-84\\-84\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-24\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\56\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\-168\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8148, 91392, 21924; 91392, 1048740, 251496; 21924, 251496, 60311],[84, 84, 1],[89, 41, 51, 3, 23; 516, 236, 291, 17, 139; -2184, -999, -1232, -72, -588],23,40]
%
% END OF LATTICE L24.40
%
\hbox{}\par\smallskip%
%
% LATTICE L24.41
%
\leavevmode%
\llap{\spacer}%
${L_{24.41}}=\pdualwitharg{2.7}({L_{24.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\13608\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13608\\-550312\\20384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\20384\\-755
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-1\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\-1\\-56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\0\\-168\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-16\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\168\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-16\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-56\\-168\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-336, 13608, -504; 13608, -550312, 20384; -504, 20384, -755],[168, 56, 1],[-3, -3, 4, 13, 42; 2, 0, -3, -1, -1; 56, 2, -84, -36, -56],23,41]
%
% END OF LATTICE L24.41
%
\hbox{}\par\smallskip%
%
% LATTICE L24.42
%
\leavevmode%
\llap{\spacer}%
${L_{24.42}}=\pdualwitharg{2.7}({L_{24.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84982800\\253512\\506856
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
253512\\-728\\-1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
506856\\-1512\\-3023
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-503
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-1\\4192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-1\\14084
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\0\\-168\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-8\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\168\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-8\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-28\\-168\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-84982800, 253512, 506856; 253512, -728, -1512; 506856, -1512, -3023],[168, 56, 1],[-3, -3, 4, 25, 84; 2, 0, -3, -1, -1; -504, -503, 672, 4192, 14084],23,42]
%
% END OF LATTICE L24.42
%
\hbox{}\par\smallskip%
%
% LATTICE L24.43
%
\leavevmode%
\llap{\spacer}%
${L_{24.43}}=\pdualwitharg{2.3.7}({L_{24.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1650936\\4200168\\74760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4200168\\-10684464\\-190176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
74760\\-190176\\-3385
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-65\\3696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\618
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\32\\-1820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\75\\-4236
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\182\\-10248
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\0\\-168\\-168\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-48\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\56\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-48\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-168\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1650936, 4200168, 74760; 4200168, -10684464, -190176; 74760, -190176, -3385],[168, 168, 1],[2, 0, -1, -1, -1; -65, -11, 32, 75, 182; 3696, 618, -1820, -4236, -10248],23,43]
%
% END OF LATTICE L24.43
%
\hbox{}\par\smallskip%
%
% LATTICE L24.44
%
\leavevmode%
\llap{\spacer}%
${L_{24.44}}=\pdualwitharg{2.3.7}({L_{24.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2184\\252504\\-85680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252504\\-28102704\\9535848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85680\\9535848\\-3235717
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-171\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-170\\-501
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\76\\224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1417\\4176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4760\\14028
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\0\\-168\\-168\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-24\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\56\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-84\\-168\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2184, 252504, -85680; 252504, -28102704, 9535848; -85680, 9535848, -3235717],[168, 168, 1],[2, 0, -1, -1, -1; -171, -170, 76, 1417, 4760; -504, -501, 224, 4176, 14028],23,44]
%
% END OF LATTICE L24.44
%

\medskip%
%------------------------------WEYL-GROUP-W25--------------------------------
%
\leavevmode\llap{}%
$W_{25\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=6$%
\hfill%
$22222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L25.1
%
\leavevmode%
\llap{\spacer}%
${L_{25.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{23\rightarrow N_{25},3,2}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{252}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1925028\\-70560\\2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70560\\2586\\-93
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2520\\-93\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-134\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\-1260\\630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\456\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\161\\-70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\252\\0\\-252\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-252\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-126\\-7\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[1925028, -70560, 2520; -70560, 2586, -93; 2520, -93, 2],[252, 3, 1],[-5, -47, 0, 17, 6; -134, -1260, 0, 456, 161; 72, 630, -1, -216, -70],24,1]
%
% END OF LATTICE L25.1
%
\hbox{}\par\smallskip%
%
% LATTICE L25.2
%
\leavevmode%
\llap{\spacer}%
${L_{25.2}}=\pfillwitharg{2.3}({L_{25.1}})$%
{${}={}$}Nikulin $25$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210\\63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\19\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-10\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-21\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-7\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-7\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-7\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[210, 63, 0; 63, 19, 0; 0, 0, -1],[21, 1, 1],[-1, -2, 1, 3, 6; 3, 7, -3, -10, -21; 0, 0, -1, -3, -7],24,2]
%
% END OF LATTICE L25.2
%
\hbox{}\par\smallskip%
%
% LATTICE L25.3
%
\leavevmode%
\llap{\spacer}%
${L_{25.3}}=\pfillwitharg{3}({L_{25.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1092\\252\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252\\-2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\18\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\56\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-21\\-70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-28\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-28\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-7\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[1092, 252, -84; 252, -2, -3; -84, -3, 2],[84, 1, 1],[1, 3, 0, -1, -1; 18, 56, 0, -20, -21; 72, 210, -1, -72, -70],24,3]
%
% END OF LATTICE L25.3
%
\hbox{}\par\smallskip%
%
% LATTICE L25.4
%
\leavevmode%
\llap{\spacer}%
${L_{25.4}}=\pfillwitharg{2}({L_{25.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{63}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\-15\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\63\\0\\-63\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-63\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-63\\-7\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[63, 63, 0; 63, -15, -9; 0, -9, -1],[63, 3, 1],[-1, 1, 1, 2, -1; 1, 0, -1, -3, 0; -9, 0, 8, 18, -7],24,4]
%
% END OF LATTICE L25.4
%
\hbox{}\par\smallskip%
%
% LATTICE L25.5
%
\leavevmode%
\llap{\spacer}%
${L_{25.5}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2898\\273\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
273\\-3\\81
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\81\\-293
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-9\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-14\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\20\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\20\\75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-7\\-21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-21\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-21\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-63\\-21\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2898, 273, -924; 273, -3, 81; -924, 81, -293],[21, 3, 1],[10, 19, -21, -22, 6; -9, -14, 20, 20, -7; -34, -63, 72, 75, -21],24,5]
%
% END OF LATTICE L25.5
%
\hbox{}\par\smallskip%
%
% LATTICE L25.6
%
\leavevmode%
\llap{\spacer}%
${L_{25.6}}=\pdualwitharg{7}(\pfillwitharg{2.3}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2562\\-651\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-651\\21\\-196
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-756\\-196\\-223
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82\\-6\\-273
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-1\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\3\\203
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\4\\203
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-21\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-7\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\14\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-7\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-7\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2562, -651, -756; -651, 21, -196; -756, -196, -223],[21, 7, 1],[82, 6, -61, -61, 3; -6, -1, 3, 4, 0; -273, -20, 203, 203, -10],24,6]
%
% END OF LATTICE L25.6
%
\hbox{}\par\smallskip%
%
% LATTICE L25.7
%
\leavevmode%
\llap{\spacer}%
${L_{25.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{126}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\63\\252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\-6\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252\\-21\\-41
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\41\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\84\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\21\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-9\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-7\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\18\\0\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-7\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-63\\-63\\0\\126
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-630, 63, 252; 63, -6, -21; 252, -21, -41],[63, 3, 1],[3, 6, -1, -1, 2; 41, 84, -12, -14, 21; -3, -7, 0, 1, 0],24,7]
%
% END OF LATTICE L25.7
%
\hbox{}\par\smallskip%
%
% LATTICE L25.8
%
\leavevmode%
\llap{\spacer}%
${L_{25.8}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10836\\420\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\-6\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-9\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-28\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\28\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\7\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-84\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-84\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-126\\-21\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10836, 420, 168; 420, -6, -9; 168, -9, -2],[84, 3, 1],[-1, -3, 1, 3, 1; -10, -28, 10, 28, 7; -40, -126, 39, 120, 42],24,8]
%
% END OF LATTICE L25.8
%
\hbox{}\par\smallskip%
%
% LATTICE L25.9
%
\leavevmode%
\llap{\spacer}%
${L_{25.9}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1228584\\-16884\\291648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16884\\232\\-4008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
291648\\-4008\\69233
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\0\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\7\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\3\\76
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\1\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
113\\0\\-476
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-14\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-14\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-28\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1228584, -16884, 291648; -16884, 232, -4008; 291648, -4008, 69233],[84, 4, 1],[47, 30, -18, 5, 113; 0, 7, 3, 1, 0; -198, -126, 76, -21, -476],24,9]
%
% END OF LATTICE L25.9
%
\hbox{}\par\smallskip%
%
% LATTICE L25.10
%
\leavevmode%
\llap{\spacer}%
${L_{25.10}}=\pdualwitharg{3.7}(\pfillwitharg{2.3}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210\\-19005\\-5859
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19005\\1726326\\532203
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5859\\532203\\164071
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-41\\133
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-234\\759
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-505\\1638
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-123\\399
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\74\\-240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-21\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-21\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\42\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-21\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-21\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[210, -19005, -5859; -19005, 1726326, 532203; -5859, 532203, 164071],[21, 21, 1],[0, -1, -2, 1, 1; -41, -234, -505, -123, 74; 133, 759, 1638, 399, -240],24,10]
%
% END OF LATTICE L25.10
%
\hbox{}\par\smallskip%
%
% LATTICE L25.11
%
\leavevmode%
\llap{\spacer}%
${L_{25.11}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\84\\84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-14\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-7\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\20\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\20\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-42\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-28\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\14\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-28\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-7\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-420, 84, 84; 84, -14, -7; 84, -7, 18],[84, 7, 1],[1, 3, 3, -1, -1; 6, 20, 20, -8, -7; 0, -6, -7, 0, 2],24,11]
%
% END OF LATTICE L25.11
%
\hbox{}\par\smallskip%
%
% LATTICE L25.12
%
\leavevmode%
\llap{\spacer}%
${L_{25.12}}=\pdualwitharg{3}({L_{25.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{126}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23940\\1008\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\30\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-756\\-27\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-26\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-28\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\24\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\24\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\21\\63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-18\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-28\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\18\\0\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-28\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-126\\-63\\0\\126
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[23940, 1008, -756; 1008, 30, -27; -756, -27, 22],[252, 3, 1],[-1, -1, 1, 1, 1; -26, -28, 24, 24, 21; -66, -70, 63, 64, 63],24,12]
%
% END OF LATTICE L25.12
%
\hbox{}\par\smallskip%
%
% LATTICE L25.13
%
\leavevmode%
\llap{\spacer}%
${L_{25.13}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
632184\\-24276\\149688
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24276\\936\\-5748
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
149688\\-5748\\35443
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\7\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\3\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\1\\-135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
179\\0\\-756
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-42\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-42\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-252\\-84\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[632184, -24276, 149688; -24276, 936, -5748; 149688, -5748, 35443],[84, 12, 1],[9, 50, 20, 32, 179; 0, 7, 3, 1, 0; -38, -210, -84, -135, -756],24,13]
%
% END OF LATTICE L25.13
%
\hbox{}\par\smallskip%
%
% LATTICE L25.14
%
\leavevmode%
\llap{\spacer}%
${L_{25.14}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{126}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
126\\315\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
315\\777\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-315\\127
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-2\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-63\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-7\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\0\\126\\0\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-7\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-63\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[126, 315, -126; 315, 777, -315; -126, -315, 127],[63, 21, 1],[3, 1, -1, -2, -1; -1, 0, 0, -2, -3; 0, 1, 0, -7, -9],24,14]
%
% END OF LATTICE L25.14
%
\hbox{}\par\smallskip%
%
% LATTICE L25.15
%
\leavevmode%
\llap{\spacer}%
${L_{25.15}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{63}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1071\\2079\\315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2079\\-3633\\-546
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
315\\-546\\-82
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\21\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\7\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-21\\-63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-63\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\14\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-63\\0\\63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-7\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1071, 2079, 315; 2079, -3633, -546; 315, -546, -82],[63, 21, 1],[1, -1, -1, 4, 1; 10, -6, -9, 21, 7; -63, 36, 56, -126, -43],24,15]
%
% END OF LATTICE L25.15
%
\hbox{}\par\smallskip%
%
% LATTICE L25.16
%
\leavevmode%
\llap{\spacer}%
${L_{25.16}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3276\\420\\1176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\42\\-147
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1176\\-147\\-422
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-2\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89\\8\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\2\\-51
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-42\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-84\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\42\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-21\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3276, 420, 1176; 420, 42, -147; 1176, -147, -422],[84, 21, 1],[5, 17, 0, -89, -18; 0, -2, -1, 8, 2; 14, 48, 0, -252, -51],24,16]
%
% END OF LATTICE L25.16
%
\hbox{}\par\smallskip%
%
% LATTICE L25.17
%
\leavevmode%
\llap{\spacer}%
${L_{25.17}}=\pdualwitharg{2.7}(\pfillwitharg{3}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\2100\\1344
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2100\\4872\\3080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1344\\3080\\1943
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-93\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\25\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\165\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\67\\-91
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-84\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-14\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\56\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-14\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-28\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[840, 2100, 1344; 2100, 4872, 3080; 1344, 3080, 1943],[84, 28, 1],[31, -8, -54, -22, 1; -93, 25, 165, 67, -3; 126, -34, -224, -91, 4],24,17]
%
% END OF LATTICE L25.17
%
\hbox{}\par\smallskip%
%
% LATTICE L25.18
%
\leavevmode%
\llap{\spacer}%
${L_{25.18}}=\pdualwitharg{2.3}({L_{25.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{504}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5516280\\-108612\\-1402380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108612\\2136\\27612
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1402380\\27612\\356521
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-1\\-114
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82\\7\\322
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\9\\432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\-59
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-577\\-21\\-2268
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-36\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-14\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\72\\0\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-14\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-252\\-252\\0\\504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5516280, -108612, -1402380; -108612, 2136, 27612; -1402380, 27612, 356521],[252, 12, 1],[-29, 82, 110, -15, -577; -1, 7, 9, 0, -21; -114, 322, 432, -59, -2268],24,18]
%
% END OF LATTICE L25.18
%
\hbox{}\par\smallskip%
%
% LATTICE L25.19
%
\leavevmode%
\llap{\spacer}%
${L_{25.19}}=\pdualwitharg{2}({L_{25.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{252}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21203784\\1589364\\-5365332
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1589364\\119064\\-402168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5365332\\-402168\\1357625
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1443\\-70\\5682
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2144\\-105\\8442
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-642\\31\\-2528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
368\\-18\\1449
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3463\\-168\\13636
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\252\\0\\-126\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-126\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-252\\-28\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[21203784, 1589364, -5365332; 1589364, 119064, -402168; -5365332, -402168, 1357625],[252, 12, 1],[1443, 2144, -642, 368, 3463; -70, -105, 31, -18, -168; 5682, 8442, -2528, 1449, 13636],24,19]
%
% END OF LATTICE L25.19
%
\hbox{}\par\smallskip%
%
% LATTICE L25.20
%
\leavevmode%
\llap{\spacer}%
${L_{25.20}}=\pdualwitharg{7}({L_{25.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{252}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252\\0\\252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252\\-105\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\42\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\15\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-12\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-42\\-252\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-252\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\14\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-252\\0\\252\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-7\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[252, 0, 252; 0, -42, -105; 252, -105, -10],[252, 21, 1],[1, 17, 6, -5, -2; 2, 42, 15, -12, -5; 0, -18, -7, 0, 2],24,20]
%
% END OF LATTICE L25.20
%
\hbox{}\par\smallskip%
%
% LATTICE L25.21
%
\leavevmode%
\llap{\spacer}%
${L_{25.21}}=\pdualwitharg{3.7}({L_{25.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{126}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1048068\\8064\\8064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8064\\-42\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8064\\-63\\-62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\40\\868
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\24\\495
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-126\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-28\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\0\\126\\0\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-28\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-63\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1048068, 8064, 8064; 8064, -42, -63; 8064, -63, -62],[252, 21, 1],[-1, -1, 1, 7, 4; -4, -6, 3, 40, 24; -126, -124, 126, 868, 495],24,21]
%
% END OF LATTICE L25.21
%
\hbox{}\par\smallskip%
%
% LATTICE L25.22
%
\leavevmode%
\llap{\spacer}%
${L_{25.22}}=\pdualwitharg{2.3.7}(\pfillwitharg{3}({L_{25.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\-5124\\-1428
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5124\\54264\\15120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1428\\15120\\4213
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-47\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-41\\147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-20\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-84\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-42\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\168\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-84\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[504, -5124, -1428; -5124, 54264, 15120; -1428, 15120, 4213],[84, 84, 1],[1, 0, -2, 0, 1; 4, -5, -47, -41, -20; -14, 18, 168, 147, 72],24,22]
%
% END OF LATTICE L25.22
%
\hbox{}\par\smallskip%
%
% LATTICE L25.23
%
\leavevmode%
\llap{\spacer}%
${L_{25.23}}=\pdualwitharg{2.3.7}({L_{25.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{504}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64980216\\1015812\\15737148
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1015812\\15960\\245952
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15737148\\245952\\3811327
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-95\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-93\\-122
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1921\\-5763\\-7560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
884\\-2651\\-3479
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1738\\-5211\\-6840
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-252\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-14\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\0\\504\\0\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-14\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-252\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[64980216, 1015812, 15737148; 1015812, 15960, 245952; 15737148, 245952, 3811327],[252, 84, 1],[32, 31, 1921, 884, 1738; -95, -93, -5763, -2651, -5211; -126, -122, -7560, -3479, -6840],24,23]
%
% END OF LATTICE L25.23
%
\hbox{}\par\smallskip%
%
% LATTICE L25.24
%
\leavevmode%
\llap{\spacer}%
${L_{25.24}}=\pdualwitharg{2.7}({L_{25.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{63}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
199080\\-66276\\99792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66276\\22008\\-33180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99792\\-33180\\49991
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\157\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-12\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57\\-166\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-234\\-315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-84\\-126\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-126\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\56\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-126\\0\\63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-28\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[199080, -66276, 99792; -66276, 22008, -33180; 99792, -33180, 49991],[252, 84, 1],[-53, 5, 57, 80, -1; 157, -12, -166, -234, 3; 210, -18, -224, -315, 4],24,24]
%
% END OF LATTICE L25.24
%

\medskip%
%------------------------------WEYL-GROUP-W26--------------------------------
%
\leavevmode\llap{}%
$W_{26\phantom{0}}$%
\qquad\llap{8} lattices, $\chi=24$%
\hfill%
$\infty22\infty22\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L26.1
%
\leavevmode%
\llap{\spacer}%
${L_{26.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{26}}%
\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{4,3}{\infty b}{88}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-19976, -9240, -880; -9240, -4274, -407; -880, -407, -38],[88, 1, 1],[-25, 61, 5, -107, -589, -17; 55, -132, -11, 231, 1276, 37; -11, 0, 2, 0, -44, -3],25,1]
%
% END OF LATTICE L26.1
%
\hbox{}\par\smallskip%
%
% LATTICE L26.2
%
\leavevmode%
\llap{\spacer}%
${L_{26.2}}=\pfillwitharg{2}({L_{26.1}})$%
{${}={}$}Nikulin $26$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{2,1}{\infty b}{22}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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22\\0\\0
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0\\0\\-1
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0\\-1\\0
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-23\\-66\\-88
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4\\11\\16
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[22, 0, 0; 0, 0, -1; 0, -1, 0],[22, 1, 1],[-1, 1, 0, -10, -23, -1; 0, 0, -1, -33, -66, -2; -11, 0, 1, -33, -88, -5],25,2]
%
% END OF LATTICE L26.2
%
\hbox{}\par\smallskip%
%
% LATTICE L26.3
%
\leavevmode%
\llap{\spacer}%
${L_{26.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{26.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{4,1}{\infty z}{11}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-14476\\242\\-6578
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\kern3pt%
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242\\-4\\110
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6578\\110\\-2989
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4839\\880\\-10560
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-77\\-15\\168
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2211\\402\\-4825
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{$\left[\!\llap{\phantom{%
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100\\-33\\-220
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\HardButStrongLineBreak\kern3pt%
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126\\22\\-275
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14476, 242, -6578; 242, -4, 110; -6578, 110, -2989],[22, 2, 1],[100, 126, 13, 21, -5, -2; -33, 22, 13, 55, 0, -11; -220, -275, -28, -44, 11, 4],25,3]
%
% END OF LATTICE L26.3
%
\hbox{}\par\smallskip%
%
% LATTICE L26.4
%
\leavevmode%
\llap{\spacer}%
${L_{26.4}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{26.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty b}{2}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\kern3pt%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[154, -242, 66; -242, -660, -121; 66, -121, 28],[22, 11, 1],[25, 3, -54, -24, -1, 55; 0, -2, -3, -1, 0, 2; -53, -2, 121, 53, 2, -121],25,4]
%
% END OF LATTICE L26.4
%
\hbox{}\par\smallskip%
%
% LATTICE L26.5
%
\leavevmode%
\llap{\spacer}%
${L_{26.5}}=\pdualwitharg{2.11}(\pfillwitharg{2}({L_{26.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{44}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[44, 5082, 2420; 5082, 584056, 278124; 2420, 278124, 132441],[22, 22, 1],[33, 38, 37, 5, 0, 1; -326, -377, -367, -41, 10, 0; 684, 791, 770, 86, -21, 0],25,5]
%
% END OF LATTICE L26.5
%
\hbox{}\par\smallskip%
%
% LATTICE L26.6
%
\leavevmode%
\llap{\spacer}%
${L_{26.6}}=\pdualwitharg{2}({L_{26.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{8,1}{\infty z}{44}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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-174064\\-25256\\-8272
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-25256\\-3664\\-1200
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-174064, -25256, -8272; -25256, -3664, -1200; -8272, -1200, -393],[88, 8, 1],[1, 5, 1, -1, -5, -1; 22, -99, -33, -165, -44, 20; -88, 198, 80, 528, 242, -40],25,6]
%
% END OF LATTICE L26.6
%
\hbox{}\par\smallskip%
%
% LATTICE L26.7
%
\leavevmode%
\llap{\spacer}%
${L_{26.7}}=\pdualwitharg{11}({L_{26.1}})$%
\nopagebreak\par%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-96536, -2728, 2024; -2728, -66, 55; 2024, 55, -42],[88, 11, 1],[2, 13, 5, 1, -1, -2; 21, 148, 61, 13, -12, -27; 122, 812, 319, 65, -64, -132],25,7]
%
% END OF LATTICE L26.7
%
\hbox{}\par\smallskip%
%
% LATTICE L26.8
%
\leavevmode%
\llap{\spacer}%
${L_{26.8}}=\pdualwitharg{2.11}({L_{26.1}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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-528\\-4664\\-2112
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-4664\\-39248\\-17776
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-528, -4664, -2112; -4664, -39248, -17776; -2112, -17776, -8051],[88, 88, 1],[1, 0, -4, -6, -5, -1; 18, -19, -119, -97, -60, 40; -40, 42, 264, 216, 134, -88],25,8]
%
% END OF LATTICE L26.8
%

\medskip%
%------------------------------WEYL-GROUP-W27--------------------------------
%
\leavevmode\llap{}%
$W_{27\phantom{0}}$%
\qquad\llap{6} lattices, $\chi=22$%
\hfill%
$423423\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L27.1
%
\leavevmode%
\llap{\spacer}%
${L_{27.1}}$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-182068, 2024, 4048; 2024, -22, -45; 4048, -45, -90],[92, 1, 1],[0, -1, 2, 30, 31, 3; 2, -2, -5, -38, -34, -1; -1, -44, 92, 1363, 1406, 135],26,1]
%
% END OF LATTICE L27.1
%
\hbox{}\par\smallskip%
%
% LATTICE L27.2
%
\leavevmode%
\llap{\spacer}%
${L_{27.2}}=\pfillwitharg{2}({L_{27.1}})$%
{${}={}$}Nikulin $27$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-23, 0, 0; 0, 1, 0; 0, 0, 1],[23, 1, 1],[0, 0, -1, -12, -6, -1; -1, 1, 0, -17, -10, -3; 1, 0, -5, -55, -27, -4],26,2]
%
% END OF LATTICE L27.2
%
\hbox{}\par\smallskip%
%
% LATTICE L27.3
%
\leavevmode%
\llap{\spacer}%
${L_{27.3}}=\pdualwitharg{2}({L_{27.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1738984, 92, -455492; 92, 8, -24; -455492, -24, 119307],[92, 4, 1],[0, -11, -439, -5136, -2557, -417; 1, 0, 18, 221, 111, 19; 0, -42, -1676, -19608, -9762, -1592],26,3]
%
% END OF LATTICE L27.3
%
\hbox{}\par\smallskip%
%
% LATTICE L27.4
%
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\llap{\spacer}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-47334, 5037, -4324; 5037, -506, 460; -4324, 460, -395],[23, 23, 1],[63, -154, -396, -2589, -1109, -25; -1, -3, -3, -5, 0, 2; -690, 1679, 4324, 28290, 12121, 276],26,4]
%
% END OF LATTICE L27.4
%
\hbox{}\par\smallskip%
%
% LATTICE L27.5
%
\leavevmode%
\llap{\spacer}%
${L_{27.5}}=\pdualwitharg{23}({L_{27.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${46}\above{1pt}{1pt}{*}{4}{92}\above{1pt}{1pt}{b}{2}{46}\above{1pt}{1pt}{+}{3}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1932\\2944\\1012
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2944\\-4462\\-1541
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1012\\-1541\\-530
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5167\\-408\\10948
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7980\\629\\-16905
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2736\\216\\-5797
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
465\\-38\\989
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
459\\-34\\966
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\-1\\69
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\-46\\-115\\-874\\-782\\-23
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\kern3pt%
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-46\\92\\0\\-782\\-920\\-138
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-115\\0\\46\\-23\\-138\\-69
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\-46\\-115
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\92\\0
\end{tabular}\endgroup%
\kern3pt%
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-115\\0\\46
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-874\\-782\\-23
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-782\\-920\\-138
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-138\\-69
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1932, 2944, 1012; 2944, -4462, -1541; 1012, -1541, -530],[92, 23, 1],[465, 459, 35, -9, -3, 41; -38, -34, -1, 2, -2, -5; 989, 966, 69, -23, 0, 92],26,5]
%
% END OF LATTICE L27.5
%
\hbox{}\par\smallskip%
%
% LATTICE L27.6
%
\leavevmode%
\llap{\spacer}%
${L_{27.6}}=\pdualwitharg{2.23}({L_{27.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${184}\above{1pt}{1pt}{*}{4}{92}\above{1pt}{1pt}{*}{2}{184}\above{1pt}{1pt}{+}{3}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
184\\-1196\\276
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1196\\34408\\-8188
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
276\\-8188\\1949
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
629\\2520\\10488
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15015\\-60061\\-249964
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3570\\14280\\59431
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
629\\2520\\10488
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
313\\1249\\5198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\199\\828
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
184\\-92\\-460\\-3496\\-1564\\-92
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\92\\0\\-1564\\-920\\-276
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-460\\0\\184\\-92\\-276\\-276
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
184\\-92\\-460
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\92\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-460\\0\\184
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3496\\-1564\\-92
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1564\\-920\\-276
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\-276\\-276
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[184, -1196, 276; -1196, 34408, -8188; 276, -8188, 1949],[92, 92, 1],[629, 313, 51, 1, 2, 54; 2520, 1249, 199, 0, 11, 221; 10488, 5198, 828, 0, 46, 920],26,6]
%
% END OF LATTICE L27.6
%

\medskip%
%------------------------------WEYL-GROUP-W28--------------------------------
%
\leavevmode\llap{}%
$W_{28\phantom{0}}$%
\qquad\llap{8} lattices, $\chi=14$%
\hfill%
$\infty2322$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L28.1
%
\leavevmode%
\llap{\spacer}%
${L_{28.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{28}}%
\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{4,1}{\infty a}{104}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{+}{3}{2}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-255855288\\3601312\\-238784
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3601312\\-50690\\3357
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-238784\\3357\\-194
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1062\\-76154\\-10621
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
343\\24596\\3432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
161\\11545\\1610
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-12047\\-1681
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-795\\-57008\\-7952
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-52\\-13\\-13\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\104\\0\\-52\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\2\\-1\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-52\\-1\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-104\\-8\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-255855288, 3601312, -238784; 3601312, -50690, 3357; -238784, 3357, -194],[104, 1, 1],[-1062, 343, 161, -168, -795; -76154, 24596, 11545, -12047, -57008; -10621, 3432, 1610, -1681, -7952],27,1]
%
% END OF LATTICE L28.1
%
\hbox{}\par\smallskip%
%
% LATTICE L28.2
%
\leavevmode%
\llap{\spacer}%
${L_{28.2}}=\pfillwitharg{2}({L_{28.1}})$%
{${}={}$}Nikulin $28$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{2,1}{\infty a}{26}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{-}{3}{2}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-26\\-13\\-13\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\26\\0\\-26\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\2\\-1\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-26\\-1\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-26\\-4\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[26, 0, 0; 0, 0, -1; 0, -1, 0],[26, 1, 1],[-1, 1, 0, -1, -1; 0, 0, -1, -3, -2; -13, 0, 1, -4, -6],27,2]
%
% END OF LATTICE L28.2
%
\hbox{}\par\smallskip%
%
% LATTICE L28.3
%
\leavevmode%
\llap{\spacer}%
${L_{28.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{28.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{4,3}{\infty z}{13}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{+}{3}{4}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77532\\3406\\-37726
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3406\\-144\\1654
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37726\\1654\\-18355
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-539\\689\\1170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\-52\\-91
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82\\-105\\-178
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-83\\104\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\127\\217
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\-26\\-26\\-26\\0
\end{tabular}\endgroup%
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-26\\13\\0\\-26\\-13
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-26\\0\\4\\-2\\-4
\end{tabular}\endgroup%
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-26\\-26\\-2\\4\\0
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0\\-13\\-4\\0\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-77532, 3406, -37726; 3406, -144, 1654; -37726, 1654, -18355],[26, 2, 1],[-539, 42, 82, -83, -100; 689, -52, -105, 104, 127; 1170, -91, -178, 180, 217],27,3]
%
% END OF LATTICE L28.3
%
\hbox{}\par\smallskip%
%
% LATTICE L28.4
%
\leavevmode%
\llap{\spacer}%
${L_{28.4}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{28.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty a}{2}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{-}{3}{26}\above{1pt}{1pt}{l}{2}{26}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264342\\-10738\\-123214
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10738\\-312\\-5005
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-123214\\-5005\\-57432
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-364\\-1\\781
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\0\\-118
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
721\\2\\-1547
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-715\\-1\\1534
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1745\\-4\\3744
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-13\\-13\\0
\end{tabular}\endgroup%
\kern3pt%
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-2\\2\\0\\-26\\-26
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-13\\0\\26\\-13\\-52
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-13\\-26\\-13\\26\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-26\\-52\\0\\26
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-264342, -10738, -123214; -10738, -312, -5005; -123214, -5005, -57432],[26, 13, 1],[-364, 55, 721, -715, -1745; -1, 0, 2, -1, -4; 781, -118, -1547, 1534, 3744],27,4]
%
% END OF LATTICE L28.4
%
\hbox{}\par\smallskip%
%
% LATTICE L28.5
%
\leavevmode%
\llap{\spacer}%
${L_{28.5}}=\pdualwitharg{2.13}(\pfillwitharg{2}({L_{28.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{+}{3}{52}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\77714\\37310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77714\\122558592\\58839612
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37310\\58839612\\28248529
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1013\\2110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-205\\427
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-4631\\9646
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-3027\\6305
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-26\\-26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-26\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\0\\52\\-26\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-26\\-26\\52\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-13\\-52\\0\\13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[52, 77714, 37310; 77714, 122558592, 58839612; 37310, 58839612, 28248529],[26, 26, 1],[3, 0, 1, 24, 13; -1013, -205, 0, -4631, -3027; 2110, 427, 0, 9646, 6305],27,5]
%
% END OF LATTICE L28.5
%
\hbox{}\par\smallskip%
%
% LATTICE L28.6
%
\leavevmode%
\llap{\spacer}%
${L_{28.6}}=\pdualwitharg{2}({L_{28.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{8,7}{\infty z}{52}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{-}{3}{16}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39338000\\3890744\\842192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3890744\\-384784\\-83216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
842192\\-83216\\-17823
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7561\\83538\\-32760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-966\\-10673\\4186
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1010\\-11159\\4376
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1933\\21357\\-8376
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
844\\9325\\-3657
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\-104\\-104\\-104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\52\\0\\-104\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\0\\16\\-8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-104\\-8\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-26\\-8\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-39338000, 3890744, 842192; 3890744, -384784, -83216; 842192, -83216, -17823],[104, 8, 1],[7561, -966, -1010, 1933, 844; 83538, -10673, -11159, 21357, 9325; -32760, 4186, 4376, -8376, -3657],27,6]
%
% END OF LATTICE L28.6
%
\hbox{}\par\smallskip%
%
% LATTICE L28.7
%
\leavevmode%
\llap{\spacer}%
${L_{28.7}}=\pdualwitharg{13}({L_{28.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,1}{\infty a}{8}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{+}{3}{26}\above{1pt}{1pt}{l}{2}{104}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56472\\-2704\\-13312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2704\\-78\\-637
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13312\\-637\\-3138
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-1\\85
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-113\\-4\\480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-1\\208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\2\\-221
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\0\\-208
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-13\\-13\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-52\\-104
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\26\\-13\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-52\\-13\\26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-104\\-104\\0\\104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-56472, -2704, -13312; -2704, -78, -637; -13312, -637, -3138],[104, 13, 1],[-20, -113, -49, 52, 49; -1, -4, -1, 2, 0; 85, 480, 208, -221, -208],27,7]
%
% END OF LATTICE L28.7
%
\hbox{}\par\smallskip%
%
% LATTICE L28.8
%
\leavevmode%
\llap{\spacer}%
${L_{28.8}}=\pdualwitharg{2.13}({L_{28.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,7}{\infty z}{4}\above{1pt}{1pt}{*}{2}{208}\above{1pt}{1pt}{-}{3}{208}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-82472\\1144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82472\\-9994192\\138632
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1144\\138632\\-1923
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\26\\1874
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\88\\6344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-13\\-936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-793
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-104\\-104\\0
\end{tabular}\endgroup%
\kern3pt%
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-8\\4\\0\\-104\\-26
\end{tabular}\endgroup%
\kern3pt%
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-104\\0\\208\\-104\\-104
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-624, -82472, 1144; -82472, -9994192, 138632; 1144, 138632, -1923],[104, 104, 1],[-1, -1, -1, 2, 0; 5, 26, 88, -13, -11; 360, 1874, 6344, -936, -793],27,8]
%
% END OF LATTICE L28.8
%

\medskip%
%------------------------------WEYL-GROUP-W29--------------------------------
%
\leavevmode\llap{}%
$W_{29\phantom{0}}$%
\qquad\llap{6} lattices, $\chi=28$%
\hfill%
$22232223\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L29.1
%
\leavevmode%
\llap{\spacer}%
${L_{29.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{29}}%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42108\\1508\\812
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1508\\-54\\-29
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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3189\\90596\\-3828
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-115\\-3267\\138
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31\\880\\-36
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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0\\58\\0\\-29
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-4\\0\\4\\0
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-5\\-29\\0\\2
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-4\\-58\\-8\\-1
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58\\-1740\\-348\\-145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-348\\-76\\-36
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-42108, 1508, 812; 1508, -54, -29; 812, -29, -14],[116, 1, 1],[2, 133, 31, 16, 3, 12, -1, -1; 56, 3770, 880, 455, 86, 348, -28, -29; -1, -145, -36, -20, -5, -29, 0, 2],28,1]
%
% END OF LATTICE L29.1
%
\hbox{}\par\smallskip%
%
% LATTICE L29.2
%
\leavevmode%
\llap{\spacer}%
${L_{29.2}}=\pfillwitharg{2}({L_{29.1}})$%
{${}={}$}Nikulin $29$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
435\\0\\-29
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
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-29\\0\\2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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8\\31\\120
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[435, 0, -29; 0, -1, 0; -29, 0, 2],[29, 1, 1],[0, 2, 0, -1, -2, -60, -6, -5; 0, 0, -1, -5, -8, -232, -23, -19; 1, 29, -1, -17, -31, -899, -89, -73],28,2]
%
% END OF LATTICE L29.2
%
\hbox{}\par\smallskip%
%
% LATTICE L29.3
%
\leavevmode%
\llap{\spacer}%
${L_{29.3}}=\pdualwitharg{2}({L_{29.1}})$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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73080\\13108\\18096
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\kern3pt%
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13108\\2408\\3248
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18096\\3248\\4481
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{$\left.\llap{\phantom{%
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0\\0\\0
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-106315\\-17342\\441844
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-20022\\-3267\\83212
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[73080, 13108, 18096; 13108, 2408, 3248; 18096, 3248, 4481],[116, 4, 1],[126, 7256, 416, 847, 156, 922, 7, -1; 21, 1189, 68, 138, 25, 145, 1, 0; -524, -30160, -1729, -3520, -648, -3828, -29, 4],28,3]
%
% END OF LATTICE L29.3
%
\hbox{}\par\smallskip%
%
% LATTICE L29.4
%
\leavevmode%
\llap{\spacer}%
${L_{29.4}}=\pdualwitharg{29}(\pfillwitharg{2}({L_{29.1}}))$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1653\\-841\\1595
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-841\\-406\\812
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1595\\812\\-1539
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112141\\-252\\-116928
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64970\\-147\\-67744
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107690\\242\\112287
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\2\\-29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-629\\0\\-655
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2366\\-2\\-2465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2782\\-5\\-2900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\0\\-58\\-145\\-116\\-58\\-116\\-29
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-29\\-58\\-60\\-174\\-145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\0\\-58\\-145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-29
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58\\0\\29\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145\\-29\\0\\58
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116\\-58\\-116\\-29
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58\\-60\\-174\\-145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116\\-174\\-551\\-522
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-145\\-522\\-580
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1653, -841, 1595; -841, -406, 812; 1595, 812, -1539],[29, 29, 1],[-29, -629, -2366, -2782, -861, -261, -304, 112; 2, 0, -2, -5, -4, -2, -4, -1; -29, -655, -2465, -2900, -899, -273, -319, 116],28,4]
%
% END OF LATTICE L29.4
%
\hbox{}\par\smallskip%
%
% LATTICE L29.5
%
\leavevmode%
\llap{\spacer}%
${L_{29.5}}=\pdualwitharg{29}({L_{29.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${58}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{116}\above{1pt}{1pt}{r}{2}{58}\above{1pt}{1pt}{+}{3}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
116\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-406\\-29
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-29\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-36\\348
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\77\\-754
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\6\\-59
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5\\46
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-36\\348
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-20\\203
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\0\\-116\\-145\\-116\\-58\\-232\\-29
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-29\\-58\\-60\\-348\\-145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116\\0\\116\\0\\-232\\-348\\-2204\\-1044
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\0\\-116\\-145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-29
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116\\0\\116\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145\\-29\\0\\58
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116\\-58\\-232\\-29
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58\\-60\\-348\\-145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-232\\-348\\-2204\\-1044
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-145\\-1044\\-580
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[116, 0, 0; 0, -406, -29; 0, -29, -2],[116, 29, 1],[-2, -3, -19, -9, -1, 0, 1, 0; -1, -5, -36, -20, -5, -1, 0, 2; 0, 46, 348, 203, 58, 12, 0, -29],28,5]
%
% END OF LATTICE L29.5
%
\hbox{}\par\smallskip%
%
% LATTICE L29.6
%
\leavevmode%
\llap{\spacer}%
${L_{29.6}}=\pdualwitharg{2.29}({L_{29.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${232}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{29}\above{1pt}{1pt}{r}{2}{232}\above{1pt}{1pt}{+}{3}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
232\\3364\\812
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3364\\47560\\11484
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
812\\11484\\2773
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\78\\-348
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
910\\909\\-4060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
221\\221\\-987
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\27\\-116
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\31\\-136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\46\\-203
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\78\\-348
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
232\\0\\-116\\-580\\-464\\-232\\-232\\-116
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-116\\-232\\-240\\-348\\-580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116\\0\\29\\0\\-232\\-348\\-551\\-1044
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-580\\-116\\0\\232\\-116\\-580\\-1044\\-2320
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
232\\0\\-116\\-580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-116
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116\\0\\29\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-580\\-116\\0\\232
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-464\\-232\\-232\\-116
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-232\\-240\\-348\\-580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-232\\-348\\-551\\-1044
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116\\-580\\-1044\\-2320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[232, 3364, 812; 3364, 47560, 11484; 812, 11484, 2773],[116, 116, 1],[14, 24, 39, 77, 12, 2, 0, 1; 27, 31, 46, 78, -1, -5, -7, 0; -116, -136, -203, -348, 0, 20, 29, 0],28,6]
%
% END OF LATTICE L29.6
%

\medskip%
%------------------------------WEYL-GROUP-W30--------------------------------
%
\leavevmode\llap{}%
$W_{30\phantom{0}}$%
\qquad\llap{48} lattices, $\chi=24$%
\hfill%
$\infty22\infty22\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L30.1
%
\leavevmode%
\llap{\spacer}%
${L_{30.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{235\rightarrow N_{30},35,25,23,5,3,2}%
\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{60,31}{\infty b}{120}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{60,19}{\infty a}{120}\above{1pt}{1pt}{b}{2}{450}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1348200\\21600\\-55800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21600\\-330\\915
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55800\\915\\-2282
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-317\\240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-131\\-2756\\2100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-483\\369
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-293\\225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\316\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\585\\-450
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-60\\-45\\-105\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\120\\0\\-240\\-1320\\-900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\18\\0\\-180\\-225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-240\\0\\30\\-60\\-225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1320\\-180\\-60\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1348200, 21600, -55800; 21600, -330, 915; -55800, 915, -2282],[1800, 15, 1],[-15, -131, -23, -14, 15, 28; -317, -2756, -483, -293, 316, 585; 240, 2100, 369, 225, -240, -450],29,1]
%
% END OF LATTICE L30.1
%
\hbox{}\par\smallskip%
%
% LATTICE L30.2
%
\leavevmode%
\llap{\spacer}%
${L_{30.2}}=\pfillwitharg{2.3.5}({L_{30.1}})$%
{${}={}$}Nikulin $30$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{2,1}{\infty b}{30}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[30, 0, 0; 0, 0, -1; 0, -1, 0],[30, 1, 1],[-1, 1, 0, -4, -11, -1; 0, 0, -1, -15, -30, -2; -15, 0, 1, -15, -60, -7],29,2]
%
% END OF LATTICE L30.2
%
\hbox{}\par\smallskip%
%
% LATTICE L30.3
%
\leavevmode%
\llap{\spacer}%
${L_{30.3}}=\pfillwitharg{3.5}({L_{30.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{4,3}{\infty a}{120}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-606120\\-296640\\-7080
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\kern3pt%
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-296640\\-145178\\-3465
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\kern3pt%
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{$\left.\llap{\phantom{%
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94639\\-193440\\3120
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\kern3pt%
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46319\\-94675\\1527
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-606120, -296640, -7080; -296640, -145178, -3465; -7080, -3465, -82],[120, 1, 1],[-154, 323, 22, -301, -2143, -113; 315, -660, -45, 615, 4380, 231; -15, 0, 2, 0, -60, -5],29,3]
%
% END OF LATTICE L30.3
%
\hbox{}\par\smallskip%
%
% LATTICE L30.4
%
\leavevmode%
\llap{\spacer}%
${L_{30.4}}=\pfillwitharg{2.5}({L_{30.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{6,1}{\infty b}{30}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-141570, -29880, -11250; -29880, -6306, -2373; -11250, -2373, -890],[90, 3, 1],[84, -29, -28, 86, 369, 130; -460, 160, 153, -475, -2030, -714; 165, -60, -54, 180, 750, 261],29,4]
%
% END OF LATTICE L30.4
%
\hbox{}\par\smallskip%
%
% LATTICE L30.5
%
\leavevmode%
\llap{\spacer}%
${L_{30.5}}=\pfillwitharg{2.3}({L_{30.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{10,9}{\infty b}{30}\above{1pt}{1pt}{r}{2}{50}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{10,1}{\infty b}{30}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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30750\\3485\\-882
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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2\\9\\105
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1\\-24\\-60
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-105\\-120\\0\\30\\-30\\-15
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-120\\-330\\-150\\-30\\30\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1072050, -121500, 30750; -121500, -13770, 3485; 30750, 3485, -882],[150, 5, 1],[2, 1, -2, -4, -5, 0; 9, -24, 5, 81, 204, 17; 105, -60, -50, 180, 630, 67],29,5]
%
% END OF LATTICE L30.5
%
\hbox{}\par\smallskip%
%
% LATTICE L30.6
%
\leavevmode%
\llap{\spacer}%
${L_{30.6}}=\pfillwitharg{5}({L_{30.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{12,7}{\infty b}{120}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-164520\\-75960\\2880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75960\\-35070\\1329
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2880\\1329\\-50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5999\\-13800\\-21600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2780\\-6395\\-10008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\253\\395
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\95\\165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-243\\560\\900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\78\\117
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-60\\-45\\-105\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\120\\0\\-240\\-1320\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\18\\0\\-180\\-45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-60\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1320\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-164520, -75960, 2880; -75960, -35070, 1329; 2880, 1329, -50],[360, 3, 1],[-41, -243, -34, -9, 43, 4; 95, 560, 78, 20, -100, -9; 165, 900, 117, 15, -180, -9],29,6]
%
% END OF LATTICE L30.6
%
\hbox{}\par\smallskip%
%
% LATTICE L30.7
%
\leavevmode%
\llap{\spacer}%
${L_{30.7}}=\pfillwitharg{3}({L_{30.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{20,11}{\infty a}{120}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{20,19}{\infty a}{120}\above{1pt}{1pt}{b}{2}{50}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12445800\\-6142200\\25800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6142200\\-3031230\\12755
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25800\\12755\\-42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1315\\2643\\-5130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7087\\14244\\-27660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-304\\611\\-1187
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103\\207\\-405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1415\\-2844\\5520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
102\\-205\\400
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-60\\-15\\-105\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\120\\0\\-240\\-1320\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\2\\0\\-60\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-240\\0\\30\\-60\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1320\\-60\\-60\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-300\\-25\\-75\\0\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-12445800, -6142200, 25800; -6142200, -3031230, 12755; 25800, 12755, -42],[600, 5, 1],[-1315, -7087, -304, -103, 1415, 102; 2643, 14244, 611, 207, -2844, -205; -5130, -27660, -1187, -405, 5520, 400],29,7]
%
% END OF LATTICE L30.7
%
\hbox{}\par\smallskip%
%
% LATTICE L30.8
%
\leavevmode%
\llap{\spacer}%
${L_{30.8}}=\pfillwitharg{2}({L_{30.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{30,1}{\infty b}{30}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{30,19}{\infty a}{30}\above{1pt}{1pt}{r}{2}{450}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-94050\\6300\\44100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6300\\-420\\-2955
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44100\\-2955\\-20678
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\41\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
121\\124\\240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\36\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\14\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-14\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-30\\-45\\-105\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\30\\0\\-120\\-330\\-450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\18\\0\\-90\\-225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-120\\0\\30\\-30\\-225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-330\\-90\\-30\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-450\\-225\\-225\\0\\450
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-94050, 6300, 44100; 6300, -420, -2955; 44100, -2955, -20678],[450, 15, 1],[45, 121, 32, 8, -15, -1; 41, 124, 36, 14, -14, -15; 90, 240, 63, 15, -30, 0],29,8]
%
% END OF LATTICE L30.8
%
\hbox{}\par\smallskip%
%
% LATTICE L30.9
%
\leavevmode%
\llap{\spacer}%
${L_{30.9}}=\pdualwitharg{2}(\pfillwitharg{2.3.5}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{4,1}{\infty z}{15}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50340\\450\\-23490
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
450\\-4\\210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23490\\210\\-10961
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5779\\6120\\-12240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-55\\108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2703\\2862\\-5725
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\-15\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
92\\90\\-195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\33\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-30\\-30\\-210\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\15\\0\\-120\\-165\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\4\\0\\-30\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-165\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-50340, 450, -23490; 450, -4, 210; -23490, 210, -10961],[30, 2, 1],[56, 92, 19, 29, -7, -2; -15, 90, 33, 105, 0, -15; -120, -195, -40, -60, 15, 4],29,9]
%
% END OF LATTICE L30.9
%
\hbox{}\par\smallskip%
%
% LATTICE L30.10
%
\leavevmode%
\llap{\spacer}%
${L_{30.10}}=\pdualwitharg{3}(\pfillwitharg{2.3.5}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{2,1}{\infty b}{10}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-33510, 450, -10050; 450, -6, 135; -10050, 135, -3014],[30, 3, 1],[12, 43, 14, 8, -3, -2; -5, 60, 33, 35, 0, -15; -40, -140, -45, -25, 10, 6],29,10]
%
% END OF LATTICE L30.10
%
\hbox{}\par\smallskip%
%
% LATTICE L30.11
%
\leavevmode%
\llap{\spacer}%
${L_{30.11}}=\pdualwitharg{5}(\pfillwitharg{2.3.5}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{2,1}{\infty b}{6}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-20010\\450\\-3330
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\kern3pt%
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450\\-10\\75
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-20010, 450, -3330; 450, -10, 75; -3330, 75, -554],[30, 5, 1],[4, 17, 10, 4, -1, -2; -3, 36, 33, 21, 0, -15; -24, -96, -55, -21, 6, 10],29,11]
%
% END OF LATTICE L30.11
%
\hbox{}\par\smallskip%
%
% LATTICE L30.12
%
\leavevmode%
\llap{\spacer}%
${L_{30.12}}=\pdualwitharg{2.3}(\pfillwitharg{2.3.5}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{4,1}{\infty z}{5}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9120\\1080\\4373
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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1821\\10\\-3800
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\HardButStrongLineBreak\kern3pt%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19020, 2250, 9120; 2250, 672, 1080; 9120, 1080, 4373],[30, 6, 1],[1821, 2662, 1567, 829, -12, 23; 10, 15, 9, 5, 0, 0; -3800, -5555, -3270, -1730, 25, -48],29,12]
%
% END OF LATTICE L30.12
%
\hbox{}\par\smallskip%
%
% LATTICE L30.13
%
\leavevmode%
\llap{\spacer}%
${L_{30.13}}=\pdualwitharg{3}(\pfillwitharg{2.5}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{6,5}{\infty b}{30}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26370, 5940, -1800; 5940, -1338, 405; -1800, 405, -112],[90, 3, 1],[-79, 9, 10, -26, -219, -31; -355, 40, 45, -115, -980, -139; -15, 0, 2, 0, -30, -5],29,13]
%
% END OF LATTICE L30.13
%
\hbox{}\par\smallskip%
%
% LATTICE L30.14
%
\leavevmode%
\llap{\spacer}%
${L_{30.14}}=\pdualwitharg{2.5}(\pfillwitharg{2.3.5}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{4,1}{\infty z}{3}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13620, 3150, 6480; 3150, 1120, 1500; 6480, 1500, 3083],[30, 10, 1],[935, 1364, 1335, 419, -10, 19; 6, 9, 9, 3, 0, 0; -1968, -2871, -2810, -882, 21, -40],29,14]
%
% END OF LATTICE L30.14
%
\hbox{}\par\smallskip%
%
% LATTICE L30.15
%
\leavevmode%
\llap{\spacer}%
${L_{30.15}}=\pdualwitharg{3}(\pfillwitharg{3.5}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{4,3}{\infty a}{40}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-42360, -9120, 1080; -9120, -1962, 231; 1080, 231, -26],[120, 3, 1],[-2, 19, 1, -18, -99, -13; 10, -100, -5, 95, 520, 68; 5, -100, -3, 95, 500, 63],29,15]
%
% END OF LATTICE L30.15
%
\hbox{}\par\smallskip%
%
% LATTICE L30.16
%
\leavevmode%
\llap{\spacer}%
${L_{30.16}}=\pdualwitharg{3.5}(\pfillwitharg{2.3.5}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty b}{2}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4050, -240, -1410; -240, 1680, 1125; -1410, 1125, 1142],[30, 15, 1],[-230, -671, -985, -103, 5, -14; 462, 1348, 1979, 207, -10, 28; -739, -2156, -3165, -331, 16, -45],29,16]
%
% END OF LATTICE L30.16
%
\hbox{}\par\smallskip%
%
% LATTICE L30.17
%
\leavevmode%
\llap{\spacer}%
${L_{30.17}}=\pdualwitharg{2}(\pfillwitharg{2.5}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{12,1}{\infty z}{15}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1440, -1710, 810; -1710, 37680, -3126; 810, -3126, 587],[90, 6, 1],[1898, 81, 97, 3957, 5774, 3416; -235, -10, -12, -490, -715, -423; -3870, -165, -198, -8070, -11775, -6966],29,17]
%
% END OF LATTICE L30.17
%
\hbox{}\par\smallskip%
%
% LATTICE L30.18
%
\leavevmode%
\llap{\spacer}%
${L_{30.18}}=\pdualwitharg{2.3}(\pfillwitharg{2.5}({L_{30.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[696240, 115290, 348930; 115290, 19104, 57780; 348930, 57780, 174871],[90, 6, 1],[-853, -66, -1, -1457, -2244, -461; -115, -10, 0, -190, -295, -61; 1740, 135, 2, 2970, 4575, 940],29,18]
%
% END OF LATTICE L30.18
%
\hbox{}\par\smallskip%
%
% LATTICE L30.19
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30840, 3360, 0; 3360, 350, 15; 0, 15, -14],[120, 5, 1],[-16, -85, -18, -1, 17, 1; 147, 780, 165, 9, -156, -9; 156, 828, 175, 9, -168, -10],29,19]
%
% END OF LATTICE L30.19
%
\hbox{}\par\smallskip%
%
% LATTICE L30.20
%
\leavevmode%
\llap{\spacer}%
${L_{30.20}}=\pdualwitharg{2.3.5}(\pfillwitharg{2.3.5}({L_{30.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L30.20
%
\hbox{}\par\smallskip%
%
% LATTICE L30.21
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-856560, -24600, 1080; -24600, -688, 40; 1080, 40, 3],[120, 8, 1],[-31, -7, 7, 71, 47, 1; 990, 225, -223, -2265, -1500, -32; -2040, -450, 464, 4680, 3090, 64],29,21]
%
% END OF LATTICE L30.21
%
\hbox{}\par\smallskip%
%
% LATTICE L30.22
%
\leavevmode%
\llap{\spacer}%
${L_{30.22}}=\pdualwitharg{3}(\pfillwitharg{5}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{12,11}{\infty a}{120}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-127080, -24120, 3240; -24120, -4578, 615; 3240, 615, -82],[360, 3, 1],[-7, 19, 1, -18, -119, -6; 35, -100, -5, 95, 620, 31; -15, 0, 2, 0, -60, -5],29,22]
%
% END OF LATTICE L30.22
%
\hbox{}\par\smallskip%
%
% LATTICE L30.23
%
\leavevmode%
\llap{\spacer}%
${L_{30.23}}=\pdualwitharg{5}(\pfillwitharg{2.5}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{6,1}{\infty b}{6}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18810, 29430, -4590; 29430, -45930, 7155; -4590, 7155, -1114],[90, 15, 1],[-32, -99, -146, -12, 11, 14; -64, -200, -297, -25, 22, 30; -279, -876, -1305, -111, 96, 135],29,23]
%
% END OF LATTICE L30.23
%
\hbox{}\par\smallskip%
%
% LATTICE L30.24
%
\leavevmode%
\llap{\spacer}%
${L_{30.24}}=\pdualwitharg{3.5}(\pfillwitharg{2.5}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{6,5}{\infty b}{6}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16290, 9630, 1800; 9630, -5520, -1035; 1800, -1035, -194],[90, 15, 1],[3, 11, 6, 2, -1, -1; -41, -160, -90, -32, 14, 17; 246, 954, 535, 189, -84, -100],29,24]
%
% END OF LATTICE L30.24
%
\hbox{}\par\smallskip%
%
% LATTICE L30.25
%
\leavevmode%
\llap{\spacer}%
${L_{30.25}}=\pdualwitharg{2}(\pfillwitharg{2.3}({L_{30.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1827600, -32250, 901650; -32250, 960, -15920; 901650, -15920, 444831],[150, 10, 1],[4111, 244, 419, 8503, 12370, 2439; -201, -12, -20, -414, -603, -119; -8340, -495, -850, -17250, -25095, -4948],29,25]
%
% END OF LATTICE L30.25
%
\hbox{}\par\smallskip%
%
% LATTICE L30.26
%
\leavevmode%
\llap{\spacer}%
${L_{30.26}}=\pdualwitharg{3.5}(\pfillwitharg{3.5}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty a}{8}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L30.26
%
\hbox{}\par\smallskip%
%
% LATTICE L30.27
%
\leavevmode%
\llap{\spacer}%
${L_{30.27}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1222650, -33000, -415650; -33000, 1440, 11205; -415650, 11205, 141304],[150, 15, 1],[-918, -109, -281, -1899, -5525, -1634; -67, -8, -20, -138, -402, -119; -2695, -320, -825, -5575, -16220, -4797],29,27]
%
% END OF LATTICE L30.27
%
\hbox{}\par\smallskip%
%
% LATTICE L30.28
%
\leavevmode%
\llap{\spacer}%
${L_{30.28}}=\pdualwitharg{2.5}(\pfillwitharg{2.5}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3916260, -385290, 1901880; -385290, 37920, -187110; 1901880, -187110, 923623],[90, 30, 1],[718, 1089, 3325, 405, 34, 44; 101, 155, 477, 59, 5, 3; -1458, -2211, -6750, -822, -69, -90],29,28]
%
% END OF LATTICE L30.28
%
\hbox{}\par\smallskip%
%
% LATTICE L30.29
%
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${L_{30.29}}=\pdualwitharg{2.3.5}(\pfillwitharg{2.5}({L_{30.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8079840, 463410, 3948120; 463410, 26580, 226440; 3948120, 226440, 1929203],[90, 30, 1],[329, 520, 544, 210, 19, -5; 40, 58, 57, 19, 1, 2; -678, -1071, -1120, -432, -39, 10],29,29]
%
% END OF LATTICE L30.29
%
\hbox{}\par\smallskip%
%
% LATTICE L30.30
%
\leavevmode%
\llap{\spacer}%
${L_{30.30}}=\pdualwitharg{2.3}(\pfillwitharg{3.5}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{8,5}{\infty z}{20}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L30.30
%
\hbox{}\par\smallskip%
%
% LATTICE L30.31
%
\leavevmode%
\llap{\spacer}%
${L_{30.31}}=\pdualwitharg{2.3}(\pfillwitharg{2.3}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-341174700, 5309250, -163451550; 5309250, -82620, 2543580; -163451550, 2543580, -78307123],[150, 30, 1],[757, -242, -1006, 1476, 2475, 1541; 3, -4, 5, 27, 34, 17; -1580, 505, 2100, -3080, -5165, -3216],29,31]
%
% END OF LATTICE L30.31
%
\hbox{}\par\smallskip%
%
% LATTICE L30.32
%
\leavevmode%
\llap{\spacer}%
${L_{30.32}}=\pdualwitharg{2.5}(\pfillwitharg{3.5}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L30.32
%
\hbox{}\par\smallskip%
%
% LATTICE L30.33
%
\leavevmode%
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${L_{30.33}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{30.1}}))$%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-269640, -876240, -156600; -876240, -2717130, -483075; -156600, -483075, -85834],[360, 15, 1],[587, -581, -1187, 568, 5201, 4652; -1847, 1828, 3735, -1787, -16364, -14637; 9324, -9228, -18855, 9021, 82608, 73890],29,33]
%
% END OF LATTICE L30.33
%
\hbox{}\par\smallskip%
%
% LATTICE L30.34
%
\leavevmode%
\llap{\spacer}%
${L_{30.34}}=\pdualwitharg{3.5}(\pfillwitharg{5}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,11}{\infty a}{24}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4680, 3240, -360; 3240, -1290, 135; -360, 135, -14],[360, 15, 1],[0, -1, 0, 1, 5, 1; 1, -20, -1, 19, 100, 21; 9, -168, -10, 156, 828, 175],29,34]
%
% END OF LATTICE L30.34
%
\hbox{}\par\smallskip%
%
% LATTICE L30.35
%
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\llap{\spacer}%
${L_{30.35}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1053450, 22050, 423450; 22050, -420, -8865; 423450, -8865, -170212],[450, 15, 1],[435, 1583, 172, 284, -145, -141; 41, 154, 17, 29, -14, -15; 1080, 3930, 427, 705, -360, -350],29,35]
%
% END OF LATTICE L30.35
%
\hbox{}\par\smallskip%
%
% LATTICE L30.36
%
\leavevmode%
\llap{\spacer}%
${L_{30.36}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{30.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-117360, -52920, 10440; -52920, -22800, 4536; 10440, 4536, -901],[360, 24, 1],[76, 47, 1, -31, -2, 26; 755, 470, 12, -310, -25, 255; 4680, 2910, 72, -1920, -150, 1584],29,36]
%
% END OF LATTICE L30.36
%
\hbox{}\par\smallskip%
%
% LATTICE L30.37
%
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${L_{30.37}}=\pdualwitharg{2.3}(\pfillwitharg{5}({L_{30.1}}))$%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2824560, -329400, -140400; -329400, -38352, -16344; -140400, -16344, -6965],[360, 24, 1],[21, 17, 1, -11, -7, 1; -1970, -1565, -87, 1015, 610, -104; 4200, 3330, 184, -2160, -1290, 224],29,37]
%
% END OF LATTICE L30.37
%
\hbox{}\par\smallskip%
%
% LATTICE L30.38
%
\leavevmode%
\llap{\spacer}%
${L_{30.38}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{20,11}{\infty a}{40}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{20,19}{\infty a}{40}\above{1pt}{1pt}{b}{2}{150}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-449400\\631800\\-151800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
631800\\-863610\\205065
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-151800\\205065\\-48446
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-703\\-1717\\-5065
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6435\\-15716\\-46360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1151\\-2811\\-8292
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-726\\-1773\\-5230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
719\\1756\\5180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1468\\3585\\10575
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-20\\-15\\-35\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\40\\0\\-80\\-440\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\6\\0\\-60\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-80\\0\\10\\-20\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-440\\-60\\-20\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-300\\-75\\-75\\0\\150
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-449400, 631800, -151800; 631800, -863610, 205065; -151800, 205065, -48446],[600, 15, 1],[-703, -6435, -1151, -726, 719, 1468; -1717, -15716, -2811, -1773, 1756, 3585; -5065, -46360, -8292, -5230, 5180, 10575],29,38]
%
% END OF LATTICE L30.38
%
\hbox{}\par\smallskip%
%
% LATTICE L30.39
%
\leavevmode%
\llap{\spacer}%
${L_{30.39}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{60,1}{\infty z}{15}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{60,49}{\infty z}{15}\above{1pt}{1pt}{r}{2}{900}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
900\\450\\450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
450\\2940\\420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
450\\420\\239
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
139\\22\\-300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\34\\-465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
134\\21\\-288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\13\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\1\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-30\\-90\\-210\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\15\\0\\-120\\-165\\-450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\36\\0\\-90\\-450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-120\\0\\60\\-30\\-450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-165\\-90\\-30\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-450\\-450\\-450\\0\\900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[900, 450, 450; 450, 2940, 420; 450, 420, 239],[450, 30, 1],[139, 216, 134, 84, 7, -1; 22, 34, 21, 13, 1, 0; -300, -465, -288, -180, -15, 0],29,39]
%
% END OF LATTICE L30.39
%
\hbox{}\par\smallskip%
%
% LATTICE L30.40
%
\leavevmode%
\llap{\spacer}%
${L_{30.40}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{60,41}{\infty z}{15}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{60,29}{\infty z}{15}\above{1pt}{1pt}{r}{2}{100}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24798600\\-342450\\12115800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-342450\\4740\\-167310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12115800\\-167310\\5919391
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1979\\22\\-4050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2734\\34\\-5595
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
517\\7\\-1058
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
777\\13\\-1590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\1\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
171\\0\\-350
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-30\\-30\\-210\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\15\\0\\-120\\-165\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\4\\0\\-30\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-120\\0\\60\\-30\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-165\\-30\\-30\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-150\\-50\\-150\\0\\100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24798600, -342450, 12115800; -342450, 4740, -167310; 12115800, -167310, 5919391],[450, 30, 1],[1979, 2734, 517, 777, 22, 171; 22, 34, 7, 13, 1, 0; -4050, -5595, -1058, -1590, -45, -350],29,40]
%
% END OF LATTICE L30.40
%
\hbox{}\par\smallskip%
%
% LATTICE L30.41
%
\leavevmode%
\llap{\spacer}%
${L_{30.41}}=\pdualwitharg{2.3.5}(\pfillwitharg{3.5}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,5}{\infty z}{4}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-947760\\-974520\\-476640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-974520\\-1002000\\-490080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-476640\\-490080\\-239699
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2183\\-69264\\137280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2247\\-71263\\141240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1099\\-34854\\69079
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\89\\-176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-89\\178
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-845\\1680
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-120\\-56\\-32\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-32\\-44\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\240\\0\\-120\\-600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-32\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-44\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-947760, -974520, -476640; -974520, -1002000, -490080; -476640, -490080, -239699],[120, 120, 1],[-3, 2, 24, 10, 5, -3; 89, -89, -845, -311, -133, 179; -176, 178, 1680, 616, 262, -360],29,41]
%
% END OF LATTICE L30.41
%
\hbox{}\par\smallskip%
%
% LATTICE L30.42
%
\leavevmode%
\llap{\spacer}%
${L_{30.42}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{40,21}{\infty z}{60}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{40,29}{\infty z}{60}\above{1pt}{1pt}{*}{2}{400}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-186387600\\-11898600\\409800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11898600\\-759440\\26160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
409800\\26160\\-901
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-93\\-16800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\26\\4848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\228\\41640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\141\\25470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\-120\\-840\\-480\\0
\end{tabular}\endgroup%
\kern3pt%
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-120\\60\\0\\-480\\-660\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\16\\0\\-120\\-200
\end{tabular}\endgroup%
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-840\\-480\\0\\240\\-120\\-600
\end{tabular}\endgroup%
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-480\\-660\\-120\\-120\\60\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-600\\-200\\-600\\0\\400
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-186387600, -11898600, 409800; -11898600, -759440, 26160; 409800, 26160, -901],[600, 40, 1],[-31, -1, 9, 77, 47, 1; -93, -6, 26, 228, 141, 5; -16800, -630, 4848, 41640, 25470, 600],29,42]
%
% END OF LATTICE L30.42
%
\hbox{}\par\smallskip%
%
% LATTICE L30.43
%
\leavevmode%
\llap{\spacer}%
${L_{30.43}}=\pdualwitharg{3}({L_{30.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{60,11}{\infty a}{120}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{60,59}{\infty a}{120}\above{1pt}{1pt}{b}{2}{50}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-412200\\-225000\\27000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-225000\\-117870\\13875
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27000\\13875\\-1618
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99\\-581\\-3330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
523\\-3068\\-17580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\-129\\-739
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-29\\-165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-107\\628\\3600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\35\\200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-60\\-15\\-105\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\120\\0\\-240\\-1320\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\2\\0\\-60\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-240\\0\\30\\-60\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1320\\-60\\-60\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-300\\-25\\-75\\0\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-412200, -225000, 27000; -225000, -117870, 13875; 27000, 13875, -1618],[1800, 15, 1],[99, 523, 22, 5, -107, -6; -581, -3068, -129, -29, 628, 35; -3330, -17580, -739, -165, 3600, 200],29,43]
%
% END OF LATTICE L30.43
%
\hbox{}\par\smallskip%
%
% LATTICE L30.44
%
\leavevmode%
\llap{\spacer}%
${L_{30.44}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,13}{\infty z}{12}\above{1pt}{1pt}{*}{2}{720}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-150480\\-125640\\23400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-125640\\-99600\\18600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23400\\18600\\-3473
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1223\\27504\\155520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1003\\22537\\127440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\-4202\\-23761
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-157\\-888
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\137\\774
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62\\1401\\7920
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-360\\-168\\-96\\0
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\kern3pt%
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-24\\12\\0\\-96\\-132\\-360
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\720\\0\\-360\\-1800
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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48\\-24\\-360
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-1800
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-150480, -125640, 23400; -125640, -99600, 18600; 23400, 18600, -3473],[360, 120, 1],[-7, 6, 62, 24, 11, -11; -157, 137, 1401, 539, 245, -255; -888, 774, 7920, 3048, 1386, -1440],29,44]
%
% END OF LATTICE L30.44
%
\hbox{}\par\smallskip%
%
% LATTICE L30.45
%
\leavevmode%
\llap{\spacer}%
${L_{30.45}}=\pdualwitharg{2.3.5}(\pfillwitharg{5}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,5}{\infty z}{12}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2648880\\-1059480\\-519840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1059480\\-423600\\-207840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-519840\\-207840\\-101977
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1067\\-128427\\256320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
424\\-50987\\101760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\-25012\\49919
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-830\\1656
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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5\\-563\\1122
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-81\\160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-120\\-168\\-96\\0
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-24\\12\\0\\-96\\-132\\-120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\80\\0\\-120\\-200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-120
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\kern3pt%
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-24\\12\\0
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-120\\0\\80
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\kern3pt%
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-168\\-96\\0
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-96\\-132\\-120
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-200
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2648880, -1059480, -519840; -1059480, -423600, -207840; -519840, -207840, -101977],[360, 120, 1],[7, 5, 1, -3, -1, 3; -830, -563, -81, 349, 82, -400; 1656, 1122, 160, -696, -162, 800],29,45]
%
% END OF LATTICE L30.45
%
\hbox{}\par\smallskip%
%
% LATTICE L30.46
%
\leavevmode%
\llap{\spacer}%
${L_{30.46}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{30.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{40,29}{\infty z}{20}\above{1pt}{1pt}{*}{2}{1200}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{40,21}{\infty z}{20}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2830800\\-265800\\-71400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-265800\\-15120\\-3840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71400\\-3840\\-967
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\-5229\\17960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-2338\\8030
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\2620\\-9000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\1514\\-5200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-1377\\4730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-2753\\9456
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-40\\-600\\-280\\-160\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\20\\0\\-160\\-220\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-600\\0\\1200\\0\\-600\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-160\\0\\80\\-40\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-220\\-600\\-40\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-600\\-120\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2830800, -265800, -71400; -265800, -15120, -3840; -71400, -3840, -967],[600, 120, 1],[38, 17, -19, -11, 10, 20; -5229, -2338, 2620, 1514, -1377, -2753; 17960, 8030, -9000, -5200, 4730, 9456],29,46]
%
% END OF LATTICE L30.46
%
\hbox{}\par\smallskip%
%
% LATTICE L30.47
%
\leavevmode%
\llap{\spacer}%
${L_{30.47}}=\pdualwitharg{2.3}({L_{30.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{120,101}{\infty z}{60}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{120,29}{\infty z}{60}\above{1pt}{1pt}{*}{2}{400}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-867600\\5400\\3600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5400\\4560\\-720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3600\\-720\\91
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-328\\-2160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-31\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\89\\584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\803\\5280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\506\\3330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\30\\200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\-120\\-840\\-480\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\60\\0\\-480\\-660\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\16\\0\\-120\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-480\\0\\240\\-120\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-660\\-120\\-120\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-600\\-200\\-600\\0\\400
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-867600, 5400, 3600; 5400, 4560, -720; 3600, -720, 91],[1800, 120, 1],[-11, -1, 3, 27, 17, 1; -328, -31, 89, 803, 506, 30; -2160, -210, 584, 5280, 3330, 200],29,47]
%
% END OF LATTICE L30.47
%
\hbox{}\par\smallskip%
%
% LATTICE L30.48
%
\leavevmode%
\llap{\spacer}%
${L_{30.48}}=\pdualwitharg{2}({L_{30.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{120,109}{\infty z}{60}\above{1pt}{1pt}{*}{2}{3600}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{120,61}{\infty z}{60}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31899600\\376200\\-36000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
376200\\-2640\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36000\\120\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-1363\\-8040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-391\\-2310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\4275\\25200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\3113\\18360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\2141\\12630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\195\\1152
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\-1800\\-840\\-480\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\60\\0\\-480\\-660\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\0\\3600\\0\\-1800\\-1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-480\\0\\240\\-120\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-660\\-1800\\-120\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-1800\\-360\\0\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-31899600, 376200, -36000; 376200, -2640, 120; -36000, 120, 11],[1800, 120, 1],[-7, -2, 22, 16, 11, 1; -1363, -391, 4275, 3113, 2141, 195; -8040, -2310, 25200, 18360, 12630, 1152],29,48]
%
% END OF LATTICE L30.48
%

\medskip%
%------------------------------WEYL-GROUP-W31--------------------------------
%
\leavevmode\llap{}%
$W_{31\phantom{0}}$%
\qquad\llap{16} lattices, $\chi=4$%
\hfill%
$6222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L31.1
%
\leavevmode%
\llap{\spacer}%
${L_{31.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{31}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-221160\\960\\1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
960\\-2\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1800\\-9\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\52\\95
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-51\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-105\\-190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\48\\96
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\6\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-221160, 960, 1800; 960, -2, -9; 1800, -9, -14],[120, 1, 1],[1, -1, -2, 1; 52, -51, -105, 48; 95, -96, -190, 96],30,1]
%
% END OF LATTICE L31.1
%
\hbox{}\par\smallskip%
%
% LATTICE L31.2
%
\leavevmode%
\llap{\spacer}%
${L_{31.2}}=\pfillwitharg{2}({L_{31.1}})$%
{${}={}$}Nikulin $31$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
510\\150\\30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
150\\44\\9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\9\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\6\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-20\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-42\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\6\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[510, 150, 30; 150, 44, 9; 30, 9, 2],[30, 1, 1],[1, -2, 6, 13; -3, 6, -20, -42; -1, 3, -5, -12],30,2]
%
% END OF LATTICE L31.2
%
\hbox{}\par\smallskip%
%
% LATTICE L31.3
%
\leavevmode%
\llap{\spacer}%
${L_{31.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{31.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{6}{12}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62880\\690\\-30750
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
690\\-4\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30750\\336\\-15037
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\52\\130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\-51\\-132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
126\\-105\\-260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\12\\33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-6\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-62880, 690, -30750; 690, -4, 336; -30750, 336, -15037],[30, 2, 1],[-63, 64, 126, -16; 52, -51, -105, 12; 130, -132, -260, 33],30,3]
%
% END OF LATTICE L31.3
%
\hbox{}\par\smallskip%
%
% LATTICE L31.4
%
\leavevmode%
\llap{\spacer}%
${L_{31.4}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{31.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74670\\900\\-24270
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
900\\-6\\291
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24270\\291\\-7888
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\52\\165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-17\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
106\\-105\\-330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\8\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-74670, 900, -24270; 900, -6, 291; -24270, 291, -7888],[30, 3, 1],[-53, 18, 106, -9; 52, -17, -105, 8; 165, -56, -330, 28],30,4]
%
% END OF LATTICE L31.4
%
\hbox{}\par\smallskip%
%
% LATTICE L31.5
%
\leavevmode%
\llap{\spacer}%
${L_{31.5}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{31.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{30}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100770\\1320\\-19590
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1320\\-10\\255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19590\\255\\-3808
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\52\\235
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\-51\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-21\\-94
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\24\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-15\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\30\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-100770, 1320, -19590; 1320, -10, 255; -19590, 255, -3808],[30, 5, 1],[-45, 46, 18, -23; 52, -51, -21, 24; 235, -240, -94, 120],30,5]
%
% END OF LATTICE L31.5
%
\hbox{}\par\smallskip%
%
% LATTICE L31.6
%
\leavevmode%
\llap{\spacer}%
${L_{31.6}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{31.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{6}{4}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4882560\\-187290\\2342190
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187290\\7188\\-89844
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2342190\\-89844\\1123561
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
190\\1\\-396
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
191\\2\\-398
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
331\\0\\-690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
130\\0\\-271
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\4\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4882560, -187290, 2342190; -187290, 7188, -89844; 2342190, -89844, 1123561],[30, 6, 1],[190, 191, 331, 130; 1, 2, 0, 0; -396, -398, -690, -271],30,6]
%
% END OF LATTICE L31.6
%
\hbox{}\par\smallskip%
%
% LATTICE L31.7
%
\leavevmode%
\llap{\spacer}%
${L_{31.7}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{31.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{6}{60}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4215840\\-224670\\2006310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224670\\11980\\-106920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2006310\\-106920\\954799
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
238\\1\\-500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
757\\6\\-1590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79\\0\\-166
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
464\\0\\-975
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-30\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\60\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4215840, -224670, 2006310; -224670, 11980, -106920; 2006310, -106920, 954799],[30, 10, 1],[238, 757, 79, 464; 1, 6, 0, 0; -500, -1590, -166, -975],30,7]
%
% END OF LATTICE L31.7
%
\hbox{}\par\smallskip%
%
% LATTICE L31.8
%
\leavevmode%
\llap{\spacer}%
${L_{31.8}}=\pdualwitharg{3}({L_{31.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-236280\\1560\\2280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1560\\-6\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2280\\-15\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-236280, 1560, 2280; 1560, -6, -15; 2280, -15, -22],[120, 3, 1],[-1, 1, 2, -1; 2, -1, -5, 0; -105, 104, 210, -104],30,8]
%
% END OF LATTICE L31.8
%
\hbox{}\par\smallskip%
%
% LATTICE L31.9
%
\leavevmode%
\llap{\spacer}%
${L_{31.9}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{31.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4737810\\291720\\-1296840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
291720\\17970\\-79845
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1296840\\-79845\\354976
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-178\\357\\-570
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-189\\380\\-605
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59\\118\\-189
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\462\\-740
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-15\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\10\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4737810, 291720, -1296840; 291720, 17970, -79845; -1296840, -79845, 354976],[30, 15, 1],[-178, -189, -59, -231; 357, 380, 118, 462; -570, -605, -189, -740],30,9]
%
% END OF LATTICE L31.9
%
\hbox{}\par\smallskip%
%
% LATTICE L31.10
%
\leavevmode%
\llap{\spacer}%
${L_{31.10}}=\pdualwitharg{5}({L_{31.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-15\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\30\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120, 0, 0; 0, -10, -5; 0, -5, -2],[120, 5, 1],[0, -1, 0, 1; 2, -3, -1, 0; -5, 0, 2, 0],30,10]
%
% END OF LATTICE L31.10
%
\hbox{}\par\smallskip%
%
% LATTICE L31.11
%
\leavevmode%
\llap{\spacer}%
${L_{31.11}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{31.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{}{6}{20}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14856960\\-8907510\\-3662910
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8907510\\-5340480\\-2196090
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3662910\\-2196090\\-903067
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\-2443\\5730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\822\\-1930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\980\\-2298
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-202\\475
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-30\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\20\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14856960, -8907510, -3662910; -8907510, -5340480, -2196090; -3662910, -2196090, -903067],[30, 30, 1],[52, -17, -21, 4; -2443, 822, 980, -202; 5730, -1930, -2298, 475],30,11]
%
% END OF LATTICE L31.11
%
\hbox{}\par\smallskip%
%
% LATTICE L31.12
%
\leavevmode%
\llap{\spacer}%
${L_{31.12}}=\pdualwitharg{2}({L_{31.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{6}{48}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-557520\\-79080\\-9360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-79080\\-11216\\-1328
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9360\\-1328\\-157
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-38\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\45\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\75\\200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-24\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\48\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-557520, -79080, -9360; -79080, -11216, -1328; -9360, -1328, -157],[120, 8, 1],[7, -8, -14, 1; -38, 45, 75, -6; -96, 96, 200, -9],30,12]
%
% END OF LATTICE L31.12
%
\hbox{}\par\smallskip%
%
% LATTICE L31.13
%
\leavevmode%
\llap{\spacer}%
${L_{31.13}}=\pdualwitharg{3.5}({L_{31.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110040\\2400\\-43320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2400\\-30\\945
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43320\\945\\-17054
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\2\\-165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-1\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-1\\66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\0\\-160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-15\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\10\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-110040, 2400, -43320; 2400, -30, 945; -43320, 945, -17054],[120, 15, 1],[65, -63, -26, 63; 2, -1, -1, 0; -165, 160, 66, -160],30,13]
%
% END OF LATTICE L31.13
%
\hbox{}\par\smallskip%
%
% LATTICE L31.14
%
\leavevmode%
\llap{\spacer}%
${L_{31.14}}=\pdualwitharg{2.3}({L_{31.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{}{6}{16}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90846960\\-74040\\883440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74040\\-48\\720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
883440\\720\\-8591
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\2\\720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-1\\-4936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-5\\-1440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\617
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\16\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-90846960, -74040, 883440; -74040, -48, 720; 883440, 720, -8591],[120, 24, 1],[7, -48, -14, 6; 2, -1, -5, 0; 720, -4936, -1440, 617],30,14]
%
% END OF LATTICE L31.14
%
\hbox{}\par\smallskip%
%
% LATTICE L31.15
%
\leavevmode%
\llap{\spacer}%
${L_{31.15}}=\pdualwitharg{2.5}({L_{31.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{}{6}{240}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\360\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-1040\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-3\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-120\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\240\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240, 360, 0; 360, -1040, -40; 0, -40, -1],[120, 40, 1],[-3, 5, 1, -1; 2, -3, -1, 0; -80, 120, 32, -15],30,15]
%
% END OF LATTICE L31.15
%
\hbox{}\par\smallskip%
%
% LATTICE L31.16
%
\leavevmode%
\llap{\spacer}%
${L_{31.16}}=\pdualwitharg{2.3.5}({L_{31.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{}{6}{80}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-24600\\5640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24600\\-2005680\\459840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5640\\459840\\-105427
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\55\\240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-376\\-1640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\47\\205
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\80\\0\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\48\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\0\\5
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-240, -24600, 5640; -24600, -2005680, 459840; 5640, 459840, -105427],[120, 120, 1],[2, -1, -1, 0; 55, -376, -22, 47; 240, -1640, -96, 205],30,16]
%
% END OF LATTICE L31.16
%

\medskip%
%------------------------------WEYL-GROUP-W32--------------------------------
%
\leavevmode\llap{}%
$W_{32\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=20$%
\hfill%
$226226\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L32.1
%
\leavevmode%
\llap{\spacer}%
${L_{32.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{32},3,2}%
\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1983564\\-559152\\9108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-559152\\-157602\\2559
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9108\\2559\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99001\\364320\\803880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27675\\101843\\224721
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
350\\-1288\\-2843
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-322\\1185\\2619
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-278\\1023\\2255
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\11\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-18\\-99\\-99\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\0\\-99\\-143\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\6\\-9\\-33\\-15
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-99\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-143\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-33\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1983564, -559152, 9108; -559152, -157602, 2559; 9108, 2559, -38],[396, 3, 1],[-322, -278, -3, 97, 3, -72; 1185, 1023, 11, -357, -11, 265; 2619, 2255, 21, -792, -22, 588],31,1]
%
% END OF LATTICE L32.1
%
\hbox{}\par\smallskip%
%
% LATTICE L32.2
%
\leavevmode%
\llap{\spacer}%
${L_{32.2}}=\pfillwitharg{2.3}({L_{32.1}})$%
{${}={}$}Nikulin $32$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
231\\-66\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\18\\11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\11\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-199\\-561\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\220\\91
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-51\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\22\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-6\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\right.$}%
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2\\0\\-6\\-11\\-33\\-3
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\kern3pt%
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0\\22\\0\\-33\\-143\\-33
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\-3\\-33\\-15
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-33\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-143\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-33\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[231, -66, -33; -66, 18, 11; -33, 11, 2],[33, 1, 1],[0, 8, -2, -18, -74, -16; 0, 22, -6, -51, -209, -45; 1, 11, -3, -22, -88, -18],31,2]
%
% END OF LATTICE L32.2
%
\hbox{}\par\smallskip%
%
% LATTICE L32.3
%
\leavevmode%
\llap{\spacer}%
${L_{32.3}}=\pfillwitharg{3}({L_{32.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3144108\\1143384\\11880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1143384\\-415802\\-4321
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11880\\-4321\\-42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
925451\\2538228\\633204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336699\\-923462\\-230373
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2907\\-7973\\-1990
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
649\\1780\\445
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1885\\5170\\1287
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\384\\93
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3144108, 1143384, 11880; 1143384, -415802, -4321; 11880, -4321, -42],[132, 1, 1],[649, 1885, 140, -136, -4, 373; 1780, 5170, 384, -373, -11, 1023; 445, 1287, 93, -94, 0, 258],31,3]
%
% END OF LATTICE L32.3
%
\hbox{}\par\smallskip%
%
% LATTICE L32.4
%
\leavevmode%
\llap{\spacer}%
${L_{32.4}}=\pfillwitharg{2}({L_{32.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-247302, 27522, -4059; 27522, -2595, 336; -4059, 336, -38],[99, 3, 1],[103, 89, 1, -31, -1, 23; 2292, 1980, 22, -690, -22, 512; 9261, 7997, 87, -2790, -88, 2070],31,4]
%
% END OF LATTICE L32.4
%
\hbox{}\par\smallskip%
%
% LATTICE L32.5
%
\leavevmode%
\llap{\spacer}%
${L_{32.5}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{32.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-368379, 57651, 120120; 57651, -8553, -18849; 120120, -18849, -39163],[33, 3, 1],[-1116, 23134, 6812, 22701, 56011, 383; -349, 7205, 2123, 7078, 17468, 120; -3255, 67485, 19871, 66219, 163383, 1117],31,5]
%
% END OF LATTICE L32.5
%
\hbox{}\par\smallskip%
%
% LATTICE L32.6
%
\leavevmode%
\llap{\spacer}%
${L_{32.6}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{32.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1132857, -27225, 21087; -27225, -570, 477; 21087, 477, -382],[99, 3, 1],[46, 434, 16, -8, -16, 22; 1605, 15147, 559, -279, -561, 767; 4543, 42867, 1581, -790, -1584, 2172],31,6]
%
% END OF LATTICE L32.6
%
\hbox{}\par\smallskip%
%
% LATTICE L32.7
%
\leavevmode%
\llap{\spacer}%
${L_{32.7}}=\pdualwitharg{11}(\pfillwitharg{2.3}({L_{32.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3871230, 46629, 1375902; 46629, -462, -16588; 1375902, -16588, -489017],[33, 11, 1],[-1737, 3259, 31691, 35220, 7902, 1792; -744, 1406, 13656, 15165, 3401, 765; -4862, 9122, 88704, 98582, 22118, 5016],31,7]
%
% END OF LATTICE L32.7
%
\hbox{}\par\smallskip%
%
% LATTICE L32.8
%
\leavevmode%
\llap{\spacer}%
${L_{32.8}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{32.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2140644, 17160, 11220; 17160, -126, -93; 11220, -93, -58],[132, 3, 1],[-4, -8, 3, 25, 85, 4; -144, -286, 108, 897, 3047, 143; -543, -1089, 407, 3396, 11550, 544],31,8]
%
% END OF LATTICE L32.8
%
\hbox{}\par\smallskip%
%
% LATTICE L32.9
%
\leavevmode%
\llap{\spacer}%
${L_{32.9}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{32.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[436211160, 10956660, 109743084; 10956660, 275208, 2756504; 109743084, 2756504, 27609437],[132, 4, 1],[3822, 14064, 2594, 63, 181, 1291; 5811, 21373, 3939, 96, 286, 1968; -15772, -58036, -10704, -260, -748, -5328],31,9]
%
% END OF LATTICE L32.9
%
\hbox{}\par\smallskip%
%
% LATTICE L32.10
%
\leavevmode%
\llap{\spacer}%
${L_{32.10}}=\pdualwitharg{3.11}(\pfillwitharg{2.3}({L_{32.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L32.10
%
\hbox{}\par\smallskip%
%
% LATTICE L32.11
%
\leavevmode%
\llap{\spacer}%
${L_{32.11}}=\pdualwitharg{3}({L_{32.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19036908, -111276, 86328; -111276, -570, 477; 86328, 477, -382],[396, 3, 1],[23, 217, 8, -4, -8, 11; 3123, 29469, 1087, -543, -1089, 1493; 9097, 85833, 3165, -1582, -3168, 4350],31,11]
%
% END OF LATTICE L32.11
%
\hbox{}\par\smallskip%
%
% LATTICE L32.12
%
\leavevmode%
\llap{\spacer}%
${L_{32.12}}=\pdualwitharg{11}(\pfillwitharg{3}({L_{32.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-288420\\24552\\12276
\end{tabular}\endgroup%
\kern3pt%
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24552\\-2090\\-1045
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\kern3pt%
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2639\\33792\\-5808
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-288420, 24552, 12276; 24552, -2090, -1045; 12276, -1045, -522],[132, 11, 1],[-4, 0, 11, 19, 5, 4; -47, -1, 129, 239, 65, 63; 0, 2, 0, -33, -13, -33],31,12]
%
% END OF LATTICE L32.12
%
\hbox{}\par\smallskip%
%
% LATTICE L32.13
%
\leavevmode%
\llap{\spacer}%
${L_{32.13}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{32.1}}))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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256394952\\-6351972\\62097420
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-6351972\\157368\\-1538412
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[256394952, -6351972, 62097420; -6351972, 157368, -1538412; 62097420, -1538412, 15039647],[132, 12, 1],[218, 288, 502, 4849, 18291, 1187; 1, 11, 7, 50, 176, 10; -900, -1188, -2072, -20016, -75504, -4900],31,13]
%
% END OF LATTICE L32.13
%
\hbox{}\par\smallskip%
%
% LATTICE L32.14
%
\leavevmode%
\llap{\spacer}%
${L_{32.14}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{32.1}}))$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-45243, 9306, 4653; 9306, -1914, -957; 4653, -957, -478],[99, 33, 1],[-8, 0, 6, 26, 2, 0; -39, -1, 29, 174, 16, 16; 0, 2, 0, -99, -13, -33],31,14]
%
% END OF LATTICE L32.14
%
\hbox{}\par\smallskip%
%
% LATTICE L32.15
%
\leavevmode%
\llap{\spacer}%
${L_{32.15}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{32.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41778\\-138798\\4653
\end{tabular}\endgroup%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-41778, -138798, 4653; -138798, -460185, 15444; 4653, 15444, -518],[99, 33, 1],[-31, -1, 85, 152, 122, 36; -8, 0, 22, 38, 30, 8; -517, -9, 1419, 2497, 1989, 561],31,15]
%
% END OF LATTICE L32.15
%
\hbox{}\par\smallskip%
%
% LATTICE L32.16
%
\leavevmode%
\llap{\spacer}%
${L_{32.16}}=\pdualwitharg{3.11}(\pfillwitharg{3}({L_{32.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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-1695276\\-147048\\16368
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-147048\\-12738\\1419
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1695276, -147048, 16368; -147048, -12738, 1419; 16368, 1419, -158],[132, 33, 1],[13, 3, 0, -4, 0, 3; 115, 29, 5, -31, -1, 23; 2376, 570, 44, -693, -9, 517],31,16]
%
% END OF LATTICE L32.16
%
\hbox{}\par\smallskip%
%
% LATTICE L32.17
%
\leavevmode%
\llap{\spacer}%
${L_{32.17}}=\pdualwitharg{2}({L_{32.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2732784120, -25054524, -687827844; -25054524, 229704, 6306096; -687827844, 6306096, 173122765],[396, 12, 1],[-8383, -10757, -2173, -491, -89, -785; -3135, -4015, -809, -186, -44, -298; -33192, -42592, -8604, -1944, -352, -3108],31,17]
%
% END OF LATTICE L32.17
%
\hbox{}\par\smallskip%
%
% LATTICE L32.18
%
\leavevmode%
\llap{\spacer}%
${L_{32.18}}=\pdualwitharg{2.3}({L_{32.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}\right.$}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[49516292232, -520313508, -12505391712; -520313508, 5467416, 131405724; -12505391712, 131405724, 3158249837],[396, 12, 1],[-9439, -101917, -6007, -99, -3001, -3531; -63, -627, -31, 0, -66, -32; -37372, -403524, -23784, -392, -11880, -13980],31,18]
%
% END OF LATTICE L32.18
%
\hbox{}\par\smallskip%
%
% LATTICE L32.19
%
\leavevmode%
\llap{\spacer}%
${L_{32.19}}=\pdualwitharg{2.11}(\pfillwitharg{3}({L_{32.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[190344, 94380, 23232; 94380, 48312, 12628; 23232, 12628, 3647],[132, 44, 1],[-53, 1, -1081, -3445, -1167, -2417; 161, -3, 3285, 10468, 3546, 7344; -220, 4, -4488, -14300, -4844, -10032],31,19]
%
% END OF LATTICE L32.19
%
\hbox{}\par\smallskip%
%
% LATTICE L32.20
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L32.20
%
\hbox{}\par\smallskip%
%
% LATTICE L32.21
%
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\llap{\spacer}%
${L_{32.21}}=\pdualwitharg{3.11}({L_{32.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L32.21
%
\hbox{}\par\smallskip%
%
% LATTICE L32.22
%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[57816, -1255452, -304128; -1255452, 27379704, 6632604; -304128, 6632604, 1606717],[132, 132, 1],[-150, -52, -38, -9, -1, -15; 13679, 4765, 3505, 799, 61, 1321; -56496, -19680, -14476, -3300, -252, -5456],31,22]
%
% END OF LATTICE L32.22
%
\hbox{}\par\smallskip%
%
% LATTICE L32.23
%
\leavevmode%
\llap{\spacer}%
${L_{32.23}}=\pdualwitharg{2.11}({L_{32.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${792}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
414216\\169884\\-23760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
169884\\81048\\-18348
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23760\\-18348\\7871
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80515\\240840\\318384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93933\\280979\\371448
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50694\\-151640\\-200465
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
701\\-2097\\-2772
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1135\\-3395\\-4488
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
792\\0\\-792\\-4356\\-396\\-396
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0\\8\\0\\-396\\-52\\-132
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-792\\0\\264\\-396\\-132\\-660
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
792\\0\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-792\\0\\264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4356\\-396\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-52\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-132\\-660
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[414216, 169884, -23760; 169884, 81048, -18348; -23760, -18348, 7871],[396, 132, 1],[701, -1, 1135, 12718, 1492, 3338; -2097, 3, -3395, -38043, -4463, -9985; -2772, 4, -4488, -50292, -5900, -13200],31,23]
%
% END OF LATTICE L32.23
%
\hbox{}\par\smallskip%
%
% LATTICE L32.24
%
\leavevmode%
\llap{\spacer}%
${L_{32.24}}=\pdualwitharg{2.3.11}({L_{32.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${88}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
529538328\\-24809004\\152252892
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24809004\\1162920\\-7133544
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
152252892\\-7133544\\43776103
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81293992\\-240818721\\-321982452
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3907993\\11576720\\15478452
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23447958\\-69460326\\-92870713
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-611\\1810\\2420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-409\\1212\\1620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12731\\37714\\50424
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\0\\-264\\-484\\-396\\-132
\end{tabular}\endgroup%
\kern3pt%
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0\\72\\0\\-396\\-468\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\264\\-132\\-396\\-660
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\0\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\72\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-484\\-396\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-468\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-396\\-660
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[529538328, -24809004, 152252892; -24809004, 1162920, -7133544; 152252892, -7133544, 43776103],[396, 132, 1],[-611, -409, -12731, -38082, -38284, -25962; 1810, 1212, 37714, 112811, 113409, 76907; 2420, 1620, 50424, 150832, 151632, 102828],31,24]
%
% END OF LATTICE L32.24
%

\medskip%
%------------------------------WEYL-GROUP-W33--------------------------------
%
\leavevmode\llap{}%
$W_{33\phantom{0}}$%
\qquad\llap{44} lattices, $\chi=15$%
\hfill%
$222224$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L33.1
%
\leavevmode%
\llap{\spacer}%
${L_{33.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{33}}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-292380\\1584\\1848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1584\\-8\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1848\\-11\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-110\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-108\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\440\\253
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\328\\192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\2178\\1287
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\108\\65
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-22\\-12\\-66\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12\\-132\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\22\\0\\-132\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\0\\4\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-132\\-132\\0\\66\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-12\\-22\\-4\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-292380, 1584, 1848; 1584, -8, -11; 1848, -11, -10],[132, 1, 1],[-1, -1, 4, 3, 20, 1; -110, -108, 440, 328, 2178, 108; -64, -66, 253, 192, 1287, 65],32,1]
%
% END OF LATTICE L33.1
%
\hbox{}\par\smallskip%
%
% LATTICE L33.2
%
\leavevmode%
\llap{\spacer}%
${L_{33.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{22}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1159752\\-577368\\7128
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-577368\\-287435\\3548
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7128\\3548\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-75\\-55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\-156\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-293\\594\\440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-223\\452\\328
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-749\\1518\\1089
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\77\\54
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-22\\-12\\-33\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-24\\-132\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\44\\0\\-132\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\0\\8\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-132\\-132\\0\\33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-22\\-4\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1159752, -577368, 7128; -577368, -287435, 3548; 7128, 3548, -43],[264, 1, 1],[37, 77, -293, -223, -749, -38; -75, -156, 594, 452, 1518, 77; -55, -108, 440, 328, 1089, 54],32,2]
%
% END OF LATTICE L33.2
%
\hbox{}\par\smallskip%
%
% LATTICE L33.3
%
\leavevmode%
\llap{\spacer}%
${L_{33.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{132}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24359016\\35904\\78936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35904\\-49\\-119
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78936\\-119\\-254
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\640\\943
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\5592\\8256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\4631\\6842
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\1436\\2124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-1122\\-1650
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-798\\-1180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-11\\-12\\-66\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-24\\-264\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\11\\0\\-132\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\0\\8\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-264\\-132\\0\\132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-24\\-22\\-8\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-24359016, 35904, 78936; 35904, -49, -119; 78936, -119, -254],[264, 1, 1],[4, 35, 29, 9, -7, -5; 640, 5592, 4631, 1436, -1122, -798; 943, 8256, 6842, 2124, -1650, -1180],32,3]
%
% END OF LATTICE L33.3
%
\hbox{}\par\smallskip%
%
% LATTICE L33.4
%
\leavevmode%
\llap{\spacer}%
${L_{33.4}}=\pfillwitharg{2}({L_{33.1}})$%
{${}={}$}Nikulin $33$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
264\\-99\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\37\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-33\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-14\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82\\-231\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-17\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-11\\-3\\-33\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-3\\-66\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\22\\0\\-132\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-3\\0\\1\\0\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-66\\-132\\0\\66\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-22\\-2\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[264, -99, 0; -99, 37, 0; 0, 0, 1],[33, 1, 1],[0, 1, -12, -5, -82, -6; 0, 3, -33, -14, -231, -17; 1, 0, -11, -3, -33, -1],32,4]
%
% END OF LATTICE L33.4
%
\hbox{}\par\smallskip%
%
% LATTICE L33.5
%
\leavevmode%
\llap{\spacer}%
${L_{33.5}}=\pfillwitharg{2}({L_{33.2}})$%
{${}={}$}Nikulin $22'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2706\\2244\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2244\\1861\\-55
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-55\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\6\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\33\\11
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\16\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\99\\66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\5\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-11\\-6\\-33\\-1
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-6\\-66\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\11\\0\\-66\\-11
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-6\\0\\2\\0\\-2
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-66\\-66\\0\\33\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-11\\-2\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[2706, 2244, -66; 2244, 1861, -55; -66, -55, 2],[66, 1, 1],[0, -5, -27, -13, -80, -4; 0, 6, 33, 16, 99, 5; 1, 0, 11, 8, 66, 5],32,5]
%
% END OF LATTICE L33.5
%
\hbox{}\par\smallskip%
%
% LATTICE L33.6
%
\leavevmode%
\llap{\spacer}%
${L_{33.6}}=\main({L_{33.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2041908\\12012\\16104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12012\\-62\\-95
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16104\\-95\\-127
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\16\\495
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\120\\3840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\187\\6072
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\26\\868
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-33\\-990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-15\\-496
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-11\\-6\\-33\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12\\-132\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\22\\0\\-132\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-12\\0\\4\\0\\-4
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-132\\-132\\0\\66\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-22\\-4\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-2041908, 12012, 16104; 12012, -62, -95; 16104, -95, -127],[132, 1, 1],[4, 31, 49, 7, -8, -4; 16, 120, 187, 26, -33, -15; 495, 3840, 6072, 868, -990, -496],32,6]
%
% END OF LATTICE L33.6
%
\hbox{}\par\smallskip%
%
% LATTICE L33.7
%
\leavevmode%
\llap{\spacer}%
${L_{33.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{33.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-652971\\11748\\222915
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11748\\-186\\-4011
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
222915\\-4011\\-76100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
327\\16\\957
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
423\\20\\1238
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4014\\187\\11748
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
287\\13\\840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-218\\-11\\-638
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-328\\-15\\-960
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-33\\-9\\-33\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-3\\-22\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\66\\0\\-132\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-3\\0\\3\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-22\\-132\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-66\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-652971, 11748, 222915; 11748, -186, -4011; 222915, -4011, -76100],[33, 3, 1],[327, 423, 4014, 287, -218, -328; 16, 20, 187, 13, -11, -15; 957, 1238, 11748, 840, -638, -960],32,7]
%
% END OF LATTICE L33.7
%
\hbox{}\par\smallskip%
%
% LATTICE L33.8
%
\leavevmode%
\llap{\spacer}%
${L_{33.8}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{33.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{22}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{66}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-181302\\2442\\-89892
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2442\\-32\\1210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89892\\1210\\-44569
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-55\\-62
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-27\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\220\\242
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\82\\93
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-607\\1089\\1254
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\54\\64
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-22\\-6\\-66\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-3\\-66\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\22\\0\\-132\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-3\\0\\1\\0\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-66\\-132\\0\\66\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-6\\-22\\-2\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-181302, 2442, -89892; 2442, -32, 1210; -89892, 1210, -44569],[66, 2, 1],[30, 16, -117, -45, -607, -31; -55, -27, 220, 82, 1089, 54; -62, -33, 242, 93, 1254, 64],32,8]
%
% END OF LATTICE L33.8
%
\hbox{}\par\smallskip%
%
% LATTICE L33.9
%
\leavevmode%
\llap{\spacer}%
${L_{33.9}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{33.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-223674\\3300\\-73854
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3300\\-48\\1089
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73854\\1089\\-24385
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-55\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-18\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73\\220\\231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\164\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-129\\363\\407
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\54\\63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-33\\-18\\-33\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-6\\-22\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\33\\0\\-66\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-6\\0\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-22\\-66\\0\\11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-33\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-223674, 3300, -73854; 3300, -48, 1089; -73854, 1089, -24385],[66, 3, 1],[19, 7, -73, -57, -129, -20; -55, -18, 220, 164, 363, 54; -60, -22, 231, 180, 407, 63],32,9]
%
% END OF LATTICE L33.9
%
\hbox{}\par\smallskip%
%
% LATTICE L33.10
%
\leavevmode%
\llap{\spacer}%
${L_{33.10}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{33.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1486452\\-45606\\537471
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45606\\-1397\\16489
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
537471\\16489\\-194338
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
152\\231\\440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
581\\897\\1683
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
166\\258\\481
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
129\\203\\374
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-42\\-81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-148\\-232\\-429
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-11\\-33\\-33\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\33\\0\\-33\\-66\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\2\\0\\-12\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-33\\0\\11\\0\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-66\\-12\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-66\\-22\\-22\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1486452, -45606, 537471; -45606, -1397, 16489; 537471, 16489, -194338],[33, 11, 1],[152, 581, 166, 129, -28, -148; 231, 897, 258, 203, -42, -232; 440, 1683, 481, 374, -81, -429],32,10]
%
% END OF LATTICE L33.10
%
\hbox{}\par\smallskip%
%
% LATTICE L33.11
%
\leavevmode%
\llap{\spacer}%
${L_{33.11}}=\pdualwitharg{3}(\main({L_{33.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2646732\\-220044\\10428
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220044\\-18294\\867
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10428\\867\\-41
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-11\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\60\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\451\\-132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\106\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\11\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-59\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-33\\-18\\-33\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12\\-44\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\66\\0\\-132\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-12\\0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-44\\-132\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-12\\-66\\-12\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2646732, -220044, 10428; -220044, -18294, 867; 10428, 867, -41],[132, 3, 1],[1, -5, -38, -9, -1, 5; -11, 60, 451, 106, 11, -59; 21, -4, -132, -48, -22, 24],32,11]
%
% END OF LATTICE L33.11
%
\hbox{}\par\smallskip%
%
% LATTICE L33.12
%
\leavevmode%
\llap{\spacer}%
${L_{33.12}}=\pdualwitharg{3}({L_{33.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-238260\\2772\\2772
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2772\\-30\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2772\\-33\\-32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\77\\264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\104\\324
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\297\\902
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\65\\192
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-66\\-36\\-66\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12\\-44\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\0\\66\\0\\-132\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-12\\0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-44\\-132\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-12\\-66\\-12\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-238260, 2772, 2772; 2772, -30, -33; 2772, -33, -32],[132, 3, 1],[-1, -1, 4, 5, 14, 3; -20, -22, 77, 104, 297, 65; -66, -64, 264, 324, 902, 192],32,12]
%
% END OF LATTICE L33.12
%
\hbox{}\par\smallskip%
%
% LATTICE L33.13
%
\leavevmode%
\llap{\spacer}%
${L_{33.13}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{33.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{66}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{22}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4512684\\-224202\\2214498
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224202\\11166\\-110022
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2214498\\-110022\\1086715
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\0\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
135\\1\\-275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4957\\33\\-10098
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1265\\8\\-2577
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5486\\33\\-11176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
913\\5\\-1860
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-66\\-18\\-66\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-3\\-22\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\0\\66\\0\\-132\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-3\\0\\3\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-22\\-132\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-6\\-66\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4512684, -224202, 2214498; -224202, 11166, -110022; 2214498, -110022, 1086715],[66, 6, 1],[53, 135, 4957, 1265, 5486, 913; 0, 1, 33, 8, 33, 5; -108, -275, -10098, -2577, -11176, -1860],32,13]
%
% END OF LATTICE L33.13
%
\hbox{}\par\smallskip%
%
% LATTICE L33.14
%
\leavevmode%
\llap{\spacer}%
${L_{33.14}}=\pdualwitharg{2}(\main({L_{33.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
546702024\\-522588\\132395736
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-522588\\500\\-126556
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132395736\\-126556\\32062495
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
929\\31\\-3836
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1076\\33\\-4443
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5573\\154\\-23012
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
325\\6\\-1342
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
959\\0\\-3960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\2\\-128
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-44\\-12\\-132\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-6\\-132\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\0\\88\\0\\-528\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-6\\0\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-132\\-528\\0\\264\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-12\\-88\\-8\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[546702024, -522588, 132395736; -522588, 500, -126556; 132395736, -126556, 32062495],[132, 4, 1],[929, 1076, 5573, 325, 959, 31; 31, 33, 154, 6, 0, 2; -3836, -4443, -23012, -1342, -3960, -128],32,14]
%
% END OF LATTICE L33.14
%
\hbox{}\par\smallskip%
%
% LATTICE L33.15
%
\leavevmode%
\llap{\spacer}%
${L_{33.15}}=\pdualwitharg{2}({L_{33.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{264}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4935216\\-270996\\1193940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270996\\14888\\-65560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1193940\\-65560\\288841
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-62
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\3\\-330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1194\\33\\-4928
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
164\\4\\-677
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4540\\99\\-18744
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
278\\5\\-1148
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-44\\-6\\-132\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-6\\-264\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\0\\88\\0\\-528\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-6\\0\\1\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-264\\-528\\0\\264\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-24\\-88\\-4\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4935216, -270996, 1193940; -270996, 14888, -65560; 1193940, -65560, 288841],[132, 4, 1],[15, 80, 1194, 164, 4540, 278; 0, 3, 33, 4, 99, 5; -62, -330, -4928, -677, -18744, -1148],32,15]
%
% END OF LATTICE L33.15
%
\hbox{}\par\smallskip%
%
% LATTICE L33.16
%
\leavevmode%
\llap{\spacer}%
${L_{33.16}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{33.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{l}{2}{66}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{22}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-625614\\-7260\\112794
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7260\\-55\\1309
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112794\\1309\\-20336
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\2\\-671
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119\\0\\-660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\-1\\244
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
361\\-6\\2002
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
218\\-3\\1209
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
119\\-1\\660
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\0\\-11\\-66\\-33\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\66\\0\\-66\\-66\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\1\\0\\-6\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-66\\0\\22\\0\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-66\\-6\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-66\\-11\\-22\\0\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-625614, -7260, 112794; -7260, -55, 1309; 112794, 1309, -20336],[66, 11, 1],[-121, -119, 44, 361, 218, 119; 2, 0, -1, -6, -3, -1; -671, -660, 244, 2002, 1209, 660],32,16]
%
% END OF LATTICE L33.16
%
\hbox{}\par\smallskip%
%
% LATTICE L33.17
%
\leavevmode%
\llap{\spacer}%
${L_{33.17}}=\pdualwitharg{3}({L_{33.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85272\\20328\\1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20328\\-4845\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1056\\-252\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\-27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\44\\198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\32\\108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\33\\88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-66\\-36\\-33\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-24\\-44\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\0\\132\\0\\-132\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-24\\0\\24\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-44\\-132\\0\\11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-12\\-66\\-12\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-85272, 20328, 1056; 20328, -4845, -252; 1056, -252, -13],[264, 3, 1],[-2, -1, 13, 9, 9, 1; -7, -4, 44, 32, 33, 4; -27, -4, 198, 108, 88, 3],32,17]
%
% END OF LATTICE L33.17
%
\hbox{}\par\smallskip%
%
% LATTICE L33.18
%
\leavevmode%
\llap{\spacer}%
${L_{33.18}}=\pdualwitharg{3}({L_{33.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{44}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1634424\\16104\\11616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16104\\-147\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11616\\-117\\-82
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\-111
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\120\\536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\451\\2046
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\212\\972
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\22\\110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-118\\-540
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-33\\-36\\-66\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-24\\-88\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\33\\0\\-132\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-24\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-88\\-132\\0\\44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-24\\-66\\-24\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1634424, 16104, 11616; 16104, -147, -117; 11616, -117, -82],[264, 3, 1],[-1, 5, 19, 9, 1, -5; -22, 120, 451, 212, 22, -118; -111, 536, 2046, 972, 110, -540],32,18]
%
% END OF LATTICE L33.18
%
\hbox{}\par\smallskip%
%
% LATTICE L33.19
%
\leavevmode%
\llap{\spacer}%
${L_{33.19}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{33.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
967065\\63129\\33363
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63129\\4125\\2178
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33363\\2178\\1151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\31\\-957
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\22\\-737
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\14\\-519
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\6\\-330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\-33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\0\\-33\\-99\\-33\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\0\\-33\\-22\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\6\\0\\-12\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-33\\0\\33\\0\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-22\\-12\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-66\\-66\\-66\\0\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[967065, 63129, 33363; 63129, 4125, 2178; 33363, 2178, 1151],[33, 33, 1],[31, 24, 17, 11, 1, 1; 31, 22, 14, 6, 0, 2; -957, -737, -519, -330, -29, -33],32,19]
%
% END OF LATTICE L33.19
%
\hbox{}\par\smallskip%
%
% LATTICE L33.20
%
\leavevmode%
\llap{\spacer}%
${L_{33.20}}=\pdualwitharg{11}({L_{33.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3828\\660\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
660\\-88\\77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\77\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\14\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\24\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-4\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-44\\132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-30\\87
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-20\\55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\-22\\-132\\-66\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\132\\0\\-132\\-132\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\2\\0\\-12\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-132\\0\\44\\0\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-132\\-12\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-132\\-22\\-44\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3828, 660, -396; 660, -88, 77; -396, 77, -38],[132, 11, 1],[7, 11, -2, -21, -14, -9; 14, 24, -4, -44, -30, -20; -44, -66, 13, 132, 87, 55],32,20]
%
% END OF LATTICE L33.20
%
\hbox{}\par\smallskip%
%
% LATTICE L33.21
%
\leavevmode%
\llap{\spacer}%
${L_{33.21}}=\pdualwitharg{2.11}(\pfillwitharg{2}({L_{33.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{22}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22440462\\28314\\10688436
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28314\\44\\13486
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10688436\\13486\\5090923
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-613\\0\\1287
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-603\\1\\1266
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1598\\4\\3355
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1789\\6\\3756
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-985\\5\\2068
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\-22\\-66\\-66\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\33\\0\\-33\\-66\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\2\\0\\-12\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-33\\0\\11\\0\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-66\\-12\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-66\\-22\\-22\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22440462, 28314, 10688436; 28314, 44, 13486; 10688436, 13486, 5090923],[66, 22, 1],[0, -613, -603, -1598, -1789, -985; 1, 0, 1, 4, 6, 5; 0, 1287, 1266, 3355, 3756, 2068],32,21]
%
% END OF LATTICE L33.21
%
\hbox{}\par\smallskip%
%
% LATTICE L33.22
%
\leavevmode%
\llap{\spacer}%
${L_{33.22}}=\pdualwitharg{11}(\main({L_{33.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{}{2}{132}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-199452\\49104\\-5808
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49104\\-12089\\1430
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5808\\1430\\-169
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-15\\44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\12\\132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\9\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\34\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-19\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-11\\-66\\-33\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\132\\0\\-132\\-132\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\2\\0\\-12\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-132\\0\\44\\0\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-132\\-12\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-132\\-22\\-44\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-199452, 49104, -5808; 49104, -12089, 1430; -5808, 1430, -169],[132, 11, 1],[-5, -1, 2, 9, 1, -5; -15, 12, 9, 34, 3, -19; 44, 132, 7, -22, -9, 11],32,22]
%
% END OF LATTICE L33.22
%
\hbox{}\par\smallskip%
%
% LATTICE L33.23
%
\leavevmode%
\llap{\spacer}%
${L_{33.23}}=\pdualwitharg{2.3}(\main({L_{33.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{264}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69960\\20724\\16368
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20724\\20652\\4440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16368\\4440\\3841
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1730\\-201\\-7140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
637\\-74\\-2629
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9659\\-1122\\-39864
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
551\\-64\\-2274
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
757\\-88\\-3124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
189\\-22\\-780
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-132\\-36\\-132\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-6\\-44\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\264\\0\\-528\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-6\\0\\12\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-44\\-528\\0\\88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\-264\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[69960, 20724, 16368; 20724, 20652, 4440; 16368, 4440, 3841],[132, 12, 1],[1730, 637, 9659, 551, 757, 189; -201, -74, -1122, -64, -88, -22; -7140, -2629, -39864, -2274, -3124, -780],32,23]
%
% END OF LATTICE L33.23
%
\hbox{}\par\smallskip%
%
% LATTICE L33.24
%
\leavevmode%
\llap{\spacer}%
${L_{33.24}}=\pdualwitharg{2.3}({L_{33.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5188920\\-167508\\1255320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-167508\\5424\\-40524
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1255320\\-40524\\303691
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\1\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-62
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1789\\22\\-7392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
281\\4\\-1161
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2769\\44\\-11440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
549\\10\\-2268
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-132\\-18\\-132\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-6\\-88\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\264\\0\\-528\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-6\\0\\3\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-88\\-528\\0\\88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\-264\\-12\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5188920, -167508, 1255320; -167508, 5424, -40524; 1255320, -40524, 303691],[132, 12, 1],[16, 15, 1789, 281, 2769, 549; 1, 0, 22, 4, 44, 10; -66, -62, -7392, -1161, -11440, -2268],32,24]
%
% END OF LATTICE L33.24
%
\hbox{}\par\smallskip%
%
% LATTICE L33.25
%
\leavevmode%
\llap{\spacer}%
${L_{33.25}}=\pdualwitharg{2}({L_{33.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{88}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{264}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1239216\\-20856\\7128
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20856\\-344\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7128\\120\\-41
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-174
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-11\\660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\338
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-33\\5280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\344
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-88\\-24\\-264\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12\\-264\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\0\\88\\0\\-528\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\0\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-264\\-528\\0\\264\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-24\\-88\\-8\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1239216, -20856, 7128; -20856, -344, 120; 7128, 120, -41],[264, 8, 1],[-1, -1, 4, 2, 31, 2; 2, 0, -11, -3, -33, -1; -168, -174, 660, 338, 5280, 344],32,25]
%
% END OF LATTICE L33.25
%
\hbox{}\par\smallskip%
%
% LATTICE L33.26
%
\leavevmode%
\llap{\spacer}%
${L_{33.26}}=\pdualwitharg{2}({L_{33.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{88}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{264}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1308912\\7656\\15048
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7656\\-40\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15048\\-88\\-173
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-87
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-11\\352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-3\\262
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\-33\\4620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\348
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-88\\-24\\-264\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-6\\-132\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\0\\88\\0\\-528\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-6\\0\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-132\\-528\\0\\264\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-12\\-88\\-8\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1308912, 7656, 15048; 7656, -40, -88; 15048, -88, -173],[264, 8, 1],[-1, -1, 4, 3, 53, 4; 2, 0, -11, -3, -33, -1; -88, -87, 352, 262, 4620, 348],32,26]
%
% END OF LATTICE L33.26
%
\hbox{}\par\smallskip%
%
% LATTICE L33.27
%
\leavevmode%
\llap{\spacer}%
${L_{33.27}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{33.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{l}{2}{22}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{66}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7630590\\-20196\\-2303796
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20196\\66\\6105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2303796\\6105\\695557
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
185\\-370\\616
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
273\\-545\\909
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1447\\-2886\\4818
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
270\\-538\\899
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
446\\-887\\1485
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\0\\-33\\-198\\-33\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\0\\-66\\-22\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\3\\0\\-6\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-66\\0\\66\\0\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-22\\-6\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-66\\-33\\-66\\0\\33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7630590, -20196, -2303796; -20196, 66, 6105; -2303796, 6105, 695557],[66, 33, 1],[0, 185, 273, 1447, 270, 446; 1, -370, -545, -2886, -538, -887; 0, 616, 909, 4818, 899, 1485],32,27]
%
% END OF LATTICE L33.27
%
\hbox{}\par\smallskip%
%
% LATTICE L33.28
%
\leavevmode%
\llap{\spacer}%
${L_{33.28}}=\pdualwitharg{11}({L_{33.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{b}{2}{264}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{88}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34584\\-4224\\-4488
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4224\\-473\\-506
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4488\\-506\\-541
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-97\\99
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\132\\-132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\100\\-102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\472\\-484
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\87\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\0\\-22\\-132\\-33\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\264\\0\\-264\\-132\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\4\\0\\-12\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-264\\0\\88\\0\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-132\\-12\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-132\\-22\\-44\\0\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-34584, -4224, -4488; -4224, -473, -506; -4488, -506, -541],[264, 11, 1],[-1, 1, 1, 5, 1, 0; -97, 132, 100, 472, 87, -12; 99, -132, -102, -484, -90, 11],32,28]
%
% END OF LATTICE L33.28
%
\hbox{}\par\smallskip%
%
% LATTICE L33.29
%
\leavevmode%
\llap{\spacer}%
${L_{33.29}}=\pdualwitharg{11}({L_{33.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{l}{2}{264}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17822904\\98472\\77616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
98472\\-539\\-429
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77616\\-429\\-338
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-30\\-1111
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\24\\-264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\218
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\68\\1980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\222
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-38\\-1100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\0\\-11\\-132\\-66\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\264\\0\\-264\\-264\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\1\\0\\-12\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-264\\0\\88\\0\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-264\\-12\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-264\\-22\\-88\\0\\44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17822904, 98472, 77616; 98472, -539, -429; 77616, -429, -338],[264, 11, 1],[-5, -1, 1, 9, 1, -5; -30, 24, 9, 68, 6, -38; -1111, -264, 218, 1980, 222, -1100],32,29]
%
% END OF LATTICE L33.29
%
\hbox{}\par\smallskip%
%
% LATTICE L33.30
%
\leavevmode%
\llap{\spacer}%
${L_{33.30}}=\pdualwitharg{3.11}({L_{33.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{132}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12540\\-21120\\2508
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21120\\-35376\\4191
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2508\\4191\\-496
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-12\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-26\\-154
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\3\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\92\\528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\27\\157
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\73\\429
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-66\\-396\\-66\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\44\\0\\-132\\-44\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\0\\6\\0\\-12\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-132\\0\\132\\0\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-44\\-12\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-132\\-66\\-132\\0\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12540, -21120, 2508; -21120, -35376, 4191; 2508, 4191, -496],[132, 33, 1],[7, 13, -2, -49, -14, -37; -12, -26, 3, 92, 27, 73; -66, -154, 15, 528, 157, 429],32,30]
%
% END OF LATTICE L33.30
%
\hbox{}\par\smallskip%
%
% LATTICE L33.31
%
\leavevmode%
\llap{\spacer}%
${L_{33.31}}=\pdualwitharg{2.3.11}(\pfillwitharg{2}({L_{33.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{66}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34279014\\-22877118\\-10570560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22877118\\-15267648\\-7054542
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10570560\\-7054542\\-3259609
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3687\\-7986
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\615\\-1331
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1332\\2886
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5500\\11913
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2221\\4810
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3658\\7920
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-66\\-198\\-66\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\0\\-33\\-22\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\0\\6\\0\\-12\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-33\\0\\33\\0\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-22\\-12\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-66\\-66\\-66\\0\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-34279014, -22877118, -10570560; -22877118, -15267648, -7054542; -10570560, -7054542, -3259609],[66, 66, 1],[2, 0, -1, -3, -1, -1; 3687, 615, -1332, -5500, -2221, -3658; -7986, -1331, 2886, 11913, 4810, 7920],32,31]
%
% END OF LATTICE L33.31
%
\hbox{}\par\smallskip%
%
% LATTICE L33.32
%
\leavevmode%
\llap{\spacer}%
${L_{33.32}}=\pdualwitharg{3.11}(\main({L_{33.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{}{2}{44}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-528\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-2046\\-1023
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-1023\\-511
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\20\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\5\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\0\\-33\\-198\\-33\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\44\\0\\-132\\-44\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\6\\0\\-12\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-132\\0\\132\\0\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-44\\-12\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-132\\-66\\-132\\0\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-132, -528, -264; -528, -2046, -1023; -264, -1023, -511],[132, 33, 1],[0, 5, 3, 7, 0, -4; 16, 20, 5, -2, -1, 1; -33, -44, -12, 0, 2, 0],32,32]
%
% END OF LATTICE L33.32
%
\hbox{}\par\smallskip%
%
% LATTICE L33.33
%
\leavevmode%
\llap{\spacer}%
${L_{33.33}}=\pdualwitharg{2.11}(\main({L_{33.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{88}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101112\\-22308\\-41316
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22308\\8492\\11792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41316\\11792\\18889
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1402\\-4321\\5764
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1501\\-4626\\6171
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
647\\-1994\\2660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
305\\-940\\1254
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\-108\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107\\-330\\440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\-44\\-132\\-132\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\33\\0\\-66\\-132\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\0\\8\\0\\-48\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-66\\0\\44\\0\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-132\\-48\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-132\\-88\\-88\\0\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[101112, -22308, -41316; -22308, 8492, 11792; -41316, 11792, 18889],[132, 44, 1],[1402, 1501, 647, 305, 35, 107; -4321, -4626, -1994, -940, -108, -330; 5764, 6171, 2660, 1254, 144, 440],32,33]
%
% END OF LATTICE L33.33
%
\hbox{}\par\smallskip%
%
% LATTICE L33.34
%
\leavevmode%
\llap{\spacer}%
${L_{33.34}}=\pdualwitharg{2.11}({L_{33.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
507408\\-43692\\-156684
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43692\\3784\\13508
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156684\\13508\\48395
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\81\\-110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\51\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\129\\-172
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\238\\-319
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-232\\705\\-948
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-194\\587\\-792
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\-44\\-66\\-132\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\132\\0\\-66\\-264\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\0\\8\\0\\-48\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-66\\0\\11\\0\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-264\\-48\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-264\\-88\\-44\\0\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[507408, -43692, -156684; -43692, 3784, 13508; -156684, 13508, 48395],[132, 44, 1],[-27, -16, -42, -78, -232, -194; 81, 51, 129, 238, 705, 587; -110, -66, -172, -319, -948, -792],32,34]
%
% END OF LATTICE L33.34
%
\hbox{}\par\smallskip%
%
% LATTICE L33.35
%
\leavevmode%
\llap{\spacer}%
${L_{33.35}}=\pdualwitharg{2.3}({L_{33.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{264}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{88}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\2904\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2904\\-15816\\1440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\1440\\-131
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-11\\-132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-11\\-176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-1\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-264\\-72\\-264\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12\\-88\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\264\\0\\-528\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-12\\0\\12\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-88\\-528\\0\\88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\-264\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-528, 2904, -264; 2904, -15816, 1440; -264, 1440, -131],[264, 24, 1],[-1, -1, 4, 4, 25, 6; 2, 0, -11, -3, -11, -1; 24, 2, -132, -42, -176, -24],32,35]
%
% END OF LATTICE L33.35
%
\hbox{}\par\smallskip%
%
% LATTICE L33.36
%
\leavevmode%
\llap{\spacer}%
${L_{33.36}}=\pdualwitharg{2.3}({L_{33.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{264}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{88}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36451536\\69432\\138600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69432\\-120\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
138600\\-264\\-527
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-263
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-11\\1056
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-3\\1842
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\-11\\12364
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-1\\3156
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-264\\-72\\-264\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-6\\-44\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\264\\0\\-528\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-6\\0\\12\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-44\\-528\\0\\88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\-264\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36451536, 69432, 138600; 69432, -120, -264; 138600, -264, -527],[264, 24, 1],[-1, -1, 4, 7, 47, 12; 2, 0, -11, -3, -11, -1; -264, -263, 1056, 1842, 12364, 3156],32,36]
%
% END OF LATTICE L33.36
%
\hbox{}\par\smallskip%
%
% LATTICE L33.37
%
\leavevmode%
\llap{\spacer}%
${L_{33.37}}=\pdualwitharg{3.11}({L_{33.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{b}{2}{88}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{s}{2}{264}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93192\\53064\\4488
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53064\\-30030\\-2541
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4488\\-2541\\-215
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-15\\198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\28\\-352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\18\\-234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\28\\-396
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\47
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-29\\363
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\0\\-66\\-396\\-33\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\88\\0\\-264\\-44\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\0\\12\\0\\-12\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-264\\0\\264\\0\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-44\\-12\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-132\\-66\\-132\\0\\33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-93192, 53064, 4488; 53064, -30030, -2541; 4488, -2541, -215],[264, 33, 1],[1, -1, -1, -3, 0, 1; -15, 28, 18, 28, -4, -29; 198, -352, -234, -396, 47, 363],32,37]
%
% END OF LATTICE L33.37
%
\hbox{}\par\smallskip%
%
% LATTICE L33.38
%
\leavevmode%
\llap{\spacer}%
${L_{33.38}}=\pdualwitharg{3.11}({L_{33.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{l}{2}{88}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{264}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17668200\\169224\\134376
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
169224\\-1617\\-1287
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
134376\\-1287\\-1022
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\32\\-2145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\40\\-2024
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\-138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-4\\1188
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\134
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\2\\-660
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\0\\-33\\-396\\-66\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\88\\0\\-264\\-88\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\3\\0\\-12\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-264\\0\\264\\0\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-88\\-12\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-264\\-66\\-264\\0\\132
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17668200, 169224, 134376; 169224, -1617, -1287; 134376, -1287, -1022],[264, 33, 1],[-16, -15, -1, 9, 1, -5; 32, 40, 5, -4, -2, 2; -2145, -2024, -138, 1188, 134, -660],32,38]
%
% END OF LATTICE L33.38
%
\hbox{}\par\smallskip%
%
% LATTICE L33.39
%
\leavevmode%
\llap{\spacer}%
${L_{33.39}}=\pdualwitharg{2.3.11}(\main({L_{33.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{264}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2244\\3300\\792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3300\\25080\\6204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
792\\6204\\1535
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-559\\2244
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-200\\803
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-257\\1032
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-115\\462
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-33\\132
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-132\\-396\\-132\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\0\\-66\\-44\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\24\\0\\-48\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-66\\0\\132\\0\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-44\\-48\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-132\\-264\\-264\\0\\264
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2244, 3300, 792; 3300, 25080, 6204; 792, 6204, 1535],[132, 132, 1],[31, 11, 14, 6, 0, 2; -559, -200, -257, -115, -1, -33; 2244, 803, 1032, 462, 4, 132],32,39]
%
% END OF LATTICE L33.39
%
\hbox{}\par\smallskip%
%
% LATTICE L33.40
%
\leavevmode%
\llap{\spacer}%
${L_{33.40}}=\pdualwitharg{2.3.11}({L_{33.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70224\\-555852\\-135828
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-555852\\4399824\\1075140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135828\\1075140\\262721
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-16\\66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\5\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\43\\-192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\66\\-297
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\51\\-232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93\\85\\-396
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-132\\-198\\-132\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\44\\0\\-66\\-88\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\24\\0\\-48\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-66\\0\\33\\0\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-88\\-48\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-264\\-264\\-132\\0\\264
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[70224, -555852, -135828; -555852, 4399824, 1075140; -135828, 1075140, 262721],[132, 132, 1],[1, -3, -31, -52, -45, -93; -16, 5, 43, 66, 51, 85; 66, -22, -192, -297, -232, -396],32,40]
%
% END OF LATTICE L33.40
%
\hbox{}\par\smallskip%
%
% LATTICE L33.41
%
\leavevmode%
\llap{\spacer}%
${L_{33.41}}=\pdualwitharg{2.11}({L_{33.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{44}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{88}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2891856\\2854104\\-795432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2854104\\-2816792\\785048
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-795432\\785048\\-218791
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
123\\2\\-440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
118\\0\\-429
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-1\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-3\\1298
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-574\\-3\\2076
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-473\\-1\\1716
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
176\\0\\-88\\-264\\-264\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\33\\0\\-66\\-132\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\0\\8\\0\\-48\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-66\\0\\44\\0\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-132\\-48\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-132\\-88\\-88\\0\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2891856, 2854104, -795432; 2854104, -2816792, 785048; -795432, 785048, -218791],[264, 88, 1],[123, 118, -45, -360, -574, -473; 2, 0, -1, -3, -3, -1; -440, -429, 160, 1298, 2076, 1716],32,41]
%
% END OF LATTICE L33.41
%
\hbox{}\par\smallskip%
%
% LATTICE L33.42
%
\leavevmode%
\llap{\spacer}%
${L_{33.42}}=\pdualwitharg{2.11}({L_{33.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{44}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{88}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276975336\\248842704\\-4341480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
248842704\\-223567432\\3900512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4341480\\3900512\\-68051
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
835\\-1\\-53328
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
917\\-3\\-58674
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
184\\-1\\-11796
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\-1\\-2354
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-79\\0\\5040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\1\\2992
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
176\\0\\-88\\-264\\-264\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\132\\0\\-132\\-264\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\0\\8\\0\\-48\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-132\\0\\44\\0\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-264\\-48\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-264\\-88\\-88\\0\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-276975336, 248842704, -4341480; 248842704, -223567432, 3900512; -4341480, 3900512, -68051],[264, 88, 1],[835, 917, 184, 36, -79, -46; -1, -3, -1, -1, 0, 1; -53328, -58674, -11796, -2354, 5040, 2992],32,42]
%
% END OF LATTICE L33.42
%
\hbox{}\par\smallskip%
%
% LATTICE L33.43
%
\leavevmode%
\llap{\spacer}%
${L_{33.43}}=\pdualwitharg{2.3.11}({L_{33.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${528}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{s}{2}{132}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{264}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-173976\\2252184\\18744
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2252184\\-29119728\\-242352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18744\\-242352\\-2017
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-35\\4224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-13\\1562
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\16\\-1932
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\86\\-10362
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\41\\-4936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\90\\-10824
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\0\\-264\\-792\\-264\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\44\\0\\-132\\-88\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\24\\0\\-48\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-792\\-132\\0\\132\\0\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-88\\-48\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-264\\-264\\-264\\0\\264
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-173976, 2252184, 18744; 2252184, -29119728, -242352; 18744, -242352, -2017],[264, 264, 1],[2, 0, -1, -3, -1, -1; -35, -13, 16, 86, 41, 90; 4224, 1562, -1932, -10362, -4936, -10824],32,43]
%
% END OF LATTICE L33.43
%
\hbox{}\par\smallskip%
%
% LATTICE L33.44
%
\leavevmode%
\llap{\spacer}%
${L_{33.44}}=\pdualwitharg{2.3.11}({L_{33.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${528}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{132}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{264}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\66792\\-30624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66792\\-3066096\\1405800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30624\\1405800\\-644557
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-121\\-264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-116\\-253
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\44\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\817\\1782
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\497\\1084
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1392\\3036
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\0\\-264\\-792\\-264\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\0\\-66\\-44\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\24\\0\\-48\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-792\\-66\\0\\132\\0\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-44\\-48\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-132\\-264\\-264\\0\\264
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1320, 66792, -30624; 66792, -3066096, 1405800; -30624, 1405800, -644557],[264, 264, 1],[2, 0, -1, -3, -1, -1; -121, -116, 44, 817, 497, 1392; -264, -253, 96, 1782, 1084, 3036],32,44]
%
% END OF LATTICE L33.44
%

\medskip%
%------------------------------WEYL-GROUP-W34--------------------------------
%
\leavevmode\llap{}%
$W_{34\phantom{0}}$%
\qquad\llap{88} lattices, $\chi=72$%
\hfill%
$\infty222222\infty222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L34.1
%
\leavevmode%
\llap{\spacer}%
${L_{34.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{23\rightarrow N_{34},3,2}%
\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{3,2}{\infty b}{132}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{396}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6574916700\\52718688\\1769328
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52718688\\-422706\\-14187
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1769328\\-14187\\-476
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-6574916700, 52718688, 1769328; 52718688, -422706, -14187; 1769328, -14187, -476],[396, 3, 1],[-311, 3, 31, 5, -41, -3505, -209, -789, -553, -106, -55, -84, -3095, -141; -36388, 352, 3627, 584, -4801, -410322, -24466, -92356, -64724, -12405, -6436, -9829, -362142, -16498; -71478, 660, 7128, 1180, -9306, -798732, -47662, -180114, -126456, -24282, -12616, -19284, -710820, -32390],33,1]
%
% END OF LATTICE L34.1
%
\hbox{}\par\smallskip%
%
% LATTICE L34.2
%
\leavevmode%
\llap{\spacer}%
${L_{34.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{3m,3,2}%
\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{12,11}{\infty a}{264}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{792}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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-2375208\\-49104\\-50688
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49104\\-1014\\-1047
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\kern3pt%
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-50688\\-1047\\-1081
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2375208, -49104, -50688; -49104, -1014, -1047; -50688, -1047, -1081],[792, 3, 1],[-5, -1, 1, 1, 1, 83, 2, 5, 1, -1, -1, -1, -83, -2; -781, -88, 156, 104, 53, 792, -21, -319, -1012, -306, -204, -178, -13992, -329; 990, 132, -198, -148, -99, -4752, -76, 66, 924, 342, 244, 219, 17424, 412],33,2]
%
% END OF LATTICE L34.2
%
\hbox{}\par\smallskip%
%
% LATTICE L34.3
%
\leavevmode%
\llap{\spacer}%
${L_{34.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{32\rightarrow N'_{21},3,m}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-35211528, 6384312, -1161864; 6384312, -1157559, 210660; -1161864, 210660, -38335],[792, 3, 1],[1076, -95, -98, 71, 673, 46759, 1336, 4732, 5903, 494, 457, 647, 22937, 512; 6457, -572, -588, 428, 4046, 281028, 8029, 28435, 35464, 2967, 2744, 3884, 137676, 3073; 2871, -264, -261, 200, 1836, 127116, 3629, 12837, 15972, 1332, 1228, 1734, 61380, 1369],33,3]
%
% END OF LATTICE L34.3
%
\hbox{}\par\smallskip%
%
% LATTICE L34.4
%
\leavevmode%
\llap{\spacer}%
${L_{34.4}}=\pfillwitharg{2.3}({L_{34.1}})$%
{${}={}$}Nikulin $34$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{2,1}{\infty}{33}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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33\\0\\0
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\kern3pt%
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0\\0\\-1
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-23\\-66\\-132
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\kern3pt%
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4\\11\\24
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\kern3pt%
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2\\6\\11
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\HardButStrongLineBreak\kern3pt%
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1\\0\\0
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\HardButStrongLineBreak\kern3pt%
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0\\-1\\1
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\HardButStrongLineBreak\kern3pt%
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-1\\-4\\-4
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\HardButStrongLineBreak\kern3pt%
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-8\\-27\\-39
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[33, 0, 0; 0, 0, -1; 0, -1, 0],[33, 1, 1],[-1, 1, 0, -1, -8, -79, -13, -43, -23, -2, -1, -2, -9, -1; 0, 0, -1, -4, -27, -253, -41, -132, -66, -5, -2, -3, -11, -1; -33, 0, 1, -4, -39, -407, -68, -231, -132, -13, -8, -21, -121, -16],33,4]
%
% END OF LATTICE L34.4
%
\hbox{}\par\smallskip%
%
% LATTICE L34.5
%
\leavevmode%
\llap{\spacer}%
${L_{34.5}}=\pfillwitharg{3.2}({L_{34.3}})$%
{${}={}$}Nikulin $21'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{2,1}{\infty b}{66}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{22}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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66\\0\\66
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0\\960\\-25889
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\kern3pt%
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66\\-25889\\698233
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6049\\-163746\\-6072
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[66, 0, 66; 0, 960, -25889; 66, -25889, 698233],[66, 1, 1],[11111, 6049, 274, 303, 362, 3973, 511, 989, 1, 1, 247, 1013, 20227, 3339; -300795, -163746, -7416, -8198, -9789, -107382, -13807, -26697, 0, -27, -6688, -27426, -547602, -90395; -11154, -6072, -275, -304, -363, -3982, -512, -990, 0, -1, -248, -1017, -20306, -3352],33,5]
%
% END OF LATTICE L34.5
%
\hbox{}\par\smallskip%
%
% LATTICE L34.6
%
\leavevmode%
\llap{\spacer}%
${L_{34.6}}=\main(\pfillwitharg{3}({L_{34.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-8580, 132, 2244; 132, -2, -33; 2244, -33, -511],[132, 1, 1],[28, 43, 3, 5, 8, 101, 7, 16, 1, -1, -1, 2, 75, 7; 3300, 4818, 321, 518, 813, 10164, 701, 1584, 66, -99, -74, 297, 9372, 853; -99, -132, -8, -12, -18, -220, -15, -33, 0, 2, 0, -12, -308, -27],33,6]
%
% END OF LATTICE L34.6
%
\hbox{}\par\smallskip%
%
% LATTICE L34.7
%
\leavevmode%
\llap{\spacer}%
${L_{34.7}}=\pfillwitharg{3}({L_{34.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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6049\\-508200\\28205100
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-8363388, 3417348, 63360; 3417348, -1396334, -25889; 63360, -25889, -480],[132, 1, 1],[-43, -23, -1, -1, -1, -9, -1, -1, 1, 0, -1, -4, -79, -13; 2706, 1848, 115, 176, 267, 3278, 448, 990, 0, -31, -8, 153, 4114, 728; -151602, -102696, -6334, -9624, -14532, -177980, -24294, -53526, 132, 1672, 300, -8778, -232276, -40974],33,7]
%
% END OF LATTICE L34.7
%
\hbox{}\par\smallskip%
%
% LATTICE L34.8
%
\leavevmode%
\llap{\spacer}%
${L_{34.8}}=\pfillwitharg{3}({L_{34.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{4,3}{\infty a}{264}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{88}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-19009320\\4488\\139392
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-19009320, 4488, 139392; 4488, -1, -33; 139392, -33, -1022],[264, 1, 1],[28, 43, 3, 5, 4, 101, 7, 16, 1, -1, -1, 1, 75, 7; 6600, 9636, 642, 1036, 813, 20328, 1402, 3168, 132, -198, -148, 297, 18744, 1706; 3597, 5544, 388, 648, 519, 13112, 909, 2079, 132, -130, -132, 126, 9592, 897],33,8]
%
% END OF LATTICE L34.8
%
\hbox{}\par\smallskip%
%
% LATTICE L34.9
%
\leavevmode%
\llap{\spacer}%
${L_{34.9}}=\pfillwitharg{3}({L_{34.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{4,3}{\infty b}{264}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-17160\\-17160\\8712
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\kern3pt%
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-17160\\-17094\\8645
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8712\\8645\\-4355
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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-27281\\54120\\52800
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-95\\189\\185
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-454\\907\\892
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-17160, -17160, 8712; -17160, -17094, 8645; 8712, 8645, -4355],[264, 1, 1],[-1688, -2663, -95, -321, -517, -6543, -454, -1040, -65, 33, 73, -103, -4369, -414; 3333, 5280, 189, 640, 1032, 13068, 907, 2079, 132, -66, -148, 198, 8580, 815; 3234, 5148, 185, 628, 1014, 12848, 892, 2046, 132, -65, -148, 186, 8272, 788],33,9]
%
% END OF LATTICE L34.9
%
\hbox{}\par\smallskip%
%
% LATTICE L34.10
%
\leavevmode%
\llap{\spacer}%
${L_{34.10}}=\pfillwitharg{2}({L_{34.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{6,5}{\infty}{33}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{99}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-117216\\-9207\\-495
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\kern3pt%
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-9207\\-723\\-39
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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2507\\-28842\\-52668
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-117216, -9207, -495; -9207, -723, -39; -495, -39, -2],[99, 3, 1],[-76, -45, -14, -3, -8, -137, -6, -12, 1, 2, -1, -12, -391, -22; 869, 517, 162, 35, 94, 1617, 71, 143, -11, -24, 11, 136, 4455, 251; 1683, 957, 279, 55, 135, 2178, 92, 165, -33, -27, 29, 285, 8910, 496],33,10]
%
% END OF LATTICE L34.10
%
\hbox{}\par\smallskip%
%
% LATTICE L34.11
%
\leavevmode%
\llap{\spacer}%
${L_{34.11}}=\pfillwitharg{2}({L_{34.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-728678610, 12410046, 832986; 12410046, -211353, -14187; 832986, -14187, -952],[198, 3, 1],[-789, -553, -106, -55, -84, -3095, -141, -311, 3, 31, 5, -41, -3505, -209; -40282, -28226, -5409, -2806, -4285, -157872, -7192, -15862, 154, 1581, 254, -2095, -178992, -10672; -90057, -63228, -12141, -6308, -9642, -355410, -16195, -35739, 330, 3564, 590, -4653, -399366, -23831],33,11]
%
% END OF LATTICE L34.11
%
\hbox{}\par\smallskip%
%
% LATTICE L34.12
%
\leavevmode%
\llap{\spacer}%
${L_{34.12}}=\main({L_{34.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{12,11}{\infty}{132}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{396}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1187604, -24552, -26136; -24552, -507, -540; -26136, -540, -575],[396, 3, 1],[-5, -1, 1, 1, 2, 83, 2, 5, 1, -1, -1, -2, -83, -2; -286, -22, 57, 30, 7, -1584, -59, -286, -550, -135, -82, -137, -5280, -123; 495, 66, -99, -74, -99, -2376, -38, 33, 462, 171, 122, 219, 8712, 206],33,12]
%
% END OF LATTICE L34.12
%
\hbox{}\par\smallskip%
%
% LATTICE L34.13
%
\leavevmode%
\llap{\spacer}%
${L_{34.13}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{34.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-132033, 891, 48906; 891, -6, -330; 48906, -330, -18115],[33, 3, 1],[-205, -114, -32, -18, -14, -218, -27, -15, 4, 2, -12, -36, -1102, -183; 957, 583, 189, 127, 117, 2046, 272, 187, -11, -33, 29, 143, 4818, 820; -572, -319, -90, -51, -40, -627, -78, -44, 11, 6, -33, -100, -3069, -510],33,13]
%
% END OF LATTICE L34.13
%
\hbox{}\par\smallskip%
%
% LATTICE L34.14
%
\leavevmode%
\llap{\spacer}%
${L_{34.14}}=\pdualwitharg{2}(\pfillwitharg{3.2}({L_{34.3}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18817788, 5126022, -9313854; 5126022, -1396334, 2537122; -9313854, 2537122, -4609887],[66, 2, 1],[77177, 26042, 3199, 2422, 7298, 44639, 12179, 26795, -49, -836, -59, 4517, 59333, 20903; 2706, 924, 115, 88, 267, 1639, 448, 990, 0, -31, -4, 153, 2057, 728; -154440, -52107, -6400, -4845, -14598, -89287, -24360, -53592, 99, 1672, 117, -9042, -118745, -41832],33,14]
%
% END OF LATTICE L34.14
%
\hbox{}\par\smallskip%
%
% LATTICE L34.15
%
\leavevmode%
\llap{\spacer}%
${L_{34.15}}=\pdualwitharg{3}(\pfillwitharg{3.2}({L_{34.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{2,1}{\infty b}{22}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{66}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-22302522\\6834696\\-7349694
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\kern3pt%
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6834696\\-2094501\\2252343
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\kern3pt%
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{$\left.\llap{\phantom{%
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3188\\115\\-9567
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4833\\176\\-14502
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\HardButStrongLineBreak\kern3pt%
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2429\\89\\-7288
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89179\\3278\\-267564
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-22302522, 6834696, -7349694; 6834696, -2094501, 2252343; -7349694, 2252343, -2422058],[66, 3, 1],[25568, 17277, 3188, 4833, 2429, 89179, 12168, 8928, -29, -836, -129, 1491, 117797, 20760; 902, 616, 115, 176, 89, 3278, 448, 330, 0, -31, -8, 51, 4114, 728; -76747, -51854, -9567, -14502, -7288, -267564, -36507, -26785, 88, 2508, 384, -4477, -353628, -62319],33,15]
%
% END OF LATTICE L34.15
%
\hbox{}\par\smallskip%
%
% LATTICE L34.16
%
\leavevmode%
\llap{\spacer}%
${L_{34.16}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{34.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{6,1}{\infty}{33}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-363330\\-30492\\-14157
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\kern3pt%
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-30492\\-2559\\-1188
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\kern3pt%
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-14157\\-1188\\-544
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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62369\\-746130\\6930
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-363330, -30492, -14157; -30492, -2559, -1188; -14157, -1188, -544],[99, 3, 1],[-1000, -606, -65, -130, -119, -691, -275, -188, 12, 11, -32, -151, -1685, -859; 11957, 7249, 778, 1557, 1426, 8283, 3297, 2255, -143, -132, 381, 1804, 20141, 10269; -99, -66, -8, -18, -18, -110, -45, -33, 0, 2, 0, -12, -154, -81],33,16]
%
% END OF LATTICE L34.16
%
\hbox{}\par\smallskip%
%
% LATTICE L34.17
%
\leavevmode%
\llap{\spacer}%
${L_{34.17}}=\pdualwitharg{11}(\pfillwitharg{2.3}({L_{34.1}}))$%
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{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{2,1}{\infty}{3}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19437, 10890, 7128; 10890, -5984, -3993; 7128, -3993, -2614],[33, 11, 1],[1001, 613, 734, 497, 1382, 735, 1077, 203, -11, -132, 105, 1628, 1673, 3137; -9, -6, -8, -6, -18, -10, -15, -3, 0, 2, 0, -12, -14, -27; 2742, 1680, 2013, 1364, 3795, 2019, 2959, 558, -30, -363, 286, 4455, 4581, 8591],33,17]
%
% END OF LATTICE L34.17
%
\hbox{}\par\smallskip%
%
% LATTICE L34.18
%
\leavevmode%
\llap{\spacer}%
${L_{34.18}}=\pdualwitharg{3}(\main(\pfillwitharg{3}({L_{34.2}})))$%
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{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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1584\\-6\\-33
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-395868, 1584, 8184; 1584, -6, -33; 8184, -33, -169],[132, 3, 1],[5, 1, -1, -3, -2, -83, -6, -5, -1, 1, 3, 2, 83, 6; 165, 22, -33, -74, -33, -792, -38, 11, 154, 57, 122, 73, 2904, 206; 209, 44, -42, -132, -92, -3960, -291, -253, -88, 36, 120, 82, 3432, 249],33,18]
%
% END OF LATTICE L34.18
%
\hbox{}\par\smallskip%
%
% LATTICE L34.19
%
\leavevmode%
\llap{\spacer}%
${L_{34.19}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{34.1}}))$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2038702644, 1706496, 16948668; 1706496, -1428, -14187; 16948668, -14187, -140902],[132, 3, 1],[-311, 3, 31, 15, -41, -3505, -627, -789, -553, -106, -165, -84, -3095, -423; -10142, 88, 1012, 520, -1298, -112024, -20074, -25322, -17820, -3430, -5356, -2732, -100760, -13778; -36388, 352, 3627, 1752, -4801, -410322, -73398, -92356, -64724, -12405, -19308, -9829, -362142, -49494],33,19]
%
% END OF LATTICE L34.19
%
\hbox{}\par\smallskip%
%
% LATTICE L34.20
%
\leavevmode%
\llap{\spacer}%
${L_{34.20}}=\pdualwitharg{2.3}(\pfillwitharg{3.2}({L_{34.3}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12673841334, 8543370, 6221703840; 8543370, 5760, 4194018; 6221703840, 4194018, 3054290933],[66, 6, 1],[-797048, -216959, -58963, -32599, -25961, -427336, -109918, -70896, -27, -215, -26579, -72669, -2176496, -718574; -100265, -27291, -7416, -4099, -3263, -53691, -13807, -8899, 0, -27, -3344, -9142, -273801, -90395; 1623754, 441991, 120120, 66411, 52888, 870573, 223926, 144430, 55, 438, 54147, 148042, 4433979, 1463886],33,20]
%
% END OF LATTICE L34.20
%
\hbox{}\par\smallskip%
%
% LATTICE L34.21
%
\leavevmode%
\llap{\spacer}%
${L_{34.21}}=\pdualwitharg{2}(\main(\pfillwitharg{3}({L_{34.2}})))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[78936, 39468, 19536; 39468, 19732, 9768; 19536, 9768, 4835],[132, 4, 1],[1975, 1061, 93, 47, 95, 215, 48, 49, -49, -1, 45, 365, 1809, 596; 363, 198, 18, 10, 24, 66, 17, 33, 0, 0, 8, 66, 330, 109; -8712, -4686, -412, -210, -432, -1001, -228, -264, 198, 4, -198, -1608, -7975, -2628],33,21]
%
% END OF LATTICE L34.21
%
\hbox{}\par\smallskip%
%
% LATTICE L34.22
%
\leavevmode%
\llap{\spacer}%
${L_{34.22}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{34.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6082971576, 3417348, 1474659648; 3417348, 1920, 828448; 1474659648, 828448, 357493217],[132, 4, 1],[-172199, -46874, -8493, -2348, -11221, -61577, -7920, -15329, -8, -31, -1914, -31399, -313479, -51748; -300795, -81873, -14832, -4099, -19578, -107382, -13807, -26697, 0, -54, -3344, -54852, -547602, -90395; 711018, 193545, 35068, 9695, 46332, 254254, 32702, 63294, 33, 128, 7903, 129648, 1294370, 213670],33,22]
%
% END OF LATTICE L34.22
%
\hbox{}\par\smallskip%
%
% LATTICE L34.23
%
\leavevmode%
\llap{\spacer}%
${L_{34.23}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{34.2}}))$%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-333745038, -32039172, 9040086; -32039172, -3075720, 867813; 9040086, 867813, -244769],[198, 3, 1],[59531, 40371, 2493, 11375, 5729, 70189, 28747, 21121, -45, -660, -377, 3436, 91115, 48239; -669713, -454168, -28046, -127968, -64451, -789624, -323403, -237611, 506, 7425, 4242, -38654, -1025024, -542679; -175758, -119196, -7361, -33588, -16917, -207262, -84888, -62370, 132, 1949, 1116, -10143, -268994, -142416],33,23]
%
% END OF LATTICE L34.23
%
\hbox{}\par\smallskip%
%
% LATTICE L34.24
%
\leavevmode%
\llap{\spacer}%
${L_{34.24}}=\pdualwitharg{11}(\pfillwitharg{3.2}({L_{34.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-548017734\\-2405634\\93741912
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-2405634\\-10560\\411499
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-548017734, -2405634, 93741912; -2405634, -10560, 411499; 93741912, 411499, -16035149],[66, 11, 1],[449, 313, 219, 341, 523, 587, 885, 179, 1, -62, -27, 262, 669, 1313; -6891, -4668, -3167, -4812, -7266, -8090, -12147, -2433, 6, 836, 150, -4389, -10558, -20487; 2448, 1710, 1199, 1870, 2871, 3224, 4862, 984, 6, -341, -154, 1419, 3640, 7150],33,24]
%
% END OF LATTICE L34.24
%
\hbox{}\par\smallskip%
%
% LATTICE L34.25
%
\leavevmode%
\llap{\spacer}%
${L_{34.25}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{34.2}}))$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{4,3}{\infty a}{88}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{264}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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1584\\-3\\-33
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16368\\-33\\-338
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-791736, 1584, 16368; 1584, -3, -33; 16368, -33, -338],[264, 3, 1],[5, 1, -1, -3, -1, -83, -6, -5, -1, 1, 3, 1, 83, 6; 330, 44, -66, -148, -33, -1584, -76, 22, 308, 114, 244, 73, 5808, 412; 209, 44, -42, -132, -46, -3960, -291, -253, -88, 36, 120, 41, 3432, 249],33,25]
%
% END OF LATTICE L34.25
%
\hbox{}\par\smallskip%
%
% LATTICE L34.26
%
\leavevmode%
\llap{\spacer}%
${L_{34.26}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{34.3}}))$%
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{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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3920399\\-21431520\\23504976
\end{tabular}\endgroup%
\kern3pt%
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650025\\-3553471\\3897261
\end{tabular}\endgroup%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\end{tabular}\endgroup%
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\end{tabular}\endgroup%
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-163\\891\\-978
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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387\\-2116\\2316
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1163\\-6358\\6970
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11737176, -1932480, 195624; -1932480, -318153, 32229; 195624, 32229, -3242],[264, 3, 1],[1795, -169, -163, 387, 1163, 80357, 6879, 8105, 10069, 838, 2313, 1087, 38443, 2571; -9812, 924, 891, -2116, -6358, -439296, -37606, -44308, -55044, -4581, -12644, -5942, -210144, -14054; 10769, -1012, -978, 2316, 6970, 481668, 41235, 48587, 60368, 5025, 13872, 6520, 230604, 15423],33,26]
%
% END OF LATTICE L34.26
%
\hbox{}\par\smallskip%
%
% LATTICE L34.27
%
\leavevmode%
\llap{\spacer}%
${L_{34.27}}=\pdualwitharg{3.11}(\pfillwitharg{2.3}({L_{34.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{2,1}{\infty}{1}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\-1089\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1089\\19041\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-33\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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293\\18\\528
\end{tabular}\endgroup%
\kern3pt%
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98\\6\\175
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\11\\320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
98\\6\\175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
293\\18\\528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
162\\10\\297
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
129\\8\\242
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
193\\12\\369
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
273\\17\\528
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-99\\-33\\0\\33\\0\\-99\\-231
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[66, -1089, 0; -1089, 19041, -33; 0, -33, 1],[33, 33, 1],[180, 98, 293, 162, 129, 193, 273, 16, 0, 1, 132, 361, 983, 1785; 11, 6, 18, 10, 8, 12, 17, 1, 0, 0, 8, 22, 60, 109; 320, 175, 528, 297, 242, 369, 528, 32, 1, 0, 231, 638, 1743, 3168],33,27]
%
% END OF LATTICE L34.27
%
\hbox{}\par\smallskip%
%
% LATTICE L34.28
%
\leavevmode%
\llap{\spacer}%
${L_{34.28}}=\pdualwitharg{3}(\main({L_{34.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{12,7}{\infty}{132}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7524\\-2376\\1980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2376\\-750\\627
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1980\\627\\-511
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2113\\6072\\-792
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-656\\1885\\-246
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
608\\-1748\\227
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\176\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-22\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-11\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\42\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\121\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-949\\2772\\-308
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-244\\711\\-81
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7524, -2376, 1980; -2376, -750, 627; 1980, 627, -511],[396, 3, 1],[-64, 7, 4, -13, -41, -949, -244, -288, -359, -20, -83, -39, -459, -92; 176, -22, -11, 42, 121, 2772, 711, 836, 1034, 57, 234, 109, 1276, 255; -33, 0, 2, 0, -12, -308, -81, -99, -132, -8, -36, -18, -220, -45],33,28]
%
% END OF LATTICE L34.28
%
\hbox{}\par\smallskip%
%
% LATTICE L34.29
%
\leavevmode%
\llap{\spacer}%
${L_{34.29}}=\pdualwitharg{3}({L_{34.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L34.29
%
\hbox{}\par\smallskip%
%
% LATTICE L34.30
%
\leavevmode%
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${L_{34.30}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{34.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26491880646, -362445534, 13109774040; -362445534, 4958772, -179359824; 13109774040, -179359824, 6487503763],[198, 6, 1],[-4804960, -1302243, -351019, -63897, -150673, -2459596, -209826, -400680, 833, -3911, -54413, -440877, -13157324, -1446514; -55363, -15004, -4044, -736, -1735, -28314, -2415, -4609, 11, -45, -627, -5080, -151602, -16667; 9708204, 2631123, 709218, 129101, 304428, 4969503, 423944, 809556, -1683, 7902, 109939, 890772, 26583777, 2922616],33,30]
%
% END OF LATTICE L34.30
%
\hbox{}\par\smallskip%
%
% LATTICE L34.31
%
\leavevmode%
\llap{\spacer}%
${L_{34.31}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{34.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[209269820628, -14403708198, 104004768582; -14403708198, 991384278, -7158482448; 104004768582, -7158482448, 51689210873],[198, 6, 1],[-3140669, -854544, -77341, -128008, -101612, -555867, -428137, -274567, 735, -278, -104849, -286483, -2859369, -2831861; 214258, 58300, 5277, 8736, 6937, 37961, 29244, 18766, -44, 19, 7152, 19543, 195063, 193188; 6349068, 1727517, 156350, 258777, 205416, 1123727, 865512, 555060, -1485, 562, 211959, 579144, 5780401, 5724792],33,31]
%
% END OF LATTICE L34.31
%
\hbox{}\par\smallskip%
%
% LATTICE L34.32
%
\leavevmode%
\llap{\spacer}%
${L_{34.32}}=\pdualwitharg{11}(\main(\pfillwitharg{3}({L_{34.2}})))$%
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{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15180, -660, -792; -660, -22, -33; -792, -33, -41],[132, 11, 1],[17, 17, 7, 5, 2, -1, -2, -1, -1, 1, 11, 38, 65, 58; 99, 102, 45, 38, 27, 12, 2, -3, -6, 3, 58, 213, 372, 334; -417, -420, -176, -132, -66, 4, 33, 21, 24, -22, -264, -924, -1588, -1419],33,32]
%
% END OF LATTICE L34.32
%
\hbox{}\par\smallskip%
%
% LATTICE L34.33
%
\leavevmode%
\llap{\spacer}%
${L_{34.33}}=\pdualwitharg{11}(\pfillwitharg{3}({L_{34.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{1,0}{\infty b}{12}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-9141396\\-205524\\-2821500
\end{tabular}\endgroup%
\kern3pt%
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-205524\\-4620\\-63437
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2821500\\-63437\\-870854
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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219673\\-1406652\\-609180
\end{tabular}\endgroup%
\kern3pt%
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4879\\-31243\\-13530
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67949\\-435102\\-188431
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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223\\-1350\\-624
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\HardButStrongLineBreak\kern3pt%
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251\\-1608\\-696
\end{tabular}\endgroup%
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243\\-1596\\-671
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\HardButStrongLineBreak\kern3pt%
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447\\-2968\\-1232
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\HardButStrongLineBreak\kern3pt%
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731\\-4878\\-2013
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829\\-5544\\-2282
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1255\\-8402\\-3454
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-198\\-132\\-88\\-132\\-198\\-220\\-330
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9141396, -205524, -2821500; -205524, -4620, -63437; -2821500, -63437, -870854],[132, 11, 1],[223, 251, 243, 447, 731, 829, 1255, 253, -13, -124, -209, -136, 123, 411; -1350, -1608, -1596, -2968, -4878, -5544, -8402, -1698, 84, 834, 1444, 1068, -552, -2266; -624, -696, -671, -1232, -2013, -2282, -3454, -696, 36, 341, 572, 363, -358, -1166],33,33]
%
% END OF LATTICE L34.33
%
\hbox{}\par\smallskip%
%
% LATTICE L34.34
%
\leavevmode%
\llap{\spacer}%
${L_{34.34}}=\pdualwitharg{2.11}(\pfillwitharg{3.2}({L_{34.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{4,1}{\infty z}{3}\above{1pt}{1pt}{}{2}{22}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{}{2}{66}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{44}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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9319728\\11957814\\4437972
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11957814\\15361126\\5694194
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4437972\\5694194\\2113323
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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117940831\\-150144\\-247270848
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168455931\\-214453\\-353179134
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{$\left.\llap{\phantom{%
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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216793\\-276\\-454521
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216016\\-275\\-452892
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\HardButStrongLineBreak\kern3pt%
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119409\\-152\\-250349
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\HardButStrongLineBreak\kern3pt%
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285209\\-363\\-597960
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\HardButStrongLineBreak\kern3pt%
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142231\\-181\\-298197
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402367\\-512\\-843590
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9319728, 11957814, 4437972; 11957814, 15361126, 5694194; 4437972, 5694194, 2113323],[66, 22, 1],[796465, 216793, 216016, 119409, 285209, 142231, 402367, 70747, 10, 787, 97394, 798806, 724981, 2632875; -1014, -276, -275, -152, -363, -181, -512, -90, 0, -1, -124, -1017, -923, -3352; -1669842, -454521, -452892, -250349, -597960, -298197, -843590, -148326, -21, -1650, -204193, -1674750, -1519971, -5519998],33,34]
%
% END OF LATTICE L34.34
%
\hbox{}\par\smallskip%
%
% LATTICE L34.35
%
\leavevmode%
\llap{\spacer}%
${L_{34.35}}=\pdualwitharg{2.3}(\main(\pfillwitharg{3}({L_{34.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{4,1}{\infty}{44}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1206744\\-90684\\291588
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\kern3pt%
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\kern3pt%
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291588\\-21912\\70457
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1206744, -90684, 291588; -90684, 6924, -21912; 291588, -21912, 70457],[132, 12, 1],[1670, 101, 29, 1101, 2995, 44830, 14799, 16414, 8941, 2437, 1365, 1115, 9422, 2463; 11, 0, 0, 8, 22, 330, 109, 121, 66, 18, 10, 8, 66, 17; -6908, -418, -120, -4554, -12388, -185427, -61212, -67892, -36982, -10080, -5646, -4612, -38973, -10188],33,35]
%
% END OF LATTICE L34.35
%
\hbox{}\par\smallskip%
%
% LATTICE L34.36
%
\leavevmode%
\llap{\spacer}%
${L_{34.36}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{34.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{2,1}{\infty z}{11}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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-232049004\\6093576\\-56192076
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8836764288, -232049004, 2139876156; -232049004, 6093576, -56192076; 2139876156, -56192076, 518184011],[132, 12, 1],[-129775, 301, -2534, -52943, -285968, -8534156, -1407359, -1558285, -422316, -227656, -62152, -97682, -1594204, -203971; -154, 0, -3, -62, -335, -9999, -1649, -1826, -495, -267, -73, -115, -1881, -241; 535898, -1243, 10464, 218625, 1180888, 35241294, 5811606, 6434846, 1743929, 940092, 256653, 403372, 6583170, 842286],33,36]
%
% END OF LATTICE L34.36
%
\hbox{}\par\smallskip%
%
% LATTICE L34.37
%
\leavevmode%
\llap{\spacer}%
${L_{34.37}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{34.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${528}\above{1pt}{1pt}{8,1}{\infty z}{132}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-60572160\\-1720568\\15601920
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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62555404\\64978155\\250027800
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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43\\64\\174
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-2425\\-2442\\-9684
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2132427000, -60572160, 549261504; -60572160, -1720568, 15601920; 549261504, 15601920, -141476449],[264, 8, 1],[-14851, 17, 464, 43, -2425, -32121, -11335, -41953, -14218, -1755, -1334, -4030, -24683, -6739; -15807, 0, 495, 64, -2442, -32835, -11621, -43197, -14751, -1836, -1405, -4263, -26169, -7153; -59400, 66, 1856, 174, -9684, -128326, -45288, -167640, -56826, -7016, -5334, -16116, -98714, -26952],33,37]
%
% END OF LATTICE L34.37
%
\hbox{}\par\smallskip%
%
% LATTICE L34.38
%
\leavevmode%
\llap{\spacer}%
${L_{34.38}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{34.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9072106032, 133284888, 70813776; 133284888, -1958152, -1040392; 70813776, -1040392, -552741],[264, 8, 1],[-15805, -1, 495, 65, -2434, -16378, -11595, -43111, -14728, -1834, -1404, -4261, -13080, -7151; -504768, -33, 15809, 2077, -77727, -523028, -370286, -1376760, -470349, -58571, -44839, -136083, -417736, -228382; -1074744, -66, 33660, 4418, -165528, -1113783, -788512, -2931720, -1001550, -124716, -95474, -289752, -889449, -486272],33,38]
%
% END OF LATTICE L34.38
%
\hbox{}\par\smallskip%
%
% LATTICE L34.39
%
\leavevmode%
\llap{\spacer}%
${L_{34.39}}=\pdualwitharg{3.11}(\pfillwitharg{3.2}({L_{34.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[61172166, 37542186, 4173642; 37542186, 23041689, 2562351; 4173642, 2562351, 285320],[66, 33, 1],[-124483, -67767, -101286, -111977, -44576, -133377, -188659, -11057, -3, -369, -91333, -124849, -679863, -1234511; 248628, 135350, 202297, 223650, 89031, 266392, 376806, 22084, 6, 737, 182418, 249359, 1357880, 2465670; -411905, -224236, -335148, -370524, -147499, -441336, -624261, -36587, -10, -1221, -302214, -413116, -2249616, -4084905],33,39]
%
% END OF LATTICE L34.39
%
\hbox{}\par\smallskip%
%
% LATTICE L34.40
%
\leavevmode%
\llap{\spacer}%
${L_{34.40}}=\pdualwitharg{3}({L_{34.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{12,7}{\infty b}{264}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{88}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-157144680, -52368624, 400752; -52368624, -17451897, 133551; 400752, 133551, -1022],[792, 3, 1],[-112, 15, 7, -29, -40, -1823, -467, -548, -675, -37, -151, -35, -817, -163; 352, -44, -22, 84, 121, 5544, 1422, 1672, 2068, 114, 468, 109, 2552, 510; 2079, 132, -130, -396, 126, 9592, 2691, 3597, 5544, 388, 1944, 519, 13112, 2727],33,40]
%
% END OF LATTICE L34.40
%
\hbox{}\par\smallskip%
%
% LATTICE L34.41
%
\leavevmode%
\llap{\spacer}%
${L_{34.41}}=\pdualwitharg{3}({L_{34.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{12,7}{\infty a}{264}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2745549576, -499033656, 3457872; -499033656, -90704823, 628506; 3457872, 628506, -4355],[792, 3, 1],[-361, 7, 11, -13, -149, -3721, -973, -1179, -1547, -46, -407, -201, -2439, -497; 2167, -44, -66, 84, 902, 22484, 5877, 7117, 9328, 277, 2448, 1208, 14652, 2985; 26103, -792, -791, 1800, 11868, 290356, 75591, 90981, 117876, 3452, 30132, 14742, 177980, 36171],33,41]
%
% END OF LATTICE L34.41
%
\hbox{}\par\smallskip%
%
% LATTICE L34.42
%
\leavevmode%
\llap{\spacer}%
${L_{34.42}}=\pdualwitharg{11}(\pfillwitharg{3}({L_{34.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2124408, 5280, 13200; 5280, -11, -33; 13200, -33, -82],[264, 11, 1],[1, 1, -1, -11, -19, -65, -58, -17, -17, -7, -5, -1, 1, 2; 18, 12, -18, -148, -243, -816, -724, -210, -204, -78, -44, 3, 48, 52; 153, 156, -154, -1716, -2970, -10168, -9075, -2661, -2664, -1100, -792, -165, 136, 297],33,42]
%
% END OF LATTICE L34.42
%
\hbox{}\par\smallskip%
%
% LATTICE L34.43
%
\leavevmode%
\llap{\spacer}%
${L_{34.43}}=\pdualwitharg{11}(\pfillwitharg{3}({L_{34.3}}))$%
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{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2602776, 31680, -798864; 31680, -385, 9724; -798864, 9724, -245193],[264, 11, 1],[-62, 19, 21, -153, -857, -1669, -1544, -482, -563, -157, -391, -503, -507, -360; -177, 48, 61, -368, -2184, -4300, -3989, -1251, -1476, -418, -1060, -1386, -1412, -1009; 195, -60, -66, 484, 2706, 5268, 4873, 1521, 1776, 495, 1232, 1584, 1596, 1133],33,43]
%
% END OF LATTICE L34.43
%
\hbox{}\par\smallskip%
%
% LATTICE L34.44
%
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${L_{34.44}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{34.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[198, -99, 0; -99, -41217, 3795; 0, 3795, -349],[99, 33, 1],[157, 84, 241, 40, 79, 95, 38, 3, -4, -1, 40, 321, 865, 522; 311, 166, 474, 78, 152, 180, 71, 5, -8, 0, 80, 638, 1716, 1035; 3378, 1803, 5148, 847, 1650, 1953, 770, 54, -87, 0, 869, 6930, 18639, 11242],33,44]
%
% END OF LATTICE L34.44
%
\hbox{}\par\smallskip%
%
% LATTICE L34.45
%
\leavevmode%
\llap{\spacer}%
${L_{34.45}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{34.1}}))$%
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{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-198, -3366, -297; -3366, -56067, -4950; -297, -4950, -437],[99, 33, 1],[42, 23, 23, 37, 27, 12, 47, 2, -1, -1, 29, 83, 76, 415; 34, 17, 14, 15, 4, -1, -12, -2, -1, 2, 33, 76, 65, 348; -417, -210, -176, -198, -66, 2, 99, 21, 12, -22, -396, -924, -794, -4257],33,45]
%
% END OF LATTICE L34.45
%
\hbox{}\par\smallskip%
%
% LATTICE L34.46
%
\leavevmode%
\llap{\spacer}%
${L_{34.46}}=\pdualwitharg{3.11}(\main(\pfillwitharg{3}({L_{34.2}})))$%
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{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-494868, -10164, 8052; -10164, -198, 165; 8052, 165, -131],[132, 33, 1],[1, -1, -2, 25, 37, 205, 187, 19, 21, 32, 35, 13, 35, 23; 3, -2, -7, 42, 69, 392, 360, 37, 42, 67, 78, 31, 88, 60; 65, -64, -132, 1584, 2354, 13056, 11913, 1211, 1340, 2046, 2244, 836, 2256, 1485],33,46]
%
% END OF LATTICE L34.46
%
\hbox{}\par\smallskip%
%
% LATTICE L34.47
%
\leavevmode%
\llap{\spacer}%
${L_{34.47}}=\pdualwitharg{3.11}(\pfillwitharg{3}({L_{34.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2503908, 72732, 69300; 72732, 2112, 2013; 69300, 2013, 1918],[132, 33, 1],[1, -3, -31, -165, -169, -835, -1473, -141, -67, -74, -45, -6, -5, 3; -2, 8, 92, 520, 542, 2696, 4766, 458, 220, 250, 164, 28, 40, 22; -34, 100, 1023, 5412, 5533, 27318, 48180, 4610, 2188, 2409, 1452, 187, 138, -132],33,47]
%
% END OF LATTICE L34.47
%
\hbox{}\par\smallskip%
%
% LATTICE L34.48
%
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\llap{\spacer}%
${L_{34.48}}=\pdualwitharg{2.3.11}(\pfillwitharg{3.2}({L_{34.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-34354858956, -22514946498, -10395102630; -22514946498, -14755491102, -6812578674; -10395102630, -6812578674, -3145352999],[66, 66, 1],[-2297, -778, -3167, -2406, -2422, -4045, -12147, -811, 1, 836, 75, -1463, -5279, -20487; 69018, 23326, 94725, 71822, 72207, 120513, 361772, 24134, -38, -24861, -1958, 44233, 158943, 616324; -141896, -47951, -194700, -147609, -148390, -247653, -743424, -49592, 79, 51084, 3993, -90970, -326811, -1267200],33,48]
%
% END OF LATTICE L34.48
%
\hbox{}\par\smallskip%
%
% LATTICE L34.49
%
\leavevmode%
\llap{\spacer}%
${L_{34.49}}=\pdualwitharg{2}(\main({L_{34.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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24542100\\400752\\-6195024
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24542100, 400752, -6195024; 400752, 6612, -101160; -6195024, -101160, 1563775],[396, 12, 1],[6195, 383, 109, 1357, 11073, 165743, 18238, 60685, 33057, 9011, 1683, 4127, 34897, 3042; 154, 11, 3, 33, 269, 4026, 443, 1474, 803, 219, 41, 101, 858, 75; 24552, 1518, 432, 5378, 43884, 656865, 72280, 240504, 131010, 35712, 6670, 16356, 138303, 12056],33,49]
%
% END OF LATTICE L34.49
%
\hbox{}\par\smallskip%
%
% LATTICE L34.50
%
\leavevmode%
\llap{\spacer}%
${L_{34.50}}=\pdualwitharg{2.3}(\main({L_{34.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4039992, -96228, -1008612; -96228, 2292, 24024; -1008612, 24024, 251807],[396, 12, 1],[-5, -17, 1, 47, 119, 578, 568, 621, 325, 27, 37, 21, 38, 20; -209, -22, 0, -102, -286, -1441, -1431, -1595, -880, -82, -144, -124, -363, -291; 0, -66, 4, 198, 504, 2453, 2412, 2640, 1386, 116, 162, 96, 187, 108],33,50]
%
% END OF LATTICE L34.50
%
\hbox{}\par\smallskip%
%
% LATTICE L34.51
%
\leavevmode%
\llap{\spacer}%
${L_{34.51}}=\pdualwitharg{2}({L_{34.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[105967522584, -1662977844, -26764213212; -1662977844, 26097648, 420018252; -26764213212, 420018252, 6759836329],[396, 12, 1],[434888, -975, 8491, 59113, 957895, 28586653, 1571401, 5219772, 1414640, 762599, 69402, 327255, 5341307, 227809; -4609, 11, -90, -627, -10160, -303204, -16667, -55363, -15004, -8088, -736, -3470, -56628, -2415; 1722138, -3861, 33624, 234085, 3793224, 113201946, 6222682, 20670078, 5601915, 3019860, 274829, 1295916, 21151350, 902114],33,51]
%
% END OF LATTICE L34.51
%
\hbox{}\par\smallskip%
%
% LATTICE L34.52
%
\leavevmode%
\llap{\spacer}%
${L_{34.52}}=\pdualwitharg{2.3}({L_{34.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31635263916432, -431868139164, -7943018067468; -431868139164, 5895638808, 108433943880; -7943018067468, 108433943880, 1994342016137],[396, 12, 1],[776433, -1649, 1572, 295519, 1615098, 16120958, 7983043, 8853803, 2409208, 436166, 361088, 573592, 3139514, 1209455; 8404, -11, 17, 3183, 17399, 173679, 86007, 95392, 25960, 4701, 3894, 6191, 33913, 13071; 3091902, -6567, 6260, 1176813, 6431628, 64196726, 31789998, 35257530, 9593925, 1736896, 1437921, 2284152, 12502138, 4816278],33,52]
%
% END OF LATTICE L34.52
%
\hbox{}\par\smallskip%
%
% LATTICE L34.53
%
\leavevmode%
\llap{\spacer}%
${L_{34.53}}=\pdualwitharg{2.11}(\main(\pfillwitharg{3}({L_{34.2}})))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[280236, -20592, -83292; -20592, 1628, 6204; -83292, 6204, 24817],[132, 44, 1],[2030, 1075, 1019, 531, 1223, 306, 869, 158, 19, 75, 563, 4247, 1882, 6789; -6057, -3207, -3039, -1583, -3645, -912, -2590, -471, -57, -225, -1681, -12675, -5616, -20258; 8328, 4410, 4180, 2178, 5016, 1255, 3564, 648, 78, 308, 2310, 17424, 7721, 27852],33,53]
%
% END OF LATTICE L34.53
%
\hbox{}\par\smallskip%
%
% LATTICE L34.54
%
\leavevmode%
\llap{\spacer}%
${L_{34.54}}=\pdualwitharg{2.11}(\pfillwitharg{3}({L_{34.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{2,1}{\infty z}{3}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{264}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[493976208, 82141884, -61123524; 82141884, 13659272, -10163956; -61123524, -10163956, 7563363],[132, 44, 1],[-126331, -33631, -64238, -16659, -74416, -35636, -49171, -8113, 20, -2984, -16952, -261694, -233252, -421383; 378531, 100770, 192479, 49916, 222975, 106777, 147332, 24309, -60, 8941, 50794, 784125, 698903, 1262608; -512262, -136371, -260480, -67551, -301752, -144502, -199386, -32898, 81, -12100, -68739, -1061148, -945818, -1708674],33,54]
%
% END OF LATTICE L34.54
%
\hbox{}\par\smallskip%
%
% LATTICE L34.55
%
\leavevmode%
\llap{\spacer}%
${L_{34.55}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{34.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{8,1}{\infty z}{44}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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10313880\\1978235\\20019072
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3927790\\753363\\7623775
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-78316392, 84098520, 32038512; 84098520, -90307536, -34403928; 32038512, -34403928, -13106659],[264, 24, 1],[-8312, 9, 835, 297, -851, -39698, -14418, -18528, -6719, -2665, -2127, -2199, -40822, -11202; -1639, 0, 165, 64, -154, -7425, -2711, -3509, -1287, -516, -415, -431, -8019, -2203; -16016, 22, 1608, 558, -1676, -77550, -28128, -36080, -13046, -5160, -4110, -4244, -78738, -21600],33,55]
%
% END OF LATTICE L34.55
%
\hbox{}\par\smallskip%
%
% LATTICE L34.56
%
\leavevmode%
\llap{\spacer}%
${L_{34.56}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{34.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6795580176, 66322344, 39389856; 66322344, -647256, -384432; 39389856, -384432, -228319],[264, 24, 1],[-1637, -1, 165, 67, -146, -3594, -2633, -3423, -1264, -510, -412, -429, -3996, -2197; -16060, -11, 1619, 661, -1423, -35123, -25742, -33484, -12375, -4997, -4039, -4207, -39193, -21550; -255376, -154, 25740, 10446, -22792, -560901, -410904, -534160, -197230, -79572, -64278, -66928, -623403, -342744],33,56]
%
% END OF LATTICE L34.56
%
\hbox{}\par\smallskip%
%
% LATTICE L34.57
%
\leavevmode%
\llap{\spacer}%
${L_{34.57}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{34.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4679334, -2250468, -197208; -2250468, 1082301, 94842; -197208, 94842, 8311],[198, 33, 1],[-1, 3, 31, 55, 169, 835, 491, 141, 67, 74, 15, 6, 5, -1; 25, -70, -699, -1214, -3706, -18264, -10731, -3077, -1456, -1590, -312, -109, -48, 49; -309, 870, 8712, 15158, 46299, 228222, 134101, 38457, 18204, 19899, 3916, 1386, 666, -583],33,57]
%
% END OF LATTICE L34.57
%
\hbox{}\par\smallskip%
%
% LATTICE L34.58
%
\leavevmode%
\llap{\spacer}%
${L_{34.58}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{34.2}}))$%
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{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12858714, -872784, -33858; -872784, -58179, -2211; -33858, -2211, -82],[198, 33, 1],[-30, 1, 15, 95, 42, 31, 100, -2, -17, -23, -143, -82, -95, -436; 838, -28, -419, -2652, -1171, -862, -2772, 58, 476, 643, 3996, 2291, 2654, 12180; -10209, 342, 5104, 32274, 14223, 10426, 33363, -747, -5820, -7843, -48708, -27918, -32338, -148401],33,58]
%
% END OF LATTICE L34.58
%
\hbox{}\par\smallskip%
%
% LATTICE L34.59
%
\leavevmode%
\llap{\spacer}%
${L_{34.59}}=\pdualwitharg{3.11}(\pfillwitharg{3}({L_{34.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-989736, -505032, 16104; -505032, -257697, 8217; 16104, 8217, -262],[264, 33, 1],[-18, -21, -35, -43, -9, -53, -37, -2, 1, 5, -17, -16, -187, -173; 74, 84, 134, 156, 31, 176, 120, 6, -4, -14, 84, 69, 784, 720; 1211, 1340, 2046, 2244, 418, 2256, 1485, 65, -64, -132, 1584, 1177, 13056, 11913],33,59]
%
% END OF LATTICE L34.59
%
\hbox{}\par\smallskip%
%
% LATTICE L34.60
%
\leavevmode%
\llap{\spacer}%
${L_{34.60}}=\pdualwitharg{3.11}(\pfillwitharg{3}({L_{34.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3565320, 20064, 10824; 20064, -99, -66; 10824, -66, -31],[264, 33, 1],[19, 21, 16, 35, 13, 35, 23, 1, -1, -1, 25, 37, 205, 187; 1365, 1508, 1148, 2508, 930, 2500, 1641, 71, -72, -71, 1800, 2660, 14732, 13437; 3719, 4112, 3135, 6864, 2552, 6876, 4521, 197, -196, -198, 4884, 7238, 40116, 36597],33,60]
%
% END OF LATTICE L34.60
%
\hbox{}\par\smallskip%
%
% LATTICE L34.61
%
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\llap{\spacer}%
${L_{34.61}}=\pdualwitharg{11}(\main({L_{34.3}}))$%
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{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1250172, -5487768, -460152; -5487768, -24065349, -2017752; -460152, -2017752, -169177],[396, 33, 1],[124, -123, -251, 1017, 4528, 25105, 7635, 2328, 2575, 3929, 1435, 1602, 4319, 947; -434, 430, 879, -3554, -15827, -87756, -26689, -8138, -9002, -13737, -5018, -5603, -15108, -3313; 4839, -4794, -9801, 39622, 176451, 978372, 297550, 90729, 100362, 153153, 55946, 62469, 168444, 36938],33,61]
%
% END OF LATTICE L34.61
%
\hbox{}\par\smallskip%
%
% LATTICE L34.62
%
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${L_{34.62}}=\pdualwitharg{3.11}(\main({L_{34.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-187308, 60192, -2772; 60192, -19338, 891; -2772, 891, -41],[396, 33, 1],[0, -1, 0, 25, 33, 59, 160, 16, 17, 8, 23, 7, 5, 8; 1, -2, -1, 42, 61, 112, 306, 31, 34, 17, 54, 19, 16, 30; 21, 24, -22, -792, -924, -1588, -4257, -417, -420, -176, -396, -66, 4, 99],33,62]
%
% END OF LATTICE L34.62
%
\hbox{}\par\smallskip%
%
% LATTICE L34.63
%
\leavevmode%
\llap{\spacer}%
${L_{34.63}}=\pdualwitharg{11}({L_{34.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9358668, -3526776, -313236; -3526776, 1329042, 118041; -313236, 118041, 10484],[396, 33, 1],[-1, 3, 31, 55, 169, 835, 491, 141, 67, 74, 15, 6, 5, -1; 16, -40, -375, -624, -1879, -9210, -5402, -1544, -724, -771, -140, -31, 42, 54; -210, 540, 5148, 8668, 26202, 128628, 75482, 21594, 10152, 10890, 2024, 528, -324, -638],33,63]
%
% END OF LATTICE L34.63
%
\hbox{}\par\smallskip%
%
% LATTICE L34.64
%
\leavevmode%
\llap{\spacer}%
${L_{34.64}}=\pdualwitharg{3.11}({L_{34.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-25717428, -19491120, 4732596; -19491120, -14772120, 3586737; 4732596, 3586737, -870854],[396, 33, 1],[1573, -57, -784, -4807, -1984, -1241, -3167, 311, 999, 1255, 7633, 4339, 5009, 22949; -3206, 116, 1598, 9804, 4052, 2544, 6534, -626, -2032, -2556, -15552, -8842, -10208, -46770; -4656, 168, 2321, 14256, 5907, 3734, 9702, -888, -2940, -3707, -22572, -12837, -14822, -67914],33,64]
%
% END OF LATTICE L34.64
%
\hbox{}\par\smallskip%
%
% LATTICE L34.65
%
\leavevmode%
\llap{\spacer}%
${L_{34.65}}=\pdualwitharg{2.11}(\pfillwitharg{2}({L_{34.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[469425330, 12821292, 232327656; 12821292, 350130, 6345504; 232327656, 6345504, 114983441],[198, 66, 1],[1703, 46, -27316, -34534, -230147, -585320, -694575, -202739, -50305, -123461, -15725, -21148, -17788, -9051; 55, 1, -891, -1123, -7480, -19020, -22569, -6587, -1634, -4008, -510, -685, -576, -293; -3444, -93, 55242, 69839, 465432, 1183707, 1404656, 410004, 101733, 249678, 31801, 42768, 35973, 18304],33,65]
%
% END OF LATTICE L34.65
%
\hbox{}\par\smallskip%
%
% LATTICE L34.66
%
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\llap{\spacer}%
${L_{34.66}}=\pdualwitharg{2.3.11}(\pfillwitharg{2}({L_{34.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40454590770, 1113520518, 20021771286; 1113520518, 30649872, 551103168; 20021771286, 551103168, 9909167731],[198, 66, 1],[-12310, 46, -2265, -77570, -199601, -177909, -1928420, -192714, -51302, -48991, -76198, -56679, -27115, -224332; 49, 1, 9, 279, 716, 638, 6915, 691, 184, 176, 276, 209, 102, 855; 24870, -93, 4576, 156717, 403260, 359435, 3896046, 389346, 103647, 98978, 153945, 114510, 54781, 453222],33,66]
%
% END OF LATTICE L34.66
%
\hbox{}\par\smallskip%
%
% LATTICE L34.67
%
\leavevmode%
\llap{\spacer}%
${L_{34.67}}=\pdualwitharg{2.3.11}(\main(\pfillwitharg{3}({L_{34.2}})))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6996, -175692, -42504; -175692, 4681512, 1132560; -42504, 1132560, 273991],[132, 132, 1],[-1, 0, -2, -16, -40, -53, -191, -19, -10, -28, -14, -10, -7, -19; 448, 15, 447, 4997, 12889, 17253, 62359, 6236, 3327, 9579, 5029, 3821, 2799, 7823; -1852, -62, -1848, -20658, -53284, -71325, -257796, -25780, -13754, -39600, -20790, -15796, -11571, -32340],33,67]
%
% END OF LATTICE L34.67
%
\hbox{}\par\smallskip%
%
% LATTICE L34.68
%
\leavevmode%
\llap{\spacer}%
${L_{34.68}}=\pdualwitharg{2.3.11}(\pfillwitharg{3}({L_{34.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5544, -42900, -10428; -42900, 2095104, 509520; -10428, 509520, 123913],[132, 132, 1],[1, 0, -33, -62, -275, -699, -1244, -121, -30, -147, -28, -25, -21, -16; -303, -9, 9694, 18401, 81768, 207968, 370187, 36019, 8938, 43880, 8386, 7522, 6328, 4831; 1246, 37, -39864, -75669, -336248, -855210, -1522290, -148118, -36755, -180444, -34485, -30932, -26022, -19866],33,68]
%
% END OF LATTICE L34.68
%
\hbox{}\par\smallskip%
%
% LATTICE L34.69
%
\leavevmode%
\llap{\spacer}%
${L_{34.69}}=\pdualwitharg{2}({L_{34.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-776677176, -63310896, 12199176; -63310896, -5160792, 994416; 12199176, 994416, -191609],[792, 24, 1],[1869, -16, -185, -8, 300, 12097, 1427, 5303, 1809, 674, 171, 515, 9419, 855; -24607, 209, 2436, 107, -3937, -158895, -18747, -69685, -23782, -8865, -2250, -6778, -123981, -11255; -8712, 66, 864, 46, -1332, -54450, -6440, -24024, -8250, -3096, -790, -2388, -43758, -3976],33,69]
%
% END OF LATTICE L34.69
%
\hbox{}\par\smallskip%
%
% LATTICE L34.70
%
\leavevmode%
\llap{\spacer}%
${L_{34.70}}=\pdualwitharg{2}({L_{34.2}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26217286128, 8814306600, 119875536; 8814306600, -2963388360, -40302408; 119875536, -40302408, -548117],[792, 24, 1],[-14435, -4, 1454, 192, -1325, -32240, -7859, -30577, -11249, -4523, -1215, -3790, -35278, -6463; -38390, -11, 3867, 511, -3521, -85701, -20892, -81290, -29909, -12027, -3231, -10079, -93819, -17188; -334224, -66, 33660, 4418, -30888, -749529, -182632, -710160, -261030, -104868, -28154, -87792, -817047, -149672],33,70]
%
% END OF LATTICE L34.70
%
\hbox{}\par\smallskip%
%
% LATTICE L34.71
%
\leavevmode%
\llap{\spacer}%
${L_{34.71}}=\pdualwitharg{2.3}({L_{34.3}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-356916337272, 29938939272, -7106001408; 29938939272, -2511345072, 596067264; -7106001408, 596067264, -141476449],[792, 24, 1],[9582, -17, -299, -65, 1611, 21176, 22384, 27554, 9301, 1143, 2597, 2609, 15960, 13064; 100133, -187, -3124, -651, 16907, 221991, 234603, 288695, 97394, 11961, 27162, 27278, 166837, 136551; -59400, 66, 1856, 522, -9684, -128326, -135864, -167640, -56826, -7016, -16002, -16116, -98714, -80856],33,71]
%
% END OF LATTICE L34.71
%
\hbox{}\par\smallskip%
%
% LATTICE L34.72
%
\leavevmode%
\llap{\spacer}%
${L_{34.72}}=\pdualwitharg{2.3}({L_{34.2}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3676593096, 318300048, -50394168; 318300048, -27556680, 4362840; -50394168, 4362840, -690733],[792, 24, 1],[-5267, -1, 165, 67, -806, -5433, -11543, -14313, -4894, -610, -1402, -1419, -4357, -7147; -109615, -22, 3434, 1398, -16765, -113025, -240141, -297781, -101827, -12693, -29175, -29530, -90673, -148737; -308088, -66, 9652, 3942, -47088, -317515, -674640, -836616, -286110, -35668, -81990, -82992, -254837, -418032],33,72]
%
% END OF LATTICE L34.72
%
\hbox{}\par\smallskip%
%
% LATTICE L34.73
%
\leavevmode%
\llap{\spacer}%
${L_{34.73}}=\pdualwitharg{2.11}(\pfillwitharg{3}({L_{34.3}}))$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-68645544, 138307488, -1634424; 138307488, -278662824, 3293048; -1634424, 3293048, -38915],[264, 88, 1],[-149, 1, 76, 97, 347, 199, 605, 121, -8, -83, -104, -382, -227, -703; -87, 0, 45, 64, 258, 165, 529, 123, 9, -36, -55, -213, -129, -403; -1104, -42, 616, 1342, 7260, 5606, 19360, 5328, 1098, 440, -286, -1980, -1382, -4576],33,73]
%
% END OF LATTICE L34.73
%
\hbox{}\par\smallskip%
%
% LATTICE L34.74
%
\leavevmode%
\llap{\spacer}%
${L_{34.74}}=\pdualwitharg{2.11}(\pfillwitharg{3}({L_{34.2}}))$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-230390160, 5988840, 5057712; 5988840, -155672, -131472; 5057712, -131472, -111031],[264, 88, 1],[-85, -1, 45, 75, 346, 122, 815, 209, 32, -14, -44, -191, -60, -381; -108, -3, 59, 117, 603, 227, 1554, 420, 81, 19, -39, -213, -71, -462; -3744, -42, 1980, 3278, 15048, 5289, 35288, 9024, 1362, -660, -1958, -8448, -2649, -16808],33,74]
%
% END OF LATTICE L34.74
%
\hbox{}\par\smallskip%
%
% LATTICE L34.75
%
\leavevmode%
\llap{\spacer}%
${L_{34.75}}=\pdualwitharg{11}({L_{34.2}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2500344, -5243832, -437976; -5243832, -10986690, -917565; -437976, -917565, -76631],[792, 33, 1],[59, -59, -119, 489, 1087, 12049, 3664, 1117, 1235, 1883, 687, 383, 2063, 452; -413, 412, 834, -3412, -7588, -84120, -25581, -7799, -8624, -13152, -4800, -2677, -14424, -3161; 4608, -4596, -9306, 38060, 84645, 938376, 285362, 87000, 96204, 146718, 53548, 29865, 160920, 35266],33,75]
%
% END OF LATTICE L34.75
%
\hbox{}\par\smallskip%
%
% LATTICE L34.76
%
\leavevmode%
\llap{\spacer}%
${L_{34.76}}=\pdualwitharg{11}({L_{34.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12698136, -45140040, -3720816; -45140040, -160262850, -13208547; -3720816, -13208547, -1088609],[792, 33, 1],[1293, -1285, -1300, 10671, 47451, 263005, 79979, 24383, 26961, 20557, 15005, 16739, 45107, 9887; -4073, 4048, 4095, -33616, -149480, -828516, -251949, -76811, -84932, -64758, -47268, -52730, -142092, -31145; 45000, -44724, -45243, 371404, 1651518, 9153792, 2783638, 848640, 938364, 715473, 522236, 582582, 1569888, 344102],33,76]
%
% END OF LATTICE L34.76
%
\hbox{}\par\smallskip%
%
% LATTICE L34.77
%
\leavevmode%
\llap{\spacer}%
${L_{34.77}}=\pdualwitharg{3.11}({L_{34.2}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8842680, -4713984, 26928; -4713984, -2512983, 14355; 26928, 14355, -82],[792, 33, 1],[-15, -17, -9, -31, -6, -11, -22, -1, 1, 1, -17, -14, -53, -146; 62, 68, 34, 108, 19, 32, 60, 2, -4, -2, 84, 61, 224, 612; 5919, 6312, 2992, 8712, 1353, 1984, 3267, 21, -372, -22, 9108, 6072, 21776, 59103],33,77]
%
% END OF LATTICE L34.77
%
\hbox{}\par\smallskip%
%
% LATTICE L34.78
%
\leavevmode%
\llap{\spacer}%
${L_{34.78}}=\pdualwitharg{3.11}({L_{34.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11152152, 1155528, 99000; 1155528, -118965, -10197; 99000, -10197, -874],[792, 33, 1],[25, 29, 8, 59, 25, 25, 53, 3, -1, -1, 25, 45, 87, 241; -3116, -3620, -1001, -7404, -3146, -3152, -6690, -380, 124, 127, -3084, -5594, -10832, -30018; 39183, 45516, 12584, 93060, 39534, 39604, 84051, 4773, -1560, -1595, 38808, 70356, 136220, 377487],33,78]
%
% END OF LATTICE L34.78
%
\hbox{}\par\smallskip%
%
% LATTICE L34.79
%
\leavevmode%
\llap{\spacer}%
${L_{34.79}}=\pdualwitharg{2.11}(\main({L_{34.3}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[58269870648, -2156887260, 16367309640; -2156887260, 79838220, -605843832; 16367309640, -605843832, 4597381493],[396, 132, 1],[3383, 113, 3379, 12585, 97379, 130348, 157042, 47113, 25135, 72365, 12663, 28861, 21140, 19694; -10867, -362, -10884, -40474, -313138, -419139, -504969, -151489, -80816, -232650, -40704, -92752, -67929, -63277; -13476, -450, -13464, -50138, -387948, -519291, -625636, -187692, -100134, -288288, -50446, -114972, -84213, -78452],33,79]
%
% END OF LATTICE L34.79
%
\hbox{}\par\smallskip%
%
% LATTICE L34.80
%
\leavevmode%
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${L_{34.80}}=\pdualwitharg{2.3.11}(\main({L_{34.3}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3150972, -42768, 827244; -42768, 1452, -11880; 827244, -11880, 217669],[396, 132, 1],[-530, -53, -211, -5447, -13859, -6163, -66763, -6666, -3545, -3387, -5347, -4131, -1033, -8795; 1571, 157, 625, 16143, 41075, 18266, 197874, 19757, 10507, 10039, 15849, 12245, 3062, 26070; 2100, 210, 836, 21582, 54912, 24419, 264528, 26412, 14046, 13420, 21186, 16368, 4093, 34848],33,80]
%
% END OF LATTICE L34.80
%
\hbox{}\par\smallskip%
%
% LATTICE L34.81
%
\leavevmode%
\llap{\spacer}%
${L_{34.81}}=\pdualwitharg{2.11}({L_{34.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[211814064, -6045732, 58078548; -6045732, 172128, -1657392; 58078548, -1657392, 15924659],[396, 132, 1],[824, 22, -26441, -16712, -222745, -566491, -336115, -98108, -24343, -119485, -7609, -20465, -17213, -4379; -2429, -65, 77937, 49261, 656575, 1669821, 990752, 289189, 71755, 352203, 22429, 60325, 50739, 12908; -3258, -87, 104544, 66077, 880704, 2239830, 1328954, 387906, 96249, 472428, 30085, 80916, 68058, 17314],33,81]
%
% END OF LATTICE L34.81
%
\hbox{}\par\smallskip%
%
% LATTICE L34.82
%
\leavevmode%
\llap{\spacer}%
${L_{34.82}}=\pdualwitharg{2.3.11}({L_{34.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[78346040256, 5915301876, 15215119524; 5915301876, 446618568, 1148775672; 15215119524, 1148775672, 2954838067],[396, 132, 1],[-1705, 13, -628, -10913, -56182, -50078, -271409, -27123, -7220, -13786, -10708, -15888, -7578, -31285; 5683, -41, 2093, 36315, 186949, 166637, 903126, 90253, 24025, 45875, 35637, 52891, 25235, 104202; 6570, -51, 2420, 42075, 216612, 193078, 1046430, 104574, 27837, 53152, 41283, 61248, 29210, 120582],33,82]
%
% END OF LATTICE L34.82
%
\hbox{}\par\smallskip%
%
% LATTICE L34.83
%
\leavevmode%
\llap{\spacer}%
${L_{34.83}}=\pdualwitharg{2.3.11}(\pfillwitharg{3}({L_{34.3}}))$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,1}{\infty z}{4}\above{1pt}{1pt}{l}{2}{264}\above{1pt}{1pt}{r}{2}{132}\above{1pt}{1pt}{s}{2}{88}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{528}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[33505296, 34045704, -261888; 34045704, 34594824, -266112; -261888, -266112, 2047],[264, 264, 1],[131, 33, 84, 35, 19, 21, 47, 1, 0, 15, 64, 146, 375, 1339; -184, -45, -107, -39, -17, -16, -30, 0, -1, -31, -99, -213, -536, -1902; -7168, -1630, -3168, -594, 220, 606, 2112, 128, -130, -2112, -4686, -9020, -21726, -76032],33,83]
%
% END OF LATTICE L34.83
%
\hbox{}\par\smallskip%
%
% LATTICE L34.84
%
\leavevmode%
\llap{\spacer}%
${L_{34.84}}=\pdualwitharg{2.3.11}(\pfillwitharg{3}({L_{34.2}}))$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[321140688, -1621224, -2457840; -1621224, 8184, 12408; -2457840, 12408, 18811],[264, 264, 1],[217, 56, 150, 68, 41, 24, 113, 3, -1, 15, 97, 234, 306, 2197; -184, -45, -107, -39, -17, -8, -30, 0, -1, -31, -99, -213, -268, -1902; 28472, 7346, 19668, 8910, 5368, 3141, 14784, 392, -130, 1980, 12738, 30712, 40155, 288288],33,84]
%
% END OF LATTICE L34.84
%
\hbox{}\par\smallskip%
%
% LATTICE L34.85
%
\leavevmode%
\llap{\spacer}%
${L_{34.85}}=\pdualwitharg{2.11}({L_{34.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L34.85
%
\hbox{}\par\smallskip%
%
% LATTICE L34.86
%
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\llap{\spacer}%
${L_{34.86}}=\pdualwitharg{2.11}({L_{34.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24552, -889416, 20592; -889416, 32224368, -746064; 20592, -746064, 17273],[792, 264, 1],[0, -1, -31, -33, -213, -268, -634, -184, -45, -107, -13, -17, -8, -10; 5, -1, 45, 75, 526, 681, 1625, 479, 122, 318, 46, 79, 45, 69; 216, -42, 1980, 3278, 22968, 29727, 70928, 20904, 5322, 13860, 2002, 3432, 1953, 2992],33,86]
%
% END OF LATTICE L34.86
%
\hbox{}\par\smallskip%
%
% LATTICE L34.87
%
\leavevmode%
\llap{\spacer}%
${L_{34.87}}=\pdualwitharg{2.3.11}({L_{34.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16739640720, -1071961704, 25522992; -1071961704, -68645544, 1634424; 25522992, 1634424, -38915],[792, 264, 1],[41, 3, -12, -55, -71, -43, -403, -29, 0, 15, 64, 86, 55, 529; 2, 17, 47, 147, 169, 98, 900, 62, -1, -31, -99, -89, -34, -228; 26976, 2682, -5896, -29898, -39468, -24086, -226512, -16416, -42, 8536, 37818, 52668, 34646, 337392],33,87]
%
% END OF LATTICE L34.87
%
\hbox{}\par\smallskip%
%
% LATTICE L34.88
%
\leavevmode%
\llap{\spacer}%
${L_{34.88}}=\pdualwitharg{2.3.11}({L_{34.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,13}{\infty z}{12}\above{1pt}{1pt}{s}{2}{88}\above{1pt}{1pt}{s}{2}{396}\above{1pt}{1pt}{l}{2}{264}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{1584}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-158841072720, -42287057208, 241160832; -42287057208, -11257763352, 64202424; 241160832, 64202424, -366143],[792, 264, 1],[127, 26, 10, -22, -49, -17, -337, -27, -1, 15, 97, 174, 67, 1387; 2, 17, 47, 147, 169, 49, 900, 62, -1, -31, -99, -89, -17, -228; 84000, 20106, 14828, 11286, -2640, -2605, -64152, -6912, -834, 4444, 46530, 99000, 41149, 873576],33,88]
%
% END OF LATTICE L34.88
%

\medskip%
%------------------------------WEYL-GROUP-W35--------------------------------
%
\leavevmode\llap{}%
$W_{35\phantom{0}}$%
\qquad\llap{18} lattices, $\chi=12$%
\hfill%
$222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L35.1
%
\leavevmode%
\llap{\spacer}%
${L_{35.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{23\rightarrow N_{35},3,2}%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-16236, 1584, 0; 1584, -150, -3; 0, -3, 2],[396, 3, 1],[1, 13, 0, -5, -19, -2; 10, 132, 0, -52, -198, -21; 18, 198, -1, -78, -286, -27],34,1]
%
% END OF LATTICE L35.1
%
\hbox{}\par\smallskip%
%
% LATTICE L35.2
%
\leavevmode%
\llap{\spacer}%
${L_{35.2}}=\pfillwitharg{2.3}({L_{35.1}})$%
{${}={}$}Nikulin $35$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[462, -165, 0; -165, 59, 0; 0, 0, -1],[33, 1, 1],[1, 4, -1, -16, -59, -4; 3, 11, -3, -45, -165, -11; 0, 0, -1, -12, -44, -3],34,2]
%
% END OF LATTICE L35.2
%
\hbox{}\par\smallskip%
%
% LATTICE L35.3
%
\leavevmode%
\llap{\spacer}%
${L_{35.3}}=\pfillwitharg{3}({L_{35.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[660, 132, 132; 132, 26, 23; 132, 23, -2],[132, 1, 1],[1, 27, 3, -1, -27, -3; -6, -154, -17, 6, 154, 17; -6, 0, 2, 0, -22, -3],34,3]
%
% END OF LATTICE L35.3
%
\hbox{}\par\smallskip%
%
% LATTICE L35.4
%
\leavevmode%
\llap{\spacer}%
${L_{35.4}}=\pfillwitharg{2}({L_{35.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[99, 99, 0; 99, 21, -9; 0, -9, -1],[99, 3, 1],[-1, 1, 1, 3, 7, -1; 1, 0, -1, -4, -11, 0; -9, 0, 8, 24, 55, -9],34,4]
%
% END OF LATTICE L35.4
%
\hbox{}\par\smallskip%
%
% LATTICE L35.5
%
\leavevmode%
\llap{\spacer}%
${L_{35.5}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{35.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-41811\\1551\\14223
\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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14223\\-528\\-4838
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{$\left.\llap{\phantom{%
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0\\0\\0
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218041\\493680\\586245
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\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-41811, 1551, 14223; 1551, -57, -528; 14223, -528, -4838],[33, 3, 1],[-17, 37, 21, -89, -1203, -127; -36, 88, 43, -204, -2728, -283; -46, 99, 57, -239, -3234, -342],34,5]
%
% END OF LATTICE L35.5
%
\hbox{}\par\smallskip%
%
% LATTICE L35.6
%
\leavevmode%
\llap{\spacer}%
${L_{35.6}}=\pdualwitharg{11}(\pfillwitharg{2.3}({L_{35.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-118767, -2904, 42735; -2904, -55, 1045; 42735, 1045, -15377],[33, 11, 1],[-487, 36, 194, -2756, -1117, -1275; 9, -1, -3, 69, 27, 29; -1353, 100, 539, -7656, -3103, -3542],34,6]
%
% END OF LATTICE L35.6
%
\hbox{}\par\smallskip%
%
% LATTICE L35.7
%
\leavevmode%
\llap{\spacer}%
${L_{35.7}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{35.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8580, 264, 132; 264, 30, -3; 132, -3, -2],[132, 3, 1],[1, 1, -1, -1, -1, 1; -2, 0, 2, 0, -22, -3; 68, 66, -69, -74, -132, 66],34,7]
%
% END OF LATTICE L35.7
%
\hbox{}\par\smallskip%
%
% LATTICE L35.8
%
\leavevmode%
\llap{\spacer}%
${L_{35.8}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{35.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[210936, -10956, 53064; -10956, 568, -2756; 53064, -2756, 13349],[132, 4, 1],[109, 293, -1, -19, 37, 31; -81, -220, 0, 12, -33, -23; -450, -1210, 4, 78, -154, -128],34,8]
%
% END OF LATTICE L35.8
%
\hbox{}\par\smallskip%
%
% LATTICE L35.9
%
\leavevmode%
\llap{\spacer}%
${L_{35.9}}=\pdualwitharg{3.11}(\pfillwitharg{2.3}({L_{35.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[66, -726, -33; -726, 12111, 561; -33, 561, 26],[33, 33, 1],[-4, 1, 4, -27, -32, -35; -9, 3, 8, -73, -85, -90; 187, -63, -165, 1529, 1779, 1881],34,9]
%
% END OF LATTICE L35.9
%
\hbox{}\par\smallskip%
%
% LATTICE L35.10
%
\leavevmode%
\llap{\spacer}%
${L_{35.10}}=\pdualwitharg{11}(\pfillwitharg{3}({L_{35.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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10428\\3828\\-528
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10428, 3828, -528; 3828, 1342, -187; -528, -187, 26],[132, 11, 1],[-1, -1, -1, 1, 1, 1; -24, -14, -11, 6, 8, 8; -198, -122, -99, 66, 78, 77],34,10]
%
% END OF LATTICE L35.10
%
\hbox{}\par\smallskip%
%
% LATTICE L35.11
%
\leavevmode%
\llap{\spacer}%
${L_{35.11}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{35.1}}))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3192024, 8580, 772332; 8580, 24, 2076; 772332, 2076, 186871],[132, 12, 1],[15, -16, 0, 136, 1677, 151; 0, 11, 1, -7, -88, -8; -62, 66, 0, -562, -6930, -624],34,11]
%
% END OF LATTICE L35.11
%
\hbox{}\par\smallskip%
%
% LATTICE L35.12
%
\leavevmode%
\llap{\spacer}%
${L_{35.12}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{35.1}}))$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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-5\\1\\0
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-4\\0\\-9
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0\\-1\\-11
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0\\9\\0\\-99\\-36\\-99
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-33\\0\\22\\0\\-11\\-99
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[198, 990, -99; 990, 4983, -495; -99, -495, 49],[99, 33, 1],[-5, -4, 0, 4, 1, -1; 1, 0, -1, -4, -1, 0; 0, -9, -11, -33, -8, 0],34,12]
%
% END OF LATTICE L35.12
%
\hbox{}\par\smallskip%
%
% LATTICE L35.13
%
\leavevmode%
\llap{\spacer}%
${L_{35.13}}=\pdualwitharg{3.11}(\pfillwitharg{3}({L_{35.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10956\\1056\\-264
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1056\\66\\-33
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-33\\14
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-1\\-35\\-99
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0\\0\\-1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-38\\-110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-38\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\12\\0\\-132\\-144\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-132\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10956, 1056, -264; 1056, 66, -33; -264, -33, 14],[132, 33, 1],[-1, -1, 0, 3, 3, 1; -38, -38, -1, 106, 106, 35; -110, -108, 0, 308, 306, 99],34,13]
%
% END OF LATTICE L35.13
%
\hbox{}\par\smallskip%
%
% LATTICE L35.14
%
\leavevmode%
\llap{\spacer}%
${L_{35.14}}=\pdualwitharg{2}({L_{35.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{396}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1339272\\-36036\\-339372
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36036\\984\\9132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-339372\\9132\\85997
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\5\\-186
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-250\\33\\-990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-150\\19\\-594
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-639\\77\\-2530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-191\\21\\-756
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-48\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\396\\0\\-396\\-1584\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-44\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-396\\0\\12\\0\\-36
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-1584\\-44\\0\\44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-396\\-36\\-36\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1339272, -36036, -339372; -36036, 984, 9132; -339372, 9132, 85997],[396, 12, 1],[-47, -250, -2, -150, -639, -191; 5, 33, 1, 19, 77, 21; -186, -990, -8, -594, -2530, -756],34,14]
%
% END OF LATTICE L35.14
%
\hbox{}\par\smallskip%
%
% LATTICE L35.15
%
\leavevmode%
\llap{\spacer}%
${L_{35.15}}=\pdualwitharg{2.11}(\pfillwitharg{3}({L_{35.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2237928\\75636\\-486816
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75636\\2552\\-16456
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-486816\\-16456\\105895
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1574\\-4125\\6600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\-166\\264
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\880\\-1409
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
206\\-573\\858
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\-96\\146
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\152\\-220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-132\\-528\\-132\\0
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\kern3pt%
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0\\4\\0\\-132\\-48\\-44
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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132\\0\\-132
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-132\\0\\88
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-132\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-48\\-44
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-44\\-132
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2237928, 75636, -486816; 75636, 2552, -16456; -486816, -16456, 105895],[132, 44, 1],[206, 35, -53, -143, -14, 74; -573, -96, 152, 408, 41, -207; 858, 146, -220, -594, -58, 308],34,15]
%
% END OF LATTICE L35.15
%
\hbox{}\par\smallskip%
%
% LATTICE L35.16
%
\leavevmode%
\llap{\spacer}%
${L_{35.16}}=\pdualwitharg{11}({L_{35.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{198}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74052\\-7128\\396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7128\\-462\\33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
396\\33\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-26\\-1056
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-338
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-297
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-66\\-528\\-132\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-396\\-144\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\0\\22\\0\\-22\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-396\\0\\132\\0\\-198
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-144\\-22\\0\\4\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-198\\-99\\-198\\0\\198
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-74052, -7128, 396; -7128, -462, 33; 396, 33, -2],[396, 33, 1],[1, 1, 0, -3, -1, -1; 8, 6, -1, -26, -8, -6; 330, 288, -22, -1056, -338, -297],34,16]
%
% END OF LATTICE L35.16
%
\hbox{}\par\smallskip%
%
% LATTICE L35.17
%
\leavevmode%
\llap{\spacer}%
${L_{35.17}}=\pdualwitharg{2.3.11}(\pfillwitharg{3}({L_{35.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
223080\\-616044\\-149160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-616044\\1701480\\411972
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-149160\\411972\\99749
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-9792\\40392
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
95\\27359\\-112860
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23\\6624\\-27325
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\27\\-110
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0\\77\\-318
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\223\\-924
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\-132\\-176\\-132\\0
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0\\12\\0\\-132\\-144\\-132
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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-132\\0\\264
\end{tabular}\endgroup%
\kern3pt%
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-176\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
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-132\\-144\\-132
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[223080, -616044, -149160; -616044, 1701480, 411972; -149160, 411972, 99749],[132, 132, 1],[1, 0, -2, 0, 1, 3; 27, 77, 223, 261, 211, 65; -110, -318, -924, -1078, -870, -264],34,17]
%
% END OF LATTICE L35.17
%
\hbox{}\par\smallskip%
%
% LATTICE L35.18
%
\leavevmode%
\llap{\spacer}%
${L_{35.18}}=\pdualwitharg{2.11}({L_{35.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{792}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50813928\\382932\\12543300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
382932\\2904\\94512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12543300\\94512\\3096295
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\-149\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
341\\-1029\\-1350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
389\\-1175\\-1540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1317\\-3973\\-5214
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
342\\-1031\\-1354
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\-597\\-792
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\right.$}%
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132\\0\\-132\\-528\\-132\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-396\\-144\\-396
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\kern3pt%
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-132\\0\\88\\0\\-44\\-396
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-528\\-396\\0\\132\\0\\-396
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-132\\-144\\-44\\0\\4\\0
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0\\-396\\-396\\-396\\0\\792
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[50813928, 382932, 12543300; 382932, 2904, 94512; 12543300, 94512, 3096295],[396, 132, 1],[50, 341, 389, 1317, 342, 200; -149, -1029, -1175, -3973, -1031, -597; -198, -1350, -1540, -5214, -1354, -792],34,18]
%
% END OF LATTICE L35.18
%

\medskip%
%------------------------------WEYL-GROUP-W36--------------------------------
%
\leavevmode\llap{}%
$W_{36\phantom{0}}$%
\qquad\llap{8} lattices, $\chi=36$%
\hfill%
$\infty222\infty222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L36.1
%
\leavevmode%
\llap{\spacer}%
${L_{36.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{36}}%
\EasyButWeakLineBreak%
{${34}\above{1pt}{1pt}{4,1}{\infty b}{136}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171224\\-40800\\-4352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40800\\-9722\\-1037
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4352\\-1037\\-110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39167\\-164832\\4896
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9336\\-39290\\1167
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
984\\-4141\\122
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\136\\-17
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\-340\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-17\\2
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\172\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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34\\-68\\-17\\-68\\-323\\-1088\\-34\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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34\\-68\\-17\\-68
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\kern3pt%
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-68\\136\\0\\-136
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-17\\0\\2\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-136\\0\\8
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-323\\-1088\\-34\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1088\\-4760\\-204\\-136
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-204\\-11\\-12
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-136\\-12\\-24
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-171224, -40800, -4352; -40800, -9722, -1037; -4352, -1037, -110],[136, 1, 1],[-32, 81, 4, -41, -376, -1713, -76, -55; 136, -340, -17, 172, 1581, 7208, 320, 232; -17, 0, 2, 0, -34, -204, -11, -12],35,1]
%
% END OF LATTICE L36.1
%
\hbox{}\par\smallskip%
%
% LATTICE L36.2
%
\leavevmode%
\llap{\spacer}%
${L_{36.2}}=\pfillwitharg{2}({L_{36.1}})$%
{${}={}$}Nikulin $36$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${34}\above{1pt}{1pt}{2,1}{\infty a}{34}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-102\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\17\\36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\9\\17
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-1\\0\\-17
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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34\\-34\\-17\\-34\\-323\\-544\\-34\\0
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\kern3pt%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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34\\-34\\-17\\-34
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\kern3pt%
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-34\\34\\0\\-34
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-17\\0\\2\\0
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-34\\-34\\0\\2
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-323\\-544\\-34\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-544\\-1190\\-102\\-34
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\kern3pt%
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-34\\-102\\-11\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-6\\-6
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[34, 0, 0; 0, 0, -1; 0, -1, 0],[34, 1, 1],[-1, 1, 0, -1, -16, -35, -3, -1; 0, 0, -1, -4, -51, -102, -8, -2; -17, 0, 1, -4, -85, -204, -19, -8],35,2]
%
% END OF LATTICE L36.2
%
\hbox{}\par\smallskip%
%
% LATTICE L36.3
%
\leavevmode%
\llap{\spacer}%
${L_{36.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{36.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${68}\above{1pt}{1pt}{4,3}{\infty z}{17}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-83164\\578\\-39134
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
578\\-4\\272
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39134\\272\\-18415
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24870\\27489\\-52360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171\\-190\\360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11723\\12957\\-24681
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
290\\221\\-612
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
331\\357\\-697
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\80\\-124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\23\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\-34\\-34\\-34\\-646\\-544\\-68\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\-34\\-34\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\17\\0\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-17\\0\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-646\\-544\\-68\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-544\\-595\\-102\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-102\\-22\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-17\\-6\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-83164, 578, -39134; 578, -4, 272; -39134, 272, -18415],[34, 2, 1],[290, 331, 59, 11, 33, -8, -2, 8; 221, 357, 80, 23, 136, 0, -17, -2; -612, -697, -124, -23, -68, 17, 4, -17],35,3]
%
% END OF LATTICE L36.3
%
\hbox{}\par\smallskip%
%
% LATTICE L36.4
%
\leavevmode%
\llap{\spacer}%
${L_{36.4}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{36.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty a}{2}\above{1pt}{1pt}{r}{2}{34}\above{1pt}{1pt}{l}{2}{34}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
442\\-578\\204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-578\\-2448\\-289
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
204\\-289\\94
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7851\\156\\-16796
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11325\\224\\-24225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3775\\75\\-8076
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-2\\189
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-301\\-6\\644
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-589\\-11\\1258
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-351\\-6\\748
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-17\\-34\\-19\\-32\\-34\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-34\\-32\\-70\\-102\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\34\\0\\-34\\-102\\-187\\-102
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-34\\0\\34\\0\\-34\\-102\\-102
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-17\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\34\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-34\\0\\34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-32\\-34\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-70\\-102\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-102\\-187\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-102\\-102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[442, -578, 204; -578, -2448, -289; 204, -289, 94],[34, 17, 1],[-88, -301, -589, -351, -63, -1, 136, 49; -2, -6, -11, -6, -1, 0, 2, 0; 189, 644, 1258, 748, 134, 2, -289, -102],35,4]
%
% END OF LATTICE L36.4
%
\hbox{}\par\smallskip%
%
% LATTICE L36.5
%
\leavevmode%
\llap{\spacer}%
${L_{36.5}}=\pdualwitharg{2}({L_{36.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${272}\above{1pt}{1pt}{8,3}{\infty z}{68}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5184592\\9112\\77384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9112\\-16\\-136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77384\\-136\\-1155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2823\\112880\\-203184
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-201\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\-1680\\3023
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-17\\136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\289\\-102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\107\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\57\\-74
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
272\\-136\\-136\\-136\\-2584\\-2176\\-272\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\68\\0\\-68\\-2176\\-2380\\-408\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\0\\16\\0\\-272\\-408\\-88\\-24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\-68\\0\\4\\0\\-68\\-24\\-12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
272\\-136\\-136\\-136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\68\\0\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136\\-68\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2584\\-2176\\-272\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2176\\-2380\\-408\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-272\\-408\\-88\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-68\\-24\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5184592, 9112, 77384; 9112, -16, -136; 77384, -136, -1155],[136, 8, 1],[2, -1, -1, -1, -19, -16, -2, 0; -17, 289, 107, 57, 697, 391, 13, -17; 136, -102, -80, -74, -1360, -1122, -136, 2],35,5]
%
% END OF LATTICE L36.5
%
\hbox{}\par\smallskip%
%
% LATTICE L36.6
%
\leavevmode%
\llap{\spacer}%
${L_{36.6}}=\pdualwitharg{2.17}(\pfillwitharg{2}({L_{36.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{r}{2}{68}\above{1pt}{1pt}{l}{2}{17}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\19074\\9248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19074\\5336368\\2587332
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9248\\2587332\\1254465
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
224\\-2523\\5202
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46875\\-525626\\1083750
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22725\\-254823\\525401
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\-776\\1600
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75\\-857\\1767
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
224\\-2523\\5202
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-437\\901
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-34\\-34\\-38\\-32\\-68\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-17\\-32\\-35\\-102\\-17
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\0\\68\\0\\-68\\-102\\-374\\-102
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-17\\0\\17\\0\\-17\\-102\\-51
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-34\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\0\\68\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-17\\0\\17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\-32\\-68\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-35\\-102\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-102\\-374\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-17\\-102\\-51
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[68, 19074, 9248; 19074, 5336368, 2587332; 9248, 2587332, 1254465],[34, 34, 1],[67, 75, 224, 41, 8, 0, 1, 34; -776, -857, -2523, -437, -65, 16, 0, -404; 1600, 1767, 5202, 901, 134, -33, 0, 833],35,6]
%
% END OF LATTICE L36.6
%
\hbox{}\par\smallskip%
%
% LATTICE L36.7
%
\leavevmode%
\llap{\spacer}%
${L_{36.7}}=\pdualwitharg{17}({L_{36.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,1}{\infty b}{8}\above{1pt}{1pt}{b}{2}{34}\above{1pt}{1pt}{b}{2}{136}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
680\\-680\\408
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-680\\-102\\85
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
408\\85\\-66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
111\\2240\\3536
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\2099\\3315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-1400\\-2211
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6\\119\\188
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29\\580\\916
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\463\\731
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\388\\612
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2\\-4\\-17\\-68\\-19\\-64\\-34\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
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2\\-4\\-17\\-68
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-136
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-17\\0\\34\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-136\\0\\136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-64\\-34\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\-280\\-204\\-136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-204\\-187\\-204
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-136\\-204\\-408
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[680, -680, 408; -680, -102, 85; 408, 85, -66],[136, 17, 1],[6, 29, 23, 19, 1, -1, -2, 9; 119, 580, 463, 388, 21, -20, -43, 172; 188, 916, 731, 612, 33, -32, -68, 272],35,7]
%
% END OF LATTICE L36.7
%
\hbox{}\par\smallskip%
%
% LATTICE L36.8
%
\leavevmode%
\llap{\spacer}%
${L_{36.8}}=\pdualwitharg{2.17}({L_{36.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,3}{\infty z}{4}\above{1pt}{1pt}{*}{2}{272}\above{1pt}{1pt}{*}{2}{68}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-272\\1768\\408
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1768\\-10064\\-2312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
408\\-2312\\-531
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\-165\\680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\857\\-3536
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\198\\-817
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\8
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1\\9\\-38
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\HardButStrongLineBreak\kern3pt%
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11\\65\\-272
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\41\\-170
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-136\\-136\\-152\\-128\\-272\\0
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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16\\-8\\-136\\-136
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-68
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-136\\-68\\0\\68
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-152\\-128\\-272\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128\\-140\\-408\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-272\\-408\\-1496\\-408
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-68\\-408\\-204
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-272, 1768, 408; 1768, -10064, -2312; 408, -2312, -531],[136, 136, 1],[-1, 1, 11, 9, 9, 7, 13, -1; -2, 9, 65, 41, 35, 24, 34, -8; 8, -38, -272, -170, -144, -98, -136, 34],35,8]
%
% END OF LATTICE L36.8
%

\medskip%
%------------------------------WEYL-GROUP-W37--------------------------------
%
\leavevmode\llap{}%
$W_{37\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=8$%
\hfill%
$22322$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L37.1
%
\leavevmode%
\llap{\spacer}%
${L_{37.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{37}}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{+}{3}{2}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-246820\\3080\\-3360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3080\\-38\\39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3360\\39\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3\\-286\\-42
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\670\\100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\191\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-287\\-43
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\-3535\\-525
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-10\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\-1\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-10\\-1\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-35\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-246820, 3080, -3360; 3080, -38, 39; -3360, 39, -26],[140, 1, 1],[-3, 7, 2, -3, -37; -286, 670, 191, -287, -3535; -42, 100, 28, -43, -525],36,1]
%
% END OF LATTICE L37.1
%
\hbox{}\par\smallskip%
%
% LATTICE L37.2
%
\leavevmode%
\llap{\spacer}%
${L_{37.2}}=\pfillwitharg{2}({L_{37.1}})$%
{${}={}$}Nikulin $37$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{-}{3}{2}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\17\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-7\\-3
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-12\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-70\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-5\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\-1\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-1\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-35\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[70, -35, 0; -35, 17, 0; 0, 0, 1],[35, 1, 1],[0, 2, -3, -5, -29; 0, 5, -7, -12, -70; 1, 0, -3, -2, 0],36,2]
%
% END OF LATTICE L37.2
%
\hbox{}\par\smallskip%
%
% LATTICE L37.3
%
\leavevmode%
\llap{\spacer}%
${L_{37.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{37.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{+}{3}{10}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1364195\\21175\\551600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21175\\-310\\-8565
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
551600\\-8565\\-223034
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-403\\-172\\-990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-269\\-111\\-661
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-472\\-191\\-1160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
470\\192\\1155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
202\\70\\497
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-15\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-5\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\10\\-5\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-5\\-5\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-35\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1364195, 21175, 551600; 21175, -310, -8565; 551600, -8565, -223034],[35, 5, 1],[-403, -269, -472, 470, 202; -172, -111, -191, 192, 70; -990, -661, -1160, 1155, 497],36,3]
%
% END OF LATTICE L37.3
%
\hbox{}\par\smallskip%
%
% LATTICE L37.4
%
\leavevmode%
\llap{\spacer}%
${L_{37.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{37.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{-}{3}{14}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1633485\\-38605\\696185
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38605\\-882\\16450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
696185\\16450\\-296711
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-614\\-149\\-1449
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1987\\-500\\-4690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-685\\-176\\-1617
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
688\\175\\1624
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
182\\55\\430
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-21\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\35\\0\\-35\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\14\\-7\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-35\\-7\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-35\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1633485, -38605, 696185; -38605, -882, 16450; 696185, 16450, -296711],[35, 7, 1],[-614, -1987, -685, 688, 182; -149, -500, -176, 175, 55; -1449, -4690, -1617, 1624, 430],36,4]
%
% END OF LATTICE L37.4
%
\hbox{}\par\smallskip%
%
% LATTICE L37.5
%
\leavevmode%
\llap{\spacer}%
${L_{37.5}}=\pdualwitharg{2}({L_{37.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{+}{3}{8}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1935080\\-518980\\-503020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-518980\\139208\\134908
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-503020\\134908\\130759
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99\\-5\\386
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
145\\-6\\564
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
257\\-11\\1000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2301\\-105\\8960
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\-4\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-20\\-4\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-140\\0\\280
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1935080, -518980, -503020; -518980, 139208, 134908; -503020, 134908, 130759],[140, 4, 1],[99, -13, 145, 257, 2301; -5, 0, -6, -11, -105; 386, -50, 564, 1000, 8960],36,5]
%
% END OF LATTICE L37.5
%
\hbox{}\par\smallskip%
%
% LATTICE L37.6
%
\leavevmode%
\llap{\spacer}%
${L_{37.6}}=\pdualwitharg{5}({L_{37.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{-}{3}{10}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89460\\-4340\\1120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4340\\-190\\55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1120\\55\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\-145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\14\\-427
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-30\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-10\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\-5\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-10\\-5\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-35\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-89460, -4340, 1120; -4340, -190, 55; 1120, 55, -14],[140, 5, 1],[-3, 1, 2, -2, -6; 8, -2, -5, 4, 14; -210, 72, 140, -145, -427],36,6]
%
% END OF LATTICE L37.6
%
\hbox{}\par\smallskip%
%
% LATTICE L37.7
%
\leavevmode%
\llap{\spacer}%
${L_{37.7}}=\pdualwitharg{7}({L_{37.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{+}{3}{14}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5279260\\-612360\\46760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-612360\\-71022\\5425
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46760\\5425\\-414
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-118\\826
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\290\\-1960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\79\\-546
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\-125\\847
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\-215\\1475
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-42\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\140\\0\\-70\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\14\\-7\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-70\\-7\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-35\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5279260, -612360, 46760; -612360, -71022, 5425; 46760, 5425, -414],[140, 7, 1],[21, -51, -14, 22, 38; -118, 290, 79, -125, -215; 826, -1960, -546, 847, 1475],36,7]
%
% END OF LATTICE L37.7
%
\hbox{}\par\smallskip%
%
% LATTICE L37.8
%
\leavevmode%
\llap{\spacer}%
${L_{37.8}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{37.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{+}{3}{70}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
586390\\6030850\\-2258235
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6030850\\62026405\\-23225615
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2258235\\-23225615\\8696767
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-166\\25428\\67865
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\5959\\15904
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-236\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-4603\\-12285
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\4905\\13091
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-105\\-70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-35\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\0\\70\\-35\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-35\\-35\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-35\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[586390, 6030850, -2258235; 6030850, 62026405, -23225615; -2258235, -23225615, 8696767],[35, 35, 1],[-166, -39, 1, 30, -32; 25428, 5959, -236, -4603, 4905; 67865, 15904, -630, -12285, 13091],36,8]
%
% END OF LATTICE L37.8
%
\hbox{}\par\smallskip%
%
% LATTICE L37.9
%
\leavevmode%
\llap{\spacer}%
${L_{37.9}}=\pdualwitharg{2.5}({L_{37.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{-}{3}{40}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11457880\\56140\\-2951200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56140\\280\\-14460
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2951200\\-14460\\760139
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-188\\-5\\-730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103\\-6\\-400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-206\\-11\\-800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-548\\-21\\-2128
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-60\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\-20\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-20\\-20\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-140\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11457880, 56140, -2951200; 56140, 280, -14460; -2951200, -14460, 760139],[140, 20, 1],[-188, -17, -103, -206, -548; -5, 0, -6, -11, -21; -730, -66, -400, -800, -2128],36,9]
%
% END OF LATTICE L37.9
%
\hbox{}\par\smallskip%
%
% LATTICE L37.10
%
\leavevmode%
\llap{\spacer}%
${L_{37.10}}=\pdualwitharg{2.7}({L_{37.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{+}{3}{56}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
165459560\\5408060\\-42683060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5408060\\176792\\-1395100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42683060\\-1395100\\11010809
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2209\\149\\8582
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1027\\70\\3990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
555\\36\\2156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1737\\115\\6748
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4103\\275\\15940
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-84\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\140\\0\\-140\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\56\\-28\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-140\\-28\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-140\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[165459560, 5408060, -42683060; 5408060, 176792, -1395100; -42683060, -1395100, 11010809],[140, 28, 1],[2209, 1027, 555, 1737, 4103; 149, 70, 36, 115, 275; 8582, 3990, 2156, 6748, 15940],36,10]
%
% END OF LATTICE L37.10
%
\hbox{}\par\smallskip%
%
% LATTICE L37.11
%
\leavevmode%
\llap{\spacer}%
${L_{37.11}}=\pdualwitharg{5.7}({L_{37.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{-}{3}{70}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38220\\45500\\-11200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45500\\-54110\\13335
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11200\\13335\\-3282
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\8\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-2\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\-5\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\4\\-175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\2\\-67
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-210\\-140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-70\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\70\\-35\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-70\\-35\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-35\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38220, 45500, -11200; 45500, -54110, 13335; -11200, 13335, -3282],[140, 35, 1],[71, -27, -47, 56, 22; 8, -2, -5, 4, 2; -210, 84, 140, -175, -67],36,11]
%
% END OF LATTICE L37.11
%
\hbox{}\par\smallskip%
%
% LATTICE L37.12
%
\leavevmode%
\llap{\spacer}%
${L_{37.12}}=\pdualwitharg{2.5.7}({L_{37.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{-}{3}{280}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1960\\147700\\-37380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
147700\\11581080\\-2931040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37380\\-2931040\\741813
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\656\\2590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\39\\154
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\461\\1820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59\\922\\3640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\308\\1216
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-420\\-280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-140\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\280\\-140\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-140\\-140\\280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-140\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1960, 147700, -37380; 147700, 11581080, -2931040; -37380, -2931040, 741813],[140, 140, 1],[-39, -2, -30, -59, -19; 656, 39, 461, 922, 308; 2590, 154, 1820, 3640, 1216],36,12]
%
% END OF LATTICE L37.12
%

\medskip%
%------------------------------WEYL-GROUP-W38--------------------------------
%
\leavevmode\llap{}%
$W_{38\phantom{0}}$%
\qquad\llap{44} lattices, $\chi=36$%
\hfill%
$2222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L38.1
%
\leavevmode%
\llap{\spacer}%
${L_{38.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{38}}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[2380, 140, -140; 140, -4, -3; -140, -3, 6],[140, 1, 1],[23, 29, 7, 0, -2, -3, -1, 1, 1, 9; 240, 294, 68, -2, -28, -40, -14, 12, 12, 98; 610, 756, 178, -3, -63, -90, -28, 30, 29, 245],37,1]
%
% END OF LATTICE L38.1
%
\hbox{}\par\smallskip%
%
% LATTICE L38.2
%
\leavevmode%
\llap{\spacer}%
${L_{38.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{25}}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-2521\\-446040\\-110880
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1\\177\\43
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-133560, 560, 840; 560, -2, -5; 840, -5, 1],[280, 1, 1],[-39, -29, -17, -1, -3, -1, 1, 1, 0, -11; -6900, -5131, -3008, -177, -532, -180, 175, 176, 0, -1946; -1720, -1274, -744, -43, -126, -40, 42, 40, -1, -490],37,2]
%
% END OF LATTICE L38.2
%
\hbox{}\par\smallskip%
%
% LATTICE L38.3
%
\leavevmode%
\llap{\spacer}%
${L_{38.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[264040, -2240, -840; -2240, 19, 7; -840, 7, -2],[280, 1, 1],[119, 86, 49, 5, 3, 1, -2, -1, 1, 18; 14120, 10206, 5816, 594, 357, 120, -238, -120, 118, 2135; -840, -595, -332, -30, -14, 0, 7, -4, -12, -133],37,3]
%
% END OF LATTICE L38.3
%
\hbox{}\par\smallskip%
%
% LATTICE L38.4
%
\leavevmode%
\llap{\spacer}%
${L_{38.4}}=\pfillwitharg{2}({L_{38.1}})$%
{${}={}$}Nikulin $38$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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420\\-175\\0
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-175\\73\\0
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0\\0\\-1
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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2\\5\\0
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3\\7\\0
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-2\\-5\\-2
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\HardButStrongLineBreak\kern3pt%
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-7\\-17\\-5
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-131\\-315\\-91
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-5\\0\\1\\0\\-42
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-7\\0\\2\\0
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-140\\-175\\-42\\0\\14
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-205\\-280\\-75\\-10\\0
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\kern3pt%
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-280\\-399\\-112\\-21\\-21
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-75\\-112\\-33\\-8\\-14
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\kern3pt%
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-10\\-21\\-8\\-4\\-14
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[420, -175, 0; -175, 73, 0; 0, 0, -1],[35, 1, 1],[2, 3, -2, -7, -131, -202, -283, -78, -13, -9; 5, 7, -5, -17, -315, -485, -679, -187, -31, -21; 0, 0, -2, -5, -91, -140, -196, -54, -9, -7],37,4]
%
% END OF LATTICE L38.4
%
\hbox{}\par\smallskip%
%
% LATTICE L38.5
%
\leavevmode%
\llap{\spacer}%
${L_{38.5}}=\pfillwitharg{2}({L_{38.2}})$%
{${}={}$}Nikulin $25'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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210\\19\\89
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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5389\\-36960\\-5390
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455\\-3121\\-455
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2268\\-15552\\-2269
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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11\\-130\\0
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3\\-70\\7
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0\\5\\-1
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{$\left[\!\llap{\phantom{%
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-10\\-21\\-8\\-2\\-7
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[2310, 210, 980; 210, 19, 89; 980, 89, 416],[70, 1, 1],[11, 3, -1, 0, 36, 129, 193, 57, 7, 13; -130, -70, 2, 5, -217, -830, -1274, -386, -53, -119; 0, 7, 2, -1, -42, -140, -203, -58, -6, -7],37,5]
%
% END OF LATTICE L38.5
%
\hbox{}\par\smallskip%
%
% LATTICE L38.6
%
\leavevmode%
\llap{\spacer}%
${L_{38.6}}=\main({L_{38.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1399\\-30800\\-9800
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17\\-372\\-108
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3\\-63\\-7
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-560\\-399\\-224\\-21\\-21
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-20\\-21\\-16\\-4\\-14
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0\\-21\\-28\\-14\\-112
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[11620, 560, 0; 560, 27, 0; 0, 0, -1],[140, 1, 1],[39, 29, 17, 2, 3, 1, -1, -1, 0, 11; -860, -637, -372, -43, -63, -20, 21, 20, -1, -245; -280, -196, -108, -9, -7, 0, 0, -4, -5, -91],37,6]
%
% END OF LATTICE L38.6
%
\hbox{}\par\smallskip%
%
% LATTICE L38.7
%
\leavevmode%
\llap{\spacer}%
${L_{38.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{38.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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9590\\242\\4709
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-32\\3\\65
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3239\\-301\\-6580
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-10\\-14\\0\\2\\0
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-140\\-350\\-42\\0\\14
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-205\\-560\\-75\\-10\\0
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-560\\-1596\\-224\\-42\\-42
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-75\\-224\\-33\\-8\\-14
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-10\\-42\\-8\\-4\\-14
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0\\-42\\-14\\-14\\-112
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19530, 490, 9590; 490, -16, 242; 9590, 242, 4709],[70, 2, 1],[854, 820, -32, -63, 3239, 6126, 18724, 2824, 761, 1647; -80, -77, 3, 6, -301, -570, -1743, -263, -71, -154; -1735, -1666, 65, 128, -6580, -12445, -38038, -5737, -1546, -3346],37,7]
%
% END OF LATTICE L38.7
%
\hbox{}\par\smallskip%
%
% LATTICE L38.8
%
\leavevmode%
\llap{\spacer}%
${L_{38.8}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{38.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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12530\\875\\-5005
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875\\60\\-350
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5005\\-350\\1999
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-11173\\-1596\\-27930
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4508\\644\\11269
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-3\\3\\-7
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-14\\35
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11\\-15\\25
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5\\-16\\10
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-137\\-70\\-350
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12530, 875, -5005; 875, 60, -350; -5005, -350, 1999],[35, 5, 1],[-3, 15, 11, 5, -137, -53, -407, -123, -33, -59; 3, -14, -15, -16, -70, -11, -42, -1, 12, 42; -7, 35, 25, 10, -350, -133, -1015, -305, -80, -140],37,8]
%
% END OF LATTICE L38.8
%
\hbox{}\par\smallskip%
%
% LATTICE L38.9
%
\leavevmode%
\llap{\spacer}%
${L_{38.9}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{38.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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31\\0\\70
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-31\\-2\\-70
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-34\\-5\\-77
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-24\\-13\\-55
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7735, 0, -3430; 0, -7, 0; -3430, 0, 1521],[35, 7, 1],[31, -4, -31, -34, -24, -151, -20, -17, 22, 12; 0, 0, -2, -5, -13, -140, -28, -54, -9, -1; 70, -9, -70, -77, -55, -350, -47, -42, 49, 27],37,9]
%
% END OF LATTICE L38.9
%
\hbox{}\par\smallskip%
%
% LATTICE L38.10
%
\leavevmode%
\llap{\spacer}%
${L_{38.10}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{38.2}}))$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18690, 910, -3850; 910, -40, -185; -3850, -185, 793],[70, 5, 1],[-239, -573, 45, 44, -2270, -1717, -13119, -3957, -533, -1153; -32, -77, 6, 6, -301, -228, -1743, -526, -71, -154; -1168, -2800, 220, 215, -11095, -8392, -64120, -19340, -2605, -5635],37,10]
%
% END OF LATTICE L38.10
%
\hbox{}\par\smallskip%
%
% LATTICE L38.11
%
\leavevmode%
\llap{\spacer}%
${L_{38.11}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{38.2}}))$%
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{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20230, 1190, -5880; 1190, -56, -343; -5880, -343, 1709],[70, 7, 1],[-2147, -147, 81, 79, -583, -15433, -3369, -7113, -958, -296; -160, -11, 6, 6, -43, -1140, -249, -526, -71, -22; -7420, -508, 280, 273, -2015, -53340, -11644, -24584, -3311, -1023],37,11]
%
% END OF LATTICE L38.11
%
\hbox{}\par\smallskip%
%
% LATTICE L38.12
%
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\llap{\spacer}%
${L_{38.12}}=\pdualwitharg{2}({L_{38.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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0\\-84\\-56\\-56\\-448
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[763280, -10780, -197120; -10780, 152, 2784; -197120, 2784, 50907],[140, 4, 1],[238, 355, 104, 24, 22, -18, -47, -16, -2, 132; 215, 308, 87, 17, 21, 5, 0, 1, 5, 133; 910, 1358, 398, 92, 84, -70, -182, -62, -8, 504],37,12]
%
% END OF LATTICE L38.12
%
\hbox{}\par\smallskip%
%
% LATTICE L38.13
%
\leavevmode%
\llap{\spacer}%
${L_{38.13}}=\pdualwitharg{2}(\main({L_{38.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10220\\-2240\\-2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2240\\332\\560
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\560\\621
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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133209\\26180\\517440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7612\\-1497\\-29568
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33908\\-6664\\-131713
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1757\\-345\\-6825
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-4880\\-959\\-18956
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-666\\-131\\-2587
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\HardButStrongLineBreak\kern3pt%
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-207\\-41\\-804
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-101\\-21\\-392
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-20\\-28\\0\\8\\0
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-280\\-700\\-84\\0\\56
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-205\\-560\\-75\\-20\\0
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-560\\-1596\\-224\\-84\\-84
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-75\\-224\\-33\\-16\\-28
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\kern3pt%
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-20\\-84\\-16\\-16\\-56
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10220, -2240, -2520; -2240, 332, 560; -2520, 560, 621],[140, 4, 1],[-1757, -4880, -666, -207, -101, 27, 36, -26, -139, -2321; -345, -959, -131, -41, -21, 5, 7, -5, -27, -455; -6825, -18956, -2587, -804, -392, 105, 140, -101, -540, -9016],37,13]
%
% END OF LATTICE L38.13
%
\hbox{}\par\smallskip%
%
% LATTICE L38.14
%
\leavevmode%
\llap{\spacer}%
${L_{38.14}}=\pdualwitharg{5}({L_{38.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\30\\-5
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\-5\\-4
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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35\\41\\2310
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-25\\-30\\-1651
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{$\left[\!\llap{\phantom{%
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\HardButStrongLineBreak\kern3pt%
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-1\\0\\-70
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3\\6\\200
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3\\5\\200
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19460, 280, 280; 280, 30, -5; 280, -5, -4],[140, 5, 1],[-1, -1, 3, 3, 37, 21, 141, 37, 2, -2; -2, 0, 6, 5, 49, 26, 168, 42, 1, -7; -68, -70, 200, 200, 2450, 1388, 9310, 2440, 130, -140],37,14]
%
% END OF LATTICE L38.14
%
\hbox{}\par\smallskip%
%
% LATTICE L38.15
%
\leavevmode%
\llap{\spacer}%
${L_{38.15}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{38.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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-75\\-1120\\-165\\-40\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-210\\-40\\-20\\-70
\end{tabular}\endgroup%
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0\\-210\\-70\\-70\\-560
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3850560, -27090, 1832180; -27090, 190, -12890; 1832180, -12890, 871791],[70, 10, 1],[-110, -167, 69, -19, -4398, -1556, -23157, -3401, -814, -1433; -13, -70, 1, 5, -217, -83, -1274, -193, -53, -119; 231, 350, -145, 40, 9240, 3269, 48650, 7145, 1710, 3010],37,15]
%
% END OF LATTICE L38.15
%
\hbox{}\par\smallskip%
%
% LATTICE L38.16
%
\leavevmode%
\llap{\spacer}%
${L_{38.16}}=\pdualwitharg{5}(\main({L_{38.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-8540\\-1260\\-560
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-185\\-80
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-560\\-80\\-29
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-729\\4368\\3640
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-100\\599\\500
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-26\\156\\129
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-3\\24\\-8
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1\\-7\\0
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\HardButStrongLineBreak\kern3pt%
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7\\-56\\20
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\HardButStrongLineBreak\kern3pt%
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4\\-33\\15
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3\\-49\\105
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-20\\0\\20\\0\\-420
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-20\\-35\\0\\10\\0
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-164\\-560\\-300\\-20\\0
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\kern3pt%
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-560\\-1995\\-1120\\-105\\-105
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-300\\-1120\\-660\\-80\\-140
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-105\\-80\\-20\\-70
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8540, -1260, -560; -1260, -185, -80; -560, -80, -29],[140, 5, 1],[-3, 1, 7, 4, 3, -5, -29, -23, -6, -17; 24, -7, -56, -33, -49, 24, 175, 152, 45, 133; -8, 0, 20, 15, 105, 48, 140, 60, -5, -35],37,16]
%
% END OF LATTICE L38.16
%
\hbox{}\par\smallskip%
%
% LATTICE L38.17
%
\leavevmode%
\llap{\spacer}%
${L_{38.17}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{38.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2262960\\-24570\\1054760
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24570\\266\\-11452
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1054760\\-11452\\491621
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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4972799\\307200\\-10662400
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-50505\\-3121\\108290
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2317791\\143184\\-4969679
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-539\\-65\\1155
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-29\\-10\\62
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62\\1\\-133
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-13\\5\\28
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2262960, -24570, 1054760; -24570, 266, -11452; 1054760, -11452, 491621],[70, 14, 1],[-539, -29, 62, -13, -582, -7231, -3079, -3170, -764, -195; -65, -10, 1, 5, -31, -415, -182, -193, -53, -17; 1155, 62, -133, 28, 1248, 15505, 6602, 6797, 1638, 418],37,17]
%
% END OF LATTICE L38.17
%
\hbox{}\par\smallskip%
%
% LATTICE L38.18
%
\leavevmode%
\llap{\spacer}%
${L_{38.18}}=\pdualwitharg{7}({L_{38.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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99\\990\\-280
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17\\168\\-50
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27\\262\\-84
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-1\\-13\\0
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1120\\-228\\-448\\-42\\-6
\end{tabular}\endgroup%
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-2100\\-448\\-924\\-112\\-28
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1260, 140, 140; 140, -14, -7; 140, -7, 32],[140, 7, 1],[99, 17, 27, -1, -1, 1, 3, 13, 6, 6; 990, 168, 262, -13, -11, 10, 32, 138, 63, 61; -280, -50, -84, 0, 2, 0, -6, -28, -14, -16],37,18]
%
% END OF LATTICE L38.18
%
\hbox{}\par\smallskip%
%
% LATTICE L38.19
%
\leavevmode%
\llap{\spacer}%
${L_{38.19}}=\pdualwitharg{7}(\main({L_{38.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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0\\-14\\-7
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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14\\-1\\98
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6\\0\\41
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-4\\-1\\-25
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-560\\-57\\-224\\-21\\-3
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-140\\-21\\-112\\-28\\-14
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[140, 0, 0; 0, -14, -7; 0, -7, -3],[140, 7, 1],[-41, -4, -15, -1, 0, 1, 0, -1, -1, -2; 0, -1, -8, -3, -1, 0, 1, 8, 3, 1; -280, -25, -84, 0, 2, 0, -3, -28, -14, -16],37,19]
%
% END OF LATTICE L38.19
%
\hbox{}\par\smallskip%
%
% LATTICE L38.20
%
\leavevmode%
\llap{\spacer}%
${L_{38.20}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{38.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35, 0, 0; 0, 80220, -30205; 0, -30205, 11373],[35, 35, 1],[0, 0, 2, 5, 13, 28, 28, 54, 9, 1; 29, 32, -145, -514, -1380, -2981, -2984, -5759, -962, -96; 77, 85, -385, -1365, -3665, -7917, -7925, -15295, -2555, -255],37,20]
%
% END OF LATTICE L38.20
%
\hbox{}\par\smallskip%
%
% LATTICE L38.21
%
\leavevmode%
\llap{\spacer}%
${L_{38.21}}=\pdualwitharg{5}({L_{38.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[113960, 48440, -1400; 48440, 20590, -595; -1400, -595, 17],[280, 5, 1],[5, -3, -17, -6, -93, -45, -137, -63, 1, 23; -12, 7, 40, 14, 210, 100, 301, 136, -3, -56; -8, 0, 0, -5, -350, -232, -840, -480, -25, -70],37,21]
%
% END OF LATTICE L38.21
%
\hbox{}\par\smallskip%
%
% LATTICE L38.22
%
\leavevmode%
\llap{\spacer}%
${L_{38.22}}=\pdualwitharg{5}({L_{38.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3080, 280, 280; 280, 15, 55; 280, 55, -58],[280, 5, 1],[3, -12, -7, 7, 136, 181, 656, 375, 39, 25; -24, 98, 56, -58, -1113, -1480, -5362, -3064, -318, -203; -8, 35, 20, -20, -385, -512, -1855, -1060, -110, -70],37,22]
%
% END OF LATTICE L38.22
%
\hbox{}\par\smallskip%
%
% LATTICE L38.23
%
\leavevmode%
\llap{\spacer}%
${L_{38.23}}=\pdualwitharg{7}({L_{38.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-117320, 840, 840; 840, 63, -7; 840, -7, -6],[280, 7, 1],[41, 3, 7, -1, -1, -1, 1, 9, 2, 3; 0, -2, -16, -3, -2, 0, 2, 16, 3, 2; 5460, 395, 896, -140, -138, -140, 137, 1232, 273, 404],37,23]
%
% END OF LATTICE L38.23
%
\hbox{}\par\smallskip%
%
% LATTICE L38.24
%
\leavevmode%
\llap{\spacer}%
${L_{38.24}}=\pdualwitharg{7}({L_{38.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[280, 0, 0; 0, -7, -7; 0, -7, -6],[280, 7, 1],[-41, -4, -15, -1, 0, 1, 0, -1, -1, -1; 0, -2, -16, -6, -1, 0, 2, 16, 6, 1; -280, -25, -84, 0, 1, 0, -3, -28, -14, -8],37,24]
%
% END OF LATTICE L38.24
%
\hbox{}\par\smallskip%
%
% LATTICE L38.25
%
\leavevmode%
\llap{\spacer}%
${L_{38.25}}=\pdualwitharg{2}({L_{38.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-37520, 4200, -1400; 4200, -376, 112; -1400, 112, -31],[280, 8, 1],[-3, -1, 1, 4, 51, 73, 197, 13, 3, -2; -90, -28, 30, 119, 1505, 2150, 5796, 382, 87, -63; -190, -56, 63, 248, 3108, 4430, 11928, 785, 176, -140],37,25]
%
% END OF LATTICE L38.25
%
\hbox{}\par\smallskip%
%
% LATTICE L38.26
%
\leavevmode%
\llap{\spacer}%
${L_{38.26}}=\pdualwitharg{2}({L_{38.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-37520, -280, 840; -280, 24, -8; 840, -8, -11],[280, 8, 1],[-1, -1, 1, 2, 23, 16, 85, 11, 1, -2; -30, -28, 30, 59, 665, 460, 2436, 314, 27, -63; -55, -56, 54, 108, 1232, 855, 4536, 586, 52, -112],37,26]
%
% END OF LATTICE L38.26
%
\hbox{}\par\smallskip%
%
% LATTICE L38.27
%
\leavevmode%
\llap{\spacer}%
${L_{38.27}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{38.2}}))$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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-70\\-35\\0\\35\\0
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-574\\-560\\-1050\\-70\\0
\end{tabular}\endgroup%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[915110, -24710, 162050; -24710, 665, -4375; 162050, -4375, 28696],[70, 35, 1],[77, 9, -47, 6, 216, 1071, 1139, 2343, 281, 71; -180, -28, 96, -7, -463, -2308, -2460, -5072, -615, -159; -462, -55, 280, -35, -1290, -6398, -6805, -14000, -1680, -425],37,27]
%
% END OF LATTICE L38.27
%
\hbox{}\par\smallskip%
%
% LATTICE L38.28
%
\leavevmode%
\llap{\spacer}%
${L_{38.28}}=\pdualwitharg{2.5}({L_{38.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4102840, -17780, -1056860; -17780, 80, 4580; -1056860, 4580, 272239],[140, 20, 1],[-17, -18, -85, -237, -4905, -1539, -10874, -3027, -541, -541; 0, 7, 2, -2, -84, -28, -203, -58, -12, -14; -66, -70, -330, -920, -19040, -5974, -42210, -11750, -2100, -2100],37,28]
%
% END OF LATTICE L38.28
%
\hbox{}\par\smallskip%
%
% LATTICE L38.29
%
\leavevmode%
\llap{\spacer}%
${L_{38.29}}=\pdualwitharg{2.5}(\main({L_{38.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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17056\\1600\\66559
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13020, -10640, -3080; -10640, 9420, 2500; -3080, 2500, 729],[140, 20, 1],[-11, 72, -32, -287, -6637, -1055, -14996, -2100, -779, -825; -1, 7, -3, -27, -623, -99, -1407, -197, -73, -77; -43, 280, -125, -1120, -25900, -4117, -58520, -8195, -3040, -3220],37,29]
%
% END OF LATTICE L38.29
%
\hbox{}\par\smallskip%
%
% LATTICE L38.30
%
\leavevmode%
\llap{\spacer}%
${L_{38.30}}=\pdualwitharg{2.7}(\main({L_{38.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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35\\0\\-35\\-140\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-28\\-100
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-35\\0\\7\\0\\-84
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-140\\-28\\0\\56\\0
\end{tabular}\endgroup%
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-1435\\-560\\-525\\-140\\0
\end{tabular}\endgroup%
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-560\\-228\\-224\\-84\\-12
\end{tabular}\endgroup%
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-525\\-224\\-231\\-112\\-28
\end{tabular}\endgroup%
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-140\\-84\\-112\\-112\\-56
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10360, 980, -2660; 980, 84, -252; -2660, -252, 683],[140, 28, 1],[-159, -62, -58, -15, 1, 9, 2, -2, -15, -31; 70, 27, 25, 6, 0, 0, 1, 3, 8, 14; -595, -232, -217, -56, 4, 35, 8, -7, -56, -116],37,30]
%
% END OF LATTICE L38.30
%
\hbox{}\par\smallskip%
%
% LATTICE L38.31
%
\leavevmode%
\llap{\spacer}%
${L_{38.31}}=\pdualwitharg{2.7}({L_{38.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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88760\\32620\\-22400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32620\\11984\\-8232
\end{tabular}\endgroup%
\kern3pt%
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-22400\\-8232\\5653
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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35869\\-1190\\140420
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10550\\-351\\41300
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-9073\\301\\-35519
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-4273\\140\\-16730
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-828\\27\\-3242
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-1541\\50\\-6034
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\HardButStrongLineBreak\kern3pt%
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-193\\6\\-756
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\HardButStrongLineBreak\kern3pt%
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1\\0\\4
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}\right.$}%
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140\\0\\-140\\-280\\-560\\-5740\\-1120\\-2100\\-280\\0
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{$\left[\!\llap{\phantom{%
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0\\4\\0\\-28\\-100
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-140\\0\\28\\0\\-168
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-280\\-28\\0\\56\\0
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-5740\\-1120\\-2100\\-280\\0
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-1120\\-228\\-448\\-84\\-12
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[88760, 32620, -22400; 32620, 11984, -8232; -22400, -8232, 5653],[140, 28, 1],[-4273, -828, -1541, -193, 1, 53, -16, -147, -229, -423; 140, 27, 50, 6, 0, 0, 1, 6, 8, 14; -16730, -3242, -6034, -756, 4, 210, -62, -574, -896, -1656],37,31]
%
% END OF LATTICE L38.31
%
\hbox{}\par\smallskip%
%
% LATTICE L38.32
%
\leavevmode%
\llap{\spacer}%
${L_{38.32}}=\pdualwitharg{2.5.7}(\pfillwitharg{2}({L_{38.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
205940\\-868070\\-424760
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\kern3pt%
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-868070\\3639090\\1780660
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-424760\\1780660\\871303
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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148785\\137083631\\-280082040
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72800\\67074560\\-137043201
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-11\\-10014\\20460
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3\\2758\\-5635
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6\\5379\\-10990
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[205940, -868070, -424760; -868070, 3639090, 1780660; -424760, 1780660, 871303],[70, 70, 1],[-16, -11, 3, 6, -43, -114, -249, -263, -71, -22; -14626, -10014, 2758, 5379, -39723, -105150, -229538, -242310, -65267, -20165; 29883, 20460, -5635, -10990, 81160, 214837, 468980, 495075, 133350, 41200],37,32]
%
% END OF LATTICE L38.32
%
\hbox{}\par\smallskip%
%
% LATTICE L38.33
%
\leavevmode%
\llap{\spacer}%
${L_{38.33}}=\pdualwitharg{5.7}(\main({L_{38.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{140}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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140\\0\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-140\\-140\\-280
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-35\\-125
\end{tabular}\endgroup%
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-140\\0\\140\\0\\-420
\end{tabular}\endgroup%
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-140\\-35\\0\\70\\0
\end{tabular}\endgroup%
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-280\\-125\\-420\\0\\10
\end{tabular}\endgroup%
\kern3pt%
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-1148\\-560\\-2100\\-140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\-285\\-1120\\-105\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2100\\-1120\\-4620\\-560\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-105\\-560\\-140\\-70
\end{tabular}\endgroup%
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0\\-15\\-140\\-70\\-80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[140, 0, 0; 0, -595, 35; 0, 35, -2],[140, 35, 1],[-1, 0, 1, 0, -3, -15, -8, -33, -4, -1; -4, -1, 0, 2, 0, -4, -3, -16, -4, -2; -84, -20, 0, 35, -25, -196, -120, -560, -105, -45],37,33]
%
% END OF LATTICE L38.33
%
\hbox{}\par\smallskip%
%
% LATTICE L38.34
%
\leavevmode%
\llap{\spacer}%
${L_{38.34}}=\pdualwitharg{5.7}({L_{38.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9660\\-2380\\140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2380\\-560\\35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\35\\-2
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-235\\-156\\-21840
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-45\\-6160
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\2\\279
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\-4\\-14
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\30
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\70
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-35
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-235
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-140\\-140\\-280\\-1148\\-1120\\-2100\\-140\\0
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0\\20\\0\\-70\\-250\\-1120\\-1140\\-2240\\-210\\-30
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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0\\20\\0\\-70\\-250
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\140\\0\\-420
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-140\\-70\\0\\70\\0
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-1148\\-1120\\-2100\\-140\\0
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-1120\\-1140\\-2240\\-210\\-30
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-2100\\-2240\\-4620\\-560\\-140
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-210\\-560\\-140\\-70
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-140\\-70\\-80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9660, -2380, 140; -2380, -560, 35; 140, 35, -2],[140, 35, 1],[1, 1, 1, -1, -3, -13, -13, -25, -2, 0; -4, -2, 0, 2, 0, -4, -6, -16, -4, -2; -14, 30, 70, -35, -235, -1106, -1150, -2310, -245, -45],37,34]
%
% END OF LATTICE L38.34
%
\hbox{}\par\smallskip%
%
% LATTICE L38.35
%
\leavevmode%
\llap{\spacer}%
${L_{38.35}}=\pdualwitharg{5.7}({L_{38.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{280}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
890680\\55720\\-7840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55720\\3465\\-490
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7840\\-490\\69
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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9311\\38024\\1330840
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
660\\2694\\94325
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-343\\-12006
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-97\\-392\\-13832
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\HardButStrongLineBreak\kern3pt%
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-49\\-199\\-6995
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\HardButStrongLineBreak\kern3pt%
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-191\\-780\\-27300
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-17\\-72\\-2450
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-150
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-280\\-280\\-280\\-2296\\-1120\\-4200\\-280\\0
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[890680, 55720, -7840; 55720, 3465, -490; -7840, -490, 69],[280, 35, 1],[-97, -49, -191, -17, -1, 1, 1, -1, -7, -11; -392, -199, -780, -72, -5, 0, 3, -4, -26, -44; -13832, -6995, -27300, -2450, -150, 112, 135, -140, -980, -1565],37,35]
%
% END OF LATTICE L38.35
%
\hbox{}\par\smallskip%
%
% LATTICE L38.36
%
\leavevmode%
\llap{\spacer}%
${L_{38.36}}=\pdualwitharg{5.7}({L_{38.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{280}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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0\\35\\0
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-8\\-3\\-135
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-33\\-16\\-560
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-2\\-2\\-35
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-1\\-2\\-20
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[280, 0, 0; 0, 35, 0; 0, 0, -1],[280, 35, 1],[-15, -8, -33, -2, -1, -1, 0, 1, 0, -3; -4, -3, -16, -2, -2, -4, -1, 0, 1, 0; -252, -135, -560, -35, -20, -28, -5, 0, 0, -50],37,36]
%
% END OF LATTICE L38.36
%
\hbox{}\par\smallskip%
%
% LATTICE L38.37
%
\leavevmode%
\llap{\spacer}%
${L_{38.37}}=\pdualwitharg{2.5}({L_{38.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{560}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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1905\\3301\\60960
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-1\\0\\0
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2\\4\\75
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8\\15\\280
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107\\189\\3500
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[560, -280, 0; -280, -6680, 360; 0, 360, -19],[280, 40, 1],[-1, -1, 2, 8, 107, 31, 421, 28, 7, -2; -2, 0, 4, 15, 189, 54, 728, 48, 11, -7; -38, 0, 75, 280, 3500, 998, 13440, 885, 200, -140],37,37]
%
% END OF LATTICE L38.37
%
\hbox{}\par\smallskip%
%
% LATTICE L38.38
%
\leavevmode%
\llap{\spacer}%
${L_{38.38}}=\pdualwitharg{2.5}({L_{38.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{560}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{l}{2}{280}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[560, -280, 0; -280, 120, -40; 0, -40, -79],[280, 40, 1],[-1, -1, 9, 18, 247, 36, 981, 131, 17, -2; -2, 0, 18, 35, 469, 68, 1848, 246, 31, -7; 1, 0, -10, -20, -280, -41, -1120, -150, -20, 0],37,38]
%
% END OF LATTICE L38.38
%
\hbox{}\par\smallskip%
%
% LATTICE L38.39
%
\leavevmode%
\llap{\spacer}%
${L_{38.39}}=\pdualwitharg{2.7}({L_{38.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[985040, 139160, -4760; 139160, 19544, -672; -4760, -672, 23],[280, 56, 1],[29, 9, 3, -2, -1, 1, 3, 3, 5, 4; 0, -2, -2, -3, -1, 0, 2, 2, 3, 1; 5950, 1784, 553, -504, -236, 210, 680, 679, 1120, 852],37,39]
%
% END OF LATTICE L38.39
%
\hbox{}\par\smallskip%
%
% LATTICE L38.40
%
\leavevmode%
\llap{\spacer}%
${L_{38.40}}=\pdualwitharg{2.7}({L_{38.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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93\\290\\-1085
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77\\242\\-904
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77\\244\\-910
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16\\53\\-196
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3\\11\\-40
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-560, -840, -280; -840, -952, -336; -280, -336, -117],[280, 56, 1],[93, 77, 77, 16, 3, 2, -1, -1, 3, 16; 290, 242, 244, 53, 11, 10, -2, -4, 7, 49; -1085, -904, -910, -196, -40, -35, 8, 14, -28, -184],37,40]
%
% END OF LATTICE L38.40
%
\hbox{}\par\smallskip%
%
% LATTICE L38.41
%
\leavevmode%
\llap{\spacer}%
${L_{38.41}}=\pdualwitharg{2.5.7}(\main({L_{38.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
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-3\\5\\20
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[280, -4900, 1260; -4900, 83860, -21560; 1260, -21560, 5543],[140, 140, 1],[-4, -3, 0, 1, -25, -32, -69, -72, -37, -11; -2, 5, 9, 0, -58, -72, -153, -157, -74, -16; -7, 20, 35, 0, -220, -273, -580, -595, -280, -60],37,41]
%
% END OF LATTICE L38.41
%
\hbox{}\par\smallskip%
%
% LATTICE L38.42
%
\leavevmode%
\llap{\spacer}%
${L_{38.42}}=\pdualwitharg{2.5.7}({L_{38.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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0\\20\\0\\-140\\-500
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-140\\0\\140\\0\\-840
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-1120\\-1140\\-2240\\-420\\-60
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[280, 4340, -1120; 4340, 217280, -56000; -1120, -56000, 14433],[140, 140, 1],[-9, -3, 1, 1, -31, -79, -85, -177, -45, -13; -256, -85, 18, 0, -958, -2412, -2583, -5354, -1334, -376; -994, -330, 70, 0, -3720, -9366, -10030, -20790, -5180, -1460],37,42]
%
% END OF LATTICE L38.42
%
\hbox{}\par\smallskip%
%
% LATTICE L38.43
%
\leavevmode%
\llap{\spacer}%
${L_{38.43}}=\pdualwitharg{2.5.7}({L_{38.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}{280}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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1960\\5320\\2240
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840\\2240\\943
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-99\\4984\\-11760
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-35\\1779\\-4200
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-29\\1458\\-3440
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-29\\1454\\-3430
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-1\\51\\-120
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-560, 1960, 840; 1960, 5320, 2240; 840, 2240, 943],[280, 280, 1],[-7, -29, -29, -6, -1, 0, 1, 1, -1, -6; 353, 1458, 1454, 297, 51, 3, -34, -30, 59, 305; -833, -3440, -3430, -700, -120, -7, 80, 70, -140, -720],37,43]
%
% END OF LATTICE L38.43
%
\hbox{}\par\smallskip%
%
% LATTICE L38.44
%
\leavevmode%
\llap{\spacer}%
${L_{38.44}}=\pdualwitharg{2.5.7}({L_{38.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{280}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-12040\\-7560\\3360
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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61\\1938\\4560
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{$\left[\!\llap{\phantom{%
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-70\\0\\35\\0\\-420
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16240, -12040, 5320; -12040, -7560, 3360; 5320, 3360, -1493],[280, 280, 1],[29, 61, 31, 14, 3, 1, -1, -1, 1, 12; 910, 1938, 997, 477, 111, 42, -34, -45, -1, 365; 2142, 4560, 2345, 1120, 260, 98, -80, -105, 0, 860],37,44]
%
% END OF LATTICE L38.44
%

\medskip%
%------------------------------WEYL-GROUP-W39--------------------------------
%
\leavevmode\llap{}%
$W_{39\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=9$%
\hfill%
$42222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L39.1
%
\leavevmode%
\llap{\spacer}%
${L_{39.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{39}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-691460\\2240\\3640
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2240\\-6\\-13
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3640\\-13\\-18
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\115\\119
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-1\\-114\\-120
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-231\\-238
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\225\\240
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15\\1708\\1792
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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2\\-2\\-7\\-5\\0
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-2\\4\\0\\-20\\-84
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\14\\0\\-56
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-20\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-56\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-691460, 2240, 3640; 2240, -6, -13; 3640, -13, -18],[140, 1, 1],[1, -1, -2, 2, 15; 115, -114, -231, 225, 1708; 119, -120, -238, 240, 1792],38,1]
%
% END OF LATTICE L39.1
%
\hbox{}\par\smallskip%
%
% LATTICE L39.2
%
\leavevmode%
\llap{\spacer}%
${L_{39.2}}=\pfillwitharg{2}({L_{39.1}})$%
{${}={}$}Nikulin $39$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-21\\-21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-7\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-10\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\14\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-28\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-35, 0, 0; 0, 1, 0; 0, 0, 1],[35, 1, 1],[0, 0, -1, -3, -5; -1, 1, 0, -10, -21; 1, 0, -7, -15, -21],38,2]
%
% END OF LATTICE L39.2
%
\hbox{}\par\smallskip%
%
% LATTICE L39.3
%
\leavevmode%
\llap{\spacer}%
${L_{39.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{39.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{4}{5}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44030\\-2275\\18060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2275\\-30\\925
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18060\\925\\-7407
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-227\\-52\\-560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75\\17\\185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1674\\385\\4130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
672\\155\\1658
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4667\\1078\\11515
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-5\\-35\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\5\\0\\-10\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\70\\0\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-140\\0\\35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-44030, -2275, 18060; -2275, -30, 925; 18060, 925, -7407],[35, 5, 1],[-227, 75, 1674, 672, 4667; -52, 17, 385, 155, 1078; -560, 185, 4130, 1658, 11515],38,3]
%
% END OF LATTICE L39.3
%
\hbox{}\par\smallskip%
%
% LATTICE L39.4
%
\leavevmode%
\llap{\spacer}%
${L_{39.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{39.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{}{4}{7}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-805\\-1680\\665
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-133\\735
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
665\\735\\-423
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-91\\-469
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\30\\154
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
209\\96\\495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2941\\1350\\6965
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
584\\268\\1383
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-7\\-7\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\7\\0\\-70\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\2\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-70\\0\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-4\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-805, -1680, 665; -1680, -133, 735; 665, 735, -423],[35, 7, 1],[-198, 65, 209, 2941, 584; -91, 30, 96, 1350, 268; -469, 154, 495, 6965, 1383],38,4]
%
% END OF LATTICE L39.4
%
\hbox{}\par\smallskip%
%
% LATTICE L39.5
%
\leavevmode%
\llap{\spacer}%
${L_{39.5}}=\pdualwitharg{2}({L_{39.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28405160\\-807380\\-7311220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-807380\\22952\\207812
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7311220\\207812\\1881839
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-177\\3\\-688
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89\\2\\-346
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-209\\0\\-812
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1333\\10\\-5180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1450\\14\\-5635
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-4\\-28\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0\\-40\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\56\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-40\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-56\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28405160, -807380, -7311220; -807380, 22952, 207812; -7311220, 207812, 1881839],[140, 4, 1],[-177, -89, -209, -1333, -1450; 3, 2, 0, 10, 14; -688, -346, -812, -5180, -5635],38,5]
%
% END OF LATTICE L39.5
%
\hbox{}\par\smallskip%
%
% LATTICE L39.6
%
\leavevmode%
\llap{\spacer}%
${L_{39.6}}=\pdualwitharg{5}({L_{39.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{*}{4}{20}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62580\\1400\\-9940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1400\\-30\\235
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9940\\235\\-1458
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-333\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\386\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\665\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-155\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-209\\-5376\\560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-10\\-35\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\20\\0\\-20\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\70\\0\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-20\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-280\\0\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-62580, 1400, -9940; 1400, -30, 235; -9940, 235, -1458],[140, 5, 1],[-13, 15, 26, -6, -209; -333, 386, 665, -155, -5376; 35, -40, -70, 16, 560],38,6]
%
% END OF LATTICE L39.6
%
\hbox{}\par\smallskip%
%
% LATTICE L39.7
%
\leavevmode%
\llap{\spacer}%
${L_{39.7}}=\pdualwitharg{7}({L_{39.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{*}{4}{28}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7420\\560\\280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
560\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\-21\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\25\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-14\\-7\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\28\\0\\-140\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\2\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-140\\0\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-8\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7420, 560, 280; 560, -42, -21; 280, -21, -10],[140, 7, 1],[0, -1, 0, 2, 1; 3, -14, -1, 25, 16; -7, 0, 2, 0, -8],38,7]
%
% END OF LATTICE L39.7
%
\hbox{}\par\smallskip%
%
% LATTICE L39.8
%
\leavevmode%
\llap{\spacer}%
${L_{39.8}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{39.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{}{4}{35}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101710\\583800\\-218505
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
583800\\3350935\\-1254190
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-218505\\-1254190\\469419
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-171\\-455
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\131\\350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\170\\450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\404\\1064
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\377\\990
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\-35\\-35\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\35\\0\\-70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\10\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-70\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-20\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[101710, 583800, -218505; 583800, 3350935, -1254190; -218505, -1254190, 469419],[35, 35, 1],[4, 0, -9, -33, -37; -171, 131, 170, 404, 377; -455, 350, 450, 1064, 990],38,8]
%
% END OF LATTICE L39.8
%
\hbox{}\par\smallskip%
%
% LATTICE L39.9
%
\leavevmode%
\llap{\spacer}%
${L_{39.9}}=\pdualwitharg{2.5}({L_{39.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{*}{4}{20}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3589320\\1969100\\-907620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1969100\\1080280\\-497920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-907620\\-497920\\229507
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-209\\3\\-820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-125\\2\\-490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-177\\0\\-700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-257\\2\\-1012
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1468\\14\\-5775
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\-20\\-140\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\20\\0\\-40\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\280\\0\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-40\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-280\\0\\35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3589320, 1969100, -907620; 1969100, 1080280, -497920; -907620, -497920, 229507],[140, 20, 1],[-209, -125, -177, -257, -1468; 3, 2, 0, 2, 14; -820, -490, -700, -1012, -5775],38,9]
%
% END OF LATTICE L39.9
%
\hbox{}\par\smallskip%
%
% LATTICE L39.10
%
\leavevmode%
\llap{\spacer}%
${L_{39.10}}=\pdualwitharg{2.7}({L_{39.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{*}{4}{28}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11480\\2100\\-2940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2100\\840\\-532
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2940\\-532\\753
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\3\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\2\\-182
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-181\\10\\-700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\2\\-151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\-28\\-28\\-140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\28\\0\\-280\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\8\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-280\\0\\280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-8\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11480, 2100, -2940; 2100, 840, -532; -2940, -532, 753],[140, 28, 1],[-65, -47, 1, -181, -39; 3, 2, 0, 10, 2; -252, -182, 4, -700, -151],38,10]
%
% END OF LATTICE L39.10
%
\hbox{}\par\smallskip%
%
% LATTICE L39.11
%
\leavevmode%
\llap{\spacer}%
${L_{39.11}}=\pdualwitharg{5.7}({L_{39.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{*}{4}{140}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5740\\-1540\\140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1540\\70\\105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\105\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\29\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-30\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-9\\65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\9\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\56\\-400
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\-70\\-35\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\140\\0\\-140\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\10\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-140\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-40\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5740, -1540, 140; -1540, 70, 105; 140, 105, 6],[140, 35, 1],[-13, 13, 4, -4, -25; 29, -30, -9, 9, 56; -210, 210, 65, -63, -400],38,11]
%
% END OF LATTICE L39.11
%
\hbox{}\par\smallskip%
%
% LATTICE L39.12
%
\leavevmode%
\llap{\spacer}%
${L_{39.12}}=\pdualwitharg{2.5.7}({L_{39.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${280}\above{1pt}{1pt}{*}{4}{140}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54040\\291340\\-75600
\end{tabular}\endgroup%
\kern3pt%
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291340\\1571080\\-407680
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\kern3pt%
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-75600\\-407680\\105789
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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0\\0\\0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[54040, 291340, -75600; 291340, 1571080, -407680; -75600, -407680, 105789],[140, 140, 1],[0, -1, 1, 5, 3; 109, 91, 5, 79, 76; 420, 350, 20, 308, 295],38,12]
%
% END OF LATTICE L39.12
%

\medskip%
%------------------------------WEYL-GROUP-W40--------------------------------
%
\leavevmode\llap{}%
$W_{40\phantom{0}}$%
\qquad\llap{8} lattices, $\chi=40$%
\hfill%
$\infty232\infty232\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L40.1
%
\leavevmode%
\llap{\spacer}%
${L_{40.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{40}}%
\EasyButWeakLineBreak%
{${38}\above{1pt}{1pt}{4,3}{\infty b}{152}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{+}{3}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1205512\\6080\\8968
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6080\\-30\\-47
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8968\\-47\\-62
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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67031\\8557752\\3202640
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\kern3pt%
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-360\\-45961\\-17200
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-441\\-56301\\-21071
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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346\\44175\\16530
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619\\79040\\29564
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2\\256\\95
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-2\\-255\\-96
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-1482\\-2812\\-19\\0
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-2812\\-5928\\-76\\-76
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-19\\-76\\-3\\-7
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1205512, 6080, 8968; 6080, -30, -47; 8968, -47, -62],[152, 1, 1],[346, 619, 2, -2, -4, 65, 7, 74; 44175, 79040, 256, -255, -513, 8284, 893, 9447; 16530, 29564, 95, -96, -190, 3116, 335, 3536],39,1]
%
% END OF LATTICE L40.1
%
\hbox{}\par\smallskip%
%
% LATTICE L40.2
%
\leavevmode%
\llap{\spacer}%
${L_{40.2}}=\pfillwitharg{2}({L_{40.1}})$%
{${}={}$}Nikulin $40$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${38}\above{1pt}{1pt}{2,1}{\infty b}{38}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{-}{3}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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-39\\-152\\-190
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\kern3pt%
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5\\19\\25
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4\\16\\19
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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1\\0\\0
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0\\-1\\1
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\HardButStrongLineBreak\kern3pt%
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-7\\-30\\-31
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[38, 0, 0; 0, 0, -1; 0, -1, 0],[38, 1, 1],[-1, 1, 0, -7, -37, -39, -1, -1; 0, 0, -1, -30, -152, -152, -3, -2; -19, 0, 1, -31, -171, -190, -6, -9],39,2]
%
% END OF LATTICE L40.2
%
\hbox{}\par\smallskip%
%
% LATTICE L40.3
%
\leavevmode%
\llap{\spacer}%
${L_{40.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{40.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${76}\above{1pt}{1pt}{4,1}{\infty z}{19}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{+}{3}{4}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-477128\\5586\\-234688
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5586\\-60\\2746
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234688\\2746\\-115437
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-477128, 5586, -234688; 5586, -60, 2746; -234688, 2746, -115437],[38, 2, 1],[28663, 12829, 168, -167, -353, 1326, 577, 6127; -18202, -8132, -103, 104, 190, -874, -371, -3896; -58710, -26277, -344, 342, 722, -2717, -1182, -12550],39,3]
%
% END OF LATTICE L40.3
%
\hbox{}\par\smallskip%
%
% LATTICE L40.4
%
\leavevmode%
\llap{\spacer}%
${L_{40.4}}=\pdualwitharg{19}(\pfillwitharg{2}({L_{40.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty b}{2}\above{1pt}{1pt}{r}{2}{38}\above{1pt}{1pt}{-}{3}{38}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2306182\\-53048\\-1094362
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53048\\-1178\\-25175
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1094362\\-25175\\-519312
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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45029375\\-4114012\\-94690680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
847872\\-77465\\-1782960
\end{tabular}\endgroup%
\kern3pt%
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21376512\\-1953019\\-44951911
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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9314\\-853\\-19586
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8323\\-764\\-17502
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1003\\-96\\-2109
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-1012\\95\\2128
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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2\\-2\\-19\\-304
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-2\\2\\0\\-266
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-304\\-266\\-19\\38
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-74\\-19\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74\\-78\\-38\\-38
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-38\\-57\\-133
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2306182, -53048, -1094362; -53048, -1178, -25175; -1094362, -25175, -519312],[38, 19, 1],[9314, 8323, 1003, -1012, -98, 893, 3605, 37876; -853, -764, -96, 95, 11, -78, -325, -3463; -19586, -17502, -2109, 2128, 206, -1878, -7581, -79648],39,4]
%
% END OF LATTICE L40.4
%
\hbox{}\par\smallskip%
%
% LATTICE L40.5
%
\leavevmode%
\llap{\spacer}%
${L_{40.5}}=\pdualwitharg{2}({L_{40.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6927856\\-42104\\98800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42104\\-240\\600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
98800\\600\\-1409
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
421609\\849243\\29922264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2800\\5639\\198720
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6020\\-12126\\-427249
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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30\\57\\2128
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-129\\-266\\-9158
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-55\\-112\\-3904
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-576\\-1161\\-40880
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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304\\-152\\-152\\-2432\\-11856\\-5624\\-152\\0
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{$\left[\!\llap{\phantom{%
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304\\-152\\-152\\-2432
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-2432\\-1064\\-8\\16
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-11856\\-5624\\-152\\0
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-5624\\-2964\\-152\\-152
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-152\\-152\\-24\\-56
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6927856, -42104, 98800; -42104, -240, 600; 98800, 600, -1409],[152, 8, 1],[30, -129, -55, -576, -2690, -1201, -15, 16; 57, -266, -112, -1161, -5415, -2413, -29, 33; 2128, -9158, -3904, -40880, -190912, -85234, -1064, 1136],39,5]
%
% END OF LATTICE L40.5
%
\hbox{}\par\smallskip%
%
% LATTICE L40.6
%
\leavevmode%
\llap{\spacer}%
${L_{40.6}}=\pdualwitharg{2.19}(\pfillwitharg{2}({L_{40.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{76}\above{1pt}{1pt}{+}{3}{76}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200260\\-8617222\\-4192388
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8617222\\370832120\\180414538
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4192388\\180414538\\87773965
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-77465\\74133469\\-152380950
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3478704\\-3329138635\\6843026700
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1692432\\-1619666622\\3329216099
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1\\-1442\\2964
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14\\-13551\\27854
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[200260, -8617222, -4192388; -8617222, 370832120, 180414538; -4192388, 180414538, 87773965],[38, 38, 1],[-352, -149, 1, 14, -16, -35, -185, -1486; 336753, 142446, -1442, -13551, 15425, 33643, 177531, 1422263; -692194, -292797, 2964, 27854, -31706, -69153, -364914, -2923454],39,6]
%
% END OF LATTICE L40.6
%
\hbox{}\par\smallskip%
%
% LATTICE L40.7
%
\leavevmode%
\llap{\spacer}%
${L_{40.7}}=\pdualwitharg{19}({L_{40.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty b}{8}\above{1pt}{1pt}{b}{2}{38}\above{1pt}{1pt}{+}{3}{38}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31768\\4256\\2128
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4256\\-570\\-285
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2128\\-285\\-142
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-31768, 4256, 2128; 4256, -570, -285; 2128, -285, -142],[152, 19, 1],[0, 3, 5, 42, 10, 17, 0, -2; -1, 24, 46, 455, 111, 196, 9, -15; 2, -4, -19, -304, -78, -148, -19, 0],39,7]
%
% END OF LATTICE L40.7
%
\hbox{}\par\smallskip%
%
% LATTICE L40.8
%
\leavevmode%
\llap{\spacer}%
${L_{40.8}}=\pdualwitharg{2.19}({L_{40.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4560\\-108376\\1976
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108376\\-2558768\\46664
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1976\\46664\\-851
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-2201\\-3120\\-176320
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\kern3pt%
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-54395\\-77143\\-4359512
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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9\\19\\1064
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4560, -108376, 1976; -108376, -2558768, 46664; 1976, 46664, -851],[152, 152, 1],[111, 49, 9, -15, -1, 6, 46, 455; 158, 71, 19, -16, -2, 7, 59, 640; 8928, 4010, 1064, -912, -112, 398, 3344, 36176],39,8]
%
% END OF LATTICE L40.8
%

\medskip%
%------------------------------WEYL-GROUP-W41--------------------------------
%
\leavevmode\llap{}%
$W_{41\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=7$%
\hfill%
$2264$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L41.1
%
\leavevmode%
\llap{\spacer}%
${L_{41.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{41}}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-235092\\2496\\4836
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2496\\-26\\-53
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4836\\-53\\-94
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\237\\72
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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0\\0\\0
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\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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-6\\0\\6\\-3
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-235092, 2496, 4836; 2496, -26, -53; 4836, -53, -94],[156, 1, 1],[-3, -3, 4, 2; -178, -182, 237, 120; -54, -52, 72, 35],40,1]
%
% END OF LATTICE L41.1
%
\hbox{}\par\smallskip%
%
% LATTICE L41.2
%
\leavevmode%
\llap{\spacer}%
${L_{41.2}}=\pfillwitharg{2}({L_{41.1}})$%
{${}={}$}Nikulin $41$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390\\-117\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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0\\0\\1
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-4\\13\\0
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\HardButStrongLineBreak\kern3pt%
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1\\-3\\-3
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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1\\0\\-3\\-1
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-3\\0\\6\\-3
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-1\\-13\\-3\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-390, -117, 0; -117, -35, 0; 0, 0, 1],[39, 1, 1],[0, -4, 1, 6; 0, 13, -3, -19; 1, 0, -3, -1],40,2]
%
% END OF LATTICE L41.2
%
\hbox{}\par\smallskip%
%
% LATTICE L41.3
%
\leavevmode%
\llap{\spacer}%
${L_{41.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{41.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12207\\741\\-3900
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
741\\-33\\237
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3900\\237\\-1246
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-1\\75
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
148\\-13\\-468
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-1\\-154
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\2\\-81
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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3\\0\\-3\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\39\\0\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-39\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12207, 741, -3900; 741, -33, 237; -3900, 237, -1246],[39, 3, 1],[-24, 148, 49, 26; -1, -13, -1, 2; 75, -468, -154, -81],40,3]
%
% END OF LATTICE L41.3
%
\hbox{}\par\smallskip%
%
% LATTICE L41.4
%
\leavevmode%
\llap{\spacer}%
${L_{41.4}}=\pdualwitharg{3}({L_{41.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1651260\\21996\\8892
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21996\\-282\\-123
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8892\\-123\\-46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-266\\-642
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-416\\-1014
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\114\\275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\227\\552
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\156\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-78\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1651260, 21996, 8892; 21996, -282, -123; 8892, -123, -46],[156, 3, 1],[-7, -11, 3, 6; -266, -416, 114, 227; -642, -1014, 275, 552],40,4]
%
% END OF LATTICE L41.4
%
\hbox{}\par\smallskip%
%
% LATTICE L41.5
%
\leavevmode%
\llap{\spacer}%
${L_{41.5}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{41.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${13}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{}{6}{26}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21567\\-1404\\-6786
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1404\\-91\\-442
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6786\\-442\\-2135
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\12\\26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-4\\-37
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
103\\-63\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-14\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\0\\-39\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\78\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-13\\-39\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21567, -1404, -6786; -1404, -91, -442; -6786, -442, -2135],[39, 13, 1],[-9, 12, 103, 5; 12, -4, -63, -14; 26, -37, -312, -13],40,5]
%
% END OF LATTICE L41.5
%
\hbox{}\par\smallskip%
%
% LATTICE L41.6
%
\leavevmode%
\llap{\spacer}%
${L_{41.6}}=\pdualwitharg{2}({L_{41.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2770872\\-302484\\-715572
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-302484\\33032\\78116
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-715572\\78116\\184795
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\1\\-74
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\0\\-182
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145\\6\\-564
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-338\\17\\-1316
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\52\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-52\\-12\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2770872, -302484, -715572; -302484, 33032, 78116; -715572, 78116, 184795],[156, 4, 1],[-19, -47, -145, -338; 1, 0, 6, 17; -74, -182, -564, -1316],40,6]
%
% END OF LATTICE L41.6
%
\hbox{}\par\smallskip%
%
% LATTICE L41.7
%
\leavevmode%
\llap{\spacer}%
${L_{41.7}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{41.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${39}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{}{6}{78}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4251\\-10686\\-3432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10686\\36582\\11739
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3432\\11739\\3767
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-512\\1599
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\1317\\-4113
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\3276\\-10231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-25\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\0\\-39\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\26\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-39\\-39\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4251, -10686, -3432; -10686, 36582, 11739; -3432, 11739, 3767],[39, 39, 1],[4, -10, -25, 0; -512, 1317, 3276, -25; 1599, -4113, -10231, 78],40,7]
%
% END OF LATTICE L41.7
%
\hbox{}\par\smallskip%
%
% LATTICE L41.8
%
\leavevmode%
\llap{\spacer}%
${L_{41.8}}=\pdualwitharg{2.3}({L_{41.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{156}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216053448\\-7188948\\-55461432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7188948\\239208\\1845420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55461432\\1845420\\14237081
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-97\\1\\-378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-380\\13\\-1482
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-236\\5\\-920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1521\\22\\-5928
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\156\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-156\\-12\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[216053448, -7188948, -55461432; -7188948, 239208, 1845420; -55461432, 1845420, 14237081],[156, 12, 1],[-97, -380, -236, -1521; 1, 13, 5, 22; -378, -1482, -920, -5928],40,8]
%
% END OF LATTICE L41.8
%
\hbox{}\par\smallskip%
%
% LATTICE L41.9
%
\leavevmode%
\llap{\spacer}%
${L_{41.9}}=\pdualwitharg{13}({L_{41.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{}{6}{26}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-839748\\17004\\10920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17004\\-338\\-221
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10920\\-221\\-142
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\-234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\12\\-481
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\0\\-78\\-26
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-839748, 17004, 10920; 17004, -338, -221; 10920, -221, -142],[156, 13, 1],[-3, 1, 4, -6; 2, -2, -3, 12; -234, 80, 312, -481],40,9]
%
% END OF LATTICE L41.9
%
\hbox{}\par\smallskip%
%
% LATTICE L41.10
%
\leavevmode%
\llap{\spacer}%
${L_{41.10}}=\pdualwitharg{3.13}({L_{41.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{}{6}{78}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-526188\\-4778436\\-368940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4778436\\-43362618\\-3348033
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-368940\\-3348033\\-258502
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
175\\1402\\-18408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-218\\2862
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-601\\7891
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
163\\1316\\-17277
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\-78\\-78
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-78
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\0\\26\\-39
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-78\\-78\\-39\\78
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-526188, -4778436, -368940; -4778436, -43362618, -3348033; -368940, -3348033, -258502],[156, 39, 1],[175, -27, -75, 163; 1402, -218, -601, 1316; -18408, 2862, 7891, -17277],40,10]
%
% END OF LATTICE L41.10
%
\hbox{}\par\smallskip%
%
% LATTICE L41.11
%
\leavevmode%
\llap{\spacer}%
${L_{41.11}}=\pdualwitharg{2.13}({L_{41.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{}{6}{104}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13778232\\-92196\\3608592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92196\\728\\-24232
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3608592\\-24232\\945175
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
247\\-740\\-962
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-57\\-74
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
721\\-2157\\-2808
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2430\\-7273\\-9464
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\0\\-156\\-52
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-52
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\0\\312\\-156
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\kern3pt%
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-52\\-52\\-156\\104
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13778232, -92196, 3608592; -92196, 728, -24232; 3608592, -24232, 945175],[156, 52, 1],[247, 19, 721, 2430; -740, -57, -2157, -7273; -962, -74, -2808, -9464],40,11]
%
% END OF LATTICE L41.11
%
\hbox{}\par\smallskip%
%
% LATTICE L41.12
%
\leavevmode%
\llap{\spacer}%
${L_{41.12}}=\pdualwitharg{2.3.13}({L_{41.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{}{6}{312}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96073224\\1309261044\\-335401092
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1309261044\\17842271304\\-4570759440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-335401092\\-4570759440\\1170918293
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-324\\-1315\\-5226
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-95\\-378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-1309\\-5200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3185\\-13194\\-52416
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\-156\\-156
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\0\\104\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-156\\-156\\312
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[96073224, 1309261044, -335401092; 1309261044, 17842271304, -4570759440; -335401092, -4570759440, 1170918293],[156, 156, 1],[-324, -25, -315, -3185; -1315, -95, -1309, -13194; -5226, -378, -5200, -52416],40,12]
%
% END OF LATTICE L41.12
%

\medskip%
%------------------------------WEYL-GROUP-W42--------------------------------
%
\leavevmode\llap{}%
$W_{42\phantom{0}}$%
\qquad\llap{104} lattices, $\chi=36$%
\hfill%
$2222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L42.1
%
\leavevmode%
\llap{\spacer}%
${L_{42.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2m\rightarrow N_{42},2\rightarrow N'_{26}}%
\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2834520\\5616\\5928
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5616\\-10\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5928\\-13\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4679\\1212120\\1085760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-2591\\-2320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-2331\\-2089
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\258\\233
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\2327\\2093
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\1812\\1626
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\2332\\2088
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\18408\\16458
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-20\\-234\\-10\\-39\\-12\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\13\\0\\-52\\-780\\-39\\-182\\-78\\-52\\-234
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\12\\0\\-156\\-12\\-78\\-48\\-48\\-312
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-52\\0\\8\\0\\-4\\-52\\-48\\-72\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-780\\-156\\0\\156\\0\\-234\\-312\\-624\\-6240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-20\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\13\\0\\-52\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\12\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-52\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-780\\-156\\0\\156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-39\\-12\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-182\\-78\\-52\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-78\\-48\\-48\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-52\\-48\\-72\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-234\\-312\\-624\\-6240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2834520, 5616, 5928; 5616, -10, -13; 5928, -13, -11],[312, 1, 1],[1, 9, 7, 9, 71, 2, 4, -1, -1, 7; 258, 2327, 1812, 2332, 18408, 519, 1040, -258, -260, 1794; 233, 2093, 1626, 2088, 16458, 463, 923, -234, -232, 1638],41,1]
%
% END OF LATTICE L42.1
%
\hbox{}\par\smallskip%
%
% LATTICE L42.2
%
\leavevmode%
\llap{\spacer}%
${L_{42.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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11\\2280\\846
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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0\\-234\\-156\\-312\\-3120\\-234\\-390\\-39\\0\\39
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3004872, 12168, 6240; 12168, -49, -26; 6240, -26, -11],[312, 1, 1],[2, 31, 11, 13, 97, 5, 4, -1, -1, 10; 414, 6422, 2280, 2696, 20124, 1038, 832, -207, -208, 2067; 155, 2392, 846, 996, 7410, 380, 299, -78, -76, 780],41,2]
%
% END OF LATTICE L42.2
%
\hbox{}\par\smallskip%
%
% LATTICE L42.3
%
\leavevmode%
\llap{\spacer}%
${L_{42.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50843832\\-1271088\\49920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1271088\\-31777\\1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49920\\1248\\-49
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47423\\-1873248\\592800
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1186\\-46848\\14825
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\1817\\-576
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-274\\152
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\HardButStrongLineBreak\kern3pt%
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49\\-1924\\910
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\HardButStrongLineBreak\kern3pt%
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8\\-315\\126
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17\\-672\\200
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58\\-2301\\468
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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\kern3pt%
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-12\\-78\\-12\\-24\\-78
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\kern3pt%
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-8\\-104\\-24\\-72\\-312
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\kern3pt%
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0\\-234\\-78\\-312\\-1560
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-50843832, -1271088, 49920; -1271088, -31777, 1248; 49920, 1248, -49],[312, 1, 1],[7, 49, 8, 17, 58, 5, 3, -2, -1, 20; -274, -1924, -315, -672, -2301, -200, -130, 78, 40, -780; 152, 910, 126, 200, 468, -2, -260, -51, 0, 507],41,3]
%
% END OF LATTICE L42.3
%
\hbox{}\par\smallskip%
%
% LATTICE L42.4
%
\leavevmode%
\llap{\spacer}%
${L_{42.4}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{624}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{208}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
169728\\84398\\-310
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-310\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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65\\-132\\-336
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\HardButStrongLineBreak\kern3pt%
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-1\\2\\0
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\HardButStrongLineBreak\kern3pt%
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0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\-468\\-624\\-624\\-12480\\-234\\-3120\\-78\\0\\156
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[341328, 169728, -624; 169728, 84398, -310; -624, -310, 1],[624, 2, 1],[201, 333, 65, -1, -77, 0, 205, 34, 135, 1345; -408, -676, -132, 2, 156, 0, -416, -69, -274, -2730; -1016, -1690, -336, 0, 312, -1, -1040, -171, -680, -6786],41,4]
%
% END OF LATTICE L42.4
%
\hbox{}\par\smallskip%
%
% LATTICE L42.5
%
\leavevmode%
\llap{\spacer}%
${L_{42.5}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{624}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{208}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7951632\\7488\\18096
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7488\\-2\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18096\\-18\\-41
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\-76\\-408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-234\\-1222
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\72\\408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\229\\1222
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
167\\12792\\68016
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\307\\1629
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\5460\\28912
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\540\\2850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\155\\814
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\546\\2847
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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0\\52\\0\\-52\\-3120\\-78\\-1456\\-156\\-52\\-234
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\0\\-624\\-24\\-624\\-96\\-48\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-78\\-24\\-2\\0\\1\\0\\-6\\-10\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-1456\\-624\\-104\\-1872\\0\\208\\0\\-104\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-156\\-96\\-24\\-624\\-6\\0\\12\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-52\\-48\\-18\\-624\\-10\\-104\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-234\\-312\\-156\\-6240\\-117\\-1560\\-78\\0\\39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-7951632, 7488, 18096; 7488, -2, -18; 18096, -18, -41],[624, 2, 1],[-1, -3, 1, 3, 167, 4, 71, 7, 2, 7; -76, -234, 72, 229, 12792, 307, 5460, 540, 155, 546; -408, -1222, 408, 1222, 68016, 1629, 28912, 2850, 814, 2847],41,5]
%
% END OF LATTICE L42.5
%
\hbox{}\par\smallskip%
%
% LATTICE L42.6
%
\leavevmode%
\llap{\spacer}%
${L_{42.6}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{624}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{208}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-501072\\1872\\2496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1872\\-2\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2496\\-12\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-76\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-78\\-143
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\72\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\151\\286
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107\\8112\\15288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\380\\714
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\3276\\6136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\153\\285
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\77\\142
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\390\\702
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-48\\-40\\-1872\\-80\\-624\\-24\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\13\\0\\-26\\-1560\\-78\\-728\\-39\\-26\\-234
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\0\\-624\\-48\\-624\\-48\\-48\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-26\\0\\2\\0\\-4\\-104\\-12\\-18\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-1560\\-624\\0\\624\\0\\-1872\\-312\\-624\\-12480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-78\\-48\\-4\\0\\4\\0\\-6\\-20\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-728\\-624\\-104\\-1872\\0\\208\\0\\-104\\-3120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-39\\-48\\-12\\-312\\-6\\0\\3\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-26\\-48\\-18\\-624\\-20\\-104\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-234\\-624\\-312\\-12480\\-468\\-3120\\-78\\0\\156
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-501072, 1872, 2496; 1872, -2, -12; 2496, -12, -11],[624, 2, 1],[-1, -1, 1, 2, 107, 5, 43, 2, 1, 5; -76, -78, 72, 151, 8112, 380, 3276, 153, 77, 390; -144, -143, 144, 286, 15288, 714, 6136, 285, 142, 702],41,6]
%
% END OF LATTICE L42.6
%
\hbox{}\par\smallskip%
%
% LATTICE L42.7
%
\leavevmode%
\llap{\spacer}%
${L_{42.7}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{624}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{208}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
243984\\-1872\\1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\14\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1872\\-12\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\1072\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\611\\91
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-156\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-154\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\156\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\154\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\3536\\520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\768\\114
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\1074\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\4446\\663
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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-80\\-52\\0\\8\\0\\-8\\-208\\-48\\-72\\-312
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-78\\-96\\-48\\-624\\-12\\0\\12\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-52\\-96\\-72\\-1248\\-40\\-208\\0\\8\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-117\\-312\\-312\\-6240\\-234\\-1560\\-78\\0\\39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[243984, -1872, 1872; -1872, 14, -12; 1872, -12, -1],[624, 2, 1],[7, 4, -1, -1, 1, 1, 23, 5, 7, 29; 1072, 611, -156, -154, 156, 154, 3536, 768, 1074, 4446; 160, 91, -24, -24, 0, 22, 520, 114, 160, 663],41,7]
%
% END OF LATTICE L42.7
%
\hbox{}\par\smallskip%
%
% LATTICE L42.8
%
\leavevmode%
\llap{\spacer}%
${L_{42.8}}=\main(\pfillwitharg{2}({L_{42.1}}))$%
{${}={}$}Nikulin $42$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
312\\117\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
117\\43\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-13\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\-419\\-176
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\-114\\-49
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\26\\13
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-6\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-41\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
370\\-1170\\-507
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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0\\26\\0\\-26\\-780\\-78\\-364\\-78\\-26\\-234
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[312, 117, -39; 117, 43, -13; -39, -13, 2],[39, 1, 1],[0, -8, 2, 13, 370, 36, 164, 34, 11, 98; 0, 26, -6, -41, -1170, -114, -520, -108, -35, -312; 1, 13, -3, -18, -507, -49, -221, -45, -14, -117],41,8]
%
% END OF LATTICE L42.8
%
\hbox{}\par\smallskip%
%
% LATTICE L42.9
%
\leavevmode%
\llap{\spacer}%
${L_{42.9}}=\pfillwitharg{2}({L_{42.1}})$%
{${}={}$}Nikulin $26'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[4602, 2262, 234; 2262, 1112, 117; 234, 117, 35],[78, 1, 1],[39, -1628, -1228, -2689, -27938, -2172, -7615, -905, -155, 209; -80, 3341, 2520, 5518, 57330, 4457, 15626, 1857, 318, -429; 7, -299, -225, -492, -5109, -397, -1391, -165, -28, 39],41,9]
%
% END OF LATTICE L42.9
%
\hbox{}\par\smallskip%
%
% LATTICE L42.10
%
\leavevmode%
\llap{\spacer}%
${L_{42.10}}=\main({L_{42.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-4361916, -109044, 4680; -109044, -2726, 117; 4680, 117, -5],[156, 1, 1],[4, 8, -1, -1, 7, 2, 18, 7, 9, 71; -161, -325, 39, 40, -273, -79, -715, -279, -360, -2847; -28, -130, -24, 0, 156, 22, 104, 18, -8, -234],41,10]
%
% END OF LATTICE L42.10
%
\hbox{}\par\smallskip%
%
% LATTICE L42.11
%
\leavevmode%
\llap{\spacer}%
${L_{42.11}}=\main({L_{42.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\26\\0\\-52\\-780\\-78\\-364\\-78\\-52\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-52\\0\\4\\0\\-4\\-52\\-24\\-36\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-780\\-78\\0\\78\\0\\-234\\-156\\-312\\-3120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-6\\-20\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\26\\0\\-52\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-78
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-52\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-780\\-78\\0\\78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-78\\-12\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-364\\-78\\-52\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-78\\-24\\-24\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-52\\-24\\-36\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-234\\-156\\-312\\-3120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-4836, -2496, 156; -2496, -1286, 79; 156, 79, -4],[156, 1, 1],[10, 43, 7, -1, -55, -7, -30, -4, 5, 94; -21, -91, -15, 2, 117, 15, 65, 9, -10, -195; -28, -130, -24, 0, 156, 22, 104, 18, -8, -234],41,11]
%
% END OF LATTICE L42.11
%
\hbox{}\par\smallskip%
%
% LATTICE L42.12
%
\leavevmode%
\llap{\spacer}%
${L_{42.12}}=\pfillwitharg{2}({L_{42.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{52}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{156}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-26\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-78\\-273
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-13\\-65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-10\\-117\\-20\\-39\\-12\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\52\\0\\-52\\-780\\-156\\-364\\-156\\-52\\-468
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\3\\0\\-39\\-12\\-39\\-24\\-12\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-52\\0\\2\\0\\-4\\-26\\-24\\-18\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-780\\-39\\0\\39\\0\\-117\\-156\\-156\\-3120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-156\\-12\\-4\\0\\4\\0\\-12\\-20\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-364\\-39\\-26\\-117\\0\\13\\0\\-26\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-156\\-24\\-24\\-156\\-12\\0\\12\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-52\\-12\\-18\\-156\\-20\\-26\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-468\\-156\\-312\\-3120\\-468\\-780\\-156\\0\\156
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[156, 0, 0; 0, 2, 0; 0, 0, -1],[156, 2, 1],[0, -3, -1, -2, -20, -3, -5, -1, 0, 1; -1, -26, -6, -9, -78, -10, -13, 0, 1, 0; -1, -52, -15, -28, -273, -40, -65, -12, 0, 0],41,12]
%
% END OF LATTICE L42.12
%
\hbox{}\par\smallskip%
%
% LATTICE L42.13
%
\leavevmode%
\llap{\spacer}%
${L_{42.13}}=\main({L_{42.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{26}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{312}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{78}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-250536\\2496\\936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2496\\-22\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
936\\-12\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\-72\\-76
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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1\\72\\72
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\HardButStrongLineBreak\kern3pt%
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2\\143\\151
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107\\7644\\8112
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\HardButStrongLineBreak\kern3pt%
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5\\357\\380
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43\\3068\\3276
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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4\\285\\306
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\71\\77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\351\\390
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-24\\-20\\-936\\-40\\-312\\-24\\-4\\0
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0\\26\\0\\-26\\-1560\\-78\\-728\\-78\\-26\\-234
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-312\\-24\\-312\\-48\\-24\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-26\\0\\1\\0\\-2\\-52\\-12\\-9\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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-40\\-78\\-24\\-2\\0\\2\\0\\-6\\-10\\-234
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-728\\-312\\-52\\-936\\0\\104\\0\\-52\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-78\\-48\\-12\\-312\\-6\\0\\6\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-26\\-24\\-9\\-312\\-10\\-52\\0\\1\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-234\\-312\\-156\\-6240\\-234\\-1560\\-78\\0\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-250536, 2496, 936; 2496, -22, -12; 936, -12, -1],[312, 2, 1],[-1, -2, 1, 2, 107, 5, 43, 4, 1, 5; -72, -143, 72, 143, 7644, 357, 3068, 285, 71, 351; -76, -156, 72, 151, 8112, 380, 3276, 306, 77, 390],41,13]
%
% END OF LATTICE L42.13
%
\hbox{}\par\smallskip%
%
% LATTICE L42.14
%
\leavevmode%
\llap{\spacer}%
${L_{42.14}}=\main({L_{42.7}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{2}{104}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{78}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{26}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{312}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13416\\6240\\312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6240\\2866\\176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
312\\176\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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289\\-585\\-702
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\HardButStrongLineBreak\kern3pt%
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73\\-148\\-176
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\HardButStrongLineBreak\kern3pt%
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173\\-351\\-416
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\HardButStrongLineBreak\kern3pt%
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65\\-132\\-156
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\HardButStrongLineBreak\kern3pt%
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-1\\2\\2
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\HardButStrongLineBreak\kern3pt%
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-461\\936\\1092
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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\kern3pt%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[13416, 6240, 312; 6240, 2866, 176; 312, 176, -19],[312, 2, 1],[-32, -307, -28, 7, 289, 73, 173, 65, -1, -461; 65, 624, 57, -14, -585, -148, -351, -132, 2, 936; 76, 728, 66, -18, -702, -176, -416, -156, 2, 1092],41,14]
%
% END OF LATTICE L42.14
%
\hbox{}\par\smallskip%
%
% LATTICE L42.15
%
\leavevmode%
\llap{\spacer}%
${L_{42.15}}=\pdualwitharg{3}(\main(\pfillwitharg{2}({L_{42.1}})))$%
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{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-29211, -156, -9516; -156, 6, -51; -9516, -51, -3100],[39, 3, 1],[172, 4650, 890, 2455, 15822, 3446, 11028, 316, -43, -144; -13, -377, -73, -203, -1313, -287, -923, -27, 3, 13; -528, -14274, -2732, -7536, -48568, -10578, -33852, -970, 132, 442],41,15]
%
% END OF LATTICE L42.15
%
\hbox{}\par\smallskip%
%
% LATTICE L42.16
%
\leavevmode%
\llap{\spacer}%
${L_{42.16}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{42.1}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3272880, 94458, -1640496; 94458, -2726, 47346; -1640496, 47346, -822281],[78, 2, 1],[-87, 2800, 2173, 2415, 50459, 3942, 13905, 1682, 155, -344; 77, -2327, -1821, -2033, -42549, -3329, -11765, -1431, -135, 273; 178, -5720, -4440, -4935, -103116, -8056, -28418, -3438, -317, 702],41,16]
%
% END OF LATTICE L42.16
%
\hbox{}\par\smallskip%
%
% LATTICE L42.17
%
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${L_{42.17}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{42.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5230758, 146250, 1739400; 146250, -4089, -48633; 1739400, -48633, -578408],[78, 3, 1],[65, -2140, -552, -3675, -12790, -2996, -10561, -425, -233, 89; 77, -2327, -607, -4066, -14183, -3329, -11765, -477, -270, 91; 189, -6240, -1609, -10710, -37271, -8730, -30771, -1238, -678, 260],41,17]
%
% END OF LATTICE L42.17
%
\hbox{}\par\smallskip%
%
% LATTICE L42.18
%
\leavevmode%
\llap{\spacer}%
${L_{42.18}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{42.1}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[623298, -63414, 286026; -63414, 6672, -29094; 286026, -29094, 131255],[78, 6, 1],[-1327, 55319, 13911, 45696, 316523, 73825, 258839, 10255, 2636, -2365; -80, 3341, 840, 2759, 19110, 4457, 15626, 619, 159, -143; 2874, -119808, -30128, -98967, -685516, -159888, -560586, -22210, -5709, 5122],41,18]
%
% END OF LATTICE L42.18
%
\hbox{}\par\smallskip%
%
% LATTICE L42.19
%
\leavevmode%
\llap{\spacer}%
${L_{42.19}}=\pdualwitharg{3}(\main({L_{42.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L42.19
%
\hbox{}\par\smallskip%
%
% LATTICE L42.20
%
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${L_{42.20}}=\pdualwitharg{3}(\main({L_{42.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1716, 780, -156; 780, 354, -75; -156, -75, -16],[156, 3, 1],[-4, 43, 7, -1, -79, -35, -212, -20, -51, -90; 9, -91, -15, 2, 169, 75, 455, 43, 110, 195; -6, 0, 2, 0, -26, -12, -78, -8, -24, -52],41,20]
%
% END OF LATTICE L42.20
%
\hbox{}\par\smallskip%
%
% LATTICE L42.21
%
\leavevmode%
\llap{\spacer}%
${L_{42.21}}=\pdualwitharg{13}(\main(\pfillwitharg{2}({L_{42.1}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-59592, -1989, -18291; -1989, -65, -611; -18291, -611, -5614],[39, 13, 1],[104, 224, 1682, 1553, 2314, 2188, 540, 610, -25, -22; -124, -258, -1926, -1771, -2634, -2486, -612, -684, 31, 24; -325, -701, -5265, -4862, -7245, -6851, -1691, -1911, 78, 69],41,21]
%
% END OF LATTICE L42.21
%
\hbox{}\par\smallskip%
%
% LATTICE L42.22
%
\leavevmode%
\llap{\spacer}%
${L_{42.22}}=\pdualwitharg{2}(\main({L_{42.3}}))$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[316680, 8892, -81432; 8892, 2380, -2340; -81432, -2340, 20941],[156, 4, 1],[-5777, -16455, -797, 10, -8761, -1633, -15655, -6613, -2480, -87569; -566, -1612, -78, 1, -858, -160, -1534, -648, -243, -8580; -22528, -64168, -3108, 39, -34164, -6368, -61048, -25788, -9671, -341484],41,22]
%
% END OF LATTICE L42.22
%
\hbox{}\par\smallskip%
%
% LATTICE L42.23
%
\leavevmode%
\llap{\spacer}%
${L_{42.23}}=\pdualwitharg{2}(\main({L_{42.2}}))$%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8\\0\\-24\\-40\\-936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\104\\0\\-104\\-3120
\end{tabular}\endgroup%
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-24\\0\\24\\0\\-312
\end{tabular}\endgroup%
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-40\\-104\\0\\4\\0
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-936\\-3120\\-312\\0\\312
\end{tabular}\endgroup%
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-80\\-312\\-48\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-1456\\-312\\-104\\-936
\end{tabular}\endgroup%
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-48\\-312\\-96\\-48\\-624
\end{tabular}\endgroup%
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-8\\-104\\-48\\-36\\-624
\end{tabular}\endgroup%
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0\\-936\\-624\\-624\\-12480
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2181192, -16692, -551460; -16692, 128, 4220; -551460, 4220, 139423],[156, 4, 1],[-67, -241, -37, -19, -401, -35, -201, -65, -49, -925; -166, -572, -78, -38, -858, -80, -494, -168, -126, -2340; -260, -936, -144, -74, -1560, -136, -780, -252, -190, -3588],41,23]
%
% END OF LATTICE L42.23
%
\hbox{}\par\smallskip%
%
% LATTICE L42.24
%
\leavevmode%
\llap{\spacer}%
${L_{42.24}}=\pdualwitharg{3}({L_{42.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1235208\\6864\\6864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6864\\-33\\-39
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6864\\-39\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4159\\104000\\642720
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-451\\-2781
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-600\\-3709
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\77\\462
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\559\\3393
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\126\\772
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\424\\2628
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\1014\\6344
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-6\\-60\\-234\\-30\\-117\\-12\\-12\\0
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0\\39\\0\\-156\\-780\\-117\\-546\\-78\\-156\\-234
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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3\\0\\-6\\-60\\-234
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0\\39\\0\\-156\\-780
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-6\\0\\4\\0\\-52
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-60\\-156\\0\\24\\0
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-234\\-780\\-52\\0\\52
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-30\\-117\\-12\\-12\\0
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-117\\-546\\-78\\-156\\-234
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-12\\-78\\-16\\-48\\-104
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-156\\-48\\-216\\-624
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1235208, 6864, 6864; 6864, -33, -39; 6864, -39, -38],[312, 3, 1],[3, 22, 5, 17, 41, 3, 4, -1, -1, 11; 77, 559, 126, 424, 1014, 73, 91, -26, -24, 286; 462, 3393, 772, 2628, 6344, 465, 624, -154, -156, 1690],41,24]
%
% END OF LATTICE L42.24
%
\hbox{}\par\smallskip%
%
% LATTICE L42.25
%
\leavevmode%
\llap{\spacer}%
${L_{42.25}}=\pdualwitharg{3}({L_{42.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-188760\\2496\\5928
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2496\\-33\\-78
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5928\\-78\\-155
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6\\467\\-6
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83\\6448\\-78
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\698\\-8
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\2244\\-24
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\5174\\-52
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\692\\-6
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\299\\0
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\HardButStrongLineBreak\kern3pt%
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-1\\-78\\1
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-1\\-76\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\936\\-13
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-6\\-60\\-234\\-60\\-117\\-6\\-12\\0
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-234\\-1560\\-52\\0\\52\\0\\-234\\-52\\-624\\-1040
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\kern3pt%
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-60\\-468\\-24\\-24\\0\\12\\0\\-6\\-120\\-234
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-117\\-1092\\-78\\-156\\-234\\0\\39\\0\\-156\\-390
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-78\\-8\\-24\\-52\\-6\\0\\1\\0\\-13
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-312\\-48\\-216\\-624\\-120\\-156\\0\\24\\0
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0\\-234\\-52\\-312\\-1040\\-234\\-390\\-13\\0\\13
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-188760, 2496, 5928; 2496, -33, -78; 5928, -78, -155],[312, 3, 1],[6, 83, 9, 29, 67, 9, 4, -1, -1, 12; 467, 6448, 698, 2244, 5174, 692, 299, -78, -76, 936; -6, -78, -8, -24, -52, -6, 0, 1, 0, -13],41,25]
%
% END OF LATTICE L42.25
%
\hbox{}\par\smallskip%
%
% LATTICE L42.26
%
\leavevmode%
\llap{\spacer}%
${L_{42.26}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{42.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{52}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{156}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76596\\6396\\-936
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\kern3pt%
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6396\\-534\\78
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\78\\-11
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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2\\21\\-27
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23\\260\\-156
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1\\12\\-1
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2\\25\\6
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2\\26\\13
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-76596, 6396, -936; 6396, -534, 78; -936, 78, -11],[156, 6, 1],[-1, -5, -1, 0, 9, 2, 23, 1, 2, 2; -12, -65, -14, -5, 78, 21, 260, 12, 25, 26; 0, -39, -16, -42, -260, -27, -156, -1, 6, 13],41,26]
%
% END OF LATTICE L42.26
%
\hbox{}\par\smallskip%
%
% LATTICE L42.27
%
\leavevmode%
\llap{\spacer}%
${L_{42.27}}=\pdualwitharg{3}({L_{42.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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0\\0\\0
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-12\\-78\\-4\\-24\\-26
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-25896, -624, -3120; -624, -15, -78; -3120, -78, -155],[312, 3, 1],[23, 153, 8, 49, 54, 13, 3, -2, -1, 24; -898, -5980, -313, -1920, -2119, -512, -130, 78, 40, -936; -12, -78, -4, -24, -26, -6, 0, 1, 0, -13],41,27]
%
% END OF LATTICE L42.27
%
\hbox{}\par\smallskip%
%
% LATTICE L42.28
%
\leavevmode%
\llap{\spacer}%
${L_{42.28}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{42.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-34301670, 66924, -5337150; 66924, -130, 10413; -5337150, 10413, -830431],[78, 13, 1],[-180, 418, 4253, 9497, 7645, 7776, 2114, 3343, 631, -49; 11, -40, -387, -840, -669, -674, -181, -276, -44, 6; 1157, -2687, -27339, -61048, -49143, -49985, -13589, -21489, -4056, 315],41,28]
%
% END OF LATTICE L42.28
%
\hbox{}\par\smallskip%
%
% LATTICE L42.29
%
\leavevmode%
\llap{\spacer}%
${L_{42.29}}=\pdualwitharg{2}(\main({L_{42.7}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-376584, -312, 2184; -312, 20, -4; 2184, -4, -11],[312, 4, 1],[4, 4, -1, -1, 7, 1, 18, 7, 9, 71; 207, 208, -51, -52, 351, 51, 923, 360, 464, 3666; 716, 715, -180, -180, 1248, 179, 3224, 1254, 1612, 12714],41,29]
%
% END OF LATTICE L42.29
%
\hbox{}\par\smallskip%
%
% LATTICE L42.30
%
\leavevmode%
\llap{\spacer}%
${L_{42.30}}=\pdualwitharg{2}(\main({L_{42.5}}))$%
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{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7872072, 207168, 94848; 207168, -5452, -2496; 94848, -2496, -1141],[312, 4, 1],[18, 73, 11, -1, -71, -9, -34, -1, 13, 94; 697, 2834, 429, -38, -2769, -352, -1339, -42, 498, 3627; -28, -130, -24, 0, 156, 22, 104, 9, -8, -117],41,30]
%
% END OF LATTICE L42.30
%
\hbox{}\par\smallskip%
%
% LATTICE L42.31
%
\leavevmode%
\llap{\spacer}%
${L_{42.31}}=\pdualwitharg{3.13}(\main(\pfillwitharg{2}({L_{42.1}})))$%
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{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11193, -58578, -18759; -58578, 316524, 101361; -18759, 101361, 32459],[39, 39, 1],[-17, -35, -87, -240, -119, -337, -83, -31, 4, 1; 3906, 8272, 20662, 57194, 28402, 80556, 19882, 7492, -886, -248; -12207, -25851, -64571, -178737, -88759, -251745, -62133, -23413, 2769, 775],41,31]
%
% END OF LATTICE L42.31
%
\hbox{}\par\smallskip%
%
% LATTICE L42.32
%
\leavevmode%
\llap{\spacer}%
${L_{42.32}}=\pdualwitharg{2.3}(\main({L_{42.3}}))$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[116376, 19812, -29328; 19812, 3396, -4992; -29328, -4992, 7391],[156, 12, 1],[-205, -485, 1, -10, -415, -167, -1127, -125, -114, -1197; 32, 78, 0, 1, 52, 22, 156, 18, 17, 182; -792, -1872, 4, -39, -1612, -648, -4368, -484, -441, -4628],41,32]
%
% END OF LATTICE L42.32
%
\hbox{}\par\smallskip%
%
% LATTICE L42.33
%
\leavevmode%
\llap{\spacer}%
${L_{42.33}}=\pdualwitharg{3}(\main({L_{42.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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2184\\138\\-30
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-312\\-30\\5
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-5\\-39\\-546
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-1\\-8\\-112
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\HardButStrongLineBreak\kern3pt%
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0\\-1\\-9
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9\\52\\832
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2\\13\\198
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\HardButStrongLineBreak\kern3pt%
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23\\156\\2340
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1\\7\\104
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1\\7\\105
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2\\13\\208
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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\kern3pt%
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-12\\-78\\-24\\-27\\-312\\-30\\-156\\0\\3\\0
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0\\-234\\-104\\-156\\-2080\\-234\\-1560\\-26\\0\\26
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17160, 2184, -312; 2184, 138, -30; -312, -30, 5],[312, 6, 1],[-1, -5, -1, 0, 9, 2, 23, 1, 1, 2; -8, -39, -8, -1, 52, 13, 156, 7, 7, 13; -108, -546, -112, -9, 832, 198, 2340, 104, 105, 208],41,33]
%
% END OF LATTICE L42.33
%
\hbox{}\par\smallskip%
%
% LATTICE L42.34
%
\leavevmode%
\llap{\spacer}%
${L_{42.34}}=\pdualwitharg{2.3}(\main({L_{42.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2019576\\-89700\\-503880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89700\\3984\\22380
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-503880\\22380\\125717
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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43055\\22126\\168636
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2016\\-1037\\-7896
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10728\\-5513\\-42019
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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325\\172\\1272
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837\\442\\3276
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\HardButStrongLineBreak\kern3pt%
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1\\0\\4
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-17\\-12\\-66
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
145\\52\\572
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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0\\312\\0\\-312\\-3120
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-24\\0\\8\\0\\-104
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-120\\-312\\0\\12\\0
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0\\-936\\-208\\-624\\-4160
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2019576, -89700, -503880; -89700, 3984, 22380; -503880, 22380, 125717],[156, 12, 1],[325, 837, 1, -17, 145, 107, 1035, 143, 305, 1727; 172, 442, 0, -12, 52, 50, 520, 74, 160, 910; 1272, 3276, 4, -66, 572, 420, 4056, 560, 1194, 6760],41,34]
%
% END OF LATTICE L42.34
%
\hbox{}\par\smallskip%
%
% LATTICE L42.35
%
\leavevmode%
\llap{\spacer}%
${L_{42.35}}=\pdualwitharg{3}(\main({L_{42.7}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{2}{78}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{104}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{312}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{26}\above{1pt}{1pt}{}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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37\\476\\1206
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26\\333\\792
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2\\25\\38
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\-234\\-104\\-312\\-2080\\-234\\-1560\\-26\\0\\26
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-28289352, 2357472, -62712; 2357472, -196458, 5226; -62712, 5226, -139],[312, 6, 1],[-1, 29, 13, 37, 239, 26, 163, 2, -3, -9; -12, 377, 168, 476, 3068, 333, 2080, 25, -40, -117; 0, 1092, 452, 1206, 7540, 792, 4680, 38, -150, -338],41,35]
%
% END OF LATTICE L42.35
%
\hbox{}\par\smallskip%
%
% LATTICE L42.36
%
\leavevmode%
\llap{\spacer}%
${L_{42.36}}=\pdualwitharg{13}(\main({L_{42.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{l}{2}{52}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18876, 4680, -312; 4680, -1157, 78; -312, 78, -5],[156, 13, 1],[-2, 0, 5, 15, 13, 14, 4, 7, 1, -1; -7, -1, 9, 36, 33, 37, 11, 21, 4, -3; 13, -19, -195, -416, -327, -325, -85, -117, 0, 15],41,36]
%
% END OF LATTICE L42.36
%
\hbox{}\par\smallskip%
%
% LATTICE L42.37
%
\leavevmode%
\llap{\spacer}%
${L_{42.37}}=\pdualwitharg{13}(\main({L_{42.2}}))$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[109356, 53196, -936; 53196, 25870, -455; -936, -455, 8],[156, 13, 1],[-1, 0, 2, 5, 4, 4, 1, 1, -1, -1; 3, -1, -15, -34, -27, -27, -7, -9, 2, 3; 52, -58, -624, -1352, -1068, -1066, -280, -390, 0, 54],41,37]
%
% END OF LATTICE L42.37
%
\hbox{}\par\smallskip%
%
% LATTICE L42.38
%
\leavevmode%
\llap{\spacer}%
${L_{42.38}}=\pdualwitharg{2.13}(\pfillwitharg{2}({L_{42.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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29430\\7\\-60034
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[101109840, -179478, 49566270; -179478, 910, -87984; 49566270, -87984, 24298477],[78, 26, 1],[29430, -94507, -926720, -1014634, -1621810, -1639107, -442052, -682957, -58484, 12133; 7, -23, -225, -246, -393, -397, -107, -165, -14, 3; -60034, 192784, 1890408, 2069743, 3308316, 3343600, 901738, 1393158, 119301, -24750],41,38]
%
% END OF LATTICE L42.38
%
\hbox{}\par\smallskip%
%
% LATTICE L42.39
%
\leavevmode%
\llap{\spacer}%
${L_{42.39}}=\pdualwitharg{2}({L_{42.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[85488, 312, -312; 312, -8, 0; -312, 0, 1],[312, 8, 1],[0, -3, -2, -1, -40, -3, -10, -1, 0, 1; 1, -91, -63, -32, -1287, -97, -325, -33, 0, 39; 4, -884, -600, -303, -12168, -916, -3068, -312, -1, 312],41,39]
%
% END OF LATTICE L42.39
%
\hbox{}\par\smallskip%
%
% LATTICE L42.40
%
\leavevmode%
\llap{\spacer}%
${L_{42.40}}=\pdualwitharg{2}({L_{42.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{104}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{312}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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0\\8\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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37322\\-1287\\-75504
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261\\-10\\-528
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18444\\-636\\-37313
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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0\\1\\0
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\HardButStrongLineBreak\kern3pt%
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257\\0\\-520
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172\\-3\\-348
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\HardButStrongLineBreak\kern3pt%
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173\\-5\\-350
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3470\\-117\\-7020
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24024, 0, -11856; 0, 8, 0; -11856, 0, -5851],[312, 8, 1],[0, 257, 172, 173, 3470, 261, 874, 89, 1, -77; 1, 0, -3, -5, -117, -10, -39, -6, -1, 0; 0, -520, -348, -350, -7020, -528, -1768, -180, -2, 156],41,40]
%
% END OF LATTICE L42.40
%
\hbox{}\par\smallskip%
%
% LATTICE L42.41
%
\leavevmode%
\llap{\spacer}%
${L_{42.41}}=\pdualwitharg{2}({L_{42.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{312}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}{104}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{312}\above{1pt}{1pt}{}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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\HardButStrongLineBreak\kern3pt%
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-2\\-5\\-38
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-40\\-117\\-780
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-3\\-10\\-60
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-1\\-6\\-24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[312, 0, 0; 0, 8, 0; 0, 0, -1],[312, 8, 1],[0, -3, -2, -2, -40, -3, -10, -1, 0, 1; 1, 0, -3, -5, -117, -10, -39, -6, -1, 0; 0, -52, -36, -38, -780, -60, -208, -24, -2, 0],41,41]
%
% END OF LATTICE L42.41
%
\hbox{}\par\smallskip%
%
% LATTICE L42.42
%
\leavevmode%
\llap{\spacer}%
${L_{42.42}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{42.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${39}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{78}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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203502\\1365\\68679
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\3\\0\\-78\\-30
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-39\\0\\13\\0\\-13
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-390\\-78\\0\\78\\0
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-117\\-30\\-13\\0\\1
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-390\\-117\\-78\\-78\\0
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-117\\-42\\-39\\-78\\-9
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\kern3pt%
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-78\\-78\\-156\\-702\\-156
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[85160946, 203502, 28992132; 203502, 1365, 68679; 28992132, 68679, 9870472],[78, 39, 1],[22973, -73772, -241132, -1584043, -421994, -1279484, -345065, -177705, -91305, 3157; -45939, 147521, 482189, 3167594, 843857, 2558571, 690023, 355355, 182582, -6313; -67158, 215661, 704912, 4630704, 1233634, 3740373, 1008744, 519493, 266916, -9229],41,42]
%
% END OF LATTICE L42.42
%
\hbox{}\par\smallskip%
%
% LATTICE L42.43
%
\leavevmode%
\llap{\spacer}%
${L_{42.43}}=\pdualwitharg{3}({L_{42.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{624}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{208}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1620528\\-6240\\4368
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6240\\6\\12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4368\\12\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\73\\471
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\1716\\10920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\76\\478
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\77\\480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\78\\481
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-76\\-480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-390\\-2418
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-80\\-488
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\624\\4160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-6\\-30\\-117\\-120\\-234\\-24\\-6\\0
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\kern3pt%
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-30\\-312\\0\\6\\0\\-24\\-156\\-48\\-54\\-624
\end{tabular}\endgroup%
\kern3pt%
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-117\\-1560\\-26\\0\\13\\0\\-234\\-104\\-156\\-2080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-1872\\-48\\-24\\0\\48\\0\\-48\\-120\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-4368\\-156\\-156\\-234\\0\\156\\0\\-156\\-3120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-624\\-32\\-48\\-104\\-48\\0\\16\\0\\-208
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\kern3pt%
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-6\\-312\\-24\\-54\\-156\\-120\\-156\\0\\6\\0
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0\\-1872\\-208\\-624\\-2080\\-1872\\-3120\\-208\\0\\208
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1620528, -6240, 4368; -6240, 6, 12; 4368, 12, -11],[624, 6, 1],[1, 23, 1, 1, 1, -1, -5, -1, 0, 9; 73, 1716, 76, 77, 78, -76, -390, -80, -5, 624; 471, 10920, 478, 480, 481, -480, -2418, -488, -12, 4160],41,43]
%
% END OF LATTICE L42.43
%
\hbox{}\par\smallskip%
%
% LATTICE L42.44
%
\leavevmode%
\llap{\spacer}%
${L_{42.44}}=\pdualwitharg{3}({L_{42.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{624}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{208}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8385936\\-4196400\\15600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4196400\\-2099910\\7806
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15600\\7806\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-13\\28\\543
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\HardButStrongLineBreak\kern3pt%
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-289\\624\\12480
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\HardButStrongLineBreak\kern3pt%
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-6\\13\\271
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-21\\46\\1080
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\HardButStrongLineBreak\kern3pt%
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-11\\26\\1066
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\HardButStrongLineBreak\kern3pt%
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19\\-40\\-552
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\-156\\-2730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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13\\-28\\-544
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\260\\4888
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-60\\-234\\-120\\-234\\-24\\-12\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\624\\0\\-624\\-3120\\-1872\\-4368\\-624\\-624\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0\\-26\\-24\\-78\\-16\\-24\\-104
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-624\\0\\24\\0\\-48\\-312\\-96\\-216\\-1248
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-3120\\-26\\0\\52\\0\\-468\\-208\\-624\\-4160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-1872\\-24\\-48\\0\\48\\0\\-48\\-240\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-4368\\-78\\-312\\-468\\0\\156\\0\\-312\\-3120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-624\\-16\\-96\\-208\\-48\\0\\16\\0\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-624\\-24\\-216\\-624\\-240\\-312\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1872\\-104\\-1248\\-4160\\-1872\\-3120\\-208\\0\\208
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8385936, -4196400, 15600; -4196400, -2099910, 7806; 15600, 7806, -29],[624, 6, 1],[-13, -289, -6, -21, -11, 19, 73, 13, -1, -121; 28, 624, 13, 46, 26, -40, -156, -28, 2, 260; 543, 12480, 271, 1080, 1066, -552, -2730, -544, 0, 4888],41,44]
%
% END OF LATTICE L42.44
%
\hbox{}\par\smallskip%
%
% LATTICE L42.45
%
\leavevmode%
\llap{\spacer}%
${L_{42.45}}=\pdualwitharg{3}({L_{42.6}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{624}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{208}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49296\\-3744\\2496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3744\\-186\\96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2496\\96\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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5\\-666\\-1098
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\-6812\\-11232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-134\\-221
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-269\\-444
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-676\\-1118
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\132\\216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\403\\663
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\136\\224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-131\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-3848\\-6344
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
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-60\\-312\\0\\6\\0\\-24\\-78\\-48\\-54\\-624
\end{tabular}\endgroup%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1872\\-24\\-24\\0\\48\\0\\-48\\-120\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-2184\\-39\\-78\\-234\\0\\39\\0\\-78\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-312\\-12\\-54\\-312\\-120\\-78\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1872\\-104\\-624\\-4160\\-1872\\-1560\\-208\\0\\208
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[49296, -3744, 2496; -3744, -186, 96; 2496, 96, -47],[624, 6, 1],[5, 51, 1, 2, 5, -1, -3, -1, 1, 29; -666, -6812, -134, -269, -676, 132, 403, 136, -131, -3848; -1098, -11232, -221, -444, -1118, 216, 663, 224, -216, -6344],41,45]
%
% END OF LATTICE L42.45
%
\hbox{}\par\smallskip%
%
% LATTICE L42.46
%
\leavevmode%
\llap{\spacer}%
${L_{42.46}}=\pdualwitharg{2.3}(\main({L_{42.7}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{156}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}{52}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-188760\\12792\\-3432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12792\\-564\\132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3432\\132\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\230\\693
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\3367\\10140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\382\\1150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\1296\\3900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\3120\\9386
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\455\\1368
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\299\\897
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-77\\-232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-76\\-228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\845\\2548
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-6\\-60\\-234\\-60\\-117\\-12\\-12\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\156\\0\\-312\\-1560\\-468\\-1092\\-156\\-312\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4\\0\\-52\\-24\\-78\\-16\\-48\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-312\\0\\24\\0\\-24\\-156\\-48\\-216\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-1560\\-52\\0\\52\\0\\-234\\-104\\-624\\-2080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-468\\-24\\-24\\0\\12\\0\\-12\\-120\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-1092\\-78\\-156\\-234\\0\\39\\0\\-156\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-156\\-16\\-48\\-104\\-12\\0\\4\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-312\\-48\\-216\\-624\\-120\\-156\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-468\\-104\\-624\\-2080\\-468\\-780\\-52\\0\\52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-188760, 12792, -3432; 12792, -564, 132; -3432, 132, -29],[312, 12, 1],[3, 44, 5, 17, 41, 6, 4, -1, -1, 11; 230, 3367, 382, 1296, 3120, 455, 299, -77, -76, 845; 693, 10140, 1150, 3900, 9386, 1368, 897, -232, -228, 2548],41,46]
%
% END OF LATTICE L42.46
%
\hbox{}\par\smallskip%
%
% LATTICE L42.47
%
\leavevmode%
\llap{\spacer}%
${L_{42.47}}=\pdualwitharg{3}({L_{42.7}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{624}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{208}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6925776\\-29952\\15600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29952\\-114\\66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15600\\66\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\154\\1626
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\2496\\26520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\152\\1622
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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11\\554\\5940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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14\\702\\7553
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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9\\448\\4848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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4\\195\\2145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-52\\-544
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-50\\-540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\468\\4888
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\624\\0\\-624\\-1560\\-1872\\-2184\\-624\\-624\\-1872
\end{tabular}\endgroup%
\kern3pt%
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-120\\-624\\0\\24\\0\\-48\\-156\\-96\\-216\\-1248
\end{tabular}\endgroup%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-2184\\-78\\-156\\-117\\0\\39\\0\\-156\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-624\\-32\\-96\\-104\\-48\\0\\16\\0\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1872\\-208\\-1248\\-2080\\-1872\\-1560\\-208\\0\\208
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6925776, -29952, 15600; -29952, -114, 66; 15600, 66, -35],[624, 6, 1],[3, 49, 3, 11, 14, 9, 4, -1, -1, 9; 154, 2496, 152, 554, 702, 448, 195, -52, -50, 468; 1626, 26520, 1622, 5940, 7553, 4848, 2145, -544, -540, 4888],41,47]
%
% END OF LATTICE L42.47
%
\hbox{}\par\smallskip%
%
% LATTICE L42.48
%
\leavevmode%
\llap{\spacer}%
${L_{42.48}}=\pdualwitharg{2.3}(\main({L_{42.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{156}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{52}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31557864\\2629848\\1280448
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2629848\\-219156\\-106704
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1280448\\-106704\\-51947
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1126\\-14651\\2340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-54\\-703\\113
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-283\\-3688\\600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-395\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
203\\2626\\-390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\481\\-76
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-405\\-5265\\832
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-31557864, 2629848, 1280448; 2629848, -219156, -106704; 1280448, -106704, -51947],[312, 12, 1],[-183, -1126, -54, -283, -259, -30, 203, 37, -1, -405; -2380, -14651, -703, -3688, -3380, -395, 2626, 481, -12, -5265; 378, 2340, 113, 600, 559, 72, -390, -76, 0, 832],41,48]
%
% END OF LATTICE L42.48
%
\hbox{}\par\smallskip%
%
% LATTICE L42.49
%
\leavevmode%
\llap{\spacer}%
${L_{42.49}}=\pdualwitharg{13}({L_{42.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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50\\1024\\8125
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-312, 2496, -312; 2496, 1313, -182; -312, -182, 25],[312, 13, 1],[1, 0, -5, -15, -13, -7, -2, -7, -1, 1; 18, -1, -108, -308, -264, -141, -40, -138, -20, 18; 143, -8, -858, -2444, -2094, -1118, -317, -1092, -156, 144],41,49]
%
% END OF LATTICE L42.49
%
\hbox{}\par\smallskip%
%
% LATTICE L42.50
%
\leavevmode%
\llap{\spacer}%
${L_{42.50}}=\pdualwitharg{13}({L_{42.2}})$%
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{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9048, 5616, -624; 5616, 1703, -208; -624, -208, 25],[312, 13, 1],[2, 1, -1, -11, -11, -13, -2, -4, -1, 1; 30, 14, -24, -184, -180, -210, -32, -63, -16, 15; 299, 140, -234, -1820, -1782, -2080, -317, -624, -156, 150],41,50]
%
% END OF LATTICE L42.50
%
\hbox{}\par\smallskip%
%
% LATTICE L42.51
%
\leavevmode%
\llap{\spacer}%
${L_{42.51}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{42.4}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-156\\-312\\-312\\-156\\-156\\0\\156\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-52\\-156\\-234\\-156\\-260\\-26\\0\\26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-156\\-312\\-240\\-468\\-60\\-156\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[156, -156, 0; -156, 182, 0; 0, 0, -1],[156, 26, 1],[0, -1, -4, -7, -5, -9, -1, -1, 1, 1; -1, -2, -6, -9, -6, -10, -1, 0, 1, 0; -13, -16, -39, -52, -33, -52, -5, 0, 0, -12],41,51]
%
% END OF LATTICE L42.51
%
\hbox{}\par\smallskip%
%
% LATTICE L42.52
%
\leavevmode%
\llap{\spacer}%
${L_{42.52}}=\pdualwitharg{13}({L_{42.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{}{2}{104}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26520\\-3432\\936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3432\\-442\\117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
936\\117\\-25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1055\\-9768\\-6864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
128\\-1185\\-832
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\185\\129
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-7\\62\\26
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-1\\8\\-2
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4\\-39\\-39
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\HardButStrongLineBreak\kern3pt%
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31\\-288\\-208
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14\\-129\\-87
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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52\\0\\-78\\-520\\-234\\-520\\-156\\-156\\-104\\0
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0\\4\\0\\-104\\-60\\-156\\-56\\-78\\-104\\-18
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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}}\right.$}%
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52\\0\\-78\\-520\\-234
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0\\4\\0\\-104\\-60
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-78\\0\\39\\0\\-39
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-520\\-104\\0\\104\\0
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-234\\-60\\-39\\0\\3
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-520\\-156\\-156\\-104\\0
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\kern3pt%
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-156\\-56\\-78\\-104\\-18
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-78\\-156\\-312\\-78
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-104\\-312\\-936\\-312
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-78\\-312\\-120
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26520, -3432, 936; -3432, -442, 117; 936, 117, -25],[312, 13, 1],[-7, -1, 4, 31, 14, 31, 9, 8, 1, -2; 62, 8, -39, -288, -129, -284, -82, -72, -8, 18; 26, -2, -39, -208, -87, -182, -50, -39, 0, 9],41,52]
%
% END OF LATTICE L42.52
%
\hbox{}\par\smallskip%
%
% LATTICE L42.53
%
\leavevmode%
\llap{\spacer}%
${L_{42.53}}=\pdualwitharg{2}({L_{42.7}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}{624}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{208}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
341328\\624\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-204\\-1832
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-5\\-338\\-3042
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-66\\-600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\78\\624
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\HardButStrongLineBreak\kern3pt%
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0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-3\\-208\\-1872
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-69\\-618
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-137\\-1228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-1365\\-12246
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-48\\-80\\-1872\\-40\\-624\\-48\\-16\\0
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\kern3pt%
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-1872\\-3120\\-624\\0\\624\\0\\-1872\\-624\\-1248\\-12480
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-1456\\-624\\-208\\-1872\\0\\208\\0\\-208\\-3120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-156\\-96\\-48\\-624\\-6\\0\\12\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-104\\-96\\-72\\-1248\\-20\\-208\\0\\8\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-468\\-624\\-624\\-12480\\-234\\-3120\\-156\\0\\156
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[341328, 624, -624; 624, -8, 0; -624, 0, 1],[624, 8, 1],[-3, -5, -1, 0, 1, 0, -3, -1, -2, -20; -204, -338, -66, 1, 78, 0, -208, -69, -137, -1365; -1832, -3042, -600, 4, 624, -1, -1872, -618, -1228, -12246],41,53]
%
% END OF LATTICE L42.53
%
\hbox{}\par\smallskip%
%
% LATTICE L42.54
%
\leavevmode%
\llap{\spacer}%
${L_{42.54}}=\pdualwitharg{2}({L_{42.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{208}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{624}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96720\\0\\-48048
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48048\\0\\-23869
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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1\\-1\\-2
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517\\-26\\-1040
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85\\-3\\-171
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338\\-9\\-680
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3373\\-78\\-6786
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505\\-10\\-1016
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840\\-13\\-1690
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167\\0\\-336
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0\\1\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-155\\0\\312
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\208\\0\\-208\\-3120\\-624\\-1456\\-624\\-208\\-1872
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-6\\0\\3\\0\\-78\\-24\\-78\\-48\\-24\\-312
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-40\\-208\\0\\8\\0\\-16\\-104\\-96\\-72\\-1248
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\kern3pt%
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-156\\-1456\\-78\\-104\\-468\\0\\52\\0\\-104\\-3120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-624\\-48\\-96\\-624\\-48\\0\\48\\0\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-208\\-24\\-72\\-624\\-80\\-104\\0\\8\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1872\\-312\\-1248\\-12480\\-1872\\-3120\\-624\\0\\624
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-96720, 0, -48048; 0, 8, 0; -48048, 0, -23869],[624, 8, 1],[1, 517, 85, 338, 3373, 505, 840, 167, 0, -155; -1, -26, -3, -9, -78, -10, -13, 0, 1, 0; -2, -1040, -171, -680, -6786, -1016, -1690, -336, 0, 312],41,54]
%
% END OF LATTICE L42.54
%
\hbox{}\par\smallskip%
%
% LATTICE L42.55
%
\leavevmode%
\llap{\spacer}%
${L_{42.55}}=\pdualwitharg{2}({L_{42.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{208}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{624}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-2
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-26\\-104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-39\\-273
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-13\\-130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-40\\-234\\-80\\-156\\-48\\-8\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\208\\0\\-208\\-1560\\-624\\-1456\\-624\\-208\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-78\\-48\\-156\\-96\\-48\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-208\\0\\8\\0\\-16\\-104\\-96\\-72\\-1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-1560\\-78\\0\\39\\0\\-234\\-312\\-312\\-6240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-624\\-48\\-16\\0\\16\\0\\-48\\-80\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-1456\\-156\\-104\\-234\\0\\52\\0\\-104\\-3120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-624\\-96\\-96\\-312\\-48\\0\\48\\0\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-208\\-48\\-72\\-312\\-80\\-104\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1872\\-624\\-1248\\-6240\\-1872\\-3120\\-624\\0\\624
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[624, 0, 0; 0, 8, 0; 0, 0, -1],[624, 8, 1],[0, -3, -1, -2, -10, -3, -5, -1, 0, 1; -1, -26, -6, -9, -39, -10, -13, 0, 1, 0; -2, -104, -30, -56, -273, -80, -130, -24, 0, 0],41,55]
%
% END OF LATTICE L42.55
%
\hbox{}\par\smallskip%
%
% LATTICE L42.56
%
\leavevmode%
\llap{\spacer}%
${L_{42.56}}=\pdualwitharg{2}({L_{42.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{624}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{208}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{156}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96720\\6864\\3744
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6864\\232\\96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3744\\96\\37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-1162\\2304
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-663\\1313
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\168\\-336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\167\\-332
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-156\\312
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-166\\330
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-3822\\7592
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-831\\1650
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-1163\\2308
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\-9633\\19110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-48\\-80\\-1872\\-80\\-624\\-48\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\13\\0\\-52\\-1560\\-78\\-728\\-78\\-52\\-234
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\0\\-624\\-48\\-624\\-96\\-96\\-624
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-52\\0\\8\\0\\-8\\-208\\-48\\-72\\-624
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-1560\\-624\\0\\624\\0\\-1872\\-624\\-1248\\-12480
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\kern3pt%
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-80\\-78\\-48\\-8\\0\\4\\0\\-12\\-40\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-728\\-624\\-208\\-1872\\0\\208\\0\\-208\\-3120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-78\\-96\\-48\\-624\\-12\\0\\12\\0\\-156
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-52\\-96\\-72\\-1248\\-40\\-208\\0\\8\\0
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\kern3pt%
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0\\-234\\-624\\-624\\-12480\\-468\\-3120\\-156\\0\\156
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-96720, 6864, 3744; 6864, 232, 96; 3744, 96, 37],[624, 8, 1],[7, 4, -1, -1, 1, 1, 23, 5, 7, 58; -1162, -663, 168, 167, -156, -166, -3822, -831, -1163, -9633; 2304, 1313, -336, -332, 312, 330, 7592, 1650, 2308, 19110],41,56]
%
% END OF LATTICE L42.56
%
\hbox{}\par\smallskip%
%
% LATTICE L42.57
%
\leavevmode%
\llap{\spacer}%
${L_{42.57}}=\pdualwitharg{3.13}(\main({L_{42.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${78}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{l}{2}{156}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1950\\117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\117\\-7
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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2\\4\\77
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-5\\-17\\-312
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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0\\-3\\-52
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\16
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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-234\\-60\\-26\\0\\2\\0\\-18\\-52\\-312\\-80
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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}}\right.$}%
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0\\6\\0\\-156\\-60
\end{tabular}\endgroup%
\kern3pt%
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-78\\0\\26\\0\\-26
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[156, 0, 0; 0, -1950, 117; 0, 117, -7],[156, 39, 1],[-5, -1, 0, 1, 0, -1, -1, -2, -9, -2; -17, -5, -3, 0, 1, 5, 1, -1, -16, -5; -312, -90, -52, 0, 16, 78, 12, -26, -312, -94],41,57]
%
% END OF LATTICE L42.57
%
\hbox{}\par\smallskip%
%
% LATTICE L42.58
%
\leavevmode%
\llap{\spacer}%
${L_{42.58}}=\pdualwitharg{3.13}(\main({L_{42.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${78}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{b}{2}{156}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48516\\22776\\-1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22776\\10686\\-585
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[48516, 22776, -1248; 22776, 10686, -585; -1248, -585, 32],[156, 39, 1],[2, 1, 1, -1, -1, -5, -2, -2, -3, 0; -21, -7, -5, 2, 3, 15, 5, 3, -10, -5; -312, -90, -52, 0, 16, 78, 12, -26, -312, -94],41,58]
%
% END OF LATTICE L42.58
%
\hbox{}\par\smallskip%
%
% LATTICE L42.59
%
\leavevmode%
\llap{\spacer}%
${L_{42.59}}=\pdualwitharg{2.3.13}(\pfillwitharg{2}({L_{42.1}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35015081088, 86208554562, 42448017846; 86208554562, -212248969530, -104508747330; 42448017846, -104508747330, -51458804689],[78, 78, 1],[11, -40, -129, -420, -223, -674, -181, -92, -22, 2; -110183, 256431, 869439, 2911732, 1562519, 4767629, 1296051, 683043, 193232, -10037; 223782, -520824, -1765868, -5913843, -3173536, -9683232, -2632326, -1387282, -392457, 20386],41,59]
%
% END OF LATTICE L42.59
%
\hbox{}\par\smallskip%
%
% LATTICE L42.60
%
\leavevmode%
\llap{\spacer}%
${L_{42.60}}=\pdualwitharg{2.3}({L_{42.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[29328, 312, -10296; 312, -24, -72; -10296, -72, 3563],[312, 24, 1],[4, -263, -62, -96, -1296, -295, -998, -35, -1, 35; 17, -1053, -249, -386, -5213, -1187, -4017, -141, -4, 143; 12, -780, -184, -285, -3848, -876, -2964, -104, -3, 104],41,60]
%
% END OF LATTICE L42.60
%
\hbox{}\par\smallskip%
%
% LATTICE L42.61
%
\leavevmode%
\llap{\spacer}%
${L_{42.61}}=\pdualwitharg{2.3}({L_{42.1}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7176, 0, 1248; 0, 24, 0; 1248, 0, -217],[312, 24, 1],[0, -107, -24, -73, -490, -111, -374, -13, -1, 9; 1, 0, -1, -5, -39, -10, -39, -2, -1, 0; 0, -624, -140, -426, -2860, -648, -2184, -76, -6, 52],41,61]
%
% END OF LATTICE L42.61
%
\hbox{}\par\smallskip%
%
% LATTICE L42.62
%
\leavevmode%
\llap{\spacer}%
${L_{42.62}}=\pdualwitharg{2.3}({L_{42.2}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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10\\-1\\-1052
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32\\-5\\-3366
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178\\-39\\-18720
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7\\-2\\-736
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2\\-1\\-210
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1\\0\\-104
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1114152, 0, 10608; 0, 24, 0; 10608, 0, 101],[312, 24, 1],[0, 43, 10, 32, 220, 51, 178, 7, 2, 1; 1, 0, -1, -5, -39, -10, -39, -2, -1, 0; 0, -4524, -1052, -3366, -23140, -5364, -18720, -736, -210, -104],41,62]
%
% END OF LATTICE L42.62
%
\hbox{}\par\smallskip%
%
% LATTICE L42.63
%
\leavevmode%
\llap{\spacer}%
${L_{42.63}}=\pdualwitharg{2.13}(\main({L_{42.3}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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213720\\-12948\\64428
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12948\\988\\-4056
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[213720, -12948, 64428; -12948, 988, -4056; 64428, -4056, 19537],[156, 52, 1],[-253, -337, -2317, -1022, -2993, -2783, -675, -719, 10, -43; 740, 986, 6780, 2991, 8760, 8146, 1976, 2106, -29, 126; 988, 1316, 9048, 3991, 11688, 10868, 2636, 2808, -39, 168],41,63]
%
% END OF LATTICE L42.63
%
\hbox{}\par\smallskip%
%
% LATTICE L42.64
%
\leavevmode%
\llap{\spacer}%
${L_{42.64}}=\pdualwitharg{13}(\main({L_{42.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{78}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{312}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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312\\4108\\-323
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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27\\184\\2364
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37\\255\\3276
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12\\84\\1079
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8\\57\\732
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[312, -4056, 312; -4056, -52234, 4108; 312, 4108, -323],[312, 26, 1],[38, 27, 37, 12, 8, -1, -1, 1, 13, 59; 257, 184, 255, 84, 57, -4, -7, 0, 86, 396; 3302, 2364, 3276, 1079, 732, -52, -90, 0, 1105, 5088],41,64]
%
% END OF LATTICE L42.64
%
\hbox{}\par\smallskip%
%
% LATTICE L42.65
%
\leavevmode%
\llap{\spacer}%
${L_{42.65}}=\pdualwitharg{13}(\main({L_{42.7}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${104}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{312}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{l}{2}{26}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{78}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-52\\-7
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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85\\-208\\-2288
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-27\\-296
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-72\\-780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\2\\26
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-2\\5\\52
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\HardButStrongLineBreak\kern3pt%
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5\\-12\\-136
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26\\-63\\-702
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55\\-134\\-1482
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43\\-105\\-1158
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9048, -312, -312; -312, 442, -52; -312, -52, -7],[312, 26, 1],[85, 11, 29, -1, -5, -2, 5, 26, 55, 43; -208, -27, -72, 2, 12, 5, -12, -63, -134, -105; -2288, -296, -780, 26, 132, 52, -136, -702, -1482, -1158],41,65]
%
% END OF LATTICE L42.65
%
\hbox{}\par\smallskip%
%
% LATTICE L42.66
%
\leavevmode%
\llap{\spacer}%
${L_{42.66}}=\pdualwitharg{2.13}(\main({L_{42.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${104}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{312}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2788656\\36972\\685776
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
685776\\9152\\168595
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
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-271\\853\\1056
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2788656, 36972, 685776; 36972, 416, 9152; 685776, 9152, 168595],[156, 52, 1],[-387, -271, -1201, -447, -197, 267, 231, 1081, 367, 77; 1219, 853, 3777, 1401, 609, -853, -731, -3411, -1157, -243; 1508, 1056, 4680, 1742, 768, -1040, -900, -4212, -1430, -300],41,66]
%
% END OF LATTICE L42.66
%
\hbox{}\par\smallskip%
%
% LATTICE L42.67
%
\leavevmode%
\llap{\spacer}%
${L_{42.67}}=\pdualwitharg{3.13}({L_{42.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19032, -32448, 2808; -32448, 42627, -3705; 2808, -3705, 322],[312, 39, 1],[-6, -2, -3, -1, 1, 3, 1, 1, -7, -3; 263, 85, 122, 40, -38, -113, -35, -22, 360, 138; 3081, 996, 1430, 468, -446, -1326, -411, -260, 4212, 1616],41,67]
%
% END OF LATTICE L42.67
%
\hbox{}\par\smallskip%
%
% LATTICE L42.68
%
\leavevmode%
\llap{\spacer}%
${L_{42.68}}=\pdualwitharg{3.13}({L_{42.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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-2\\105\\1261
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\HardButStrongLineBreak\kern3pt%
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-1\\52\\624
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\HardButStrongLineBreak\kern3pt%
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1\\-51\\-613
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5\\-250\\-3006
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5\\-242\\-2912
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5\\-180\\-2184
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[219336, -149448, 12792; -149448, 96915, -8307; 12792, -8307, 712],[312, 39, 1],[-9, -2, -2, -1, 1, 6, 5, 5, 5, -1; 514, 109, 105, 52, -51, -305, -250, -242, -180, 70; 6162, 1308, 1261, 624, -613, -3666, -3006, -2912, -2184, 836],41,68]
%
% END OF LATTICE L42.68
%
\hbox{}\par\smallskip%
%
% LATTICE L42.69
%
\leavevmode%
\llap{\spacer}%
${L_{42.69}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{42.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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0\\78\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-3\\-2\\-65
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9516, 0, 468; 0, 78, 0; 468, 0, -23],[156, 78, 1],[-37, -5, -5, 0, 1, 2, -1, -3, -22, -6; -10, -1, 0, 1, 0, -1, -2, -2, -9, -2; -780, -105, -104, 0, 20, 39, -24, -65, -468, -127],41,69]
%
% END OF LATTICE L42.69
%
\hbox{}\par\smallskip%
%
% LATTICE L42.70
%
\leavevmode%
\llap{\spacer}%
${L_{42.70}}=\pdualwitharg{3.13}({L_{42.3}})$%
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{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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312\\0\\0
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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1\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-312\\-60
\end{tabular}\endgroup%
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-1560\\-312\\0\\312\\0
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\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[312, 0, 0; 0, -44499, 1248; 0, 1248, -35],[312, 39, 1],[-5, -1, 0, 1, 0, -1, -1, -1, -9, -1; -28, -10, -4, 0, 2, 22, 8, 3, -8, -3; -1014, -360, -143, 0, 71, 780, 282, 104, -312, -110],41,70]
%
% END OF LATTICE L42.70
%
\hbox{}\par\smallskip%
%
% LATTICE L42.71
%
\leavevmode%
\llap{\spacer}%
${L_{42.71}}=\pdualwitharg{2.3}({L_{42.7}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{624}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{l}{2}{208}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-209040\\30576\\-2496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30576\\-4440\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2496\\360\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\14\\87
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\312\\1872
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\13\\74
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2\\23\\108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\13\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-20\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-5\\-78\\-546
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-14\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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0\\1\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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9\\130\\832
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-6\\-60\\-234\\-120\\-234\\-24\\-12\\0
\end{tabular}\endgroup%
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\end{tabular}\endgroup%
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\end{tabular}\endgroup%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-1872\\-48\\-48\\0\\48\\0\\-48\\-240\\-1872
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0\\-1872\\-208\\-1248\\-4160\\-1872\\-3120\\-208\\0\\208
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-209040, 30576, -2496; 30576, -4440, 360; -2496, 360, -29],[624, 24, 1],[1, 23, 1, 2, 2, -1, -5, -1, 0, 9; 14, 312, 13, 23, 13, -20, -78, -14, 1, 130; 87, 1872, 74, 108, -26, -168, -546, -88, 12, 832],41,71]
%
% END OF LATTICE L42.71
%
\hbox{}\par\smallskip%
%
% LATTICE L42.72
%
\leavevmode%
\llap{\spacer}%
${L_{42.72}}=\pdualwitharg{2.3}({L_{42.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{624}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{208}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-45863376\\3822000\\-22464
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3822000\\-318504\\1872
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22464\\1872\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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-1\\-14\\-344
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\HardButStrongLineBreak\kern3pt%
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0\\-5\\-864
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\78\\-5200
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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\kern3pt%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-45863376, 3822000, -22464; 3822000, -318504, 1872; -22464, 1872, -11],[624, 24, 1],[2, 23, 1, 2, 1, -1, -5, -1, 0, 9; 21, 260, 12, 25, 13, -12, -65, -14, -5, 78; -522, -2808, -2, 168, 169, 0, -858, -344, -864, -5200],41,72]
%
% END OF LATTICE L42.72
%
\hbox{}\par\smallskip%
%
% LATTICE L42.73
%
\leavevmode%
\llap{\spacer}%
${L_{42.73}}=\pdualwitharg{2.3}({L_{42.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{624}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{b}{2}{156}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{208}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12250992\\-1018368\\3744
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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139\\1638\\-9256
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-624\\-24\\-216\\-624\\-240\\-312\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1872\\-104\\-1248\\-4160\\-1872\\-3120\\-208\\0\\208
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12250992, -1018368, 3744; -1018368, 84648, -312; 3744, -312, 1],[624, 24, 1],[14, 75, 0, -5, -11, -1, 21, 9, 23, 139; 165, 884, 0, -59, -130, -12, 247, 106, 271, 1638; -930, -4992, -1, 312, 650, 24, -1482, -608, -1536, -9256],41,73]
%
% END OF LATTICE L42.73
%
\hbox{}\par\smallskip%
%
% LATTICE L42.74
%
\leavevmode%
\llap{\spacer}%
${L_{42.74}}=\pdualwitharg{2.3}({L_{42.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{624}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}{208}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10349040\\-71760\\-16848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71760\\-456\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16848\\-96\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-814\\1626
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-13286\\26520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-813\\1622
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-2979\\5940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-7579\\15106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-2434\\4848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1079\\2145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\272\\-544
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\271\\-540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-2444\\4888
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-120\\-468\\-240\\-234\\-48\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\624\\0\\-624\\-3120\\-1872\\-2184\\-624\\-624\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-52\\-48\\-78\\-32\\-48\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-624\\0\\24\\0\\-48\\-156\\-96\\-216\\-1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-3120\\-52\\0\\52\\0\\-234\\-208\\-624\\-4160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1872\\-48\\-48\\0\\48\\0\\-48\\-240\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-2184\\-78\\-156\\-234\\0\\39\\0\\-156\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-624\\-32\\-96\\-208\\-48\\0\\16\\0\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-624\\-48\\-216\\-624\\-240\\-156\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1872\\-208\\-1248\\-4160\\-1872\\-1560\\-208\\0\\208
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10349040, -71760, -16848; -71760, -456, -96; -16848, -96, -17],[624, 24, 1],[3, 49, 3, 11, 28, 9, 4, -1, -1, 9; -814, -13286, -813, -2979, -7579, -2434, -1079, 272, 271, -2444; 1626, 26520, 1622, 5940, 15106, 4848, 2145, -544, -540, 4888],41,74]
%
% END OF LATTICE L42.74
%
\hbox{}\par\smallskip%
%
% LATTICE L42.75
%
\leavevmode%
\llap{\spacer}%
${L_{42.75}}=\pdualwitharg{13}({L_{42.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{624}\above{1pt}{1pt}{s}{2}{104}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19344\\-1872\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\182\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-88\\-104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-18\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-66\\-104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-48\\-66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\0\\-624\\-1040\\-1872\\-520\\-624\\-312\\-208\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-104\\-240\\-78\\-112\\-78\\-104\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\0\\624\\0\\-624\\-312\\-624\\-624\\-1248\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1040\\-104\\0\\104\\0\\-52\\-208\\-312\\-936\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-240\\-624\\0\\48\\0\\-144\\-312\\-1248\\-960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-78\\-312\\-52\\0\\13\\0\\-39\\-260\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-112\\-624\\-208\\-144\\0\\16\\0\\-208\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-78\\-624\\-312\\-312\\-39\\0\\39\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-104\\-1248\\-936\\-1248\\-260\\-208\\0\\104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-624\\-624\\-960\\-234\\-240\\-78\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19344, -1872, 0; -1872, 182, 0; 0, 0, -1],[624, 26, 1],[-9, -1, -1, 1, 1, 0, -1, -2, -7, -5; -88, -10, -12, 10, 12, 1, -8, -18, -66, -48; -104, -10, 0, 0, -24, -13, -32, -39, -104, -66],41,75]
%
% END OF LATTICE L42.75
%
\hbox{}\par\smallskip%
%
% LATTICE L42.76
%
\leavevmode%
\llap{\spacer}%
${L_{42.76}}=\pdualwitharg{2.13}(\main({L_{42.7}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{156}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{s}{2}{104}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
248664\\36504\\-2496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36504\\5252\\-364
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2496\\-364\\25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-57\\-2236
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-8\\-317
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-27\\-1092
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\143
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-24\\-858
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-64\\-2444
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-54\\-2094
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\0\\-156\\-520\\-468\\-260\\-156\\-312\\-104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-52\\-60\\-39\\-28\\-78\\-52\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\0\\156\\0\\-156\\-156\\-156\\-624\\-624\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-52\\0\\104\\0\\-52\\-104\\-624\\-936\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-60\\-156\\0\\12\\0\\-36\\-312\\-624\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-260\\-39\\-156\\-52\\0\\13\\0\\-78\\-260\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-28\\-156\\-104\\-36\\0\\4\\0\\-104\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-78\\-624\\-624\\-312\\-78\\0\\156\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-52\\-624\\-936\\-624\\-260\\-104\\0\\104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-312\\-624\\-480\\-234\\-120\\-156\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[248664, 36504, -2496; 36504, 5252, -364; -2496, -364, 25],[312, 52, 1],[-14, -2, -7, -1, 1, 1, 0, -5, -15, -13; -57, -8, -27, -4, 3, 3, -1, -24, -64, -54; -2236, -317, -1092, -156, 144, 143, -16, -858, -2444, -2094],41,76]
%
% END OF LATTICE L42.76
%
\hbox{}\par\smallskip%
%
% LATTICE L42.77
%
\leavevmode%
\llap{\spacer}%
${L_{42.77}}=\pdualwitharg{13}({L_{42.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{624}\above{1pt}{1pt}{l}{2}{26}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{l}{2}{26}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7689552\\24336\\24336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24336\\-26\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24336\\-78\\-77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\73\\4030
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\336\\18288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\109\\5889
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\156\\8368
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\216\\11466
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\71\\3718
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\24\\1239
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\0\\-624\\-520\\-1872\\-520\\-624\\-624\\-104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-52\\-240\\-78\\-112\\-156\\-52\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\0\\624\\0\\-624\\-312\\-624\\-1248\\-624\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-52\\0\\26\\0\\-26\\-104\\-312\\-234\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-240\\-624\\0\\48\\0\\-144\\-624\\-624\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-78\\-312\\-26\\0\\13\\0\\-78\\-130\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-112\\-624\\-104\\-144\\0\\16\\0\\-104\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-156\\-1248\\-312\\-624\\-78\\0\\156\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-52\\-624\\-234\\-624\\-130\\-104\\0\\26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-312\\-156\\-480\\-117\\-120\\-78\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7689552, 24336, 24336; 24336, -26, -78; 24336, -78, -77],[624, 26, 1],[-1, -1, 1, 13, 59, 19, 27, 37, 12, 4; -4, -6, 0, 73, 336, 109, 156, 216, 71, 24; -312, -310, 312, 4030, 18288, 5889, 8368, 11466, 3718, 1239],41,77]
%
% END OF LATTICE L42.77
%
\hbox{}\par\smallskip%
%
% LATTICE L42.78
%
\leavevmode%
\llap{\spacer}%
${L_{42.78}}=\pdualwitharg{13}({L_{42.6}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{624}\above{1pt}{1pt}{}{2}{26}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{}{2}{26}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\0\\624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-26\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\-156\\-311
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\151\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
119\\696\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\452\\-78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\324\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\225\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\149\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\102\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\0\\-624\\-520\\-1872\\-1040\\-624\\-312\\-104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-26\\-120\\-78\\-56\\-39\\-26\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\0\\624\\0\\-624\\-624\\-624\\-624\\-624\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-26\\0\\26\\0\\-52\\-104\\-156\\-234\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-120\\-624\\0\\48\\0\\-144\\-312\\-624\\-960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1040\\-78\\-624\\-52\\0\\52\\0\\-78\\-260\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-56\\-624\\-104\\-144\\0\\16\\0\\-104\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-39\\-624\\-156\\-312\\-78\\0\\39\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-26\\-624\\-234\\-624\\-260\\-104\\0\\26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-624\\-312\\-960\\-468\\-240\\-78\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[624, 0, 624; 0, -26, -156; 624, -156, -311],[624, 26, 1],[-1, -1, 1, 26, 119, 77, 55, 38, 25, 17; -4, -6, 0, 151, 696, 452, 324, 225, 149, 102; 0, 1, 0, -26, -120, -78, -56, -39, -26, -18],41,78]
%
% END OF LATTICE L42.78
%
\hbox{}\par\smallskip%
%
% LATTICE L42.79
%
\leavevmode%
\llap{\spacer}%
${L_{42.79}}=\pdualwitharg{13}({L_{42.7}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{624}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
386256\\42432\\-3120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42432\\4550\\-338
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-338\\25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-408\\-9152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-28\\-629
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-180\\-4056
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\286
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-102\\-2262
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-242\\-5408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-99\\-2217
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\0\\-624\\-1040\\-1872\\-1040\\-624\\-624\\-208\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-52\\-120\\-78\\-56\\-78\\-52\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\0\\624\\0\\-624\\-624\\-624\\-1248\\-1248\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1040\\-52\\0\\104\\0\\-104\\-208\\-624\\-936\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-120\\-624\\0\\48\\0\\-144\\-624\\-1248\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1040\\-78\\-624\\-104\\0\\52\\0\\-156\\-520\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-56\\-624\\-208\\-144\\0\\16\\0\\-208\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-78\\-1248\\-624\\-624\\-156\\0\\156\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-52\\-1248\\-936\\-1248\\-520\\-208\\0\\104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-312\\-312\\-480\\-234\\-120\\-78\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[386256, 42432, -3120; 42432, 4550, -338; -3120, -338, 25],[624, 26, 1],[-29, -2, -13, -1, 1, 1, -1, -7, -17, -7; -408, -28, -180, -14, 12, 12, -16, -102, -242, -99; -9152, -629, -4056, -312, 288, 286, -344, -2262, -5408, -2217],41,79]
%
% END OF LATTICE L42.79
%
\hbox{}\par\smallskip%
%
% LATTICE L42.80
%
\leavevmode%
\llap{\spacer}%
${L_{42.80}}=\pdualwitharg{2.13}(\main({L_{42.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{156}\above{1pt}{1pt}{}{2}{104}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{2}{104}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-702312\\-351312\\-173160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-351312\\-175708\\-86580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-173160\\-86580\\-42637
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\3775\\-2548
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-331\\992\\-670
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-461\\1383\\-936
\end{tabular}\endgroup%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-702312, -351312, -173160; -351312, -175708, -86580; -173160, -86580, -42637],[312, 52, 1],[-1260, -331, -461, 1, 65, 63, -68, -368, -1597, -631; 3775, 992, 1383, -2, -195, -190, 203, 1101, 4782, 1890; -2548, -670, -936, 0, 132, 130, -136, -741, -3224, -1275],41,80]
%
% END OF LATTICE L42.80
%
\hbox{}\par\smallskip%
%
% LATTICE L42.81
%
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${L_{42.81}}=\pdualwitharg{2.3.13}(\main({L_{42.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1092, -18252, 4680; -18252, 304824, -78156; 4680, -78156, 20039],[156, 156, 1],[-86, -20, -6, 0, -2, -16, -14, -28, -34, -32; -5, 5, 13, 10, 5, -1, -7, -15, -12, -7; 0, 24, 52, 39, 20, 0, -24, -52, -39, -20],41,81]
%
% END OF LATTICE L42.81
%
\hbox{}\par\smallskip%
%
% LATTICE L42.82
%
\leavevmode%
\llap{\spacer}%
${L_{42.82}}=\pdualwitharg{3.13}(\main({L_{42.5}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-132600, -30576, -14040; -30576, -6162, -2808; -14040, -2808, -1279],[312, 78, 1],[26, 19, 9, 9, 2, -1, -1, -1, 8, 13; -2189, -1608, -767, -773, -173, 76, 87, 100, -661, -1088; 4524, 3324, 1586, 1599, 358, -156, -180, -208, 1365, 2248],41,82]
%
% END OF LATTICE L42.82
%
\hbox{}\par\smallskip%
%
% LATTICE L42.83
%
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${L_{42.83}}=\pdualwitharg{3.13}(\main({L_{42.7}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7800, 15288, -312; 15288, 27066, -546; -312, -546, 11],[312, 78, 1],[7, 1, 1, -1, -1, -2, -1, 0, 3, 1; -48, -7, -8, 2, 4, 9, 4, -1, -22, -7; -2184, -318, -364, 78, 172, 390, 168, -52, -1014, -320],41,83]
%
% END OF LATTICE L42.83
%
\hbox{}\par\smallskip%
%
% LATTICE L42.84
%
\leavevmode%
\llap{\spacer}%
${L_{42.84}}=\pdualwitharg{2.3.13}(\main({L_{42.2}}))$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4992, -47580, 12168; -47580, 405912, -103740; 12168, -103740, 26513],[156, 156, 1],[94, 16, -6, -18, -2, 20, 22, 44, 94, 40; 1885, 327, -107, -341, -35, 401, 435, 869, 1865, 797; 7332, 1272, -416, -1326, -136, 1560, 1692, 3380, 7254, 3100],41,84]
%
% END OF LATTICE L42.84
%
\hbox{}\par\smallskip%
%
% LATTICE L42.85
%
\leavevmode%
\llap{\spacer}%
${L_{42.85}}=\pdualwitharg{2.13}({L_{42.3}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-565968, -141960, 2496; -141960, -35048, 624; 2496, 624, -11],[312, 104, 1],[27, 10, 29, 5, 7, 0, -1, -2, 4, 20; 29, 11, 33, 6, 9, 1, -1, -3, 4, 21; 7748, 2884, 8424, 1469, 2088, 52, -284, -624, 1131, 5712],41,85]
%
% END OF LATTICE L42.85
%
\hbox{}\par\smallskip%
%
% LATTICE L42.86
%
\leavevmode%
\llap{\spacer}%
${L_{42.86}}=\pdualwitharg{2.13}({L_{42.1}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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0\\8\\0\\-104\\-240
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-312\\0\\312\\0\\-312
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1989624, 0, -986232; 0, 104, 0; -986232, 0, -488863],[312, 104, 1],[-3093, -1146, -3325, -1147, -803, 0, 115, 232, -915, -2290; -10, -3, -6, -1, 0, 1, 0, -3, -5, -9; 6240, 2312, 6708, 2314, 1620, 0, -232, -468, 1846, 4620],41,86]
%
% END OF LATTICE L42.86
%
\hbox{}\par\smallskip%
%
% LATTICE L42.87
%
\leavevmode%
\llap{\spacer}%
${L_{42.87}}=\pdualwitharg{2.13}({L_{42.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${104}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{312}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{104}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{312}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-213096\\0\\2808
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0\\104\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2808\\0\\-37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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27\\-10\\2028
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10\\-3\\752
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29\\-6\\2184
\end{tabular}\endgroup%
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10\\-1\\754
\end{tabular}\endgroup%
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7\\0\\528
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0\\1\\0
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-1\\0\\-76
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-2\\-3\\-156
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8\\-5\\598
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20\\-9\\1500
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
104\\0\\-312\\-520\\-936\\-1040\\-312\\-624\\-104\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-213096, 0, 2808; 0, 104, 0; 2808, 0, -37],[312, 104, 1],[27, 10, 29, 10, 7, 0, -1, -2, 8, 20; -10, -3, -6, -1, 0, 1, 0, -3, -5, -9; 2028, 752, 2184, 754, 528, 0, -76, -156, 598, 1500],41,87]
%
% END OF LATTICE L42.87
%
\hbox{}\par\smallskip%
%
% LATTICE L42.88
%
\leavevmode%
\llap{\spacer}%
${L_{42.88}}=\pdualwitharg{3.13}({L_{42.4}})$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${624}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{208}\above{1pt}{1pt}{s}{2}{312}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{r}{2}{312}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[112944, 53040, -2496; 53040, 24882, -1170; -2496, -1170, 55],[624, 78, 1],[7, 1, 1, -1, -1, -1, -1, 0, 3, 1; -88, -12, -12, 2, 4, 4, 0, -3, -50, -14; -1560, -210, -208, 0, 40, 39, -48, -65, -936, -254],41,88]
%
% END OF LATTICE L42.88
%
\hbox{}\par\smallskip%
%
% LATTICE L42.89
%
\leavevmode%
\llap{\spacer}%
${L_{42.89}}=\pdualwitharg{2.3.13}(\main({L_{42.7}}))$%
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{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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-312\\-78\\-208\\-624\\-104\\-78\\0\\52\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-156\\-624\\-2808\\-624\\-780\\-312\\0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-104\\-624\\-160\\-234\\-120\\-52\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[219336, -1077960, 21216; -1077960, 5159388, -101556; 21216, -101556, 1999],[312, 156, 1],[-12, -2, -3, -1, 1, 3, 2, 1, -7, -3; 673, 109, 157, 52, -49, -146, -91, -30, 456, 176; 34320, 5559, 8008, 2652, -2500, -7449, -4644, -1534, 23244, 8974],41,89]
%
% END OF LATTICE L42.89
%
\hbox{}\par\smallskip%
%
% LATTICE L42.90
%
\leavevmode%
\llap{\spacer}%
${L_{42.90}}=\pdualwitharg{3.13}({L_{42.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${624}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{s}{2}{208}\above{1pt}{1pt}{l}{2}{78}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{l}{2}{78}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-265200\\-127920\\2496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-127920\\-59358\\1170
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2496\\1170\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-414
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\43\\3042
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\72\\5056
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\73\\5109
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\108\\7536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\52\\3614
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\53\\3666
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\413
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\0\\-624\\-1560\\-1872\\-1560\\-1872\\-624\\-312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-156\\-240\\-234\\-336\\-156\\-156\\-18
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\0\\208\\0\\-208\\-312\\-624\\-416\\-624\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-156\\0\\78\\0\\-78\\-312\\-312\\-702\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-240\\-208\\0\\16\\0\\-144\\-208\\-624\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-234\\-312\\-78\\0\\39\\0\\-78\\-390\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-336\\-624\\-312\\-144\\0\\48\\0\\-312\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-156\\-416\\-312\\-208\\-78\\0\\52\\0\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-156\\-624\\-702\\-624\\-390\\-312\\0\\78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-104\\-156\\-160\\-117\\-120\\-26\\0\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-265200, -127920, 2496; -127920, -59358, 1170; 2496, 1170, -23],[624, 78, 1],[-1, -1, -1, 8, 13, 13, 19, 9, 9, 1; -4, -6, -8, 43, 72, 73, 108, 52, 53, 6; -312, -414, -520, 3042, 5056, 5109, 7536, 3614, 3666, 413],41,90]
%
% END OF LATTICE L42.90
%
\hbox{}\par\smallskip%
%
% LATTICE L42.91
%
\leavevmode%
\llap{\spacer}%
${L_{42.91}}=\pdualwitharg{3.13}({L_{42.6}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${624}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{208}\above{1pt}{1pt}{}{2}{78}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{}{2}{78}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4450992\\-541632\\21216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-541632\\-65442\\2574
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21216\\2574\\-101
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-6\\-363
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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21\\121\\7488
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\192\\11816
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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65\\380\\23322
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\276\\16896
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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11\\65\\3965
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\131\\7956
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\30\\1814
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\0\\-624\\-1560\\-1872\\-3120\\-1872\\-312\\-312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-78\\-120\\-234\\-168\\-39\\-78\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\0\\208\\0\\-208\\-624\\-624\\-208\\-624\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-78\\0\\78\\0\\-156\\-312\\-156\\-702\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-120\\-208\\0\\16\\0\\-144\\-104\\-624\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-234\\-624\\-156\\0\\156\\0\\-78\\-780\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-168\\-624\\-312\\-144\\0\\48\\0\\-312\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-39\\-208\\-156\\-104\\-78\\0\\13\\0\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-78\\-624\\-702\\-624\\-780\\-312\\0\\78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-208\\-312\\-320\\-468\\-240\\-26\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4450992, -541632, 21216; -541632, -65442, 2574; 21216, 2574, -101],[624, 78, 1],[-1, -1, -1, 21, 33, 65, 47, 11, 22, 5; -4, -6, -8, 121, 192, 380, 276, 65, 131, 30; -312, -363, -416, 7488, 11816, 23322, 16896, 3965, 7956, 1814],41,91]
%
% END OF LATTICE L42.91
%
\hbox{}\par\smallskip%
%
% LATTICE L42.92
%
\leavevmode%
\llap{\spacer}%
${L_{42.92}}=\pdualwitharg{3.13}({L_{42.7}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${624}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{208}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1691664\\-127296\\6240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-127296\\-9438\\468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6240\\468\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-27\\-128\\-9984
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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-5\\-20\\-1768
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\HardButStrongLineBreak\kern3pt%
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-1\\-2\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\4\\352
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\HardButStrongLineBreak\kern3pt%
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3\\10\\1014
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\HardButStrongLineBreak\kern3pt%
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1\\0\\264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-8\\-442
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-70\\-4992
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-10\\-751
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\0\\-624\\-3120\\-1872\\-3120\\-1872\\-624\\-624\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-156\\-120\\-234\\-168\\-78\\-156\\-9
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\0\\208\\0\\-208\\-624\\-624\\-416\\-1248\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-156\\0\\312\\0\\-312\\-624\\-624\\-2808\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-120\\-208\\0\\16\\0\\-144\\-208\\-1248\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-234\\-624\\-312\\0\\156\\0\\-156\\-1560\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-168\\-624\\-624\\-144\\0\\48\\0\\-624\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-78\\-416\\-624\\-208\\-156\\0\\52\\0\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-156\\-1248\\-2808\\-1248\\-1560\\-624\\0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-104\\-312\\-160\\-234\\-120\\-26\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1691664, -127296, 6240; -127296, -9438, 468; 6240, 468, -23],[624, 78, 1],[-27, -2, -5, -1, 1, 3, 1, -1, -13, -2; -128, -9, -20, -2, 4, 10, 0, -8, -70, -10; -9984, -729, -1768, -312, 352, 1014, 264, -442, -4992, -751],41,92]
%
% END OF LATTICE L42.92
%
\hbox{}\par\smallskip%
%
% LATTICE L42.93
%
\leavevmode%
\llap{\spacer}%
${L_{42.93}}=\pdualwitharg{2.3.13}(\main({L_{42.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{52}\above{1pt}{1pt}{}{2}{312}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{312}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
312\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3900\\-1716
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1716\\-755
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\139\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\40\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\23\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-34\\78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\140\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\21\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\-156\\-1560\\-468\\-1560\\-468\\-156\\-312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-312\\-120\\-468\\-168\\-78\\-312\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\0\\52\\0\\-52\\-312\\-156\\-104\\-624\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-312\\0\\312\\0\\-312\\-312\\-312\\-2808\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-120\\-52\\0\\4\\0\\-36\\-52\\-624\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-468\\-312\\-312\\0\\156\\0\\-78\\-1560\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-168\\-156\\-312\\-36\\0\\12\\0\\-312\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-78\\-104\\-312\\-52\\-78\\0\\13\\0\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\-624\\-2808\\-624\\-1560\\-312\\0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-52\\-312\\-80\\-234\\-60\\-13\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[312, 0, 0; 0, -3900, -1716; 0, -1716, -755],[312, 156, 1],[-5, -1, 0, 1, 0, -1, -1, -1, -9, -1; 139, 40, 23, 0, -7, -34, -5, 6, 140, 21; -312, -90, -52, 0, 16, 78, 12, -13, -312, -47],41,93]
%
% END OF LATTICE L42.93
%
\hbox{}\par\smallskip%
%
% LATTICE L42.94
%
\leavevmode%
\llap{\spacer}%
${L_{42.94}}=\pdualwitharg{2.13}({L_{42.7}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{624}\above{1pt}{1pt}{b}{2}{104}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{b}{2}{104}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19344\\-11232\\1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11232\\3848\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1872\\-624\\101
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-448\\-2600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-50\\-290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-54\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\45\\260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\42\\240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-52\\-304
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-201\\-1170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-349\\-2028
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-249\\-1446
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\0\\-624\\-1040\\-1872\\-520\\-624\\-624\\-208\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-104\\-240\\-78\\-112\\-156\\-104\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\0\\624\\0\\-624\\-312\\-624\\-1248\\-1248\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1040\\-104\\0\\104\\0\\-52\\-208\\-624\\-936\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-240\\-624\\0\\48\\0\\-144\\-624\\-1248\\-960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-78\\-312\\-52\\0\\13\\0\\-78\\-260\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-112\\-624\\-208\\-144\\0\\16\\0\\-208\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-156\\-1248\\-624\\-624\\-78\\0\\156\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-104\\-1248\\-936\\-1248\\-260\\-208\\0\\104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-624\\-624\\-960\\-234\\-240\\-156\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19344, -11232, 1872; -11232, 3848, -624; 1872, -624, 101],[624, 104, 1],[-9, -1, -1, 1, 1, 0, -1, -4, -7, -5; -448, -50, -54, 45, 42, -2, -52, -201, -349, -249; -2600, -290, -312, 260, 240, -13, -304, -1170, -2028, -1446],41,94]
%
% END OF LATTICE L42.94
%
\hbox{}\par\smallskip%
%
% LATTICE L42.95
%
\leavevmode%
\llap{\spacer}%
${L_{42.95}}=\pdualwitharg{2.13}({L_{42.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{l}{2}{624}\above{1pt}{1pt}{}{2}{104}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{2}{104}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\-624\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\728\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-10\\-104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-9\\-104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-6\\-66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\0\\-624\\-1040\\-1872\\-1040\\-624\\-312\\-208\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-104\\-240\\-156\\-112\\-78\\-104\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\0\\624\\0\\-624\\-624\\-624\\-624\\-1248\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1040\\-104\\0\\104\\0\\-104\\-208\\-312\\-936\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-240\\-624\\0\\48\\0\\-144\\-312\\-1248\\-960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1040\\-156\\-624\\-104\\0\\52\\0\\-78\\-520\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-112\\-624\\-208\\-144\\0\\16\\0\\-208\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-78\\-624\\-312\\-312\\-78\\0\\39\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-104\\-1248\\-936\\-1248\\-520\\-208\\0\\104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-624\\-624\\-960\\-468\\-240\\-78\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[624, -624, 0; -624, 728, 0; 0, 0, -1],[624, 104, 1],[-9, -1, -1, 1, 1, 0, -1, -2, -7, -5; -10, -1, 0, 1, 0, -1, -2, -3, -9, -6; -104, -10, 0, 0, -24, -26, -32, -39, -104, -66],41,95]
%
% END OF LATTICE L42.95
%
\hbox{}\par\smallskip%
%
% LATTICE L42.96
%
\leavevmode%
\llap{\spacer}%
${L_{42.96}}=\pdualwitharg{2.13}({L_{42.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{b}{2}{624}\above{1pt}{1pt}{l}{2}{104}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{l}{2}{104}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
711984\\-711984\\4992
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-711984\\712088\\-4992
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4992\\-4992\\35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-10\\5824
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-1\\710
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\1560
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-3\\-1\\286
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[711984, -711984, 4992; -711984, 712088, -4992; 4992, -4992, 35],[624, 104, 1],[-51, -6, -11, 1, -1, -3, -7, -23, -39, -14; -10, -1, 0, 1, 0, -1, -2, -6, -9, -3; 5824, 710, 1560, 0, 144, 286, 712, 2418, 4264, 1563],41,96]
%
% END OF LATTICE L42.96
%
\hbox{}\par\smallskip%
%
% LATTICE L42.97
%
\leavevmode%
\llap{\spacer}%
${L_{42.97}}=\pdualwitharg{2.13}({L_{42.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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624\\-1976\\25
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15600, -49296, 624; -49296, 156104, -1976; 624, -1976, 25],[624, 104, 1],[-29, -2, -13, -1, 1, 1, -1, -7, -17, -14; -30, -2, -12, -1, 0, 0, -2, -9, -19, -15; -1664, -109, -624, -52, -24, -26, -136, -546, -1092, -846],41,97]
%
% END OF LATTICE L42.97
%
\hbox{}\par\smallskip%
%
% LATTICE L42.98
%
\leavevmode%
\llap{\spacer}%
${L_{42.98}}=\pdualwitharg{2.3.13}({L_{42.3}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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49296\\287976\\-1872
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
287976\\1679496\\-10920
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[49296, 287976, -1872; 287976, 1679496, -10920; -1872, -10920, 71],[312, 312, 1],[3, 2, 3, 2, 1, 0, -1, -2, -1, 0; 29, 11, 11, 6, 3, 1, -1, -1, 4, 7; 4524, 1740, 1768, 975, 488, 156, -180, -208, 585, 1072],41,98]
%
% END OF LATTICE L42.98
%
\hbox{}\par\smallskip%
%
% LATTICE L42.99
%
\leavevmode%
\llap{\spacer}%
${L_{42.99}}=\pdualwitharg{2.3.13}({L_{42.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
312\\0\\-312\\-1560\\-936\\-3120\\-936\\-624\\-312\\0
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-936\\-240\\-104\\0\\8\\0\\-72\\-208\\-624\\-320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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312\\0\\-312\\-1560\\-936
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-312\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\0\\104\\0\\-104
\end{tabular}\endgroup%
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-1560\\-312\\0\\156\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-240\\-104\\0\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-936\\-624\\-312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-336\\-312\\-312\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-312\\-416\\-624\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\-624\\-1404\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-208\\-624\\-320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[312, 0, 0; 0, -3432, -1560; 0, -1560, -709],[312, 312, 1],[-10, -3, -2, -1, 0, 1, 0, -1, -5, -3; 3, -10, -23, -35, -9, 0, 11, 24, 37, 10; 0, 24, 52, 78, 20, 0, -24, -52, -78, -20],41,99]
%
% END OF LATTICE L42.99
%
\hbox{}\par\smallskip%
%
% LATTICE L42.100
%
\leavevmode%
\llap{\spacer}%
${L_{42.100}}=\pdualwitharg{2.3.13}({L_{42.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${312}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{104}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{312}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}{104}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-222456\\0\\1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1248\\0\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-10\\468
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\702
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-364
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
312\\0\\-312\\-1560\\-936\\-3120\\-936\\-624\\-312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-312\\-240\\-936\\-336\\-312\\-312\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\0\\104\\0\\-104\\-624\\-312\\-416\\-624\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-312\\0\\156\\0\\-312\\-312\\-624\\-1404\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-240\\-104\\0\\8\\0\\-72\\-208\\-624\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-936\\-624\\-312\\0\\312\\0\\-312\\-1560\\-936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-336\\-312\\-312\\-72\\0\\24\\0\\-312\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-312\\-416\\-624\\-208\\-312\\0\\104\\0\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\-624\\-1404\\-624\\-1560\\-312\\0\\156\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-208\\-624\\-320\\-936\\-240\\-104\\0\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-222456, 0, 1248; 0, 312, 0; 1248, 0, -7],[312, 312, 1],[3, 2, 3, 4, 1, 0, -1, -2, -2, 0; -10, -3, -2, -1, 0, 1, 0, -1, -5, -3; 468, 336, 520, 702, 176, 0, -180, -364, -390, -20],41,100]
%
% END OF LATTICE L42.100
%
\hbox{}\par\smallskip%
%
% LATTICE L42.101
%
\leavevmode%
\llap{\spacer}%
${L_{42.101}}=\pdualwitharg{2.3.13}({L_{42.7}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${624}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{208}\above{1pt}{1pt}{b}{2}{312}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{b}{2}{312}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112944\\382512\\-2496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
382512\\1290120\\-8424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2496\\-8424\\55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\-44\\-8424
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-5\\-6\\-1146
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-6\\-1144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\2\\352
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\HardButStrongLineBreak\kern3pt%
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1\\2\\351
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\HardButStrongLineBreak\kern3pt%
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-1\\0\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-3\\-3\\-598
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-22\\-25\\-4836
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-7\\-1346
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\0\\-624\\-3120\\-1872\\-1560\\-1872\\-624\\-624\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-312\\-240\\-234\\-336\\-156\\-312\\-36
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-312\\0\\312\\0\\-156\\-624\\-624\\-2808\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-240\\-208\\0\\16\\0\\-144\\-208\\-1248\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-234\\-312\\-156\\0\\39\\0\\-78\\-780\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-336\\-624\\-624\\-144\\0\\48\\0\\-624\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-156\\-416\\-624\\-208\\-78\\0\\52\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-312\\-1248\\-2808\\-1248\\-780\\-624\\0\\312\\0
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\kern3pt%
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0\\-36\\-208\\-624\\-320\\-234\\-240\\-52\\0\\4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[112944, 382512, -2496; 382512, 1290120, -8424; -2496, -8424, 55],[624, 312, 1],[-37, -5, -5, 0, 1, 1, -1, -3, -22, -6; -44, -6, -6, 1, 2, 2, 0, -3, -25, -7; -8424, -1146, -1144, 156, 352, 351, -48, -598, -4836, -1346],41,101]
%
% END OF LATTICE L42.101
%
\hbox{}\par\smallskip%
%
% LATTICE L42.102
%
\leavevmode%
\llap{\spacer}%
${L_{42.102}}=\pdualwitharg{2.3.13}({L_{42.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{312}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{624}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{l}{2}{208}\above{1pt}{1pt}{}{2}{312}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38064\\0\\-18096
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18096\\0\\-8603
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-2\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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31\\-1\\-65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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446\\-9\\-936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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121\\-2\\-254
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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743\\-10\\-1560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\-1\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99\\0\\-208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\-78\\-1560\\-468\\-3120\\-468\\-624\\-312\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-624\\-240\\-1872\\-336\\-624\\-624\\-144
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\0\\13\\0\\-26\\-312\\-78\\-208\\-312\\-104
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-624\\0\\312\\0\\-624\\-312\\-1248\\-2808\\-1248
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-240\\-26\\0\\4\\0\\-36\\-208\\-624\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-1872\\-312\\-624\\0\\624\\0\\-624\\-3120\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-336\\-78\\-312\\-36\\0\\12\\0\\-312\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-624\\-208\\-1248\\-208\\-624\\0\\208\\0\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-624\\-312\\-2808\\-624\\-3120\\-312\\0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-104\\-1248\\-320\\-1872\\-240\\-208\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38064, 0, -18096; 0, 312, 0; -18096, 0, -8603],[624, 312, 1],[-37, 23, 31, 446, 121, 743, 100, 99, 0, -19; -1, -2, -1, -9, -2, -10, -1, 0, 1, 0; 78, -48, -65, -936, -254, -1560, -210, -208, 0, 40],41,102]
%
% END OF LATTICE L42.102
%
\hbox{}\par\smallskip%
%
% LATTICE L42.103
%
\leavevmode%
\llap{\spacer}%
${L_{42.103}}=\pdualwitharg{2.3.13}({L_{42.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{l}{2}{312}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{624}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{b}{2}{208}\above{1pt}{1pt}{l}{2}{312}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2861040\\0\\8112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8112\\0\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\702
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-2\\-1066
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-9\\-7800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\-1063
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\-10\\-13104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-1\\-1770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\-1768
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\352
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\-156\\-1560\\-234\\-3120\\-468\\-624\\-312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-624\\-120\\-1872\\-336\\-624\\-624\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\0\\52\\0\\-26\\-624\\-156\\-416\\-624\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-624\\0\\312\\0\\-624\\-312\\-1248\\-2808\\-1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-120\\-26\\0\\1\\0\\-18\\-104\\-312\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-1872\\-624\\-624\\0\\624\\0\\-624\\-3120\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-336\\-156\\-312\\-18\\0\\12\\0\\-312\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-624\\-416\\-1248\\-104\\-624\\0\\208\\0\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-624\\-624\\-2808\\-312\\-3120\\-312\\0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-208\\-1248\\-160\\-1872\\-240\\-208\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2861040, 0, 8112; 0, 312, 0; 8112, 0, -23],[624, 312, 1],[2, -1, -3, -22, -3, -37, -5, -5, 0, 1; -1, -2, -2, -9, -1, -10, -1, 0, 1, 0; 702, -360, -1066, -7800, -1063, -13104, -1770, -1768, 0, 352],41,103]
%
% END OF LATTICE L42.103
%
\hbox{}\par\smallskip%
%
% LATTICE L42.104
%
\leavevmode%
\llap{\spacer}%
${L_{42.104}}=\pdualwitharg{2.3.13}({L_{42.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${624}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{208}\above{1pt}{1pt}{r}{2}{312}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{s}{2}{312}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38064\\-184080\\-91104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-184080\\-889512\\-440232
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-91104\\-440232\\-217877
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\1550\\-3120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\91\\-183
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\104\\-208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-77\\156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-20\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\38\\-78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\178\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\219\\-442
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\1161\\-2340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\281\\-566
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\0\\-624\\-3120\\-1872\\-3120\\-1872\\-624\\-624\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-156\\-120\\-234\\-168\\-78\\-156\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\0\\208\\0\\-208\\-624\\-624\\-416\\-1248\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-156\\0\\312\\0\\-312\\-624\\-624\\-2808\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-120\\-208\\0\\16\\0\\-144\\-208\\-1248\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-234\\-624\\-312\\0\\156\\0\\-156\\-1560\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-168\\-624\\-624\\-144\\0\\48\\0\\-624\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-78\\-416\\-624\\-208\\-156\\0\\52\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-156\\-1248\\-2808\\-1248\\-1560\\-624\\0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-208\\-624\\-320\\-468\\-240\\-52\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38064, -184080, -91104; -184080, -889512, -440232; -91104, -440232, -217877],[624, 312, 1],[-27, -2, -5, -1, 1, 3, 1, -1, -13, -4; 1550, 91, 104, -77, -20, 38, 178, 219, 1161, 281; -3120, -183, -208, 156, 40, -78, -360, -442, -2340, -566],41,104]
%
% END OF LATTICE L42.104
%

\medskip%
%------------------------------WEYL-GROUP-W43--------------------------------
%
\leavevmode\llap{}%
$W_{43\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=24$%
\hfill%
$22222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L43.1
%
\leavevmode%
\llap{\spacer}%
${L_{43.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{43},3,2}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-805428\\2808\\4212
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2808\\-6\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4212\\-15\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1561\\-21840\\-285480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\97\\1281
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\112\\1463
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-128\\-1644
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-195\\-2561
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-66\\-918
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-36\\-24\\-204\\-312\\-108\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\26\\0\\-26\\-312\\-533\\-234\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\36\\0\\-108\\-234\\-144\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-26\\0\\2\\0\\-13\\-18\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-36\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\26\\0\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-26\\0\\2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\-312\\-108\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-533\\-234\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-234\\-144\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-13\\-18\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-805428, 2808, 4212; 2808, -6, -15; 4212, -15, -22],[468, 3, 1],[-9, -14, -5, 0, 1, 1, -1, -1; -128, -195, -66, 1, 16, 13, -18, -15; -1644, -2561, -918, -1, 180, 182, -180, -182],42,1]
%
% END OF LATTICE L43.1
%
\hbox{}\par\smallskip%
%
% LATTICE L43.2
%
\leavevmode%
\llap{\spacer}%
${L_{43.2}}=\pfillwitharg{2.3}({L_{43.1}})$%
{${}={}$}Nikulin $43$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\-78\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-30\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-13\\-2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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1351\\-4563\\4563
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
440\\-1486\\1485
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\-135\\134
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
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\HardButStrongLineBreak\kern3pt%
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8\\-26\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\14\\-19
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\right.$}%
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3\\0\\-3\\-12
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\26\\0\\-26
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-26\\0\\2
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-156\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-533\\-39\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-39\\-4\\-3
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0\\-13\\-3\\-5
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-195, -78, -39; -78, -30, -13; -39, -13, -2],[39, 1, 1],[-1, 8, 1, -4, -31, -112, -9, -4; 3, -26, -3, 14, 105, 377, 30, 13; 0, 13, -1, -19, -108, -364, -26, -8],42,2]
%
% END OF LATTICE L43.2
%
\hbox{}\par\smallskip%
%
% LATTICE L43.3
%
\leavevmode%
\llap{\spacer}%
${L_{43.3}}=\pfillwitharg{3}({L_{43.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61620\\-20280\\624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20280\\-6674\\205
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\205\\-6
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-365\\1092\\-1092
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119\\356\\-357
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-9\\8
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-19\\60\\72
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-22\\-24
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-19\\-26
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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12\\0\\-12\\-24
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0\\26\\0\\-26
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-12\\0\\4\\0
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-24\\-26\\0\\2
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-204\\-312\\-36\\0
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-312\\-533\\-78\\-13
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-36\\-78\\-16\\-6
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0\\-13\\-6\\-5
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-61620, -20280, 624; -20280, -6674, 205; 624, 205, -6],[156, 1, 1],[-19, -4, 7, 6, 11, -9, -9, -7; 60, 13, -22, -19, -36, 26, 28, 22; 72, 26, -24, -26, -96, -65, 18, 23],42,3]
%
% END OF LATTICE L43.3
%
\hbox{}\par\smallskip%
%
% LATTICE L43.4
%
\leavevmode%
\llap{\spacer}%
${L_{43.4}}=\pfillwitharg{2}({L_{43.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
202995\\-16848\\1053
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16848\\1398\\-87
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1053\\-87\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1715\\21450\\10296
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-1751\\-840
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6\\75\\35
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3\\38\\30
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\HardButStrongLineBreak\kern3pt%
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-1\\-13\\-13
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-51\\-45
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-13\\-17
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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3\\0\\-9\\-12\\-51\\-156\\-27\\0
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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3\\0\\-9\\-12
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0\\26\\0\\-26
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-9\\0\\9\\0
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-12\\-26\\0\\2
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-51\\-156\\-27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-533\\-117\\-13
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-117\\-36\\-9
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-13\\-9\\-5
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[202995, -16848, 1053; -16848, 1398, -87; 1053, -87, 5],[117, 3, 1],[3, -1, -4, -1, 5, 27, 10, 3; 38, -13, -51, -13, 62, 338, 126, 38; 30, -13, -45, -17, 18, 169, 81, 29],42,4]
%
% END OF LATTICE L43.4
%
\hbox{}\par\smallskip%
%
% LATTICE L43.5
%
\leavevmode%
\llap{\spacer}%
${L_{43.5}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{43.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12246\\234\\3939
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
234\\-15\\75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3939\\75\\1267
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2183\\273\\-6825
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\17\\-450
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
704\\88\\-2201
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-1\\56
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\-78
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\1\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\0\\-90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-12\\-17\\-156\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\78\\0\\-78\\-156\\-1599\\-117\\-39
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\78\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-78\\0\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-156\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-1599\\-117\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12246, 234, 3939; 234, -15, 75; 3939, 75, 1267],[39, 3, 1],[-18, 25, 27, 29, -14, -337, -51, -53; -1, 0, 1, 0, -3, -39, -4, -3; 56, -78, -84, -90, 44, 1053, 159, 165],42,5]
%
% END OF LATTICE L43.5
%
\hbox{}\par\smallskip%
%
% LATTICE L43.6
%
\leavevmode%
\llap{\spacer}%
${L_{43.6}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{43.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{234}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
234\\-117\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\57\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\39\\117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-17\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\4\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-12\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-36\\-51\\-468\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\234\\0\\-234\\-468\\-4797\\-117\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0\\-9\\-117\\-4\\-9
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-234\\0\\18\\0\\-117\\-9\\-45
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\234\\0\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-234\\0\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-468\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-4797\\-117\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-117\\-4\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-117\\-9\\-45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[234, -117, 0; -117, 57, -3; 0, -3, -5],[117, 3, 1],[2, -1, -1, -5, -2, 1, 1, 5; 4, 0, -2, -12, -8, -39, 1, 9; -3, 0, 1, 0, -9, -117, -4, -9],42,6]
%
% END OF LATTICE L43.6
%
\hbox{}\par\smallskip%
%
% LATTICE L43.7
%
\leavevmode%
\llap{\spacer}%
${L_{43.7}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{43.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1092\\312\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
312\\18\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-15\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\13\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-19\\-45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-24\\-68\\-312\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\78\\0\\-78\\-312\\-1599\\-234\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-78\\0\\6\\0\\-39\\-18\\-15
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\78\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-78\\0\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-312\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-1599\\-234\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-234\\-48\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-39\\-18\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1092, 312, -156; 312, 18, -15; -156, -15, 10],[156, 3, 1],[1, 1, -1, -1, -1, -1, 1, 1; 16, 13, -18, -19, -24, -52, 12, 16; 40, 39, -42, -45, -56, -117, 30, 39],42,7]
%
% END OF LATTICE L43.7
%
\hbox{}\par\smallskip%
%
% LATTICE L43.8
%
\leavevmode%
\llap{\spacer}%
${L_{43.8}}=\pdualwitharg{13}(\pfillwitharg{2.3}({L_{43.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${39}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12090\\897\\3666
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
897\\65\\273
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3666\\273\\1111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\408\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\120\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\30\\39
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-3
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-15\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-17\\-39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\0\\-39\\-156\\-663\\-156\\-117\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-26\\-156\\-41\\-39\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\13\\0\\-117\\-39\\-52\\-39
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-26\\0\\26\\0\\-13\\-39\\-65
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\0\\-39\\-156
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-26
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\13\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-26\\0\\26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-663\\-156\\-117\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-41\\-39\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-39\\-52\\-39
\end{tabular}\endgroup%
\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12090, 897, 3666; 897, 65, 273; 3666, 273, 1111],[39, 13, 1],[-14, 1, 9, 13, 10, -2, -10, -14; 30, -1, -15, -17, 18, 13, 27, 29; 39, -3, -26, -39, -39, 3, 26, 39],42,8]
%
% END OF LATTICE L43.8
%
\hbox{}\par\smallskip%
%
% LATTICE L43.9
%
\leavevmode%
\llap{\spacer}%
${L_{43.9}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{43.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4509336\\-47580\\-1141140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47580\\504\\12040
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-1141140\\12040\\288779
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-3524\\-4641\\-13728
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271\\356\\1056
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813\\1071\\3167
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-19\\-26\\-74
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{$\left[\!\llap{\phantom{%
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-51\\-312\\-18\\0
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-312\\-2132\\-156\\-52
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4509336, -47580, -1141140; -47580, 504, 12040; -1141140, 12040, 288779],[156, 4, 1],[-67, -227, -19, -155, -475, -3296, -252, -116; -90, -312, -26, -204, -624, -4329, -331, -153; -261, -884, -74, -604, -1851, -12844, -982, -452],42,9]
%
% END OF LATTICE L43.9
%
\hbox{}\par\smallskip%
%
% LATTICE L43.10
%
\leavevmode%
\llap{\spacer}%
${L_{43.10}}=\pdualwitharg{3}({L_{43.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{234}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2290860\\2340\\-10296
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2340\\-6\\-9
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10296\\-9\\46
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-2029\\-81120\\-468468
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38\\1521\\8775
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3\\122\\694
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4\\165\\927
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2290860, 2340, -10296; 2340, -6, -9; -10296, -9, 46],[468, 3, 1],[5, 38, 3, 4, 3, 1, -1, -1; 196, 1521, 122, 165, 124, 39, -42, -45; 1152, 8775, 694, 927, 696, 234, -232, -234],42,10]
%
% END OF LATTICE L43.10
%
\hbox{}\par\smallskip%
%
% LATTICE L43.11
%
\leavevmode%
\llap{\spacer}%
${L_{43.11}}=\pdualwitharg{3.13}(\pfillwitharg{2.3}({L_{43.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${13}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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7566\\-24921\\-7995
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\kern3pt%
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-24921\\79131\\25389
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-7995\\25389\\8146
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7566, -24921, -7995; -24921, 79131, 25389; -7995, 25389, 8146],[39, 39, 1],[1, 1, 0, -7, -13, -10, -9, -2; -129, -120, -25, 636, 1193, 918, 823, 162; 403, 375, 78, -1989, -3731, -2871, -2574, -507],42,11]
%
% END OF LATTICE L43.11
%
\hbox{}\par\smallskip%
%
% LATTICE L43.12
%
\leavevmode%
\llap{\spacer}%
${L_{43.12}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{43.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
824616\\-29172\\-212004
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29172\\1032\\7500
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-212004\\7500\\54505
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
389\\-312\\1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\11\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\80\\-401
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-201\\-13\\-780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-1\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\7\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-24\\-17\\-312\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\312\\0\\-312\\-312\\-6396\\-468\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\312\\0\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-312\\0\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-312\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-6396\\-468\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-468\\-48\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-156\\-36\\-60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[824616, -29172, -212004; -29172, 1032, 7500; -212004, 7500, 54505],[156, 12, 1],[-9, -201, -17, -9, -1, 6, 2, -6; 0, -13, -1, 7, 8, 169, 13, 5; -35, -780, -66, -36, -5, 0, 6, -24],42,12]
%
% END OF LATTICE L43.12
%
\hbox{}\par\smallskip%
%
% LATTICE L43.13
%
\leavevmode%
\llap{\spacer}%
${L_{43.13}}=\pdualwitharg{13}(\pfillwitharg{3}({L_{43.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-26\\13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\13\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-96\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-61\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\24\\77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\-156\\-312\\-2652\\-312\\-468\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-26\\-312\\-41\\-78\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\0\\52\\0\\-468\\-78\\-208\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-26\\0\\26\\0\\-13\\-78\\-65
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\-156\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\0\\52\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-26\\0\\26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2652\\-312\\-468\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-41\\-78\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-78\\-208\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-13\\-78\\-65
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[156, 0, 0; 0, -26, 13; 0, 13, -6],[156, 13, 1],[1, 0, -1, -2, -17, -2, -3, 0; 0, 1, 2, -7, -96, -13, -26, -5; 0, 2, 0, -26, -312, -41, -78, -13],42,13]
%
% END OF LATTICE L43.13
%
\hbox{}\par\smallskip%
%
% LATTICE L43.14
%
\leavevmode%
\llap{\spacer}%
${L_{43.14}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{43.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${39}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{234}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6786\\6786\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6786\\-6747\\117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\117\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
179\\240\\4680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\-257\\-4992
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\4\\77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-12\\-351
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\0\\-39\\-468\\-663\\-468\\-117\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-234\\-468\\-369\\-117\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\13\\0\\-117\\-117\\-52\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-234\\0\\234\\0\\-117\\-117\\-585
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\0\\-39\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\13\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-234\\0\\234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-663\\-468\\-117\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-369\\-117\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-117\\-52\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-117\\-117\\-585
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6786, 6786, -117; 6786, -6747, 117; -117, 117, -2],[117, 39, 1],[1, 1, 0, -7, -13, -10, -3, -2; 1, 0, -1, -12, -17, -12, -3, 0; 0, -63, -65, -351, -312, -171, -13, 117],42,14]
%
% END OF LATTICE L43.14
%
\hbox{}\par\smallskip%
%
% LATTICE L43.15
%
\leavevmode%
\llap{\spacer}%
${L_{43.15}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{43.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${39}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{117}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
117\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1950\\117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\117\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-6\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-35\\-663
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\0\\-117\\-156\\-663\\-156\\-351\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-26\\-156\\-41\\-117\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\0\\117\\0\\-351\\-117\\-468\\-117
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-26\\0\\26\\0\\-13\\-117\\-65
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\0\\-117\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\0\\117\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-26\\0\\26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-663\\-156\\-351\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-41\\-117\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[117, 0, 0; 0, -1950, 117; 0, 117, -7],[117, 39, 1],[-1, 0, 1, 0, -3, -1, -4, -1; 5, 1, 0, -3, -13, -3, -6, 1; 78, 16, 0, -52, -234, -55, -117, 13],42,15]
%
% END OF LATTICE L43.15
%
\hbox{}\par\smallskip%
%
% LATTICE L43.16
%
\leavevmode%
\llap{\spacer}%
${L_{43.16}}=\pdualwitharg{2}({L_{43.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
113841000\\-248508\\28211508
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-248508\\552\\-61584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28211508\\-61584\\6991235
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-7930417\\-48867\\32000904
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15904\\97\\-64176
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1965280\\-12110\\7930319
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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2130\\13\\-8595
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14201\\91\\-57304
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3011\\21\\-12150
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341\\3\\-1376
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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3\\0\\-18\\-24\\-51\\-312\\-54\\0
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{$\left[\!\llap{\phantom{%
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-51\\-312\\-54\\0
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-312\\-2132\\-468\\-52
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-468\\-144\\-36
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[113841000, -248508, 28211508; -248508, 552, -61584; 28211508, -61584, 6991235],[468, 12, 1],[2130, 14201, 3011, 341, 142, 567, 397, 795; 13, 91, 21, 3, 1, 0, 0, 4; -8595, -57304, -12150, -1376, -573, -2288, -1602, -3208],42,16]
%
% END OF LATTICE L43.16
%
\hbox{}\par\smallskip%
%
% LATTICE L43.17
%
\leavevmode%
\llap{\spacer}%
${L_{43.17}}=\pdualwitharg{2.3}({L_{43.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{936}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
723987576\\-3414996\\179422776
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3414996\\16104\\-846324
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
179422776\\-846324\\44465587
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-13084735\\61971\\52799292
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59120\\-281\\-238560
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-3242732\\15358\\13085015
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-64253\\273\\259272
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-1535\\7\\6194
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-1249\\9\\5040
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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3\\0\\-6\\-72\\-51\\-936\\-18\\0
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-51\\-936\\-18\\0
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-936\\-19188\\-468\\-468
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[723987576, -3414996, 179422776; -3414996, 16104, -846324; 179422776, -846324, 44465587],[468, 12, 1],[-3156, -64253, -1535, -1249, 200, 6611, 57, -3185; 13, 273, 7, 9, 1, 0, 0, 12; 12735, 259272, 6194, 5040, -807, -26676, -230, 12852],42,17]
%
% END OF LATTICE L43.17
%
\hbox{}\par\smallskip%
%
% LATTICE L43.18
%
\leavevmode%
\llap{\spacer}%
${L_{43.18}}=\pdualwitharg{3.13}(\pfillwitharg{3}({L_{43.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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468\\1248\\-468
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-468\\-273\\106
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1\\29\\78
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[468, 1248, -468; 1248, 702, -273; -468, -273, 106],[156, 39, 1],[-3, -1, -1, 1, 3, 1, 1, -1; -136, -47, -58, 29, 96, 32, 28, -44; -364, -126, -156, 78, 260, 87, 78, -117],42,18]
%
% END OF LATTICE L43.18
%
\hbox{}\par\smallskip%
%
% LATTICE L43.19
%
\leavevmode%
\llap{\spacer}%
${L_{43.19}}=\pdualwitharg{2.13}(\pfillwitharg{3}({L_{43.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[27768, 2964, 4836; 2964, 312, 520; 4836, 520, 839],[156, 52, 1],[-10, -1, 7, 15, 14, 3, -5, -13; 30, 3, -23, -57, -72, -24, 2, 36; 39, 4, -26, -52, -39, -4, 26, 52],42,19]
%
% END OF LATTICE L43.19
%
\hbox{}\par\smallskip%
%
% LATTICE L43.20
%
\leavevmode%
\llap{\spacer}%
${L_{43.20}}=\pdualwitharg{3.13}({L_{43.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{b}{2}{234}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-107172\\26676\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26676\\-6630\\117
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\kern3pt%
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-468\\117\\-2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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-3\\-12\\-26
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-6\\117
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-107172, 26676, -468; 26676, -6630, 117; -468, 117, -2],[468, 39, 1],[-9, -4, -3, -2, 1, 1, 1, -1; -44, -18, -12, -6, 4, 3, 2, -9; -624, -171, -26, 117, 0, -63, -130, -351],42,20]
%
% END OF LATTICE L43.20
%
\hbox{}\par\smallskip%
%
% LATTICE L43.21
%
\leavevmode%
\llap{\spacer}%
${L_{43.21}}=\pdualwitharg{13}({L_{43.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{468}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3872700\\58032\\-18252
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58032\\858\\-273
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18252\\-273\\86
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-205\\-1428\\-47736
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{$\left[\!\llap{\phantom{%
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0\\-2\\-7
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5\\36\\1170
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2\\18\\481
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{$\left.\llap{\phantom{%
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-2652\\-312\\-1404\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3872700, 58032, -18252; 58032, 858, -273; -18252, -273, 86],[468, 39, 1],[-3, 0, 5, 2, 11, 1, 1, -1; -44, -2, 36, 18, 100, 9, 6, -11; -780, -7, 1170, 481, 2652, 241, 234, -247],42,21]
%
% END OF LATTICE L43.21
%
\hbox{}\par\smallskip%
%
% LATTICE L43.22
%
\leavevmode%
\llap{\spacer}%
${L_{43.22}}=\pdualwitharg{2.3.13}(\pfillwitharg{3}({L_{43.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${13}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
169416\\-438204\\112476
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-438204\\1133496\\-290940
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112476\\-290940\\74677
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-795\\-3120
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[169416, -438204, 112476; -438204, 1133496, -290940; 112476, -290940, 74677],[156, 156, 1],[1, 4, 8, 8, 1, -1, -5, -5; -63, -97, -97, -77, -43, -62, -62, -82; -247, -384, -390, -312, -169, -240, -234, -312],42,22]
%
% END OF LATTICE L43.22
%
\hbox{}\par\smallskip%
%
% LATTICE L43.23
%
\leavevmode%
\llap{\spacer}%
${L_{43.23}}=\pdualwitharg{2.13}({L_{43.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${39}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{468}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
142241112\\2004444\\-33761052
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2004444\\27768\\-476112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33761052\\-476112\\8012951
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-1374364\\4120245\\-5545800
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\kern3pt%
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-13531\\40564\\-54600
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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3953\\-11852\\15951
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2844\\-8527\\11476
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11888\\-35643\\47970
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2616\\-7843\\10556
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-312\\-164\\-468\\-52
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[142241112, 2004444, -33761052; 2004444, 27768, -476112; -33761052, -476112, 8012951],[468, 156, 1],[3953, 2844, 11888, 2616, -87, -911, -6089, -683; -11852, -8527, -35643, -7843, 262, 2732, 18258, 2048; 15951, 11476, 47970, 10556, -351, -3676, -24570, -2756],42,23]
%
% END OF LATTICE L43.23
%
\hbox{}\par\smallskip%
%
% LATTICE L43.24
%
\leavevmode%
\llap{\spacer}%
${L_{43.24}}=\pdualwitharg{2.3.13}({L_{43.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${39}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{*}{2}{936}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3286296\\-55692\\-856440
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55692\\1560\\14976
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-856440\\14976\\223543
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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614066\\-1858317\\2477124
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5211\\-15770\\21021
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7184\\-21741\\28980
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1708\\-5169\\6890
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-116\\351\\-468
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{$\left.\llap{\phantom{%
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-663\\-936\\-234\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3286296, -55692, -856440; -55692, 1560, 14976; -856440, 14976, 223543],[468, 156, 1],[5211, 7184, 1708, -116, -29, 589, 883, 7889; -15770, -21741, -5169, 351, 88, -1782, -2672, -23874; 21021, 28980, 6890, -468, -117, 2376, 3562, 31824],42,24]
%
% END OF LATTICE L43.24
%

\medskip%
%------------------------------WEYL-GROUP-W44--------------------------------
%
\leavevmode\llap{}%
$W_{44\phantom{0}}$%
\qquad\llap{32} lattices, $\chi=16$%
\hfill%
$\infty2622$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L44.1
%
\leavevmode%
\llap{\spacer}%
${L_{44.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{44},3,2}%
\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1218168\\-337176\\-519624
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\kern3pt%
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-337176\\-91686\\-159513
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-519624\\-159513\\-71678
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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42396\\-131908\\-13797
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-28393\\88340\\9240
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13671\\-42535\\-4449
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
131419\\-408888\\-42768
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\right.$}%
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42\\-84\\-63\\-21\\0
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-84\\168\\0\\-84\\-504
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-63\\0\\18\\-9\\-144
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-21\\-84\\-9\\6\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-504\\-144\\0\\72
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1218168, -337176, -519624; -337176, -91686, -159513; -519624, -159513, -71678],[504, 3, 1],[42396, -28393, -12611, 13671, 131419; -131908, 88340, 39237, -42535, -408888; -13797, 9240, 4104, -4449, -42768],43,1]
%
% END OF LATTICE L44.1
%
\hbox{}\par\smallskip%
%
% LATTICE L44.2
%
\leavevmode%
\llap{\spacer}%
${L_{44.2}}=\pfillwitharg{2.3}({L_{44.1}})$%
{${}={}$}Nikulin $44$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{2,1}{\infty a}{42}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\-42\\-21\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\42\\0\\-42\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\2\\-3\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-42\\-3\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-8\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[42, 0, 0; 0, 0, -1; 0, -1, 0],[42, 1, 1],[-1, 1, 0, -1, -1; 0, 0, -1, -3, -2; -21, 0, 1, -6, -10],43,2]
%
% END OF LATTICE L44.2
%
\hbox{}\par\smallskip%
%
% LATTICE L44.3
%
\leavevmode%
\llap{\spacer}%
${L_{44.3}}=\pfillwitharg{3}({L_{44.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{4,1}{\infty a}{168}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-446287800\\-7827456\\229488
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7827456\\-137286\\4025
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
229488\\4025\\-118
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-903\\315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115\\-6468\\3024
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-393\\208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\282\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\168\\-104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\-84\\-21\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\168\\0\\-84\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\2\\-3\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-84\\-3\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-16\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-446287800, -7827456, 229488; -7827456, -137286, 4025; 229488, 4025, -118],[168, 1, 1],[16, 115, 7, -5, -3; -903, -6468, -393, 282, 168; 315, 3024, 208, -105, -104],43,3]
%
% END OF LATTICE L44.3
%
\hbox{}\par\smallskip%
%
% LATTICE L44.4
%
\leavevmode%
\llap{\spacer}%
${L_{44.4}}=\pfillwitharg{2}({L_{44.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{6,1}{\infty a}{42}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8661366\\-575946\\-188622
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-575946\\-38298\\-12543
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-188622\\-12543\\-4106
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-172\\2401\\567
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\-742\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\-726\\-171
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\727\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-259\\3618\\846
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\-42\\-63\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\42\\0\\-42\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\0\\18\\-9\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-42\\-9\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-126\\-72\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-8661366, -575946, -188622; -575946, -38298, -12543; -188622, -12543, -4106],[126, 3, 1],[-172, 53, 52, -52, -259; 2401, -742, -726, 727, 3618; 567, -168, -171, 168, 846],43,4]
%
% END OF LATTICE L44.4
%
\hbox{}\par\smallskip%
%
% LATTICE L44.5
%
\leavevmode%
\llap{\spacer}%
${L_{44.5}}=\pdualwitharg{2}(\pfillwitharg{2.3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{4,3}{\infty z}{21}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{}{6}{12}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26001780\\-1618302\\-12960738
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1618302\\-100720\\-806652
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12960738\\-806652\\-6460355
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-509\\-231\\1050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1667\\-630\\3423
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-508\\-183\\1042
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
163\\72\\-336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61\\21\\-125
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-42\\-42\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\21\\0\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\4\\-6\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-6\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-8\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26001780, -1618302, -12960738; -1618302, -100720, -806652; -12960738, -806652, -6460355],[42, 2, 1],[-509, -1667, -508, 163, 61; -231, -630, -183, 72, 21; 1050, 3423, 1042, -336, -125],43,5]
%
% END OF LATTICE L44.5
%
\hbox{}\par\smallskip%
%
% LATTICE L44.6
%
\leavevmode%
\llap{\spacer}%
${L_{44.6}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{2,1}{\infty a}{14}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37334514\\-2374974\\12484122
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2374974\\-151080\\794157
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12484122\\794157\\-4174510
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143\\-77\\413
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
963\\-420\\2800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
443\\-183\\1290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\24\\-133
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-107\\42\\-312
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-14\\-21\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\14\\0\\-14\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\6\\-3\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-14\\-3\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-24\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-37334514, -2374974, 12484122; -2374974, -151080, 794157; 12484122, 794157, -4174510],[42, 3, 1],[143, 963, 443, -46, -107; -77, -420, -183, 24, 42; 413, 2800, 1290, -133, -312],43,6]
%
% END OF LATTICE L44.6
%
\hbox{}\par\smallskip%
%
% LATTICE L44.7
%
\leavevmode%
\llap{\spacer}%
${L_{44.7}}=\pdualwitharg{2.3}(\pfillwitharg{2.3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{4,3}{\infty z}{7}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{}{6}{4}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2130240\\147042\\902580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
147042\\10392\\62286
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
902580\\62286\\382423
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2853\\-427\\-6664
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7537\\-1127\\-17605
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7385\\-1104\\-17250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\1\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\-21\\-327
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-14\\-42\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\7\\0\\-14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\12\\-6\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-14\\-6\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-24\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2130240, 147042, 902580; 147042, 10392, 62286; 902580, 62286, 382423],[42, 6, 1],[2853, 7537, 7385, -6, 140; -427, -1127, -1104, 1, -21; -6664, -17605, -17250, 14, -327],43,7]
%
% END OF LATTICE L44.7
%
\hbox{}\par\smallskip%
%
% LATTICE L44.8
%
\leavevmode%
\llap{\spacer}%
${L_{44.8}}=\pdualwitharg{7}(\pfillwitharg{2.3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{2,1}{\infty a}{6}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28714602\\-157878\\-3647574
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-157878\\-868\\-20055
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3647574\\-20055\\-463346
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\27\\-135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91\\222\\-726
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
92\\248\\-735
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\-63\\294
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-62\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-21\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-42\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\14\\-21\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-42\\-21\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-56\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-28714602, -157878, -3647574; -157878, -868, -20055; -3647574, -20055, -463346],[42, 7, 1],[17, 91, 92, -37, -21; 27, 222, 248, -63, -62; -135, -726, -735, 294, 168],43,8]
%
% END OF LATTICE L44.8
%
\hbox{}\par\smallskip%
%
% LATTICE L44.9
%
\leavevmode%
\llap{\spacer}%
${L_{44.9}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{6,5}{\infty a}{42}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-476406\\3906\\7686
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3906\\-30\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7686\\-63\\-124
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-7\\189
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-14\\1554
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\248
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\-42\\-21\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\42\\0\\-42\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\2\\-3\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-42\\-3\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-8\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-476406, 3906, 7686; 3906, -30, -63; 7686, -63, -124],[126, 3, 1],[3, 25, 4, -1, -1; -7, -14, -1, 2, 0; 189, 1554, 248, -63, -62],43,9]
%
% END OF LATTICE L44.9
%
\hbox{}\par\smallskip%
%
% LATTICE L44.10
%
\leavevmode%
\llap{\spacer}%
${L_{44.10}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{4,1}{\infty a}{56}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216216\\44688\\-9576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44688\\-6762\\933
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9576\\933\\-106
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\-3304\\-18151
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\2212\\12152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\983\\5400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-1065\\-5851
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-375\\-10240\\-56256
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-28\\-21\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\56\\0\\-28\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\6\\-3\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-28\\-3\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-48\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[216216, 44688, -9576; 44688, -6762, 933; -9576, 933, -106],[168, 3, 1],[-121, 81, 36, -39, -375; -3304, 2212, 983, -1065, -10240; -18151, 12152, 5400, -5851, -56256],43,10]
%
% END OF LATTICE L44.10
%
\hbox{}\par\smallskip%
%
% LATTICE L44.11
%
\leavevmode%
\llap{\spacer}%
${L_{44.11}}=\pdualwitharg{2.7}(\pfillwitharg{2.3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{4,3}{\infty z}{3}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{}{6}{84}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5578104\\431298\\2682876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
431298\\35728\\207438
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2682876\\207438\\1290371
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6967\\9\\14484
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18406\\24\\38265
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42082\\55\\87486
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\0\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-798\\1\\1659
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-42\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\28\\-42\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-42\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-56\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5578104, 431298, 2682876; 431298, 35728, 207438; 2682876, 207438, 1290371],[42, 14, 1],[-6967, -18406, -42082, 101, -798; 9, 24, 55, 0, 1; 14484, 38265, 87486, -210, 1659],43,11]
%
% END OF LATTICE L44.11
%
\hbox{}\par\smallskip%
%
% LATTICE L44.12
%
\leavevmode%
\llap{\spacer}%
${L_{44.12}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{12,7}{\infty z}{21}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{}{6}{12}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43725780\\-5530266\\21647808
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5530266\\699528\\-2737932
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21647808\\-2737932\\10717421
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20577\\-889\\-41790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3071\\-133\\-6237
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62\\3\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5247\\-226\\-10656
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13627\\-588\\-27675
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-42\\-126\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\21\\0\\-42\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\0\\36\\-18\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-18\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-63\\-72\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[43725780, -5530266, 21647808; -5530266, 699528, -2737932; 21647808, -2737932, 10717421],[126, 6, 1],[20577, 3071, -62, 5247, 13627; -889, -133, 3, -226, -588; -41790, -6237, 126, -10656, -27675],43,12]
%
% END OF LATTICE L44.12
%
\hbox{}\par\smallskip%
%
% LATTICE L44.13
%
\leavevmode%
\llap{\spacer}%
${L_{44.13}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{12,11}{\infty z}{21}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{}{6}{12}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18976860\\7686\\9332820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7686\\12\\3780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9332820\\3780\\4589881
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1859\\7\\-3780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4947\\28\\-10059
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1619\\10\\-3292
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\0\\-61
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-42\\-42\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\21\\0\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\4\\-6\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-6\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-8\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18976860, 7686, 9332820; 7686, 12, 3780; 9332820, 3780, 4589881],[126, 6, 1],[1859, 4947, 1619, 0, 30; 7, 28, 10, 1, 0; -3780, -10059, -3292, 0, -61],43,13]
%
% END OF LATTICE L44.13
%
\hbox{}\par\smallskip%
%
% LATTICE L44.14
%
\leavevmode%
\llap{\spacer}%
${L_{44.14}}=\pdualwitharg{3.7}(\pfillwitharg{2.3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty a}{2}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{}{6}{14}\above{1pt}{1pt}{l}{2}{42}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2818242\\-367332\\1245762
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-367332\\53592\\-166383
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1245762\\-166383\\553484
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1864\\3731\\5317
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9849\\19714\\28094
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33777\\67609\\96348
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-54\\-77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1281\\2564\\3654
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-21\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-14\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\42\\-21\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-14\\-21\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-168\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2818242, -367332, 1245762; -367332, 53592, -166383; 1245762, -166383, 553484],[42, 21, 1],[-1864, -9849, -33777, 27, -1281; 3731, 19714, 67609, -54, 2564; 5317, 28094, 96348, -77, 3654],43,14]
%
% END OF LATTICE L44.14
%
\hbox{}\par\smallskip%
%
% LATTICE L44.15
%
\leavevmode%
\llap{\spacer}%
${L_{44.15}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{4,1}{\infty a}{24}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45816456\\5079144\\-768264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5079144\\-563066\\85169
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-768264\\85169\\-12882
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\-297\\243
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-349\\-2796\\2328
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-167\\-1337\\1120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
86\\690\\-567
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\664\\-560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-21\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-84\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\14\\-21\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-84\\-21\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-112\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-45816456, 5079144, -768264; 5079144, -563066, 85169; -768264, 85169, -12882],[168, 7, 1],[-37, -349, -167, 86, 83; -297, -2796, -1337, 690, 664; 243, 2328, 1120, -567, -560],43,15]
%
% END OF LATTICE L44.15
%
\hbox{}\par\smallskip%
%
% LATTICE L44.16
%
\leavevmode%
\llap{\spacer}%
${L_{44.16}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{8,7}{\infty z}{84}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{}{6}{48}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109990608\\18359544\\-332808
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18359544\\-3064560\\55552
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-332808\\55552\\-1007
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\105\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
139\\966\\7350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\342\\2672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-33\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-21\\-167
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-168\\-168\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\84\\0\\-168\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\16\\-24\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-24\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-16\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-109990608, 18359544, -332808; 18359544, -3064560, 55552; -332808, 55552, -1007],[168, 8, 1],[16, 139, 49, -5, -3; 105, 966, 342, -33, -21; 504, 7350, 2672, -168, -167],43,16]
%
% END OF LATTICE L44.16
%
\hbox{}\par\smallskip%
%
% LATTICE L44.17
%
\leavevmode%
\llap{\spacer}%
${L_{44.17}}=\pdualwitharg{3}({L_{44.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{12,5}{\infty b}{168}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22837752\\27720\\51912
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27720\\-30\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51912\\-63\\-118
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-7\\1323
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-28\\12768
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-441
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\-84\\-21\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\168\\0\\-84\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\2\\-3\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-84\\-3\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-16\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-22837752, 27720, 51912; 27720, -30, -63; 51912, -63, -118],[504, 3, 1],[3, 29, 2, -1, -1; -7, -28, -1, 2, 0; 1323, 12768, 880, -441, -440],43,17]
%
% END OF LATTICE L44.17
%
\hbox{}\par\smallskip%
%
% LATTICE L44.18
%
\leavevmode%
\llap{\spacer}%
${L_{44.18}}=\pdualwitharg{2.3.7}(\pfillwitharg{2.3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{}{6}{28}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7928894484\\19221780018\\9341858778
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19221780018\\-46598782188\\-22647186786
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9341858778\\-22647186786\\-11006619599
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\1973\\-4052
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\5042\\-10343
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
248\\30149\\-61824
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-4268\\8764
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-3391\\6951
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-42\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\84\\-42\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-14\\-42\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-168\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7928894484, 19221780018, 9341858778; 19221780018, -46598782188, -22647186786; 9341858778, -22647186786, -11006619599],[42, 42, 1],[9, 37, 248, -21, -31; 1973, 5042, 30149, -4268, -3391; -4052, -10343, -61824, 8764, 6951],43,18]
%
% END OF LATTICE L44.18
%
\hbox{}\par\smallskip%
%
% LATTICE L44.19
%
\leavevmode%
\llap{\spacer}%
${L_{44.19}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{6,5}{\infty a}{6}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65127258\\65248092\\-1867572
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65248092\\-65369136\\1871037
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1867572\\1871037\\-53554
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-1\\-837
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
195\\-2\\-6870
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
219\\-1\\-7672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\2\\1953
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\0\\1918
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-21\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-42\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\14\\-21\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-42\\-21\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-56\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-65127258, 65248092, -1867572; 65248092, -65369136, 1871037; -1867572, 1871037, -53554],[126, 21, 1],[23, 195, 219, -54, -55; -1, -2, -1, 2, 0; -837, -6870, -7672, 1953, 1918],43,19]
%
% END OF LATTICE L44.19
%
\hbox{}\par\smallskip%
%
% LATTICE L44.20
%
\leavevmode%
\llap{\spacer}%
${L_{44.20}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{6,1}{\infty a}{6}\above{1pt}{1pt}{r}{2}{126}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{l}{2}{126}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80394426\\26747658\\1773198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26747658\\-8899086\\-589953
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1773198\\-589953\\-39110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\139\\1077
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\-106\\-804
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-102\\-819
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
370\\739\\5628
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1213\\2418\\18522
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-63\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-42\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\0\\126\\-63\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-42\\-63\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-126\\-504\\0\\126
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-80394426, 26747658, 1773198; 26747658, -8899086, -589953; 1773198, -589953, -39110],[126, 21, 1],[70, -53, -52, 370, 1213; 139, -106, -102, 739, 2418; 1077, -804, -819, 5628, 18522],43,20]
%
% END OF LATTICE L44.20
%
\hbox{}\par\smallskip%
%
% LATTICE L44.21
%
\leavevmode%
\llap{\spacer}%
${L_{44.21}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,1}{\infty a}{8}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{}{6}{14}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6552\\-22008\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22008\\-47334\\1995
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\1995\\-82
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-17\\-547
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\4\\128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\47\\1512
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-12\\-385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\-256\\-8232
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-21\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-28\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\42\\-21\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-28\\-21\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-336\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6552, -22008, 840; -22008, -47334, 1995; 840, 1995, -82],[168, 21, 1],[-13, 3, 36, -9, -195; -17, 4, 47, -12, -256; -547, 128, 1512, -385, -8232],43,21]
%
% END OF LATTICE L44.21
%
\hbox{}\par\smallskip%
%
% LATTICE L44.22
%
\leavevmode%
\llap{\spacer}%
${L_{44.22}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{8,7}{\infty z}{28}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{}{6}{16}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-370660080\\-27901944\\-183022224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27901944\\-2100336\\-13777248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-183022224\\-13777248\\-90371573
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12969\\-7\\-26264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2855\\0\\5782
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3105\\2\\6288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5716\\-1\\-11576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5841\\-2\\-11829
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\-56\\-168\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\28\\0\\-56\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\48\\-24\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-56\\-24\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-48\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-370660080, -27901944, -183022224; -27901944, -2100336, -13777248; -183022224, -13777248, -90371573],[168, 24, 1],[12969, -2855, -3105, 5716, 5841; -7, 0, 2, -1, -2; -26264, 5782, 6288, -11576, -11829],43,22]
%
% END OF LATTICE L44.22
%
\hbox{}\par\smallskip%
%
% LATTICE L44.23
%
\leavevmode%
\llap{\spacer}%
${L_{44.23}}=\pdualwitharg{2.3.7}(\pfillwitharg{2}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,11}{\infty z}{3}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{}{6}{84}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3096389772\\-259686\\1524941586
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-259686\\84\\-127890
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1524941586\\-127890\\751018771
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8989\\823\\-18252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23916\\2191\\-48561
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54787\\5020\\-111244
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1017\\93\\-2065
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-42\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\28\\-42\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-42\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-56\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3096389772, -259686, 1524941586; -259686, 84, -127890; 1524941586, -127890, 751018771],[126, 42, 1],[8989, 23916, 54787, 0, 1017; 823, 2191, 5020, 1, 93; -18252, -48561, -111244, 0, -2065],43,23]
%
% END OF LATTICE L44.23
%
\hbox{}\par\smallskip%
%
% LATTICE L44.24
%
\leavevmode%
\llap{\spacer}%
${L_{44.24}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,7}{\infty z}{3}\above{1pt}{1pt}{r}{2}{252}\above{1pt}{1pt}{}{6}{84}\above{1pt}{1pt}{l}{2}{63}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
673239420\\-33981066\\330466752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33981066\\1715448\\-16679964
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
330466752\\-16679964\\162213131
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22979\\-775\\46734
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-13\\783
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62\\3\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65795\\-2218\\133812
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-134347\\-4530\\273231
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-126\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-42\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\0\\252\\-126\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-126\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-63\\-504\\0\\63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[673239420, -33981066, 330466752; -33981066, 1715448, -16679964; 330466752, -16679964, 162213131],[126, 42, 1],[-22979, -385, 62, -65795, -134347; -775, -13, 3, -2218, -4530; 46734, 783, -126, 133812, 273231],43,24]
%
% END OF LATTICE L44.24
%
\hbox{}\par\smallskip%
%
% LATTICE L44.25
%
\leavevmode%
\llap{\spacer}%
${L_{44.25}}=\pdualwitharg{2.7}(\pfillwitharg{3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{8,7}{\infty z}{12}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{}{6}{336}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\83496\\-24360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83496\\-3948112\\1151864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24360\\1151864\\-336057
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\63\\216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\912\\3126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2320\\7952
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-147\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-145\\-497
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-168\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-168\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\112\\-168\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-168\\336\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-112\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1680, 83496, -24360; 83496, -3948112, 1151864; -24360, 1151864, -336057],[168, 56, 1],[-1, -1, -1, 2, 0; 63, 912, 2320, -147, -145; 216, 3126, 7952, -504, -497],43,25]
%
% END OF LATTICE L44.25
%
\hbox{}\par\smallskip%
%
% LATTICE L44.26
%
\leavevmode%
\llap{\spacer}%
${L_{44.26}}=\pdualwitharg{3.7}({L_{44.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,5}{\infty b}{24}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5277384\\35280\\-1498392
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35280\\-210\\10017
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1498392\\10017\\-425434
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\-1\\243
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-661\\-4\\2328
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-318\\-1\\1120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
161\\2\\-567
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
159\\0\\-560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-21\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-84\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\14\\-21\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-84\\-21\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-112\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5277384, 35280, -1498392; 35280, -210, 10017; -1498392, 10017, -425434],[504, 21, 1],[-69, -661, -318, 161, 159; -1, -4, -1, 2, 0; 243, 2328, 1120, -567, -560],43,26]
%
% END OF LATTICE L44.26
%
\hbox{}\par\smallskip%
%
% LATTICE L44.27
%
\leavevmode%
\llap{\spacer}%
${L_{44.27}}=\pdualwitharg{7}({L_{44.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,1}{\infty a}{24}\above{1pt}{1pt}{b}{2}{126}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{l}{2}{504}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59976\\-252000\\-29232
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252000\\-531930\\-53361
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29232\\-53361\\-5090
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
488\\-967\\7335
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-107\\212\\-1608
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1358\\2691\\-20412
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
320\\-634\\4809
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7243\\-14352\\108864
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-63\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-84\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\0\\126\\-63\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-84\\-63\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-504\\-1008\\0\\504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-59976, -252000, -29232; -252000, -531930, -53361; -29232, -53361, -5090],[504, 21, 1],[488, -107, -1358, 320, 7243; -967, 212, 2691, -634, -14352; 7335, -1608, -20412, 4809, 108864],43,27]
%
% END OF LATTICE L44.27
%
\hbox{}\par\smallskip%
%
% LATTICE L44.28
%
\leavevmode%
\llap{\spacer}%
${L_{44.28}}=\pdualwitharg{2}({L_{44.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{24,7}{\infty z}{84}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{}{6}{48}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6103888560\\-984629016\\-126012096
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-984629016\\-158832240\\-20327232
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126012096\\-20327232\\-2601463
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3848\\26929\\-24024
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
839\\-5873\\5250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
923\\-6459\\5760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1685\\11794\\-10536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1727\\12087\\-10791
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-168\\-504\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\84\\0\\-168\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\0\\144\\-72\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-72\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-126\\-144\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6103888560, -984629016, -126012096; -984629016, -158832240, -20327232; -126012096, -20327232, -2601463],[504, 24, 1],[-3848, 839, 923, -1685, -1727; 26929, -5873, -6459, 11794, 12087; -24024, 5250, 5760, -10536, -10791],43,28]
%
% END OF LATTICE L44.28
%
\hbox{}\par\smallskip%
%
% LATTICE L44.29
%
\leavevmode%
\llap{\spacer}%
${L_{44.29}}=\pdualwitharg{2.3}({L_{44.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{24,23}{\infty z}{84}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{}{6}{48}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-254779056\\253512\\506520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
253512\\-240\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
506520\\-504\\-1007
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-7\\1512
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\-7\\22134
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-1\\8048
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-503
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-168\\-168\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\84\\0\\-168\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\16\\-24\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-24\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-16\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-254779056, 253512, 506520; 253512, -240, -504; 506520, -504, -1007],[504, 24, 1],[3, 44, 16, -1, -1; -7, -7, -1, 2, 0; 1512, 22134, 8048, -504, -503],43,29]
%
% END OF LATTICE L44.29
%
\hbox{}\par\smallskip%
%
% LATTICE L44.30
%
\leavevmode%
\llap{\spacer}%
${L_{44.30}}=\pdualwitharg{2.3.7}(\pfillwitharg{3}({L_{44.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,7}{\infty z}{4}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{}{6}{112}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-453757584\\-590510088\\-219855888
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-590510088\\-768476688\\-286115592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-219855888\\-286115592\\-106525187
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2703\\-1\\-5576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1127\\0\\2326
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1305\\2\\2688
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15820\\-1\\-32648
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12528\\-2\\-25851
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-168\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-56\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\336\\-168\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-56\\-168\\112\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-336\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-453757584, -590510088, -219855888; -590510088, -768476688, -286115592; -219855888, -286115592, -106525187],[168, 168, 1],[2703, -1127, -1305, 15820, 12528; -1, 0, 2, -1, -2; -5576, 2326, 2688, -32648, -25851],43,30]
%
% END OF LATTICE L44.30
%
\hbox{}\par\smallskip%
%
% LATTICE L44.31
%
\leavevmode%
\llap{\spacer}%
${L_{44.31}}=\pdualwitharg{2.7}({L_{44.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,7}{\infty z}{12}\above{1pt}{1pt}{*}{2}{1008}\above{1pt}{1pt}{}{6}{336}\above{1pt}{1pt}{l}{2}{63}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2305430064\\-236102328\\48489840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-236102328\\-24179568\\4965912
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48489840\\4965912\\-1019881
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-493\\-2448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\191\\930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\291\\1512
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2716\\-13272
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-2193\\-10773
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-504\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-168\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\0\\1008\\-504\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-504\\336\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-126\\-1008\\0\\63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2305430064, -236102328, 48489840; -236102328, -24179568, 4965912; 48489840, 4965912, -1019881],[504, 168, 1],[-1, 0, 2, -1, -2; -493, 191, 291, -2716, -2193; -2448, 930, 1512, -13272, -10773],43,31]
%
% END OF LATTICE L44.31
%
\hbox{}\par\smallskip%
%
% LATTICE L44.32
%
\leavevmode%
\llap{\spacer}%
${L_{44.32}}=\pdualwitharg{2.3.7}({L_{44.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,23}{\infty z}{12}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{}{6}{336}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35533008\\35282520\\-15299928
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35282520\\-35033712\\15192072
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15299928\\15192072\\-6587897
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-94\\-1\\216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1347\\-1\\3126
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3425\\-1\\7952
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
219\\2\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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214\\0\\-497
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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-24\\12\\0\\-168\\-42
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35533008, 35282520, -15299928; 35282520, -35033712, 15192072; -15299928, 15192072, -6587897],[504, 168, 1],[-94, -1347, -3425, 219, 214; -1, -1, -1, 2, 0; 216, 3126, 7952, -504, -497],43,32]
%
% END OF LATTICE L44.32
%

\medskip%
%------------------------------WEYL-GROUP-W45--------------------------------
%
\leavevmode\llap{}%
$W_{45\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=12$%
\hfill%
$222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L45.1
%
\leavevmode%
\llap{\spacer}%
${L_{45.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{23\rightarrow N_{45},3,2}%
\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{126}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20664\\3024\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3024\\-438\\-3
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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7\\48\\72
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1\\7\\14
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-4\\-27\\-36
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-13\\-88\\-128
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\HardButStrongLineBreak\kern3pt%
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-31\\-210\\-315
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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72\\0\\-72\\-216\\-504\\0
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0\\14\\0\\-56\\-189\\-7
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-72\\0\\18\\0\\-63\\-9
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0\\-7\\-9\\-8\\0\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-20664, 3024, 0; 3024, -438, -3; 0, -3, 2],[504, 3, 1],[7, 1, -4, -13, -31, 0; 48, 7, -27, -88, -210, 0; 72, 14, -36, -128, -315, -1],44,1]
%
% END OF LATTICE L45.1
%
\hbox{}\par\smallskip%
%
% LATTICE L45.2
%
\leavevmode%
\llap{\spacer}%
${L_{45.2}}=\pfillwitharg{2.3}({L_{45.1}})$%
{${}={}$}Nikulin $45$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
714\\336\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\158\\-21
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-21\\-10
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{$\left.\llap{\phantom{%
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0\\0\\0
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4703\\-9912\\1344
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2296\\-4839\\656
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476\\-1003\\135
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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30\\-63\\7
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[714, 336, -42; 336, 158, -21; -42, -21, -10],[42, 1, 1],[-1, 30, -8, -111, -226, -20; 2, -63, 17, 234, 476, 42; 0, 7, -3, -32, -63, -5],44,2]
%
% END OF LATTICE L45.2
%
\hbox{}\par\smallskip%
%
% LATTICE L45.3
%
\leavevmode%
\llap{\spacer}%
${L_{45.3}}=\pfillwitharg{3}({L_{45.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\336\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\134\\-71
\end{tabular}\endgroup%
\kern3pt%
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-168\\-71\\-2
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-1\\0\\168
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\kern3pt%
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0\\0\\1
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[840, 336, -168; 336, 134, -71; -168, -71, -2],[168, 1, 1],[-27, -3, 8, 27, 3, -8; 64, 7, -19, -64, -7, 19; -8, 0, 2, 0, -7, -3],44,3]
%
% END OF LATTICE L45.3
%
\hbox{}\par\smallskip%
%
% LATTICE L45.4
%
\leavevmode%
\llap{\spacer}%
${L_{45.4}}=\pfillwitharg{2}({L_{45.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{126}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-6\\-119\\56
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15\\298\\-112
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[516474, -25452, 1386; -25452, 1254, -69; 1386, -69, 2],[126, 3, 1],[-13, -6, 8, 15, 74, 0; -258, -119, 159, 298, 1470, 0; 108, 56, -54, -112, -567, -1],44,4]
%
% END OF LATTICE L45.4
%
\hbox{}\par\smallskip%
%
% LATTICE L45.5
%
\leavevmode%
\llap{\spacer}%
${L_{45.5}}=\pdualwitharg{2}(\pfillwitharg{2.3}({L_{45.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1344, -126, -630; -126, 16, -62; -630, -62, -295],[42, 2, 1],[12, -20, -22, -10, 27, 23; -2, 7, 5, 2, -7, -5; -25, 42, 46, 21, -56, -48],44,5]
%
% END OF LATTICE L45.5
%
\hbox{}\par\smallskip%
%
% LATTICE L45.6
%
\leavevmode%
\llap{\spacer}%
${L_{45.6}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{45.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-546, -168, 210; -168, 24, 57; 210, 57, -80],[42, 3, 1],[-19, 15, 17, 15, -22, -18; -4, 7, 5, 4, -7, -5; -54, 42, 48, 42, -63, -51],44,6]
%
% END OF LATTICE L45.6
%
\hbox{}\par\smallskip%
%
% LATTICE L45.7
%
\leavevmode%
\llap{\spacer}%
${L_{45.7}}=\pdualwitharg{2.3}(\pfillwitharg{2.3}({L_{45.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L45.7
%
\hbox{}\par\smallskip%
%
% LATTICE L45.8
%
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\llap{\spacer}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7014, 42, 1092; 42, -14, 7; 1092, 7, 170],[42, 7, 1],[11, -2, -12, -11, 2, 12; -4, 0, 2, 0, -1, -3; -70, 13, 77, 70, -13, -77],44,8]
%
% END OF LATTICE L45.8
%
\hbox{}\par\smallskip%
%
% LATTICE L45.9
%
\leavevmode%
\llap{\spacer}%
${L_{45.9}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{45.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2856, -1176, 504; -1176, -438, 159; 504, 159, -38],[168, 3, 1],[-37, -5, 10, 29, -2, -11; 152, 21, -41, -120, 7, 45; 144, 21, -39, -120, 0, 42],44,9]
%
% END OF LATTICE L45.9
%
\hbox{}\par\smallskip%
%
% LATTICE L45.10
%
\leavevmode%
\llap{\spacer}%
${L_{45.10}}=\pdualwitharg{2.7}(\pfillwitharg{2.3}({L_{45.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L45.10
%
\hbox{}\par\smallskip%
%
% LATTICE L45.11
%
\leavevmode%
\llap{\spacer}%
${L_{45.11}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{45.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4777416, 449694, 2358342; 449694, 42288, 221988; 2358342, 221988, 1164181],[126, 6, 1],[-527, -3727, -4969, -1769, -9238, 684; -18, -126, -168, -60, -315, 23; 1071, 7574, 10098, 3595, 18774, -1390],44,11]
%
% END OF LATTICE L45.11
%
\hbox{}\par\smallskip%
%
% LATTICE L45.12
%
\leavevmode%
\llap{\spacer}%
${L_{45.12}}=\pdualwitharg{3.7}(\pfillwitharg{2.3}({L_{45.1}}))$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[449778, 4830, 153846; 4830, 42, 1659; 153846, 1659, 52618],[42, 21, 1],[-279, -86, 0, 367, 108, 22; 562, 173, 1, -730, -215, -43; 798, 246, 0, -1050, -309, -63],44,12]
%
% END OF LATTICE L45.12
%
\hbox{}\par\smallskip%
%
% LATTICE L45.13
%
\leavevmode%
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${L_{45.13}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{45.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1848, -168, -168; -168, 14, 7; -168, 7, -38],[168, 7, 1],[-1, -1, 0, 25, 4, 3; -16, -13, 1, 328, 52, 38; 0, 2, 0, -56, -9, -7],44,13]
%
% END OF LATTICE L45.13
%
\hbox{}\par\smallskip%
%
% LATTICE L45.14
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\leavevmode%
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${L_{45.14}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{45.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-97104, -10248, 1512; -10248, -976, 152; 1512, 152, -23],[168, 8, 1],[1, -1, -3, -1, 1, 3; 9, -7, -27, -10, 0, 26; 125, -112, -376, -133, 56, 368],44,14]
%
% END OF LATTICE L45.14
%
\hbox{}\par\smallskip%
%
% LATTICE L45.15
%
\leavevmode%
\llap{\spacer}%
${L_{45.15}}=\pdualwitharg{2.3.7}(\pfillwitharg{2.3}({L_{45.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L45.15
%
\hbox{}\par\smallskip%
%
% LATTICE L45.16
%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-22554, -12600, 882; -12600, -6846, 483; 882, 483, -34],[126, 21, 1],[-1, -2, 0, 19, 2, 4; -6, -9, 1, 90, 9, 15; -112, -180, 14, 1764, 179, 315],44,16]
%
% END OF LATTICE L45.16
%
\hbox{}\par\smallskip%
%
% LATTICE L45.17
%
\leavevmode%
\llap{\spacer}%
${L_{45.17}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{45.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L45.17
%
\hbox{}\par\smallskip%
%
% LATTICE L45.18
%
\leavevmode%
\llap{\spacer}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L45.18
%
\hbox{}\par\smallskip%
%
% LATTICE L45.19
%
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\llap{\spacer}%
${L_{45.19}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{45.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12500712, -214830, 6148926; -214830, 3696, -105672; 6148926, -105672, 3024571],[126, 42, 1],[141, 115, 117, 1517, 285, 619; -10, -6, -4, -84, -17, -45; -287, -234, -238, -3087, -580, -1260],44,19]
%
% END OF LATTICE L45.19
%
\hbox{}\par\smallskip%
%
% LATTICE L45.20
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26544, -13272, 672; -13272, -6608, 336; 672, 336, -17],[168, 56, 1],[2, 3, 3, 0, -1, -1; 0, -1, -3, -1, 0, 2; 77, 96, 56, -21, -40, 0],44,20]
%
% END OF LATTICE L45.20
%
\hbox{}\par\smallskip%
%
% LATTICE L45.21
%
\leavevmode%
\llap{\spacer}%
${L_{45.21}}=\pdualwitharg{7}({L_{45.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{504}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{126}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-471240\\56952\\-4032
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56952\\-6846\\483
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4032\\483\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-21\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\408\\3528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\21\\179
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\39\\315
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-56\\-1512\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-504\\-27\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\14\\0\\-7\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1512\\-504\\0\\504\\0\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-27\\-7\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-63\\-63\\-504\\0\\126
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-471240, 56952, -4032; 56952, -6846, 483; -4032, 483, -34],[504, 21, 1],[-1, -1, 0, 19, 1, 2; -24, -21, 1, 408, 21, 39; -224, -180, 14, 3528, 179, 315],44,21]
%
% END OF LATTICE L45.21
%
\hbox{}\par\smallskip%
%
% LATTICE L45.22
%
\leavevmode%
\llap{\spacer}%
${L_{45.22}}=\pdualwitharg{2}({L_{45.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{1008}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-101808\\2520\\504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2520\\2256\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\240\\25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\49\\-451
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\546\\-5040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-48\\440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-48\\441
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\49\\-448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\243\\-2232
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-8\\-27\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1008\\0\\-504\\-1512\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\16\\0\\-56\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-504\\0\\9\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-1512\\-56\\0\\112\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-504\\-72\\-72\\0\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-101808, 2520, 504; 2520, 2256, 240; 504, 240, 25],[504, 24, 1],[-1, -11, 1, 1, -1, -5; 49, 546, -48, -48, 49, 243; -451, -5040, 440, 441, -448, -2232],44,22]
%
% END OF LATTICE L45.22
%
\hbox{}\par\smallskip%
%
% LATTICE L45.23
%
\leavevmode%
\llap{\spacer}%
${L_{45.23}}=\pdualwitharg{2.3.7}(\pfillwitharg{3}({L_{45.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19824\\-700056\\-12264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-700056\\-24617040\\-431256
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12264\\-431256\\-7555
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-1000\\57120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-685\\-34251\\1956360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-600\\34271
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-71\\4053
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-87\\4968
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-47\\2688
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-168\\-189\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-168\\-216\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\336\\0\\-168\\-504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-189\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-216\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19824, -700056, -12264; -700056, -24617040, -431256; -12264, -431256, -7555],[168, 168, 1],[0, -1, -3, -1, 0, 2; -71, -87, -47, 21, 37, -3; 4053, 4968, 2688, -1197, -2112, 168],44,23]
%
% END OF LATTICE L45.23
%
\hbox{}\par\smallskip%
%
% LATTICE L45.24
%
\leavevmode%
\llap{\spacer}%
${L_{45.24}}=\pdualwitharg{2.7}({L_{45.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${63}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{1008}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\5544\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5544\\-38640\\-2184
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-2184\\247
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
314\\6\\693
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\3\\344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
458\\9\\1008
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-163\\-3\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-1\\-112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\0\\-504\\-189\\-504\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-56\\-216\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\0\\1008\\0\\-504\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-189\\-56\\0\\7\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-216\\-504\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-504\\-56\\0\\112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1008, 5544, -504; 5544, -38640, -2184; -504, -2184, 247],[504, 168, 1],[314, 156, 458, -3, -163, -51; 6, 3, 9, 0, -3, -1; 693, 344, 1008, -7, -360, -112],44,24]
%
% END OF LATTICE L45.24
%

\medskip%
%------------------------------WEYL-GROUP-W46--------------------------------
%
\leavevmode\llap{}%
$W_{46\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=16$%
\hfill%
$222262$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L46.1
%
\leavevmode%
\llap{\spacer}%
${L_{46.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{23\rightarrow N_{46},3,2}%
\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{*}{2}{68}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{306}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85866660\\-89964\\332928
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89964\\-66\\333
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
332928\\333\\-1282
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\2892\\5166
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-10710\\-19142
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-3571\\-6381
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-215\\-36567\\-65331
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-1361\\-2431
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\1190\\2127
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-18\\-306\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\68\\0\\-612\\-68\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\6\\0\\-6\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-306\\-612\\0\\306\\0\\-153
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-68\\-6\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-102\\-21\\-153\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-85866660, -89964, 332928; -89964, -66, 333; 332928, 333, -1282],[612, 3, 1],[17, -63, -21, -215, -8, 7; 2892, -10710, -3571, -36567, -1361, 1190; 5166, -19142, -6381, -65331, -2431, 2127],45,1]
%
% END OF LATTICE L46.1
%
\hbox{}\par\smallskip%
%
% LATTICE L46.2
%
\leavevmode%
\llap{\spacer}%
${L_{46.2}}=\pfillwitharg{2.3}({L_{46.1}})$%
{${}={}$}Nikulin $46$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{17}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{34}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-714\\255\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
255\\-91\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\17\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\-153\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-49\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-72\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-17\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\17\\0\\-102\\-34\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\0\\-6\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-102\\0\\34\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-34\\-6\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-51\\-21\\-51\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-714, 255, 0; 255, -91, 0; 0, 0, 1],[51, 1, 1],[0, 6, -1, -53, -17, -25; 0, 17, -3, -153, -49, -72; 1, 0, -3, -17, -3, 0],45,2]
%
% END OF LATTICE L46.2
%
\hbox{}\par\smallskip%
%
% LATTICE L46.3
%
\leavevmode%
\llap{\spacer}%
${L_{46.3}}=\pfillwitharg{3}({L_{46.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{68}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{34}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109956\\-12444\\816
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12444\\-1282\\95
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
816\\95\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\136\\-6426
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\30\\-1431
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\68\\-3281
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-15\\714
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-34\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\68\\0\\-204\\-68\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-6\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-204\\0\\34\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-68\\-6\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-102\\-21\\-51\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-109956, -12444, 816; -12444, -1282, 95; 816, 95, -6],[204, 1, 1],[1, -63, -14, -32, -1, 7; -2, 136, 30, 68, 2, -15; 104, -6426, -1431, -3281, -105, 714],45,3]
%
% END OF LATTICE L46.3
%
\hbox{}\par\smallskip%
%
% LATTICE L46.4
%
\leavevmode%
\llap{\spacer}%
${L_{46.4}}=\pfillwitharg{2}({L_{46.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{17}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{306}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49266\\-9027\\612
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9027\\-1335\\126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
612\\126\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-15\\342
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\85\\-1972
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\49\\-1131
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-215\\459\\-10557
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\15\\-343
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-17\\393
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-9\\-153\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\17\\0\\-306\\-34\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\6\\0\\-6\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-153\\-306\\0\\306\\0\\-153
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-34\\-6\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-51\\-21\\-153\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-49266, -9027, 612; -9027, -1335, 126; 612, 126, -7],[153, 3, 1],[7, -40, -23, -215, -7, 8; -15, 85, 49, 459, 15, -17; 342, -1972, -1131, -10557, -343, 393],45,4]
%
% END OF LATTICE L46.4
%
\hbox{}\par\smallskip%
%
% LATTICE L46.5
%
\leavevmode%
\llap{\spacer}%
${L_{46.5}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{51}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{102}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4128195\\140964\\1372512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140964\\-4806\\-46869
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1372512\\-46869\\-456322
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6895\\6275\\20094
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
138233\\125817\\402849
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12866\\11711\\37495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61198\\55709\\178347
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-736\\-669\\-2145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
94\\85\\274
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-51\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\51\\0\\-306\\-102\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0\\-6\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-306\\0\\102\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-102\\-6\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-51\\-7\\-51\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4128195, 140964, 1372512; 140964, -4806, -46869; 1372512, -46869, -456322],[51, 3, 1],[6895, 138233, 12866, 61198, -736, 94; 6275, 125817, 11711, 55709, -669, 85; 20094, 402849, 37495, 178347, -2145, 274],45,5]
%
% END OF LATTICE L46.5
%
\hbox{}\par\smallskip%
%
% LATTICE L46.6
%
\leavevmode%
\llap{\spacer}%
${L_{46.6}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{153}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{34}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1224459\\91341\\-9639
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91341\\-6753\\690
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9639\\690\\-62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\258\\539
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
587\\10098\\21114
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\671\\1404
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-17\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-843\\-1764
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\17\\33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-17\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\153\\0\\-306\\-306\\-153
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\0\\-18\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-306\\0\\34\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-306\\-18\\0\\18\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-153\\-21\\-51\\-9\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1224459, 91341, -9639; 91341, -6753, 690; -9639, 690, -62],[153, 3, 1],[15, 587, 39, -1, -49, 1; 258, 10098, 671, -17, -843, 17; 539, 21114, 1404, -34, -1764, 33],45,6]
%
% END OF LATTICE L46.6
%
\hbox{}\par\smallskip%
%
% LATTICE L46.7
%
\leavevmode%
\llap{\spacer}%
${L_{46.7}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{204}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{102}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3567756\\27948\\18360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27948\\-198\\-147
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18360\\-147\\-94
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\454\\3000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
179\\4284\\28254
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\96\\631
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-238\\-1581
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-120\\-789
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\119\\790
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-102\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\204\\0\\-612\\-204\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-6\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\-612\\0\\102\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-204\\-6\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-102\\-7\\-51\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3567756, 27948, 18360; 27948, -198, -147; 18360, -147, -94],[204, 3, 1],[19, 179, 4, -10, -5, 5; 454, 4284, 96, -238, -120, 119; 3000, 28254, 631, -1581, -789, 790],45,7]
%
% END OF LATTICE L46.7
%
\hbox{}\par\smallskip%
%
% LATTICE L46.8
%
\leavevmode%
\llap{\spacer}%
${L_{46.8}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{68}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{136}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
653938728\\280092\\-166821000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280092\\120\\-71452
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-166821000\\-71452\\42556351
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\0\\-98
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1084\\-425\\-4250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-404\\-189\\-1584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1214\\-663\\-4760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-152\\-95\\-596
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-101\\-48\\-396
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-68\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\68\\0\\-408\\-136\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0\\-24\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-408\\0\\136\\0\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-136\\-24\\0\\8\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-204\\-84\\-204\\-12\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[653938728, 280092, -166821000; 280092, 120, -71452; -166821000, -71452, 42556351],[204, 4, 1],[-25, -1084, -404, -1214, -152, -101; 0, -425, -189, -663, -95, -48; -98, -4250, -1584, -4760, -596, -396],45,8]
%
% END OF LATTICE L46.8
%
\hbox{}\par\smallskip%
%
% LATTICE L46.9
%
\leavevmode%
\llap{\spacer}%
${L_{46.9}}=\pdualwitharg{17}(\pfillwitharg{2.3}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${17}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{102}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{34}\above{1pt}{1pt}{}{6}{102}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30725358\\68493\\10790427
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68493\\-119\\-24055
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10790427\\-24055\\-3789486
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40999\\3349\\116722
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48356\\3949\\137667
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
229549\\18744\\653514
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21411\\1748\\60956
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4371\\-358\\-12444
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1666\\138\\4743
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\0\\-51\\-17\\-51\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-6\\-34\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\0\\102\\0\\-102\\-357
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-6\\0\\2\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-34\\-102\\0\\34\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-51\\-357\\-51\\-51\\102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30725358, 68493, 10790427; 68493, -119, -24055; 10790427, -24055, -3789486],[51, 17, 1],[40999, 48356, 229549, 21411, -4371, 1666; 3349, 3949, 18744, 1748, -358, 138; 116722, 137667, 653514, 60956, -12444, 4743],45,9]
%
% END OF LATTICE L46.9
%
\hbox{}\par\smallskip%
%
% LATTICE L46.10
%
\leavevmode%
\llap{\spacer}%
${L_{46.10}}=\pdualwitharg{3}({L_{46.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{612}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{34}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10451491812\\104683824\\893520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
104683824\\-1048530\\-8949
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
893520\\-8949\\-62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-243\\-24270\\1078
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9511\\-949926\\42228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-316\\-31561\\1404
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\799\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
397\\39651\\-1764
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-799\\33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-34\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\612\\0\\-612\\-612\\-306
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-18\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-612\\0\\34\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-612\\-18\\0\\18\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-306\\-21\\-51\\-9\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10451491812, 104683824, 893520; 104683824, -1048530, -8949; 893520, -8949, -62],[612, 3, 1],[-243, -9511, -316, 8, 397, -8; -24270, -949926, -31561, 799, 39651, -799; 1078, 42228, 1404, -34, -1764, 33],45,10]
%
% END OF LATTICE L46.10
%
\hbox{}\par\smallskip%
%
% LATTICE L46.11
%
\leavevmode%
\llap{\spacer}%
${L_{46.11}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{204}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{408}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1205952120\\-11047620\\-307553256
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11047620\\101208\\2817468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-307553256\\2817468\\78435125
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2367\\20\\-9282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19430\\187\\-76194
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-714\\9\\-2800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\17\\-1224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-254\\1\\-996
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2137\\16\\-8380
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-204\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\204\\0\\-1224\\-408\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-24\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\-1224\\0\\408\\0\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-408\\-24\\0\\24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-204\\-28\\-204\\-12\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1205952120, -11047620, -307553256; -11047620, 101208, 2817468; -307553256, 2817468, 78435125],[204, 12, 1],[-2367, -19430, -714, -312, -254, -2137; 20, 187, 9, 17, 1, 16; -9282, -76194, -2800, -1224, -996, -8380],45,11]
%
% END OF LATTICE L46.11
%
\hbox{}\par\smallskip%
%
% LATTICE L46.12
%
\leavevmode%
\llap{\spacer}%
${L_{46.12}}=\pdualwitharg{3.17}(\pfillwitharg{2.3}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${51}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{34}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{102}\above{1pt}{1pt}{}{6}{34}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
856341\\1600482\\50235
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1600482\\2991354\\93891
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50235\\93891\\2947
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\1148\\-34833
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-124\\1379\\-41817
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-199\\2201\\-66725
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\625\\-18939
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-95\\2907
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\0\\-51\\-51\\-153\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-18\\-102\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\0\\34\\0\\-102\\-119
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-18\\0\\6\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-153\\-102\\-102\\0\\102\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-51\\-119\\-51\\-51\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[856341, 1600482, 50235; 1600482, 2991354, 93891; 50235, 93891, 2947],[51, 51, 1],[-102, -124, -199, -57, 7, 1; 1148, 1379, 2201, 625, -95, 0; -34833, -41817, -66725, -18939, 2907, -17],45,12]
%
% END OF LATTICE L46.12
%
\hbox{}\par\smallskip%
%
% LATTICE L46.13
%
\leavevmode%
\llap{\spacer}%
${L_{46.13}}=\pdualwitharg{17}(\pfillwitharg{3}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${68}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{102}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{34}\above{1pt}{1pt}{}{6}{102}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145860\\89352\\-25500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89352\\-54638\\15623
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25500\\15623\\-4458
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-2\\136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\8\\-378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
277\\30\\-1479
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\4\\-209
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\2\\-153
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-134\\-15\\714
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\0\\-102\\-34\\-102\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12\\-68\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\0\\102\\0\\-102\\-357
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-12\\0\\2\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\-68\\-102\\0\\34\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-102\\-357\\-51\\-51\\102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-145860, 89352, -25500; 89352, -54638, 15623; -25500, 15623, -4458],[204, 17, 1],[-25, 71, 277, 39, 28, -134; -2, 8, 30, 4, 2, -15; 136, -378, -1479, -209, -153, 714],45,13]
%
% END OF LATTICE L46.13
%
\hbox{}\par\smallskip%
%
% LATTICE L46.14
%
\leavevmode%
\llap{\spacer}%
${L_{46.14}}=\pdualwitharg{2}({L_{46.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{b}{2}{68}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{1224}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120506066856\\-1340061516\\29926984824
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1340061516\\14901864\\-332796528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29926984824\\-332796528\\7432193615
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
380\\3\\-1530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25187\\51\\101422
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11667\\19\\46980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128731\\153\\518364
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6441\\5\\25936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3448\\1\\13884
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-36\\-612\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\68\\0\\-1224\\-136\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\24\\0\\-24\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-612\\-1224\\0\\1224\\0\\-612
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-136\\-24\\0\\8\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-204\\-84\\-612\\-12\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120506066856, -1340061516, 29926984824; -1340061516, 14901864, -332796528; 29926984824, -332796528, 7432193615],[612, 12, 1],[380, -25187, -11667, -128731, -6441, -3448; 3, 51, 19, 153, 5, 1; -1530, 101422, 46980, 518364, 25936, 13884],45,14]
%
% END OF LATTICE L46.14
%
\hbox{}\par\smallskip%
%
% LATTICE L46.15
%
\leavevmode%
\llap{\spacer}%
${L_{46.15}}=\pdualwitharg{2.3}({L_{46.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{612}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{136}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54810488664\\12713048460\\13655491740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12713048460\\2948735256\\3167330424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13655491740\\3167330424\\3402130855
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84875\\1141\\-341734
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1987777\\26724\\-8003430
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63211\\850\\-254508
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15217\\-204\\61268
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28111\\378\\-113184
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
237075\\3187\\-954540
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-68\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\612\\0\\-1224\\-1224\\-612
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0\\-72\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-1224\\0\\136\\0\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-1224\\-72\\0\\72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-612\\-84\\-204\\-36\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[54810488664, 12713048460, 13655491740; 12713048460, 2948735256, 3167330424; 13655491740, 3167330424, 3402130855],[612, 12, 1],[84875, 1987777, 63211, -15217, 28111, 237075; 1141, 26724, 850, -204, 378, 3187; -341734, -8003430, -254508, 61268, -113184, -954540],45,15]
%
% END OF LATTICE L46.15
%
\hbox{}\par\smallskip%
%
% LATTICE L46.16
%
\leavevmode%
\llap{\spacer}%
${L_{46.16}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${153}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{102}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{s}{2}{34}\above{1pt}{1pt}{}{6}{102}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31518\\-26775\\-36261
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26775\\-22695\\-30753
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36261\\-30753\\-41666
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\153\\-153
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-17\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71\\-262\\255
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-174\\171
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-120\\119
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\104\\-102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
153\\0\\-153\\-153\\-153\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-18\\-34\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-153\\0\\102\\0\\-102\\-357
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-153\\-18\\0\\18\\0\\-153
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-153\\-34\\-102\\0\\34\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-51\\-357\\-153\\-51\\102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-31518, -26775, -36261; -26775, -22695, -30753; -36261, -30753, -41666],[153, 51, 1],[46, -4, -71, -49, -35, 29; 153, -17, -262, -174, -120, 104; -153, 16, 255, 171, 119, -102],45,16]
%
% END OF LATTICE L46.16
%
\hbox{}\par\smallskip%
%
% LATTICE L46.17
%
\leavevmode%
\llap{\spacer}%
${L_{46.17}}=\pdualwitharg{3.17}(\pfillwitharg{2}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${17}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{102}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{306}\above{1pt}{1pt}{}{6}{102}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-227817\\33048\\16524
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33048\\-4794\\-2397
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16524\\-2397\\-1198
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\29\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\57\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\55\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-69\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\25\\-51
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\0\\-51\\-17\\-153\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-18\\-306\\-153
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\0\\102\\0\\-306\\-357
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-18\\0\\2\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-153\\-306\\-306\\0\\306\\-153
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-153\\-357\\-51\\-153\\102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-227817, 33048, 16524; 33048, -4794, -2397; 16524, -2397, -1198],[153, 51, 1],[3, 7, 8, 0, -10, 0; 29, 57, 55, -1, -69, 25; -17, -18, 0, 2, 0, -51],45,17]
%
% END OF LATTICE L46.17
%
\hbox{}\par\smallskip%
%
% LATTICE L46.18
%
\leavevmode%
\llap{\spacer}%
${L_{46.18}}=\pdualwitharg{3.17}(\pfillwitharg{3}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${204}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{34}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{102}\above{1pt}{1pt}{}{6}{34}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3056940\\-523056\\34884
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-523056\\-89454\\5967
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34884\\5967\\-398
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\104\\2346
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\74\\1722
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\39\\935
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-49\\-1173
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\119
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
204\\0\\-102\\-102\\-306\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-36\\-204\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\0\\34\\0\\-102\\-119
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\-36\\0\\6\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-306\\-204\\-102\\0\\102\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-102\\-119\\-51\\-51\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3056940, -523056, 34884; -523056, -89454, 5967; 34884, 5967, -398],[204, 51, 1],[9, 7, 4, 0, -5, 0; 104, 74, 39, -1, -49, 8; 2346, 1722, 935, -15, -1173, 119],45,18]
%
% END OF LATTICE L46.18
%
\hbox{}\par\smallskip%
%
% LATTICE L46.19
%
\leavevmode%
\llap{\spacer}%
${L_{46.19}}=\pdualwitharg{2.17}(\pfillwitharg{3}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${68}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{408}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{136}\above{1pt}{1pt}{}{6}{408}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
128114040\\-509796\\32774028
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-509796\\2040\\-130424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32774028\\-130424\\8384231
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\-123\\-170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
482\\-1405\\-1906
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3508\\-10233\\-13872
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
702\\-2049\\-2776
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1668\\-4871\\-6596
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
877\\-2559\\-3468
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\0\\-204\\-68\\-204\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-24\\-136\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\0\\408\\0\\-408\\-1428
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-24\\0\\8\\0\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\-136\\-408\\0\\136\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-204\\-1428\\-204\\-204\\408
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[128114040, -509796, 32774028; -509796, 2040, -130424; 32774028, -130424, 8384231],[204, 68, 1],[43, 482, 3508, 702, 1668, 877; -123, -1405, -10233, -2049, -4871, -2559; -170, -1906, -13872, -2776, -6596, -3468],45,19]
%
% END OF LATTICE L46.19
%
\hbox{}\par\smallskip%
%
% LATTICE L46.20
%
\leavevmode%
\llap{\spacer}%
${L_{46.20}}=\pdualwitharg{17}({L_{46.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${612}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{102}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{b}{2}{34}\above{1pt}{1pt}{}{6}{102}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9307908\\-9191628\\507348
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9191628\\-9076470\\500973
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
507348\\500973\\-27650
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-323\\648\\5814
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\-142\\-1270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
401\\-803\\-7191
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
241\\-483\\-4329
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
152\\-305\\-2737
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-134\\268\\2397
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
612\\0\\-306\\-306\\-306\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\4\\0\\-36\\-68\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-306\\0\\102\\0\\-102\\-357
\end{tabular}\endgroup%
\kern3pt%
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-306\\-36\\0\\18\\0\\-153
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-306\\-68\\-102\\0\\34\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-102\\-357\\-153\\-51\\102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9307908, -9191628, 507348; -9191628, -9076470, 500973; 507348, 500973, -27650],[612, 51, 1],[-323, 71, 401, 241, 152, -134; 648, -142, -803, -483, -305, 268; 5814, -1270, -7191, -4329, -2737, 2397],45,20]
%
% END OF LATTICE L46.20
%
\hbox{}\par\smallskip%
%
% LATTICE L46.21
%
\leavevmode%
\llap{\spacer}%
${L_{46.21}}=\pdualwitharg{3.17}({L_{46.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${68}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{b}{2}{102}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{306}\above{1pt}{1pt}{}{6}{102}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-911268\\66096\\33048
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66096\\-4794\\-2397
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33048\\-2397\\-1198
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\58\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\114\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\55\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-69\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\25\\-51
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\0\\-102\\-34\\-306\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-36\\-612\\-306
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\0\\102\\0\\-306\\-357
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-36\\0\\2\\0\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-306\\-612\\-306\\0\\306\\-153
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-306\\-357\\-51\\-153\\102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-911268, 66096, 33048; 66096, -4794, -2397; 33048, -2397, -1198],[612, 51, 1],[3, 7, 4, 0, -5, 0; 58, 114, 55, -1, -69, 25; -34, -36, 0, 2, 0, -51],45,21]
%
% END OF LATTICE L46.21
%
\hbox{}\par\smallskip%
%
% LATTICE L46.22
%
\leavevmode%
\llap{\spacer}%
${L_{46.22}}=\pdualwitharg{2.3.17}(\pfillwitharg{3}({L_{46.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${204}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{136}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{408}\above{1pt}{1pt}{}{6}{136}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
231336\\-5430684\\1384752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5430684\\127612200\\-32539428
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1384752\\-32539428\\8297125
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119\\-8822\\-34578
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-4199\\-16458
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-2533\\-9928
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-49\\-192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-1093\\-4284
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109\\-8102\\-31756
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
204\\0\\-204\\-204\\-612\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-72\\-408\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\0\\136\\0\\-408\\-476
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\-72\\0\\24\\0\\-204
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-612\\-408\\-408\\0\\408\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-204\\-476\\-204\\-204\\136
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[231336, -5430684, 1384752; -5430684, 127612200, -32539428; 1384752, -32539428, 8297125],[204, 204, 1],[-119, -57, -35, -1, -15, -109; -8822, -4199, -2533, -49, -1093, -8102; -34578, -16458, -9928, -192, -4284, -31756],45,22]
%
% END OF LATTICE L46.22
%
\hbox{}\par\smallskip%
%
% LATTICE L46.23
%
\leavevmode%
\llap{\spacer}%
${L_{46.23}}=\pdualwitharg{2.17}({L_{46.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${612}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{408}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{*}{2}{136}\above{1pt}{1pt}{}{6}{408}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16441085016\\458366580\\-3746364732
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
458366580\\12778968\\-104446164
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3746364732\\-104446164\\853669255
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\-225\\306
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1725\\-5004\\6958
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12947\\-37552\\52224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7943\\-23034\\32040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6389\\-18524\\25772
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3186\\-9235\\12852
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
612\\0\\-612\\-612\\-612\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-72\\-136\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-612\\0\\408\\0\\-408\\-1428
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-612\\-72\\0\\72\\0\\-612
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-612\\-136\\-408\\0\\136\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-204\\-1428\\-612\\-204\\408
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16441085016, 458366580, -3746364732; 458366580, 12778968, -104446164; -3746364732, -104446164, 853669255],[612, 204, 1],[76, 1725, 12947, 7943, 6389, 3186; -225, -5004, -37552, -23034, -18524, -9235; 306, 6958, 52224, 32040, 25772, 12852],45,23]
%
% END OF LATTICE L46.23
%
\hbox{}\par\smallskip%
%
% LATTICE L46.24
%
\leavevmode%
\llap{\spacer}%
${L_{46.24}}=\pdualwitharg{2.3.17}({L_{46.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${68}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}{408}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{1224}\above{1pt}{1pt}{}{6}{408}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
735624\\307836\\47124
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
307836\\157080\\40800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47124\\40800\\18743
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2539\\7634\\-10234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3577\\10755\\-14418
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2075\\6239\\-8364
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-911\\2739\\-3672
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[735624, 307836, 47124; 307836, 157080, 40800; 47124, 40800, 18743],[612, 204, 1],[-2539, -3577, -2075, 1, -911, -7035; 7634, 10755, 6239, -3, 2739, 21152; -10234, -14418, -8364, 4, -3672, -28356],45,24]
%
% END OF LATTICE L46.24
%

\medskip%
%------------------------------WEYL-GROUP-W47--------------------------------
%
\leavevmode\llap{}%
$W_{47\phantom{0}}$%
\qquad\llap{44} lattices, $\chi=48$%
\hfill%
$222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L47.1
%
\leavevmode%
\llap{\spacer}%
${L_{47.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{47}}%
\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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-16\\-66\\-408\\-20\\-36\\-68
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-84\\-612\\-36\\-84\\-204
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1142196, 4488, 4080; 4488, -16, -17; 4080, -17, -14],[204, 1, 1],[13, 37, 191, 7, 7, 5, -1, -1, 13, 2, 11, 46; 1308, 3726, 19244, 706, 708, 510, -100, -102, 1292, 200, 1104, 4624; 2194, 6240, 32198, 1179, 1176, 833, -170, -168, 2210, 339, 1860, 7769],46,1]
%
% END OF LATTICE L47.1
%
\hbox{}\par\smallskip%
%
% LATTICE L47.2
%
\leavevmode%
\llap{\spacer}%
${L_{47.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{28}}%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8568\\-26\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4488\\-17\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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13871\\2115480\\3745440
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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0\\0\\0
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25\\3813\\6750
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2521848, 8568, 4488; 8568, -26, -17; 4488, -17, -7],[408, 1, 1],[17, 25, 263, 5, 11, 5, -1, -1, 9, 1, 13, 29; 2592, 3813, 40120, 763, 1680, 765, -152, -153, 1360, 152, 1980, 4420; 4592, 6750, 70992, 1349, 2964, 1343, -272, -270, 2448, 271, 3516, 7837],46,2]
%
% END OF LATTICE L47.2
%
\hbox{}\par\smallskip%
%
% LATTICE L47.3
%
\leavevmode%
\llap{\spacer}%
${L_{47.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-6404376, 3014712, 21624; 3014712, -1419106, -10179; 21624, -10179, -73],[408, 1, 1],[201, 283, 2903, 105, 101, 63, -17, -7, 225, 33, 175, 719; 428, 603, 6188, 224, 216, 136, -36, -15, 476, 70, 372, 1530; -144, -258, -2992, -134, -204, -306, -16, 18, 272, 14, -36, -374],46,3]
%
% END OF LATTICE L47.3
%
\hbox{}\par\smallskip%
%
% LATTICE L47.4
%
\leavevmode%
\llap{\spacer}%
${L_{47.4}}=\pfillwitharg{2}({L_{47.1}})$%
{${}={}$}Nikulin $47$%
\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[714, -255, 0; -255, 91, 0; 0, 0, 1],[51, 1, 1],[0, 1, -24, -7, -23, -213, -32, -97, -520, -41, -25, -59; 0, 3, -68, -20, -66, -612, -92, -279, -1496, -118, -72, -170; 1, 0, -17, -4, -9, -68, -9, -24, -119, -8, -3, 0],46,4]
%
% END OF LATTICE L47.4
%
\hbox{}\par\smallskip%
%
% LATTICE L47.5
%
\leavevmode%
\llap{\spacer}%
${L_{47.5}}=\pfillwitharg{2}({L_{47.2}})$%
{${}={}$}Nikulin $28'$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[6630, 3264, 4386; 3264, 1607, 2159; 4386, 2159, 2902],[102, 1, 1],[-1, -2, -95, -11, -65, -293, -87, -262, -1401, -55, -67, -81; 2, 0, 102, 13, 84, 391, 118, 360, 1938, 77, 96, 119; 0, 3, 68, 7, 36, 153, 44, 129, 680, 26, 30, 34],46,5]
%
% END OF LATTICE L47.5
%
\hbox{}\par\smallskip%
%
% LATTICE L47.6
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-455532, -225012, 2652; -225012, -111145, 1309; 2652, 1309, -14],[204, 1, 1],[-641, -913, -9431, -346, -347, -250, 49, 25, -633, -98, -541, -2266; 1308, 1863, 19244, 706, 708, 510, -100, -51, 1292, 200, 1104, 4624; 868, 1233, 12716, 465, 462, 323, -68, -33, 884, 135, 738, 3077],46,6]
%
% END OF LATTICE L47.6
%
\hbox{}\par\smallskip%
%
% LATTICE L47.7
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-217668, 3111, 74817; 3111, -42, -1071; 74817, -1071, -25715],[51, 3, 1],[-1571, -134, 523, 140, -1163, -49412, -8829, -10266, -177323, -15740, -4037, -38988; -3056, -260, 1020, 271, -2265, -96203, -17188, -19984, -345168, -30637, -7857, -75871; -4443, -379, 1479, 396, -3289, -139740, -24969, -29033, -501483, -44514, -11417, -110262],46,7]
%
% END OF LATTICE L47.7
%
\hbox{}\par\smallskip%
%
% LATTICE L47.8
%
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${L_{47.8}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{47.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-762450, 7242, -379542; 7242, -64, 3604; -379542, 3604, -188933],[102, 2, 1],[59, 116, -767, -235, -644, -5377, -759, -4316, -11133, -815, -406, -573; -25, -51, 323, 100, 276, 2312, 327, 1863, 4811, 353, 177, 255; -119, -234, 1547, 474, 1299, 10846, 1531, 8706, 22457, 1644, 819, 1156],46,8]
%
% END OF LATTICE L47.8
%
\hbox{}\par\smallskip%
%
% LATTICE L47.9
%
\leavevmode%
\llap{\spacer}%
${L_{47.9}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{47.2}}))$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-978078\\9996\\-324462
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33\\-17\\-100
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-978078, 9996, -324462; 9996, -96, 3315; -324462, 3315, -107635],[102, 3, 1],[101, 33, -1313, -201, -367, -4595, -1297, -1229, -19019, -696, -231, -488; -50, -17, 646, 100, 184, 2312, 654, 621, 9622, 353, 118, 255; -306, -100, 3978, 609, 1112, 13923, 3930, 3724, 57630, 2109, 700, 1479],46,9]
%
% END OF LATTICE L47.9
%
\hbox{}\par\smallskip%
%
% LATTICE L47.10
%
\leavevmode%
\llap{\spacer}%
${L_{47.10}}=\pdualwitharg{3}(\main({L_{47.3}}))$%
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{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-216852, -408, 2244; -408, 6, 3; 2244, 3, -23],[204, 3, 1],[7, 2, 41, 0, -1, -4, 1, 2, 95, 6, 5, 38; 124, 36, 748, 1, -18, -85, 12, 32, 1564, 101, 86, 663; 696, 199, 4080, 0, -100, -408, 96, 197, 9384, 594, 496, 3774],46,10]
%
% END OF LATTICE L47.10
%
\hbox{}\par\smallskip%
%
% LATTICE L47.11
%
\leavevmode%
\llap{\spacer}%
${L_{47.11}}=\pdualwitharg{3}({L_{47.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1009596, -469404, 25296; -469404, -218244, 11763; 25296, 11763, -632],[204, 3, 1],[-499, -247, -2249, 27, 49, -359, -313, -565, -11555, -636, -455, -3092; 1018, 504, 4590, -55, -100, 731, 638, 1152, 23562, 1297, 928, 6307; -1026, -506, -4590, 57, 100, -765, -654, -1174, -23970, -1317, -940, -6375],46,11]
%
% END OF LATTICE L47.11
%
\hbox{}\par\smallskip%
%
% LATTICE L47.12
%
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${L_{47.12}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{47.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[282253380, -1649442, 139459806; -1649442, 9642, -814980; 139459806, -814980, 68906305],[102, 6, 1],[126, 251, 14792, 3353, 3228, 86535, 12788, 25577, 204746, 16018, 3229, 23234; 1, 0, 51, 13, 14, 391, 59, 120, 969, 77, 16, 119; -255, -508, -29937, -6786, -6533, -175134, -25881, -51764, -414375, -32418, -6535, -47022],46,12]
%
% END OF LATTICE L47.12
%
\hbox{}\par\smallskip%
%
% LATTICE L47.13
%
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${L_{47.13}}=\pdualwitharg{2}(\main({L_{47.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[231635676, -1725636, -58198752; -1725636, 12856, 433568; -58198752, 433568, 14622509],[204, 4, 1],[-1109, -6683, -17873, -2808, -1712, -4144, -13, -49, -1201, -558, -1654, -14930; -2189, -13182, -35241, -5533, -3369, -8143, -25, -102, -2397, -1109, -3273, -29495; -4349, -26208, -70091, -11012, -6714, -16252, -51, -192, -4709, -2188, -6486, -58548],46,13]
%
% END OF LATTICE L47.13
%
\hbox{}\par\smallskip%
%
% LATTICE L47.14
%
\leavevmode%
\llap{\spacer}%
${L_{47.14}}=\pdualwitharg{2}({L_{47.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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90775920\\-1080180\\-23149716
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-1080180\\12856\\275468
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[90775920, -1080180, -23149716; -1080180, 12856, 275468; -23149716, 275468, 5903651],[204, 4, 1],[-3362, -10037, -53426, -4160, -1246, -5912, -26, -127, -4170, -922, -1295, -22886; 59, 180, 969, 77, 24, 119, 1, 0, 51, 13, 21, 391; -13186, -39366, -209542, -16316, -4887, -23188, -102, -498, -16354, -3616, -5079, -89760],46,14]
%
% END OF LATTICE L47.14
%
\hbox{}\par\smallskip%
%
% LATTICE L47.15
%
\leavevmode%
\llap{\spacer}%
${L_{47.15}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{47.1}}))$%
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{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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-20808\\-1139\\7310
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-304725, -20808, 107049; -20808, -1139, 7310; 107049, 7310, -37606],[51, 17, 1],[-19866, -5071, 390, 1762, -44170, -36786, -111730, -389717, -131986, -199156, -153224, -29012; 49, 12, -1, -4, 111, 92, 279, 972, 329, 496, 381, 72; -56542, -14433, 1110, 5015, -125715, -104699, -318002, -1109199, -375654, -566831, -436101, -82573],46,15]
%
% END OF LATTICE L47.15
%
\hbox{}\par\smallskip%
%
% LATTICE L47.16
%
\leavevmode%
\llap{\spacer}%
${L_{47.16}}=\pdualwitharg{3}({L_{47.3}})$%
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{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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-2090592\\-1045803\\6939
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4179144, -2090592, 13872; -2090592, -1045803, 6939; 13872, 6939, -46],[408, 3, 1],[-11, -30, -1469, -95, -81, -625, -117, -34, -707, -1, 17, 81; 24, 64, 3128, 202, 172, 1326, 248, 72, 1496, 2, -36, -170; 300, 605, 28764, 1818, 1516, 11526, 2124, 607, 12444, 0, -304, -1224],46,16]
%
% END OF LATTICE L47.16
%
\hbox{}\par\smallskip%
%
% LATTICE L47.17
%
\leavevmode%
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${L_{47.17}}=\pdualwitharg{3}({L_{47.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-489192, 0, 4896; 0, 3, 0; 4896, 0, -49],[408, 3, 1],[1, 2, 95, 3, 5, 19, 7, 2, 41, 0, -1, -2; -16, -11, -408, -10, -12, -34, -8, -1, 0, 1, 0, -17; 96, 197, 9384, 297, 496, 1887, 696, 199, 4080, 0, -100, -204],46,17]
%
% END OF LATTICE L47.17
%
\hbox{}\par\smallskip%
%
% LATTICE L47.18
%
\leavevmode%
\llap{\spacer}%
${L_{47.18}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{47.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4453422, -24276, 782952; -24276, -119, 4267; 782952, 4267, -137650],[102, 17, 1],[-167, -170, 127, 334, 1843, 454, 2183, 6218, 1889, 1178, 1181, 50; -68, -66, 52, 135, 738, 181, 868, 2466, 748, 465, 462, 19; -952, -969, 724, 1904, 10506, 2588, 12444, 35445, 10768, 6715, 6732, 285],46,18]
%
% END OF LATTICE L47.18
%
\hbox{}\par\smallskip%
%
% LATTICE L47.19
%
\leavevmode%
\llap{\spacer}%
${L_{47.19}}=\pdualwitharg{2.3}(\main({L_{47.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22991208, -254388, -5865204; -254388, 2820, 64896; -5865204, 64896, 1496251],[204, 12, 1],[-411, -449, -2485, -101, -25, -3279, -385, -1143, -11047, -2287, -771, -10253; -2, -1, 0, 2, 0, -34, -4, -11, -102, -20, -6, -68; -1611, -1760, -9741, -396, -98, -12852, -1509, -4480, -43299, -8964, -3022, -40188],46,19]
%
% END OF LATTICE L47.19
%
\hbox{}\par\smallskip%
%
% LATTICE L47.20
%
\leavevmode%
\llap{\spacer}%
${L_{47.20}}=\pdualwitharg{2.3}({L_{47.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28236048, -1739916, -6913152; -1739916, 107232, 425988; -6913152, 425988, 1692577],[204, 12, 1],[-2135, -1054, -10501, 3, -12, -9431, -2285, -3366, -64639, -6633, -1093, -28139; -1439, -711, -7089, 1, -8, -6341, -1537, -2265, -43503, -4465, -736, -18955; -8358, -4126, -41106, 12, -47, -36924, -8946, -13178, -253062, -25968, -4279, -110160],46,20]
%
% END OF LATTICE L47.20
%
\hbox{}\par\smallskip%
%
% LATTICE L47.21
%
\leavevmode%
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${L_{47.21}}=\pdualwitharg{3.17}(\pfillwitharg{2}({L_{47.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[51, -3468, -102; -3468, 250869, 7395; -102, 7395, 218],[51, 51, 1],[-359, -30, 7, 30, -269, -670, -2033, -2362, -2399, -3618, -927, -526; -192, -15, 4, 14, -148, -366, -1108, -1285, -1304, -1964, -502, -284; 6324, 493, -132, -459, 4879, 12063, 36516, 42347, 42972, 64719, 16541, 9357],46,21]
%
% END OF LATTICE L47.21
%
\hbox{}\par\smallskip%
%
% LATTICE L47.22
%
\leavevmode%
\llap{\spacer}%
${L_{47.22}}=\pdualwitharg{2}({L_{47.3}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3216264, -44064, 14688; -44064, -584, 200; 14688, 200, -67],[408, 8, 1],[11, 61, 155, 11, 5, 5, -1, -1, 15, 4, 10, 80; 179, 987, 2499, 176, 78, 68, -17, -15, 255, 67, 165, 1309; 2942, 16296, 41378, 2932, 1326, 1292, -270, -264, 4046, 1076, 2682, 21420],46,22]
%
% END OF LATTICE L47.22
%
\hbox{}\par\smallskip%
%
% LATTICE L47.23
%
\leavevmode%
\llap{\spacer}%
${L_{47.23}}=\pdualwitharg{2}({L_{47.2}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11832, 816, 2040; 816, -56, -136; 2040, -136, -271],[408, 8, 1],[10, 109, 137, 19, 8, 5, -1, -1, 16, 8, 19, 148; 166, 1803, 2261, 312, 129, 68, -17, -15, 272, 135, 318, 2465; -9, -96, -119, -16, -6, 0, 1, 0, -17, -8, -18, -136],46,23]
%
% END OF LATTICE L47.23
%
\hbox{}\par\smallskip%
%
% LATTICE L47.24
%
\leavevmode%
\llap{\spacer}%
${L_{47.24}}=\pdualwitharg{17}({L_{47.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[269484, 18768, -3672; 18768, 1292, -255; -3672, -255, 50],[204, 17, 1],[-5, -1, 1, 2, -1, -2, -15, -59, -21, -17, -29, -3; -28, -6, 4, 8, -24, -16, -108, -402, -140, -110, -180, -18; -510, -102, 94, 187, -204, -231, -1666, -6426, -2270, -1819, -3060, -313],46,24]
%
% END OF LATTICE L47.24
%
\hbox{}\par\smallskip%
%
% LATTICE L47.25
%
\leavevmode%
\llap{\spacer}%
${L_{47.25}}=\pdualwitharg{2.17}(\pfillwitharg{2}({L_{47.2}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[75051294, -2720646, 36392172; -2720646, 98668, -1319234; 36392172, -1319234, 17646467],[102, 34, 1],[-305, -1088, -1619, -6413, -19066, -10135, -25611, -154408, -24297, -32461, -19808, -2823; 0, 3, 2, 7, 18, 9, 22, 129, 20, 26, 15, 2; 629, 2244, 3339, 13226, 39321, 20902, 52819, 318444, 50109, 66946, 40851, 5822],46,25]
%
% END OF LATTICE L47.25
%
\hbox{}\par\smallskip%
%
% LATTICE L47.26
%
\leavevmode%
\llap{\spacer}%
${L_{47.26}}=\pdualwitharg{17}(\main({L_{47.3}}))$%
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{$\left[\!\llap{\phantom{%
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-136\\-51\\0\\34\\0\\-34
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-272\\-561\\-408\\-340\\-612\\-68
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1428, -408, 0; -408, 119, 0; 0, 0, -1],[204, 17, 1],[1, -1, -1, -1, 1, 1, 7, 13, 9, 7, 11, 1; 4, -3, -4, -5, -6, 1, 12, 27, 20, 17, 30, 3; 0, 0, -8, -17, -102, -27, -136, -204, -128, -85, -102, -7],46,26]
%
% END OF LATTICE L47.26
%
\hbox{}\par\smallskip%
%
% LATTICE L47.27
%
\leavevmode%
\llap{\spacer}%
${L_{47.27}}=\pdualwitharg{3.17}(\pfillwitharg{2}({L_{47.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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35179698\\2281434\\-9571986
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\kern3pt%
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2281434\\148002\\-620721
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{$\left.\llap{\phantom{%
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0\\0\\0
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170\\-339\\544
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1521\\-3038\\4866
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3013\\-6019\\9639
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5973\\-11934\\19108
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4763\\-9517\\15237
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{$\left.\llap{\phantom{%
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-102\\-238\\-306\\-306\\-238\\-102
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[35179698, 2281434, -9571986; 2281434, 148002, -620721; -9571986, -620721, 2604445],[102, 51, 1],[287, 170, 1521, 3013, 5973, 4763, 24073, 24190, 22839, 15257, 6207, 1327; -574, -339, -3038, -6019, -11934, -9517, -48102, -48337, -45638, -30488, -12404, -2652; 918, 544, 4866, 9639, 19108, 15237, 77010, 77384, 73062, 48807, 19856, 4245],46,27]
%
% END OF LATTICE L47.27
%
\hbox{}\par\smallskip%
%
% LATTICE L47.28
%
\leavevmode%
\llap{\spacer}%
${L_{47.28}}=\pdualwitharg{17}({L_{47.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1496136\\8568\\-6120
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-1\\-9\\-255
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1\\8\\256
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1\\8\\255
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1496136, 8568, -6120; 8568, 17, -34; -6120, -34, 25],[408, 17, 1],[-9, -1, 1, 1, -11, -3, -39, -71, -49, -19, -61, -3; -80, -9, 8, 8, -108, -28, -360, -651, -448, -173, -552, -27; -2312, -255, 256, 255, -2856, -775, -10064, -18309, -12632, -4896, -15708, -772],46,28]
%
% END OF LATTICE L47.28
%
\hbox{}\par\smallskip%
%
% LATTICE L47.29
%
\leavevmode%
\llap{\spacer}%
${L_{47.29}}=\pdualwitharg{17}({L_{47.3}})$%
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{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${136}\above{1pt}{1pt}{b}{2}{102}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{*}{2}{68}\above{1pt}{1pt}{s}{2}{408}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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-4488\\-85\\17
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-65688, -4488, 408; -4488, -85, 17; 408, 17, -2],[408, 17, 1],[-1, 1, 1, 1, -1, -1, -7, -13, -9, -7, -11, -1; -16, 18, 16, 14, -36, -22, -144, -258, -176, -134, -204, -18; -340, 357, 332, 306, -612, -418, -2788, -5049, -3460, -2652, -4080, -364],46,29]
%
% END OF LATTICE L47.29
%
\hbox{}\par\smallskip%
%
% LATTICE L47.30
%
\leavevmode%
\llap{\spacer}%
${L_{47.30}}=\pdualwitharg{2.3}({L_{47.3}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-408, 4488, -408; 4488, -48624, 4416; -408, 4416, -401],[408, 24, 1],[10, 10, 46, -1, -1, 13, 6, 22, 226, 25, 9, 123; 7, 9, 51, 1, -1, -17, -1, 3, 51, 8, 4, 68; 66, 88, 510, 12, -10, -204, -18, 8, 306, 60, 34, 612],46,30]
%
% END OF LATTICE L47.30
%
\hbox{}\par\smallskip%
%
% LATTICE L47.31
%
\leavevmode%
\llap{\spacer}%
${L_{47.31}}=\pdualwitharg{2.3}({L_{47.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4120392, -203592, 20400; -203592, -10032, 1008; 20400, 1008, -101],[408, 24, 1],[5, 10, 23, -1, -1, 13, 3, 22, 113, 25, 9, 123; -1, -1, 0, 1, 0, -17, -2, -11, -51, -10, -3, -34; 999, 2008, 4641, -192, -202, 2448, 585, 4328, 22287, 4944, 1786, 24480],46,31]
%
% END OF LATTICE L47.31
%
\hbox{}\par\smallskip%
%
% LATTICE L47.32
%
\leavevmode%
\llap{\spacer}%
${L_{47.32}}=\pdualwitharg{3.17}(\main({L_{47.3}}))$%
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{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[132396, -7752, -816; -7752, 102, 51; -816, 51, 5],[204, 51, 1],[-19, -8, -13, -6, 1, 2, 11, 4, 5, 0, -5, -4; -24, -10, -16, -7, 2, 3, 16, 6, 8, 1, -6, -5; -2856, -1207, -1968, -918, 136, 294, 1632, 595, 744, 0, -748, -600],46,32]
%
% END OF LATTICE L47.32
%
\hbox{}\par\smallskip%
%
% LATTICE L47.33
%
\leavevmode%
\llap{\spacer}%
${L_{47.33}}=\pdualwitharg{3.17}({L_{47.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[296004, 3468, 4284; 3468, -408, 51; 4284, 51, 62],[204, 51, 1],[167, 231, 253, 212, 129, 44, 93, 29, 7, -17, 1, 21; -16, -22, -24, -20, -12, -4, -8, -2, 0, 2, 0, -2; -11526, -15946, -17466, -14637, -8908, -3039, -6426, -2006, -486, 1173, -68, -1449],46,33]
%
% END OF LATTICE L47.33
%
\hbox{}\par\smallskip%
%
% LATTICE L47.34
%
\leavevmode%
\llap{\spacer}%
${L_{47.34}}=\pdualwitharg{2.3.17}(\pfillwitharg{2}({L_{47.2}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-672573414, -403681218, -191281926; -403681218, -242291004, -114808140; -191281926, -114808140, -54401149],[102, 102, 1],[-34, -22, 26, 135, 123, 181, 434, 822, 374, 465, 77, 19; 5784, 3871, -4406, -23086, -21173, -31254, -75082, -142467, -64892, -80861, -13476, -3395; -12087, -8092, 9207, 48246, 44251, 65322, 156927, 297772, 135633, 169014, 28169, 7098],46,34]
%
% END OF LATTICE L47.34
%
\hbox{}\par\smallskip%
%
% LATTICE L47.35
%
\leavevmode%
\llap{\spacer}%
${L_{47.35}}=\pdualwitharg{2.17}(\main({L_{47.3}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10200, -204, -2448; -204, 476, -408; -2448, -408, -319],[204, 68, 1],[13, 52, -15, -173, -545, -287, -359, -2128, -331, -865, -493, -59; -38, -153, 44, 508, 1602, 844, 1056, 6261, 974, 2546, 1452, 174; -51, -204, 59, 680, 2142, 1128, 1411, 8364, 1301, 3400, 1938, 232],46,35]
%
% END OF LATTICE L47.35
%
\hbox{}\par\smallskip%
%
% LATTICE L47.36
%
\leavevmode%
\llap{\spacer}%
${L_{47.36}}=\pdualwitharg{2.17}({L_{47.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12380352, 198900, 3006144; 198900, 3128, 48348; 3006144, 48348, 729899],[204, 68, 1],[26, 911, 74, -590, -1757, -2162, -5822, -18299, -5870, -8104, -2590, -736; -77, -2748, -223, 1781, 5304, 6527, 17577, 55248, 17723, 24469, 7821, 2223; -102, -3570, -290, 2312, 6885, 8472, 22814, 71706, 23002, 31756, 10149, 2884],46,36]
%
% END OF LATTICE L47.36
%
\hbox{}\par\smallskip%
%
% LATTICE L47.37
%
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${L_{47.37}}=\pdualwitharg{3.17}({L_{47.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1346808, 655248, -3672; 655248, 318597, -1785; -3672, -1785, 10],[408, 51, 1],[-5, -2, -3, -1, 1, 1, 5, 2, 3, 1, -1, -1; 32, 12, 16, 2, -12, -10, -48, -20, -32, -14, 4, 6; 3876, 1411, 1764, 0, -1768, -1416, -6732, -2839, -4620, -2142, 340, 702],46,37]
%
% END OF LATTICE L47.37
%
\hbox{}\par\smallskip%
%
% LATTICE L47.38
%
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${L_{47.38}}=\pdualwitharg{3.17}({L_{47.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-913512, 0, 6120; 0, 51, 0; 6120, 0, -41],[408, 51, 1],[11, 4, 5, 0, -5, -2, -19, -8, -13, -3, 1, 1; -8, -1, 0, 1, 0, -1, -16, -11, -24, -10, -12, -2; 1632, 595, 744, 0, -748, -300, -2856, -1207, -1968, -459, 136, 147],46,38]
%
% END OF LATTICE L47.38
%
\hbox{}\par\smallskip%
%
% LATTICE L47.39
%
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${L_{47.39}}=\pdualwitharg{2.3.17}(\main({L_{47.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3876, -415140, 105876; -415140, 44125608, -11253660; 105876, -11253660, 2870099],[204, 204, 1],[-4, -11, -6, -20, -6, -4, -2, -1, 0, 2, 0, -2; 65, 815, 547, 2237, 737, 407, 39, -607, -443, -1925, -633, -303; 255, 3196, 2145, 8772, 2890, 1596, 153, -2380, -1737, -7548, -2482, -1188],46,39]
%
% END OF LATTICE L47.39
%
\hbox{}\par\smallskip%
%
% LATTICE L47.40
%
\leavevmode%
\llap{\spacer}%
${L_{47.40}}=\pdualwitharg{2.3.17}({L_{47.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[408, -253164, 64464; -253164, 148640928, -37848936; 64464, -37848936, 9637601],[204, 204, 1],[-21, -31, -35, -61, -10, -15, -19, -9, -5, 1, 0, -5; -26, -1619, -2498, -5766, -1277, -2362, -3818, -2225, -1346, 0, 316, 440; -102, -6358, -9810, -22644, -5015, -9276, -14994, -8738, -5286, 0, 1241, 1728],46,40]
%
% END OF LATTICE L47.40
%
\hbox{}\par\smallskip%
%
% LATTICE L47.41
%
\leavevmode%
\llap{\spacer}%
${L_{47.41}}=\pdualwitharg{2.17}({L_{47.3}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[816, 11832, 2856; 11832, 170680, 41208; 2856, 41208, 9949],[408, 136, 1],[-3, -35, -7, -13, -13, -3, -3, -1, 1, 4, 4, 0; 140, 792, 120, 149, 75, 7, -8, 0, 12, 49, 123, 59; -578, -3264, -494, -612, -306, -28, 34, 0, -50, -204, -510, -244],46,41]
%
% END OF LATTICE L47.41
%
\hbox{}\par\smallskip%
%
% LATTICE L47.42
%
\leavevmode%
\llap{\spacer}%
${L_{47.42}}=\pdualwitharg{2.17}({L_{47.2}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${17}\above{1pt}{1pt}{r}{2}{816}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{136}\above{1pt}{1pt}{r}{2}{204}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[816, 6120, -816; 6120, 45832, -6120; -816, -6120, 817],[408, 136, 1],[3, 13, 0, -5, -10, -3, -2, -1, 1, 8, 13, 4; -16, -216, -18, -53, -39, -7, -2, 0, 0, -1, -15, -11; -119, -1632, -137, -408, -306, -56, -17, 0, 1, 0, -102, -80],46,42]
%
% END OF LATTICE L47.42
%
\hbox{}\par\smallskip%
%
% LATTICE L47.43
%
\leavevmode%
\llap{\spacer}%
${L_{47.43}}=\pdualwitharg{2.3.17}({L_{47.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4080, -4488, 408; -4488, -4488, 408; 408, 408, -37],[408, 408, 1],[-1, -1, 0, 1, 0, -1, -2, -11, -3, -10, -3, -2; 6, 14, 2, -1, -3, -3, -2, 2, 2, 13, 7, 7; 51, 136, 21, 0, -34, -48, -51, -136, -21, 0, 34, 48],46,43]
%
% END OF LATTICE L47.43
%
\hbox{}\par\smallskip%
%
% LATTICE L47.44
%
\leavevmode%
\llap{\spacer}%
${L_{47.44}}=\pdualwitharg{2.3.17}({L_{47.3}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[64464, 325992, -1224; 325992, 1628328, -6120; -1224, -6120, 23],[408, 408, 1],[7, 9, 3, 1, -1, -1, -1, 3, 3, 8, 4, 4; 12, 14, 4, -1, -3, -3, -4, 2, 4, 13, 7, 7; 3570, 4216, 1230, -204, -850, -852, -1122, 680, 1218, 3876, 2074, 2076],46,44]
%
% END OF LATTICE L47.44
%

\medskip%
%------------------------------WEYL-GROUP-W48--------------------------------
%
\leavevmode\llap{}%
$W_{48\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=18$%
\hfill%
$422422\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L48.1
%
\leavevmode%
\llap{\spacer}%
${L_{48.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{48}}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-13414020\\36108\\52020
\end{tabular}\endgroup%
\kern3pt%
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36108\\-94\\-145
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\kern3pt%
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52020\\-145\\-194
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-13414020, 36108, 52020; 36108, -94, -145; 52020, -145, -194],[204, 1, 1],[6, 33, 235, 7, -7, -14; 1157, 6366, 45339, 1351, -1350, -2703; 744, 4090, 29121, 867, -868, -1734],47,1]
%
% END OF LATTICE L48.1
%
\hbox{}\par\smallskip%
%
% LATTICE L48.2
%
\leavevmode%
\llap{\spacer}%
${L_{48.2}}=\pfillwitharg{2}({L_{48.1}})$%
{${}={}$}Nikulin $48$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-51, 0, 0; 0, 1, 0; 0, 0, 1],[51, 1, 1],[0, 0, -7, -1, -1, -10; -1, 1, 0, -2, -4, -51; 1, 0, -51, -7, -6, -51],47,2]
%
% END OF LATTICE L48.2
%
\hbox{}\par\smallskip%
%
% LATTICE L48.3
%
\leavevmode%
\llap{\spacer}%
${L_{48.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{48.1}}))$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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-2075445\\-77109\\728115
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\kern3pt%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-55862034, -2075445, 19597719; -2075445, -77109, 728115; 19597719, 728115, -6875342],[51, 3, 1],[-215, 114, 3327, 1169, 840, 2079; -332, 161, 5644, 2048, 1555, 4080; -648, 342, 10081, 3549, 2559, 6358],47,3]
%
% END OF LATTICE L48.3
%
\hbox{}\par\smallskip%
%
% LATTICE L48.4
%
\leavevmode%
\llap{\spacer}%
${L_{48.4}}=\pdualwitharg{3}({L_{48.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L48.4
%
\hbox{}\par\smallskip%
%
% LATTICE L48.5
%
\leavevmode%
\llap{\spacer}%
${L_{48.5}}=\pdualwitharg{2}({L_{48.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2047556568, -64542948, -522260808; -64542948, 2034520, 16462672; -522260808, 16462672, 133210655],[204, 4, 1],[-1073, -4693, -80413, -3749, -129, -1561; 10, 55, 969, 47, 2, 0; -4208, -18406, -315384, -14704, -506, -6120],47,5]
%
% END OF LATTICE L48.5
%
\hbox{}\par\smallskip%
%
% LATTICE L48.6
%
\leavevmode%
\llap{\spacer}%
${L_{48.6}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{48.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3315, 57477, -1224; 57477, -871522, 16014; -1224, 16014, -235],[51, 17, 1],[-1975, 966, 5876, 12050, 9109, 4199; -233, 114, 693, 1421, 1074, 495; -5593, 2737, 16632, 34102, 25772, 11877],47,6]
%
% END OF LATTICE L48.6
%
\hbox{}\par\smallskip%
%
% LATTICE L48.7
%
\leavevmode%
\llap{\spacer}%
${L_{48.7}}=\pdualwitharg{2.3}({L_{48.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8997727224, -133446396, -2272583460; -133446396, 1979160, 33704964; -2272583460, 33704964, 573993349],[204, 12, 1],[3477, 133, -295, 639, 3983, 23619; 2083, 78, -204, 374, 2379, 14127; 13644, 522, -1156, 2508, 15630, 92684],47,7]
%
% END OF LATTICE L48.7
%
\hbox{}\par\smallskip%
%
% LATTICE L48.8
%
\leavevmode%
\llap{\spacer}%
${L_{48.8}}=\pdualwitharg{3.17}(\pfillwitharg{2}({L_{48.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L48.8
%
\hbox{}\par\smallskip%
%
% LATTICE L48.9
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L48.9
%
\hbox{}\par\smallskip%
%
% LATTICE L48.10
%
\leavevmode%
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${L_{48.10}}=\pdualwitharg{3.17}({L_{48.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L48.10
%
\hbox{}\par\smallskip%
%
% LATTICE L48.11
%
\leavevmode%
\llap{\spacer}%
${L_{48.11}}=\pdualwitharg{2.17}({L_{48.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[144556440, 7679988, 31132032; 7679988, 408136, 1653896; 31132032, 1653896, 6704735],[204, 68, 1],[10702, 477, 49, 2359, 12584, 13012; -31363, -1398, -144, -6914, -36879, -38133; -41956, -1870, -192, -9248, -49334, -51012],47,11]
%
% END OF LATTICE L48.11
%
\hbox{}\par\smallskip%
%
% LATTICE L48.12
%
\leavevmode%
\llap{\spacer}%
${L_{48.12}}=\pdualwitharg{2.3.17}({L_{48.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[192984, 95268, -23868; 95268, 59160, -14892; -23868, -14892, 3749],[204, 204, 1],[47, 2, 0, 10, 55, 19; -4235, -183, 1, -889, -4941, -1709; -16524, -714, 4, -3468, -19278, -6668],47,12]
%
% END OF LATTICE L48.12
%

\medskip%
%------------------------------WEYL-GROUP-W49--------------------------------
%
\leavevmode\llap{}%
$W_{49\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=30$%
\hfill%
$22242224\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L49.1
%
\leavevmode%
\llap{\spacer}%
${L_{49.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{49}}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1472020\\-81400\\9240
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81400\\-4494\\511
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9240\\511\\-58
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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23099\\-23100\\3474240
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1290\\-1291\\194016
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33\\-30\\4990
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57\\-55\\8591
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-27\\4061
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-20\\-40\\-22\\-2
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-40\\-120\\-110\\-30
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0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1472020, -81400, 9240; -81400, -4494, 511; 9240, 511, -58],[220, 1, 1],[3, 33, 57, 27, 17, 17, -2, -2; -2, -30, -55, -27, -18, -20, 0, 2; 460, 4990, 8591, 4061, 2548, 2530, -319, -301],48,1]
%
% END OF LATTICE L49.1
%
\hbox{}\par\smallskip%
%
% LATTICE L49.2
%
\leavevmode%
\llap{\spacer}%
${L_{49.2}}=\pfillwitharg{2}({L_{49.1}})$%
{${}={}$}Nikulin $49$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{4}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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330\\55\\0
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\kern3pt%
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55\\9\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
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{$\left.\llap{\phantom{%
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0\\0\\0
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8\\-43\\-9
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[330, 55, 0; 55, 9, 0; 0, 0, 1],[55, 1, 1],[0, -1, 6, 8, 6, 16, 27, 7; 0, 5, -33, -43, -32, -85, -143, -37; 1, 0, -11, -9, -5, -10, -11, -1],48,2]
%
% END OF LATTICE L49.2
%
\hbox{}\par\smallskip%
%
% LATTICE L49.3
%
\leavevmode%
\llap{\spacer}%
${L_{49.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{49.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1325170\\11495\\-261800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11495\\-90\\2270
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-261800\\2270\\-51721
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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5\\0\\-55\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\0\\110\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-10\\0\\10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-10\\-55\\-5
\end{tabular}\endgroup%
\kern3pt%
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-10\\-6\\-55\\-15
\end{tabular}\endgroup%
\kern3pt%
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-55\\-55\\-715\\-275
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1325170, 11495, -261800; 11495, -90, 2270; -261800, 2270, -51721],[55, 5, 1],[-119, 209, 6013, 3685, 1641, 552, 2358, 120; 60, -113, -3201, -1951, -864, -289, -1221, -59; 605, -1063, -30580, -18740, -8345, -2807, -11990, -610],48,3]
%
% END OF LATTICE L49.3
%
\hbox{}\par\smallskip%
%
% LATTICE L49.4
%
\leavevmode%
\llap{\spacer}%
${L_{49.4}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{49.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{}{2}{55}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2407405\\-32450\\-439780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32450\\-429\\-5929
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-439780\\-5929\\-80338
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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16075324\\-10711225\\-87200575
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
187506\\-124939\\-1017126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2940435\\-1959255\\-15950386
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
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-142\\99\\770
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1328\\-865\\-7205
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
685\\-453\\-3716
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4595\\-3055\\-24926
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\right.$}%
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11\\0\\-11\\-99\\-55\\-110\\-11\\-11
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{$\left[\!\llap{\phantom{%
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-55\\-110\\-11\\-11
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-110\\-330\\-55\\-165
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2407405, -32450, -439780; -32450, -429, -5929; -439780, -5929, -80338],[55, 11, 1],[-142, 1328, 685, 4595, 2036, 3407, 262, 140; 99, -865, -453, -3055, -1361, -2290, -178, -100; 770, -7205, -3716, -24926, -11044, -18480, -1421, -759],48,4]
%
% END OF LATTICE L49.4
%
\hbox{}\par\smallskip%
%
% LATTICE L49.5
%
\leavevmode%
\llap{\spacer}%
${L_{49.5}}=\pdualwitharg{2}({L_{49.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
557859720\\64460\\-142069620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64460\\8\\-16416
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-142069620\\-16416\\36180739
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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{$\left[\!\llap{\phantom{%
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-46178221\\-10387080\\-181331040
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-5735\\-1291\\-22520
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-11743\\-2629\\-46112
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-5735\\-1291\\-22520
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{$\left.\llap{\phantom{%
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-20\\-40\\-44\\-4
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-40\\-120\\-220\\-60
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[557859720, 64460, -142069620; 64460, 8, -16416; -142069620, -16416, 36180739],[220, 4, 1],[-355, -3359, -11743, -5735, -1939, -2376, -874, 0; -76, -745, -2629, -1291, -440, -545, -209, 1; -1394, -13190, -46112, -22520, -7614, -9330, -3432, 0],48,5]
%
% END OF LATTICE L49.5
%
\hbox{}\par\smallskip%
%
% LATTICE L49.6
%
\leavevmode%
\llap{\spacer}%
${L_{49.6}}=\pdualwitharg{5}({L_{49.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-118324580\\-185460\\941160
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\kern3pt%
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-185460\\-290\\1475
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941160\\1475\\-7486
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175\\5291\\23045
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81\\2441\\10665
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-118324580, -185460, 941160; -185460, -290, 1475; 941160, 1475, -7486],[220, 5, 1],[11, 21, 175, 81, 49, 9, -10, -6; 340, 638, 5291, 2441, 1468, 266, -319, -181; 1450, 2766, 23045, 10665, 6450, 1184, -1320, -790],48,6]
%
% END OF LATTICE L49.6
%
\hbox{}\par\smallskip%
%
% LATTICE L49.7
%
\leavevmode%
\llap{\spacer}%
${L_{49.7}}=\pdualwitharg{11}({L_{49.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-10454620\\-238480\\47740
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\kern3pt%
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-238480\\-5434\\1089
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47740\\1089\\-218
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{$\left.\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10454620, -238480, 47740; -238480, -5434, 1089; 47740, 1089, -218],[220, 11, 1],[11, 63, 8, 36, 17, 7, -1, -1; -2, -30, -5, -27, -18, -20, 0, 2; 2398, 13640, 1726, 7744, 3630, 1430, -219, -209],48,7]
%
% END OF LATTICE L49.7
%
\hbox{}\par\smallskip%
%
% LATTICE L49.8
%
\leavevmode%
\llap{\spacer}%
${L_{49.8}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{49.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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345510\\-5080845\\-968110
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5080845\\74731030\\14239335
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-968110\\14239335\\2713179
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[345510, -5080845, -968110; -5080845, 74731030, 14239335; -968110, 14239335, 2713179],[55, 55, 1],[9, -74, -159, -991, -405, -124, -39, 1; -3049, 25509, 54745, 341007, 139259, 42596, 13360, -482; 16005, -133903, -287370, -1790030, -731005, -223597, -70130, 2530],48,8]
%
% END OF LATTICE L49.8
%
\hbox{}\par\smallskip%
%
% LATTICE L49.9
%
\leavevmode%
\llap{\spacer}%
${L_{49.9}}=\pdualwitharg{2.5}({L_{49.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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840637160\\-18207860\\-214100040
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-18207860\\394520\\4637320
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[840637160, -18207860, -214100040; -18207860, 394520, 4637320; -214100040, 4637320, 54528671],[220, 20, 1],[6808, 13427, 237187, 116533, 39740, 9847, 18827, -163; 9, 17, 297, 145, 49, 12, 22, 0; 26730, 52718, 931260, 457540, 156030, 38662, 73920, -640],48,9]
%
% END OF LATTICE L49.9
%
\hbox{}\par\smallskip%
%
% LATTICE L49.10
%
\leavevmode%
\llap{\spacer}%
${L_{49.10}}=\pdualwitharg{2.11}({L_{49.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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300208920\\45100\\-76442300
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-57195\\-486\\-224620
\end{tabular}\endgroup%
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21539637\\182651\\84591891
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-5417\\-45\\-21274
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-11329\\-96\\-44492
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-57195\\-486\\-224620
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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\GramMatrix{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[300208920, 45100, -76442300; 45100, 88, -11484; -76442300, -11484, 19464529],[220, 44, 1],[-5417, -39129, -11329, -57195, -17461, -18066, -110, 0; -45, -330, -96, -486, -149, -155, -1, 1; -21274, -153670, -44492, -224620, -68574, -70950, -432, 0],48,10]
%
% END OF LATTICE L49.10
%
\hbox{}\par\smallskip%
%
% LATTICE L49.11
%
\leavevmode%
\llap{\spacer}%
${L_{49.11}}=\pdualwitharg{5.11}({L_{49.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${220}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2014540\\2248620\\-412280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2248620\\-2509870\\460185
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-412280\\460185\\-84374
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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234191\\4080\\-1122000
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\kern3pt%
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-262031\\-4566\\1255375
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47929\\835\\-229626
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-45\\-4\\198
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44\\0\\-215
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36\\2\\-165
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2014540, 2248620, -412280; 2248620, -2509870, 460185; -412280, 460185, -84374],[220, 55, 1],[-673, -45, 44, 36, -475, -529, -331, -1471; -18, -4, 0, 2, -2, -6, -5, -27; 3190, 198, -215, -165, 2310, 2552, 1590, 7040],48,11]
%
% END OF LATTICE L49.11
%
\hbox{}\par\smallskip%
%
% LATTICE L49.12
%
\leavevmode%
\llap{\spacer}%
${L_{49.12}}=\pdualwitharg{2.5.11}({L_{49.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${220}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{440}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
440\\67980\\-17160
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67980\\193728920\\-48915680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17160\\-48915680\\12350989
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[440, 67980, -17160; 67980, 193728920, -48915680; -17160, -48915680, 12350989],[220, 220, 1],[-1395, -289, -9, 1, -431, -624, -904, -4566; 31469, 6516, 202, 0, 9749, 14093, 20407, 103045; 124630, 25806, 800, 0, 38610, 55814, 80820, 408100],48,12]
%
% END OF LATTICE L49.12
%

\medskip%
%------------------------------WEYL-GROUP-W50--------------------------------
%
\leavevmode\llap{}%
$W_{50\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=24$%
\hfill%
$22222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L50.1
%
\leavevmode%
\llap{\spacer}%
${L_{50.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{50}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-604340\\110220\\5280
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110220\\-20102\\-963
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5280\\-963\\-46
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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11\\60\\6
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9\\50\\-15
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-132\\-352
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\0\\-5\\-10\\-35\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-616\\-132\\0\\44\\0\\-44\\-440\\-7788
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-30\\-616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-616\\-132\\0\\44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-14\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-8\\-10\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\-35\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-44\\-440\\-7788
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-604340, 110220, 5280; 110220, -20102, -963; 5280, -963, -46],[220, 1, 1],[21, 11, 9, -27, -7, -7, 1, 203; 114, 60, 50, -132, -37, -38, 5, 1100; 23, 6, -15, -352, -30, -8, 10, 264],49,1]
%
% END OF LATTICE L50.1
%
\hbox{}\par\smallskip%
%
% LATTICE L50.2
%
\leavevmode%
\llap{\spacer}%
${L_{50.2}}=\pfillwitharg{2}({L_{50.1}})$%
{${}={}$}Nikulin $50$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-935\\-165\\55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-29\\10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\10\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5389\\-30030\\-9240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\-5617\\-1728
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-133\\741\\227
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\-278\\-81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-33\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-10\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\11\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-30\\-308\\-51\\-7\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-33\\-7\\-2\\-5\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\0\\-5\\-5\\-35\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\-33\\0\\11\\0\\-11\\-220\\-1947
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-30\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\-33\\0\\11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-7\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-2\\-5\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-5\\-35\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-220\\-1947
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-935, -165, 55; -165, -29, 10; 55, 10, -2],[55, 1, 1],[50, 6, 2, -2, -1, 1, 33, 310; -278, -33, -10, 11, 5, -6, -185, -1727; -81, -7, 5, 0, -3, -5, -65, -528],49,2]
%
% END OF LATTICE L50.2
%
\hbox{}\par\smallskip%
%
% LATTICE L50.3
%
\leavevmode%
\llap{\spacer}%
${L_{50.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{50.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12815\\1705\\-2475
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1705\\-105\\330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2475\\330\\-478
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2729\\-88\\14080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
310\\9\\-1600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-527\\-17\\2719
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-191\\-7\\985
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-2\\325
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\-1\\191
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-532\\-11\\2750
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-30\\-1540\\-255\\-35\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-165\\-35\\-10\\-5\\-55
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\2\\0\\-5\\-5\\-7\\-220
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1540\\-165\\0\\55\\0\\-55\\-220\\-9735
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-30\\-1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1540\\-165\\0\\55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-255\\-35\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-10\\-5\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-5\\-7\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-55\\-220\\-9735
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12815, 1705, -2475; 1705, -105, 330; -2475, 330, -478],[55, 5, 1],[-191, -63, -37, -532, -26, 32, 6, -832; -7, -2, -1, -11, 0, 1, 0, -33; 985, 325, 191, 2750, 135, -165, -31, 4290],49,3]
%
% END OF LATTICE L50.3
%
\hbox{}\par\smallskip%
%
% LATTICE L50.4
%
\leavevmode%
\llap{\spacer}%
${L_{50.4}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{50.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40865\\-1210\\-7755
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1210\\-33\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7755\\-231\\-1471
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
944\\-2205\\-4620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-64\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\-420\\-881
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\131\\253
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\43\\88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\125\\275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\32\\79
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\0\\-330\\-308\\-561\\-77\\-55\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\0\\-33\\-77\\-22\\-55\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\0\\110\\0\\-55\\-55\\-385\\-220
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\-33\\0\\1\\0\\-11\\-220\\-177
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\0\\-330\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\0\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\0\\110\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\-33\\0\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-561\\-77\\-55\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\-22\\-55\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-55\\-385\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-220\\-177
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-40865, -1210, -7755; -1210, -33, -231; -7755, -231, -1471],[55, 11, 1],[-52, -18, -56, -16, -11, 9, 11, -20; 131, 43, 125, 32, 16, -22, -20, 52; 253, 88, 275, 79, 55, -44, -55, 97],49,4]
%
% END OF LATTICE L50.4
%
\hbox{}\par\smallskip%
%
% LATTICE L50.5
%
\leavevmode%
\llap{\spacer}%
${L_{50.5}}=\pdualwitharg{2}({L_{50.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1218360\\40260\\-302720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40260\\1432\\-9984
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-302720\\-9984\\75219
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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187604\\-139095\\736560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4169\\-3092\\16368
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46996\\34844\\-184513
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1136\\843\\-4460
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\HardButStrongLineBreak\kern3pt%
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-352\\261\\-1382
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1788\\1325\\-7020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5567\\4125\\-21857
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-120\\0\\40\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-616\\-66\\0\\11
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\-28\\-20\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-8\\-20\\-22
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-20\\-140\\-440
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1218360, 40260, -302720; 40260, 1432, -9984; -302720, -9984, 75219],[220, 4, 1],[-1136, -352, -1788, -5567, -1517, -27, -107, -2771; 843, 261, 1325, 4125, 1124, 20, 80, 2057; -4460, -1382, -7020, -21857, -5956, -106, -420, -10879],49,5]
%
% END OF LATTICE L50.5
%
\hbox{}\par\smallskip%
%
% LATTICE L50.6
%
\leavevmode%
\llap{\spacer}%
${L_{50.6}}=\pdualwitharg{5}({L_{50.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{220}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18585380\\-1579820\\27940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1579820\\-134290\\2375
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27940\\2375\\-42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40039\\-464464\\360360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3405\\-39499\\30645
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\696\\-541
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2\\23\\-30
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\-100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-47\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
439\\-5104\\3300
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-30\\-3080\\-255\\-70\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\right.$}%
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10\\0\\-30\\-3080
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-660
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\2\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3080\\-660\\0\\220
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-255\\-70\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-40\\-10\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\-7\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-220\\-440\\-38940
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18585380, -1579820, 27940; -1579820, -134290, 2375; 27940, 2375, -42],[220, 5, 1],[-2, -1, 4, 439, 37, 11, 1, 1; 23, 10, -47, -5104, -429, -126, -11, 0; -30, -100, 2, 3300, 345, 190, 43, 660],49,6]
%
% END OF LATTICE L50.6
%
\hbox{}\par\smallskip%
%
% LATTICE L50.7
%
\leavevmode%
\llap{\spacer}%
${L_{50.7}}=\pdualwitharg{11}({L_{50.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84260\\6160\\-1100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6160\\418\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1100\\-77\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-40\\-220
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\11
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3\\53\\528
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1\\18\\176
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-25\\-220
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-68\\-612
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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22\\0\\-330\\-616\\-561\\-154\\-55\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\0\\-330\\-616
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\44\\0\\-132
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-616\\-132\\0\\4
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-561\\-154\\-55\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-154\\-88\\-110\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-110\\-385\\-440
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-44\\-440\\-708
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[84260, 6160, -1100; 6160, 418, -77; -1100, -77, 14],[220, 11, 1],[3, 1, -1, -3, -3, -1, 1, 3; 53, 18, -25, -68, -67, -22, 15, 52; 528, 176, -220, -612, -605, -198, 165, 524],49,7]
%
% END OF LATTICE L50.7
%
\hbox{}\par\smallskip%
%
% LATTICE L50.8
%
\leavevmode%
\llap{\spacer}%
${L_{50.8}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{50.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27610\\-113740\\-21725
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-113740\\471955\\90145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21725\\90145\\17218
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\9030\\-47355
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
267\\-38271\\200695
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\-7310\\38334
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-1646\\8635
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2\\-342\\1793
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5\\-929\\4870
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[27610, -113740, -21725; -113740, 471955, 90145; -21725, 90145, 17218],[55, 55, 1],[14, 3, 2, 5, 7, 0, 1, 7; -1646, -409, -342, -929, -1364, -21, -88, -791; 8635, 2145, 1793, 4870, 7150, 110, 462, 4150],49,8]
%
% END OF LATTICE L50.8
%
\hbox{}\par\smallskip%
%
% LATTICE L50.9
%
\leavevmode%
\llap{\spacer}%
${L_{50.9}}=\pdualwitharg{2.5}({L_{50.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5281320\\86460\\-1345960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
86460\\5240\\-22100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1345960\\-22100\\343023
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-2786301\\-187275\\-10945000
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-587656\\-39499\\-2308400
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-835\\-56\\-3280
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-863\\-58\\-3390
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\HardButStrongLineBreak\kern3pt%
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-8629\\-580\\-33896
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\HardButStrongLineBreak\kern3pt%
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-213733\\-14366\\-839575
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5281320, 86460, -1345960; 86460, 5240, -22100; -1345960, -22100, 343023],[220, 20, 1],[-835, -863, -8629, -213733, -70089, -9269, -1503, -9171; -56, -58, -580, -14366, -4711, -623, -101, -616; -3280, -3390, -33896, -839575, -275320, -36410, -5904, -36025],49,9]
%
% END OF LATTICE L50.9
%
\hbox{}\par\smallskip%
%
% LATTICE L50.10
%
\leavevmode%
\llap{\spacer}%
${L_{50.10}}=\pdualwitharg{2.11}({L_{50.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${88}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{*}{2}{440}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13049960\\-281820\\-3323100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-281820\\6072\\71764
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3323100\\71764\\846209
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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90\\5\\353
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13049960, -281820, -3323100; -281820, 6072, 71764; -3323100, 71764, 846209],[220, 44, 1],[-381, 185, 897, 90, -11, -241, -1177, -202; -2, 6, 30, 5, 9, -5, -25, -3; -1496, 726, 3520, 353, -44, -946, -4620, -793],49,10]
%
% END OF LATTICE L50.10
%
\hbox{}\par\smallskip%
%
% LATTICE L50.11
%
\leavevmode%
\llap{\spacer}%
${L_{50.11}}=\pdualwitharg{5.11}({L_{50.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2420\\-440\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\-1430\\55
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\55\\-2
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-330\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3080\\-660\\0\\20
\end{tabular}\endgroup%
\kern3pt%
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-2805\\-770\\-55\\0
\end{tabular}\endgroup%
\kern3pt%
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-770\\-440\\-110\\-220
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2420, -440, 0; -440, -1430, 55; 0, 55, -2],[220, 55, 1],[1, -1, -3, -25, -22, -5, 0, 1; 1, -14, -25, -196, -169, -34, 1, 4; 0, -440, -748, -5820, -5005, -990, 33, 100],49,11]
%
% END OF LATTICE L50.11
%
\hbox{}\par\smallskip%
%
% LATTICE L50.12
%
\leavevmode%
\llap{\spacer}%
${L_{50.12}}=\pdualwitharg{2.5.11}({L_{50.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${440}\above{1pt}{1pt}{*}{2}{220}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4840\\-129580\\33000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-129580\\3463240\\-881980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33000\\-881980\\224613
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-1609\\22800\\89760
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\kern3pt%
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47101\\-667851\\-2629220
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}}\!\right]$}%
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0\\0\\0
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-74\\1073\\4224
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-11220\\-1540\\-220\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4840, -129580, 33000; -129580, 3463240, -881980; 33000, -881980, 224613],[220, 220, 1],[-34, -34, -74, -131, -435, -33, 7, -3; 559, 531, 1073, 1853, 6091, 419, -123, 47; 2200, 2090, 4224, 7295, 23980, 1650, -484, 185],49,12]
%
% END OF LATTICE L50.12
%

\medskip%
%------------------------------WEYL-GROUP-W51--------------------------------
%
\leavevmode\llap{}%
$W_{51\phantom{0}}$%
\qquad\llap{18} lattices, $\chi=20$%
\hfill%
$226226\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L51.1
%
\leavevmode%
\llap{\spacer}%
${L_{51.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{51},3,2}%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9620460\\18468\\25992
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18468\\-30\\-51
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25992\\-51\\-70
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-1\\-65\\-324
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-2\\-133\\-646
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\HardButStrongLineBreak\kern3pt%
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2\\129\\648
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16\\1046\\5175
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101\\6612\\32661
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\66\\323
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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6\\0\\-18\\-87\\-513\\-3
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0\\38\\0\\-171\\-1197\\-19
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\kern3pt%
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-18\\0\\18\\-9\\-171\\-9
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-87\\-171\\-9\\6\\0\\-6
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-513\\-1197\\-171\\0\\342\\0
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-3\\-19\\-9\\-6\\0\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-9620460, 18468, 25992; 18468, -30, -51; 25992, -51, -70],[684, 3, 1],[-1, -2, 2, 16, 101, 1; -65, -133, 129, 1046, 6612, 66; -324, -646, 648, 5175, 32661, 323],50,1]
%
% END OF LATTICE L51.1
%
\hbox{}\par\smallskip%
%
% LATTICE L51.2
%
\leavevmode%
\llap{\spacer}%
${L_{51.2}}=\pfillwitharg{2.3}({L_{51.1}})$%
{${}={}$}Nikulin $51$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{38}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
627\\0\\399
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
399\\0\\254
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-514\\285\\798
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-45\\24\\70
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-315\\175\\489
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-12\\0\\19
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[627, 0, 399; 0, -1, 0; 399, 0, 254],[57, 1, 1],[-2, -12, 2, 83, 183, 7; 0, 0, -2, -45, -95, -3; 3, 19, -3, -129, -285, -11],50,2]
%
% END OF LATTICE L51.2
%
\hbox{}\par\smallskip%
%
% LATTICE L51.3
%
\leavevmode%
\llap{\spacer}%
${L_{51.3}}=\pfillwitharg{3}({L_{51.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{b}{2}{38}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-853548780\\11382900\\-246924
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11382900\\-151802\\3293
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\kern3pt%
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-246924\\3293\\-70
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11097215\\831757680\\-16965936
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-147992\\-11092286\\226257
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-853548780, 11382900, -246924; 11382900, -151802, 3293; -246924, 3293, -70],[228, 1, 1],[62, 127, -41, -998, -2103, -63; 4647, 9519, -3073, -74802, -157624, -4722; -96, -190, 64, 1527, 3211, 95],50,3]
%
% END OF LATTICE L51.3
%
\hbox{}\par\smallskip%
%
% LATTICE L51.4
%
\leavevmode%
\llap{\spacer}%
${L_{51.4}}=\pfillwitharg{2}({L_{51.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2405115\\9234\\12996
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\kern3pt%
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9234\\-30\\-51
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12996\\-51\\-70
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{$\left.\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-2405115, 9234, 12996; 9234, -30, -51; 12996, -51, -70],[171, 3, 1],[-2, -4, 4, 32, 202, 2; -65, -133, 129, 1046, 6612, 66; -324, -646, 648, 5175, 32661, 323],50,4]
%
% END OF LATTICE L51.4
%
\hbox{}\par\smallskip%
%
% LATTICE L51.5
%
\leavevmode%
\llap{\spacer}%
${L_{51.5}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{51.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1235019\\313329\\-406296
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\kern3pt%
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313329\\-79449\\103077
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206\\-27\\-647
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1235019, 313329, -406296; 313329, -79449, 103077; -406296, 103077, -133663],[57, 3, 1],[206, 23488, 6994, 18871, 97333, -635; -27, -3135, -935, -2526, -13034, 84; -647, -73815, -21981, -59311, -305919, 1995],50,5]
%
% END OF LATTICE L51.5
%
\hbox{}\par\smallskip%
%
% LATTICE L51.6
%
\leavevmode%
\llap{\spacer}%
${L_{51.6}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{51.1}}))$%
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-8578500\\-53808\\42864
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\kern3pt%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8578500, -53808, 42864; -53808, -318, 267; 42864, 267, -214],[228, 3, 1],[-3, -6, 8, 45, 272, 6; -65, -133, 173, 977, 5909, 131; -682, -1368, 1818, 10231, 61845, 1365],50,6]
%
% END OF LATTICE L51.6
%
\hbox{}\par\smallskip%
%
% LATTICE L51.7
%
\leavevmode%
\llap{\spacer}%
${L_{51.7}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{51.1}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[379221000, 90203868, 94345488; 90203868, 21456968, 22441612; 94345488, 22441612, 23471989],[228, 4, 1],[-4091, -11219, -11103, -232060, -487322, -15136; -222, -608, -602, -12585, -26429, -821; 16656, 45676, 45204, 944796, 1984056, 61624],50,7]
%
% END OF LATTICE L51.7
%
\hbox{}\par\smallskip%
%
% LATTICE L51.8
%
\leavevmode%
\llap{\spacer}%
${L_{51.8}}=\pdualwitharg{19}(\pfillwitharg{2.3}({L_{51.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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4616775\\-384626\\-14212000
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17896062, -154755, -5809326; -154755, -1330, -50236; -5809326, -50236, -1885793],[57, 19, 1],[1037, 2107, 11937, 96730, 8756, -1074; -96, -178, -1000, -8049, -727, 95; -3192, -6486, -36746, -297768, -26954, 3306],50,8]
%
% END OF LATTICE L51.8
%
\hbox{}\par\smallskip%
%
% LATTICE L51.9
%
\leavevmode%
\llap{\spacer}%
${L_{51.9}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{51.1}}))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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3068533896\\-15685716\\753433980
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-15685716\\80184\\-3851400
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753433980\\-3851400\\184994783
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293381393\\-2435933\\-1194914256
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-1501152\\12463\\6114048
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3068533896, -15685716, 753433980; -15685716, 80184, -3851400; 753433980, -3851400, 184994783],[228, 12, 1],[55, 2519, 2413, 14759, 91303, 2525; 0, -38, -24, -123, -741, -17; -224, -10260, -9828, -60112, -371868, -10284],50,9]
%
% END OF LATTICE L51.9
%
\hbox{}\par\smallskip%
%
% LATTICE L51.10
%
\leavevmode%
\llap{\spacer}%
${L_{51.10}}=\pdualwitharg{3.19}(\pfillwitharg{2.3}({L_{51.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}\right.$}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[127110, -8541165, -2771169; -8541165, 574518219, 186401628; -2771169, 186401628, 60477746],[57, 57, 1],[1, -71, -441, -1276, -354, 16; -1874, 107468, 668536, 1936190, 537304, -23764; 5776, -331236, -2060550, -5967691, -1656069, 73245],50,10]
%
% END OF LATTICE L51.10
%
\hbox{}\par\smallskip%
%
% LATTICE L51.11
%
\leavevmode%
\llap{\spacer}%
${L_{51.11}}=\pdualwitharg{19}(\pfillwitharg{3}({L_{51.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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-2\\-45\\-228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-11\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-25\\-76
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
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114\\0\\-114
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0\\2\\0
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-1653\\-171\\-57
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5700, 2052, -456; 2052, -646, 133; -456, 133, -26],[228, 19, 1],[-2, -1, -3, 2, 1, 3; -45, -11, -25, 18, 8, 22; -228, -42, -76, 57, 23, 57],50,11]
%
% END OF LATTICE L51.11
%
\hbox{}\par\smallskip%
%
% LATTICE L51.12
%
\leavevmode%
\llap{\spacer}%
${L_{51.12}}=\pdualwitharg{2}({L_{51.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{2}{152}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{2}{1368}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2916369432\\-10491876\\-733409424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10491876\\37752\\2638500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-733409424\\2638500\\184438013
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-341\\2\\-1356
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1529\\0\\-6080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3911\\6\\-15552
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25180\\61\\-100128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-157046\\399\\-624492
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1532\\5\\-6092
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-72\\-348\\-2052\\-12
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\152\\0\\-684\\-4788\\-76
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\72\\-36\\-684\\-36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-348\\-684\\-36\\24\\0\\-24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2052\\-4788\\-684\\0\\1368\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-76\\-36\\-24\\0\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2916369432, -10491876, -733409424; -10491876, 37752, 2638500; -733409424, 2638500, 184438013],[684, 12, 1],[-341, -1529, -3911, -25180, -157046, -1532; 2, 0, 6, 61, 399, 5; -1356, -6080, -15552, -100128, -624492, -6092],50,12]
%
% END OF LATTICE L51.12
%
\hbox{}\par\smallskip%
%
% LATTICE L51.13
%
\leavevmode%
\llap{\spacer}%
${L_{51.13}}=\pdualwitharg{19}(\pfillwitharg{2}({L_{51.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${114}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{342}\above{1pt}{1pt}{}{6}{114}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{s}{2}{38}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40185\\4788\\2394
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4788\\-570\\-285
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2394\\-285\\-142
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-17\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\33\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\197\\-171
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\63\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
114\\0\\-342\\-1653\\-513\\-57
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-171\\-63\\-19
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-342\\0\\342\\-171\\-171\\-171
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-1653\\-171\\-171\\114\\0\\-114
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-513\\-63\\-171\\0\\18\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-19\\-171\\-114\\0\\38
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-40185, 4788, 2394; 4788, -570, -285; 2394, -285, -142],[171, 57, 1],[-2, 0, 4, 14, 4, 0; -17, -1, 33, 197, 63, 9; 0, 2, 0, -171, -63, -19],50,13]
%
% END OF LATTICE L51.13
%
\hbox{}\par\smallskip%
%
% LATTICE L51.14
%
\leavevmode%
\llap{\spacer}%
${L_{51.14}}=\pdualwitharg{3.19}(\pfillwitharg{3}({L_{51.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${38}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{114}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64524\\-205200\\5700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-205200\\-650370\\18069
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5700\\18069\\-502
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{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1671\\10792\\407208
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5137\\33156\\1251093
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-143\\-923\\-34828
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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36\\230\\8683
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\72\\2715
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\26\\969
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{$\left.\llap{\phantom{%
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0\\0\\0
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38\\0\\-114\\-551\\-171\\-57
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\6\\0
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-551\\-171\\-57
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-171\\-63\\-57
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-57\\-57\\-171
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-64524, -205200, 5700; -205200, -650370, 18069; 5700, 18069, -502],[228, 57, 1],[36, 11, 3, -3, 0, 8; 230, 72, 26, -17, -1, 45; 8683, 2715, 969, -646, -36, 1710],50,14]
%
% END OF LATTICE L51.14
%
\hbox{}\par\smallskip%
%
% LATTICE L51.15
%
\leavevmode%
\llap{\spacer}%
${L_{51.15}}=\pdualwitharg{2.19}(\pfillwitharg{3}({L_{51.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${456}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{152}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2178312\\208164\\-380304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208164\\19912\\-36328
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-380304\\-36328\\66407
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46925\\-142824\\190608
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2133\\-6493\\8664
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-9954\\30296\\-40433
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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6511\\-19815\\26448
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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456\\0\\-456\\-6612\\-684\\-228
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0\\8\\0\\-684\\-84\\-76
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-456\\0\\152\\-228\\-76\\-228
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
456\\0\\-456
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-456\\0\\152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6612\\-684\\-228
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-684\\-84\\-76
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-228\\-76\\-228
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2178312, 208164, -380304; 208164, 19912, -36328; -380304, -36328, 66407],[228, 76, 1],[6511, 645, 131, -112, 66, 580; -19815, -1963, -399, 339, -201, -1765; 26448, 2620, 532, -456, 268, 2356],50,15]
%
% END OF LATTICE L51.15
%
\hbox{}\par\smallskip%
%
% LATTICE L51.16
%
\leavevmode%
\llap{\spacer}%
${L_{51.16}}=\pdualwitharg{19}({L_{51.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${114}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{342}\above{1pt}{1pt}{}{6}{114}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{s}{2}{38}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160740\\9576\\4788
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9576\\-570\\-285
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4788\\-285\\-142
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-17\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\33\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\197\\-171
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\63\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
114\\0\\-342\\-1653\\-513\\-57
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-171\\-63\\-19
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-342\\0\\342\\-171\\-171\\-171
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1653\\-171\\-171\\114\\0\\-114
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-513\\-63\\-171\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-19\\-171\\-114\\0\\38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-160740, 9576, 4788; 9576, -570, -285; 4788, -285, -142],[684, 57, 1],[-1, 0, 2, 7, 2, 0; -17, -1, 33, 197, 63, 9; 0, 2, 0, -171, -63, -19],50,16]
%
% END OF LATTICE L51.16
%
\hbox{}\par\smallskip%
%
% LATTICE L51.17
%
\leavevmode%
\llap{\spacer}%
${L_{51.17}}=\pdualwitharg{2.3.19}(\pfillwitharg{3}({L_{51.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${152}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{456}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
98952\\-3687444\\-905388
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3687444\\137416056\\33740124
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-905388\\33740124\\8284301
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33156\\-98477\\404700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1230545\\3654744\\-15019500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-302149\\897389\\-3687901
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1389\\4124\\-16948
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-465\\1390\\-5712
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-287\\888\\-3648
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
152\\0\\-456\\-2204\\-684\\-228
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0\\24\\0\\-684\\-252\\-228
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-456\\0\\456\\-228\\-228\\-684
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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152\\0\\-456
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0\\24\\0
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-456\\0\\456
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-2204\\-684\\-228
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-684\\-252\\-228
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-228\\-228\\-684
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[98952, -3687444, -905388; -3687444, 137416056, 33740124; -905388, 33740124, 8284301],[228, 228, 1],[-1389, -465, -287, -12, -2, -180; 4124, 1390, 888, 37, -3, 499; -16948, -5712, -3648, -152, 12, -2052],50,17]
%
% END OF LATTICE L51.17
%
\hbox{}\par\smallskip%
%
% LATTICE L51.18
%
\leavevmode%
\llap{\spacer}%
${L_{51.18}}=\pdualwitharg{2.19}({L_{51.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${456}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{1368}\above{1pt}{1pt}{}{6}{456}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{s}{2}{152}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
198360\\41724\\18468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41724\\33288\\-14592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18468\\-14592\\15647
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
459\\-1375\\-1824
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1205\\-3609\\-4788
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12795\\-38324\\-50844
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4421\\-13242\\-17568
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1071\\-3208\\-4256
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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456\\0\\-1368\\-6612\\-2052\\-228
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-684\\-252\\-76
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1368\\0\\1368\\-684\\-684\\-684
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6612\\-684\\-684\\456\\0\\-456
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2052\\-252\\-684\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-228\\-76\\-684\\-456\\0\\152
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[198360, 41724, 18468; 41724, 33288, -14592; 18468, -14592, 15647],[684, 228, 1],[459, -1, 1205, 12795, 4421, 1071; -1375, 3, -3609, -38324, -13242, -3208; -1824, 4, -4788, -50844, -17568, -4256],50,18]
%
% END OF LATTICE L51.18
%

\medskip%
%------------------------------WEYL-GROUP-W52--------------------------------
%
\leavevmode\llap{}%
$W_{52\phantom{0}}$%
\qquad\llap{44} lattices, $\chi=54$%
\hfill%
$222422222422\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L52.1
%
\leavevmode%
\llap{\spacer}%
${L_{52.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{52}}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{114}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{*}{2}{76}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-833340\\-411084\\2736
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-411084\\-202786\\1349
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2736\\1349\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-684191\\1393080\\904020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-337109\\686387\\445422
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1662\\-3384\\-2197
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1967\\4005\\2598
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-609\\1240\\808
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4619\\9405\\6156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138\\281\\186
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\336\\226
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-877\\1786\\1216
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-228\\-30\\-246\\-3192\\-300\\-84\\-570\\-12\\-6\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-4\\-60\\-836\\-84\\-28\\-228\\-8\\-12\\-76
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-833340, -411084, 2736; -411084, -202786, 1349; 2736, 1349, -8],[228, 1, 1],[-1967, -609, -4619, -138, -165, -877, 28, 55, -644, -139, -1497, -20175; 4005, 1240, 9405, 281, 336, 1786, -57, -112, 1311, 283, 3048, 41078; 2598, 808, 6156, 186, 226, 1216, -36, -76, 798, 180, 1970, 26600],51,1]
%
% END OF LATTICE L52.1
%
\hbox{}\par\smallskip%
%
% LATTICE L52.2
%
\leavevmode%
\llap{\spacer}%
${L_{52.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\spacer%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1477896, -246696, 4560; -246696, -41179, 761; 4560, 761, -14],[456, 1, 1],[29, 1, -26, -3, -7, -97, -1, 7, 64, 5, 19, 401; -174, -4, 171, 19, 44, 608, 6, -44, -399, -31, -116, -2432; -42, 100, 798, 55, 111, 1444, 0, -112, -855, -58, -129, -1900],51,2]
%
% END OF LATTICE L52.2
%
\hbox{}\par\smallskip%
%
% LATTICE L52.3
%
\leavevmode%
\llap{\spacer}%
${L_{52.3}}$%
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{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\spacer%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-508748712, -3325152, 93024; -3325152, -21733, 608; 93024, 608, -17],[456, 1, 1],[50, 29, 205, 5, 2, 13, -1, -1, 61, 9, 40, 1051; -7650, -4436, -31350, -764, -305, -1976, 153, 152, -9348, -1378, -6121, -160816; -21, 24, 456, 34, 35, 456, 0, -36, -570, -40, -53, -912],51,3]
%
% END OF LATTICE L52.3
%
\hbox{}\par\smallskip%
%
% LATTICE L52.4
%
\leavevmode%
\llap{\spacer}%
${L_{52.4}}=\pfillwitharg{2}({L_{52.1}})$%
{${}={}$}Nikulin $52$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-912, -285, 0; -285, -89, 0; 0, 0, 1],[57, 1, 1],[727, 95, 1189, 17, 0, -6, 1, 9, 787, 87, 312, 3958; -2265, -296, -3705, -53, 0, 19, -3, -28, -2451, -271, -972, -12331; -123, -15, -171, -1, 1, 0, -3, -3, -171, -17, -55, -684],51,4]
%
% END OF LATTICE L52.4
%
\hbox{}\par\smallskip%
%
% LATTICE L52.5
%
\leavevmode%
\llap{\spacer}%
${L_{52.5}}=\pfillwitharg{2}({L_{52.2}})$%
{${}={}$}Nikulin $29'$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[62130, -20748, 10146; -20748, 6929, -3388; 10146, -3388, 1657],[114, 1, 1],[59, 347, 13958, 1518, 10736, 135849, 12443, 3225, 19976, 268, -20, -113; 102, 592, 23769, 2584, 18269, 231154, 21171, 5486, 33972, 455, -35, -190; -153, -916, -36936, -4019, -28437, -359860, -32964, -8546, -52953, -712, 51, 304],51,5]
%
% END OF LATTICE L52.5
%
\hbox{}\par\smallskip%
%
% LATTICE L52.6
%
\leavevmode%
\llap{\spacer}%
${L_{52.6}}=\main({L_{52.3}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-63156, -21204, 684; -21204, -7118, 229; 684, 229, -7],[228, 1, 1],[29, 1, -52, -6, -7, -97, -1, 7, 128, 10, 19, 401; -87, -2, 171, 19, 22, 304, 3, -22, -399, -31, -58, -1216; -42, 24, 456, 34, 35, 456, 0, -36, -570, -40, -53, -912],51,6]
%
% END OF LATTICE L52.6
%
\hbox{}\par\smallskip%
%
% LATTICE L52.7
%
\leavevmode%
\llap{\spacer}%
${L_{52.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{52.1}}))$%
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{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-71592, 3135, -23427; 3135, -111, 1026; -23427, 1026, -7666],[57, 3, 1],[18, 56, -404, -270, -1471, -19852, -1292, -602, -3054, -276, -167, -896; 0, 1, 0, -2, -15, -209, -14, -7, -38, -4, -3, -19; -55, -171, 1235, 825, 4494, 60648, 3947, 1839, 9329, 843, 510, 2736],51,7]
%
% END OF LATTICE L52.7
%
\hbox{}\par\smallskip%
%
% LATTICE L52.8
%
\leavevmode%
\llap{\spacer}%
${L_{52.8}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{52.2}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3428094, -22344, -1697004; -22344, -142, -11060; -1697004, -11060, -840065],[114, 2, 1],[1939, 5707, 458968, 49911, 352961, 2233073, 409066, 53008, 656617, 8804, -671, -1903; 990, 2910, 233985, 25444, 179929, 1138347, 208527, 27021, 334704, 4487, -343, -969; -3930, -11567, -930240, -101160, -715384, -4526009, -829098, -107437, -1330836, -17844, 1360, 3857],51,8]
%
% END OF LATTICE L52.8
%
\hbox{}\par\smallskip%
%
% LATTICE L52.9
%
\leavevmode%
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${L_{52.9}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{52.2}}))$%
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{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3382266, -27132, 1141254; -27132, -213, 9156; 1141254, 9156, -385085],[114, 3, 1],[-499, -8813, -118128, -38538, -272534, -3448475, -105285, -81859, -169000, -6798, 518, 2939; 330, 5820, 77995, 25444, 179929, 2276694, 69509, 54042, 111568, 4487, -343, -1938; -1471, -25980, -348232, -113607, -803409, -10165836, -310372, -241314, -498199, -20040, 1527, 8664],51,9]
%
% END OF LATTICE L52.9
%
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%
% LATTICE L52.10
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2498196, -19836, 13224; -19836, -156, 105; 13224, 105, -70],[228, 3, 1],[2, 7, 54, 15, 91, 1117, 33, 23, 41, 0, -1, 23; -4, -8, -38, -8, -34, -380, -10, -4, 0, 2, -2, -76; 371, 1308, 10127, 2817, 17112, 210102, 6209, 4332, 7733, 3, -192, 4218],51,10]
%
% END OF LATTICE L52.10
%
\hbox{}\par\smallskip%
%
% LATTICE L52.11
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30888870, 1133160, 14601804; 1133160, 41574, 535668; 14601804, 535668, 6902573],[114, 6, 1],[-1929, -17155, -460548, -150295, -1063173, -6726733, -410770, -159716, -659635, -26576, 1947, 5635; 34, 296, 7923, 2584, 18269, 115577, 7057, 2743, 11324, 455, -35, -95; 4078, 36267, 973636, 317736, 2247636, 14220873, 868402, 337653, 1394524, 56184, -4116, -11913],51,11]
%
% END OF LATTICE L52.11
%
\hbox{}\par\smallskip%
%
% LATTICE L52.12
%
\leavevmode%
\llap{\spacer}%
${L_{52.12}}=\pdualwitharg{3}(\main({L_{52.3}}))$%
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{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-410172, 0, 5472; 0, 3, 0; 5472, 0, -73],[228, 3, 1],[4, 11, 73, 19, 54, 1307, 38, 25, 41, -1, 0, 61; -1, -6, -57, -17, -55, -1368, -41, -30, -57, -1, 1, 0; 299, 822, 5453, 1419, 4032, 97584, 2837, 1866, 3059, -75, 0, 4560],51,12]
%
% END OF LATTICE L52.12
%
\hbox{}\par\smallskip%
%
% LATTICE L52.13
%
\leavevmode%
\llap{\spacer}%
${L_{52.13}}=\pdualwitharg{2}(\main({L_{52.3}}))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12027912, -73644, 2977680; -73644, 452, -18232; 2977680, -18232, 737167],[228, 4, 1],[3601, 503, 6937, 159, 107, 471, 109, 27, 3133, 371, 1483, 9599; -5268, -736, -10146, -232, -153, -665, -150, -38, -4560, -542, -2169, -14041; -14676, -2050, -28272, -648, -436, -1919, -444, -110, -12768, -1512, -6044, -39121],51,13]
%
% END OF LATTICE L52.13
%
\hbox{}\par\smallskip%
%
% LATTICE L52.14
%
\leavevmode%
\llap{\spacer}%
${L_{52.14}}=\pdualwitharg{2}({L_{52.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[324413448, -2158932, 80740728; -2158932, 14368, -537320; 80740728, -537320, 20094929],[228, 4, 1],[25175, 1788, 49775, 1163, 580, 3611, 53, 14, 18701, 2441, 10232, 133341; -51000, -3623, -100890, -2360, -1181, -7372, -114, -28, -37848, -4942, -20725, -270104; -102516, -7281, -202692, -4736, -2362, -14706, -216, -57, -76152, -9940, -41666, -542982],51,14]
%
% END OF LATTICE L52.14
%
\hbox{}\par\smallskip%
%
% LATTICE L52.15
%
\leavevmode%
\llap{\spacer}%
${L_{52.15}}=\pdualwitharg{19}(\pfillwitharg{2}({L_{52.1}}))$%
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{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-277077, -6498, -88293; -6498, -152, -2071; -88293, -2071, -28135],[57, 19, 1],[19, 13, -23, -83, -423, -298, -1097, -167, -131, -71, -39, -10; -36, -38, 42, 180, 985, 700, 2598, 404, 324, 186, 113, 32; -57, -38, 69, 247, 1254, 883, 3249, 494, 387, 209, 114, 29],51,15]
%
% END OF LATTICE L52.15
%
\hbox{}\par\smallskip%
%
% LATTICE L52.16
%
\leavevmode%
\llap{\spacer}%
${L_{52.16}}=\pdualwitharg{3}({L_{52.3}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-820344, -16416, 10944; -16416, -327, 219; 10944, 219, -146],[456, 3, 1],[2, 11, 73, 19, 54, 1307, 19, 25, 41, -1, 0, 61; -1, -12, -114, -34, -110, -2736, -41, -60, -114, -2, 2, 0; 148, 804, 5282, 1368, 3867, 93480, 1357, 1776, 2888, -78, 3, 4560],51,16]
%
% END OF LATTICE L52.16
%
\hbox{}\par\smallskip%
%
% LATTICE L52.17
%
\leavevmode%
\llap{\spacer}%
${L_{52.17}}=\pdualwitharg{3}({L_{52.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13400472, 912, 44232; 912, 3, -3; 44232, -3, -146],[456, 3, 1],[5, 17, 65, 18, 109, 2675, 79, 55, 49, 0, -1, 61; -2, -12, -57, -17, -110, -2736, -82, -60, -57, -1, 2, 0; 1514, 5148, 19684, 5451, 33009, 810084, 23924, 16656, 14839, 0, -303, 18468],51,17]
%
% END OF LATTICE L52.17
%
\hbox{}\par\smallskip%
%
% LATTICE L52.18
%
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\llap{\spacer}%
${L_{52.18}}=\pdualwitharg{19}(\pfillwitharg{2}({L_{52.2}}))$%
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{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5041194, -128022, -791616; -128022, -3211, -20102; -791616, -20102, -124307],[114, 19, 1],[-2476, -14553, -30793, -63621, -449898, -299615, -521404, -135127, -44047, -11219, 858, 255; -483, -2846, -6024, -12447, -88025, -58622, -102018, -26440, -8619, -2196, 167, 50; 15846, 93138, 197073, 407170, 2879317, 1917516, 3336951, 864804, 281898, 71801, -5491, -1632],51,18]
%
% END OF LATTICE L52.18
%
\hbox{}\par\smallskip%
%
% LATTICE L52.19
%
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${L_{52.19}}=\pdualwitharg{2.3}({L_{52.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[68234928, -684, 16752984; -684, 24, -168; 16752984, -168, 4113179],[228, 12, 1],[2630, 1904, 55120, 14958, 44737, 547720, 32272, 5575, 39614, 0, 137, 20684; -31, -22, -627, -169, -502, -6137, -361, -62, -437, 1, -2, -247; -10712, -7755, -224504, -60924, -182214, -2230866, -131444, -22707, -161348, 0, -558, -84246],51,19]
%
% END OF LATTICE L52.19
%
\hbox{}\par\smallskip%
%
% LATTICE L52.20
%
\leavevmode%
\llap{\spacer}%
${L_{52.20}}=\pdualwitharg{2.3}(\main({L_{52.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20951148, 0, 5140716; 0, 12, 0; 5140716, 0, 1261361],[228, 12, 1],[1059, 1615, 24093, 6619, 20016, 122810, 14509, 5057, 18163, 53, 0, 3958; -1, -3, -57, -17, -55, -342, -41, -15, -57, -1, 1, 0; -4316, -6582, -98192, -26976, -81576, -500517, -59132, -20610, -74024, -216, 0, -16131],51,20]
%
% END OF LATTICE L52.20
%
\hbox{}\par\smallskip%
%
% LATTICE L52.21
%
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${L_{52.21}}=\pdualwitharg{3.19}(\pfillwitharg{2}({L_{52.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[92454, -199272, -64695; -199272, 441921, 143469; -64695, 143469, 46577],[57, 57, 1],[1, 0, 3, 23, 100, 69, 83, 36, 9, 13, 8, 3; -456, -37, -1618, -12314, -53120, -36607, -43980, -19007, -4730, -6730, -4012, -1481; 1406, 114, 4988, 37962, 163761, 112854, 135584, 58596, 14582, 20748, 12369, 4566],51,21]
%
% END OF LATTICE L52.21
%
\hbox{}\par\smallskip%
%
% LATTICE L52.22
%
\leavevmode%
\llap{\spacer}%
${L_{52.22}}=\pdualwitharg{2}({L_{52.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8721912, -108072, 35568; -108072, -1328, 440; 35568, 440, -145],[456, 8, 1],[71, 11, 167, 5, 6, 16, -1, -1, 23, 5, 54, 364; 1341, 209, 3192, 97, 119, 323, -18, -20, 399, 92, 1015, 6859; 21468, 3330, 50616, 1520, 1832, 4902, -300, -306, 6840, 1504, 16312, 110010],51,22]
%
% END OF LATTICE L52.22
%
\hbox{}\par\smallskip%
%
% LATTICE L52.23
%
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${L_{52.23}}=\pdualwitharg{2}({L_{52.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-40584, -3192, 1368; -3192, -112, 72; 1368, 72, -37],[456, 8, 1],[71, 11, 167, 5, 6, 8, -1, -1, 23, 5, 54, 182; 1299, 202, 3078, 93, 113, 152, -18, -19, 399, 90, 985, 3325; 5112, 794, 12084, 364, 440, 589, -72, -74, 1596, 356, 3880, 13091],51,23]
%
% END OF LATTICE L52.23
%
\hbox{}\par\smallskip%
%
% LATTICE L52.24
%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[110580, 35568, -1596; 35568, 11438, -513; -1596, -513, 23],[228, 19, 1],[29, 7, 2, 0, -1, -1, -1, 1, 2, 4, 13, 17; -339, -86, -27, -5, 4, 4, 3, -10, -21, -43, -148, -196; -5586, -1444, -468, -114, 19, 20, 0, -152, -330, -684, -2413, -3212],51,24]
%
% END OF LATTICE L52.24
%
\hbox{}\par\smallskip%
%
% LATTICE L52.25
%
\leavevmode%
\llap{\spacer}%
${L_{52.25}}=\pdualwitharg{19}({L_{52.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-24852\\6156\\-228
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6156\\-1520\\57
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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21\\56\\-912
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7\\18\\-327
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2\\4\\-133
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-1\\-6\\-76
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-1\\-4\\-6
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24852, 6156, -228; 6156, -1520, 57; -228, 57, -2],[228, 19, 1],[79, 21, 7, 2, -1, -1, -2, 1, 4, 9, 67, 45; 216, 56, 18, 4, -6, -4, -6, 4, 12, 26, 186, 124; -3249, -912, -327, -133, -76, -6, 57, 0, -129, -323, -2660, -1818],51,25]
%
% END OF LATTICE L52.25
%
\hbox{}\par\smallskip%
%
% LATTICE L52.26
%
\leavevmode%
\llap{\spacer}%
${L_{52.26}}=\pdualwitharg{2.19}(\pfillwitharg{2}({L_{52.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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221741\\-458\\-449483
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939067\\-1944\\-1903548
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1940508\\-4019\\-3933532
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13724641\\-28437\\-27820712
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1045794390, 8113608, 515907798; 8113608, 62966, 4002578; 515907798, 4002578, 254505913],[114, 38, 1],[75304, 221741, 939067, 1940508, 13724641, 4570188, 15907033, 2061450, 1344124, 342683, -25495, -3806; -153, -458, -1944, -4019, -28437, -9470, -32964, -4273, -2787, -712, 51, 8; -152646, -449483, -1903548, -3933532, -27820712, -9264059, -32244558, -4178689, -2724624, -694640, 51680, 7715],51,26]
%
% END OF LATTICE L52.26
%
\hbox{}\par\smallskip%
%
% LATTICE L52.27
%
\leavevmode%
\llap{\spacer}%
${L_{52.27}}=\pdualwitharg{3.19}(\pfillwitharg{2}({L_{52.2}}))$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${19}\above{1pt}{1pt}{}{2}{114}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{57}\above{1pt}{1pt}{}{4}{114}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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837139278\\-8890632\\290079384
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290079384\\-3080736\\100516201
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{$\left.\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[837139278, -8890632, 290079384; -8890632, 94449, -3080736; 290079384, -3080736, 100516201],[114, 57, 1],[19327, 341465, 241016, 1494122, 10567486, 7037765, 4082617, 3174487, 344976, 263854, -19630, -5861; -38705, -683846, -482680, -2992263, -21163409, -14094470, -8176222, -6357520, -690881, -528420, 39311, 11738; -56962, -1006392, -710341, -4403592, -31145313, -20742246, -12032605, -9356094, -1016740, -777651, 57855, 17274],51,27]
%
% END OF LATTICE L52.27
%
\hbox{}\par\smallskip%
%
% LATTICE L52.28
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[221160, -35568, 3192; -35568, 5719, -513; 3192, -513, 46],[456, 19, 1],[-29, -7, -1, 0, 1, 1, 1, -1, -1, -2, -13, -17; -678, -172, -27, -5, 8, 8, 6, -20, -21, -43, -296, -392; -5586, -1444, -234, -57, 19, 20, 0, -152, -165, -342, -2413, -3212],51,28]
%
% END OF LATTICE L52.28
%
\hbox{}\par\smallskip%
%
% LATTICE L52.29
%
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${L_{52.29}}=\pdualwitharg{19}({L_{52.3}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8331576, -925680, 9576; -925680, -102847, 1064; 9576, 1064, -11],[456, 19, 1],[-76, -37, -11, -1, 2, 1, -1, -7, -11, -21, -68, -89; 558, 268, 78, 4, -17, -8, 9, 56, 84, 158, 503, 656; -12369, -6384, -2064, -494, 95, 96, 0, -684, -1470, -3040, -10697, -14232],51,29]
%
% END OF LATTICE L52.29
%
\hbox{}\par\smallskip%
%
% LATTICE L52.30
%
\leavevmode%
\llap{\spacer}%
${L_{52.30}}=\pdualwitharg{2.3}({L_{52.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-899688, 0, -446424; 0, 24, 0; -446424, 0, -221515],[456, 24, 1],[-129, -259, -4148, -1167, -7133, -43837, -2594, -908, -3243, 0, 131, -509; -2, -2, -19, -4, -17, -95, -5, -1, 0, 1, -1, -19; 260, 522, 8360, 2352, 14376, 88350, 5228, 1830, 6536, 0, -264, 1026],51,30]
%
% END OF LATTICE L52.30
%
\hbox{}\par\smallskip%
%
% LATTICE L52.31
%
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${L_{52.31}}=\pdualwitharg{2.3}({L_{52.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-475152, -18696, 1824; -18696, -696, 72; 1824, 72, -7],[456, 24, 1],[1, 2, 32, 9, 55, 169, 20, 7, 25, 0, -1, 2; -1, -3, -57, -17, -110, -342, -41, -15, -57, -1, 2, 0; 248, 486, 7676, 2148, 13056, 40071, 4736, 1650, 5852, -12, -240, 513],51,31]
%
% END OF LATTICE L52.31
%
\hbox{}\par\smallskip%
%
% LATTICE L52.32
%
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${L_{52.32}}=\pdualwitharg{2.3.19}(\pfillwitharg{2}({L_{52.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10910623422, 26952853104, 13335577272; 26952853104, -66582471666, -32943291474; 13335577272, -32943291474, -16299491833],[114, 114, 1],[-161, -1423, -2008, -12447, -88025, -29311, -34006, -13220, -2873, -2196, 167, 25; -725518, -6396340, -9022739, -55925240, -395476951, -131686309, -152777755, -59390777, -12906300, -9861877, 754249, 112075; 1466230, 12926631, 18234430, 113021652, 799235874, 266130357, 308754940, 120025299, 26082880, 19930278, -1524294, -226497],51,32]
%
% END OF LATTICE L52.32
%
\hbox{}\par\smallskip%
%
% LATTICE L52.33
%
\leavevmode%
\llap{\spacer}%
${L_{52.33}}=\pdualwitharg{3.19}({L_{52.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2296644, 549708, -10716; 549708, 131556, -2565; -10716, -2565, 50],[228, 57, 1],[3, 3, -1, -10, -91, -63, -38, -33, -4, -5, 1, 3; -4, -8, -2, -8, -34, -20, -10, -4, 0, 2, -2, -4; 437, 228, -319, -2565, -21318, -14574, -8683, -7296, -859, -969, 114, 438],51,33]
%
% END OF LATTICE L52.33
%
\hbox{}\par\smallskip%
%
% LATTICE L52.34
%
\leavevmode%
\llap{\spacer}%
${L_{52.34}}=\pdualwitharg{3.19}(\main({L_{52.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-66804, 0, 684; 0, 57, 0; 684, 0, -7],[228, 57, 1],[1, -1, -2, -14, -54, -73, -43, -35, -4, -4, 0, 1; -1, -6, -3, -17, -55, -72, -41, -30, -3, -1, 1, 0; 95, -114, -205, -1425, -5472, -7392, -4351, -3534, -403, -399, 0, 96],51,34]
%
% END OF LATTICE L52.34
%
\hbox{}\par\smallskip%
%
% LATTICE L52.35
%
\leavevmode%
\llap{\spacer}%
${L_{52.35}}=\pdualwitharg{2.19}(\main({L_{52.3}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1280220, 45600, -281352; 45600, 1292, -10260; -281352, -10260, 61661],[228, 76, 1],[14993, 2303, 1799, 951, 317, -13, -895, -289, 215, 1063, 5725, 2027; -43947, -6749, -5271, -2785, -928, 38, 2619, 845, -633, -3119, -16784, -5942; 61104, 9386, 7332, 3876, 1292, -53, -3648, -1178, 876, 4332, 23332, 8261],51,35]
%
% END OF LATTICE L52.35
%
\hbox{}\par\smallskip%
%
% LATTICE L52.36
%
\leavevmode%
\llap{\spacer}%
${L_{52.36}}=\pdualwitharg{2.19}({L_{52.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[390336, -13452, -106020; -13452, 760, 3876; -106020, 3876, 28963],[228, 76, 1],[27118, 1947, 2884, 1326, 663, 202, -56, -14, 982, 2540, 10935, 7530; -82785, -5944, -8805, -4049, -2025, -617, 171, 43, -2997, -7753, -33381, -22987; 110352, 7923, 11736, 5396, 2698, 822, -228, -57, 3996, 10336, 44498, 30642],51,36]
%
% END OF LATTICE L52.36
%
\hbox{}\par\smallskip%
%
% LATTICE L52.37
%
\leavevmode%
\llap{\spacer}%
${L_{52.37}}=\pdualwitharg{3.19}({L_{52.3}})$%
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{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-133608, -27816, 1368; -27816, -5757, 285; 1368, 285, -14],[456, 57, 1],[1, 5, 1, 3, 1, -1, -1, -5, -1, -3, -1, 1; -1, -12, -6, -34, -110, -144, -41, -60, -6, -2, 2, 0; 76, 228, -34, -456, -2337, -3288, -1007, -1824, -232, -342, -57, 96],51,37]
%
% END OF LATTICE L52.37
%
\hbox{}\par\smallskip%
%
% LATTICE L52.38
%
\leavevmode%
\llap{\spacer}%
${L_{52.38}}=\pdualwitharg{3.19}({L_{52.2}})$%
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{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1485192, -129960, 4560; -129960, -11343, 399; 4560, 399, -14],[456, 57, 1],[1, -1, -1, -7, -54, -73, -43, -35, -2, -2, 0, 1; -2, -12, -3, -17, -110, -144, -82, -60, -3, -1, 2, 0; 266, -684, -416, -2793, -20919, -28140, -16492, -13224, -743, -684, 57, 324],51,38]
%
% END OF LATTICE L52.38
%
\hbox{}\par\smallskip%
%
% LATTICE L52.39
%
\leavevmode%
\llap{\spacer}%
${L_{52.39}}=\pdualwitharg{2.3.19}(\main({L_{52.3}}))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[228, 0, 0; 0, 3502308, 859560; 0, 859560, 210959],[228, 228, 1],[-1, -3, -3, -17, -55, -18, -41, -15, -3, -1, 1, 0; 429, 1315, 1557, 9289, 31896, 10565, 24379, 9317, 1987, 1343, 0, 67; -1748, -5358, -6344, -37848, -129960, -43047, -99332, -37962, -8096, -5472, 0, -273],51,39]
%
% END OF LATTICE L52.39
%
\hbox{}\par\smallskip%
%
% LATTICE L52.40
%
\leavevmode%
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${L_{52.40}}=\pdualwitharg{2.3.19}({L_{52.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[456, 216372, 53124; 216372, 39634608, 9730584; 53124, 9730584, 2388929],[228, 228, 1],[35, 52, 95, 521, 1617, 1049, 1183, 209, 79, 1, -51, -5; -1978, -2953, -5384, -29502, -91445, -59306, -66860, -11798, -4450, 0, 2939, 302; 8056, 12027, 21928, 120156, 372438, 241542, 272308, 48051, 18124, 0, -11970, -1230],51,40]
%
% END OF LATTICE L52.40
%
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%
% LATTICE L52.41
%
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${L_{52.41}}=\pdualwitharg{2.19}({L_{52.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L52.41
%
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%
% LATTICE L52.42
%
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${L_{52.42}}=\pdualwitharg{2.19}({L_{52.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11856, -2280, 912; -2280, 5624, -2128; 912, -2128, 805],[456, 152, 1],[1, -1, -2, -3, -11, -3, -8, 0, 1, 2, 5, 1; 87, -101, -219, -347, -1506, -440, -1305, -73, 45, 173, 462, 92; 228, -266, -576, -912, -3952, -1154, -3420, -190, 120, 456, 1216, 242],51,42]
%
% END OF LATTICE L52.42
%
\hbox{}\par\smallskip%
%
% LATTICE L52.43
%
\leavevmode%
\llap{\spacer}%
${L_{52.43}}=\pdualwitharg{2.3.19}({L_{52.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-1319208\\-275880\\4104
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6302832, -1319208, 19608; -1319208, -275880, 4104; 19608, 4104, -61],[456, 456, 1],[-81, -34, -8, -9, 1, 1, 4, 1, -3, -24, -199, -34; -41, -15, -3, -1, 2, 0, -1, -3, -3, -17, -110, -18; -28880, -11970, -2780, -2964, 456, 321, 1216, 114, -1172, -8892, -71592, -12177],51,43]
%
% END OF LATTICE L52.43
%
\hbox{}\par\smallskip%
%
% LATTICE L52.44
%
\leavevmode%
\llap{\spacer}%
${L_{52.44}}=\pdualwitharg{2.3.19}({L_{52.2}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[912, 13224, -5016; 13224, 175560, -66576; -5016, -66576, 25247],[456, 456, 1],[-5, -1, 0, 1, -1, -1, -2, -2, -1, -4, -17, -5; -2045, -821, -185, -173, 0, 16, 29, -43, -103, -691, -5184, -1744; -5396, -2166, -488, -456, 0, 42, 76, -114, -272, -1824, -13680, -4602],51,44]
%
% END OF LATTICE L52.44
%

\medskip%
%------------------------------WEYL-GROUP-W53--------------------------------
%
\leavevmode\llap{}%
$W_{53\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=28$%
\hfill%
$22232223\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L53.1
%
\leavevmode%
\llap{\spacer}%
${L_{53.1}}$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{53}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-527020, 3120, 4680; 3120, -18, -29; 4680, -29, -38],[260, 1, 1],[-1, -2, 7, 4, 26, 67, 43, 1; -109, -221, 760, 436, 2839, 7319, 4700, 110; -40, -78, 280, 159, 1030, 2652, 1700, 39],52,1]
%
% END OF LATTICE L53.1
%
\hbox{}\par\smallskip%
%
% LATTICE L53.2
%
\leavevmode%
\llap{\spacer}%
${L_{53.2}}=\pfillwitharg{2}({L_{53.1}})$%
{${}={}$}Nikulin $53$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[1430, 585, -65; 585, 239, -26; -65, -26, 2],[65, 1, 1],[0, -10, 6, 11, 90, 244, 84, 7; 0, 26, -15, -28, -230, -624, -215, -18; 1, 13, -5, -11, -91, -247, -85, -7],52,2]
%
% END OF LATTICE L53.2
%
\hbox{}\par\smallskip%
%
% LATTICE L53.3
%
\leavevmode%
\llap{\spacer}%
${L_{53.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{53.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{s}{2}{130}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{-}{3}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-995085\\9685\\-395200
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-995085, 9685, -395200; 9685, -90, 3845; -395200, 3845, -156954],[65, 5, 1],[-128, -282, 83, 492, 3238, 8368, 539, 130; 110, 208, -80, -449, -2900, -7462, -478, -109; 325, 715, -211, -1250, -8225, -21255, -1369, -330],52,3]
%
% END OF LATTICE L53.3
%
\hbox{}\par\smallskip%
%
% LATTICE L53.4
%
\leavevmode%
\llap{\spacer}%
${L_{53.4}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{53.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1921465\\-32305\\-740675
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32305\\-494\\-12454
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-740675\\-12454\\-285511
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{$\left.\llap{\phantom{%
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19275999\\-1175520\\-49953280
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2182\\-120\\-5655
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2453\\-146\\-6357
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-676\\-130\\-520\\-13
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\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1921465, -32305, -740675; -32305, -494, -12454; -740675, -12454, -285511],[65, 13, 1],[-602, -88, 2182, 2453, 15852, 3138, 13068, 597; 39, 7, -120, -146, -969, -193, -810, -40; 1560, 228, -5655, -6357, -41080, -8132, -33865, -1547],52,4]
%
% END OF LATTICE L53.4
%
\hbox{}\par\smallskip%
%
% LATTICE L53.5
%
\leavevmode%
\llap{\spacer}%
${L_{53.5}}=\pdualwitharg{2}({L_{53.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{-}{3}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13041080\\-613340\\3204500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-613340\\28856\\-150712
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3204500\\-150712\\787421
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6791589\\173550\\-27606020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-320502\\-8191\\1302756
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1668841\\42645\\-6783399
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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63\\2\\-256
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115\\0\\-468
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
444\\10\\-1805
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
862\\21\\-3504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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8\\0\\-20\\-36\\-208\\-520\\-80\\-4
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-520\\-80\\-4
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-1456\\-260\\-52
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-260\\-55\\-20
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-52\\-20\\-16
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13041080, -613340, 3204500; -613340, 28856, -150712; 3204500, -150712, 787421],[260, 4, 1],[63, 115, 444, 862, 5807, 15147, 2491, 312; 2, 0, 10, 21, 148, 390, 65, 9; -256, -468, -1805, -3504, -23604, -61568, -10125, -1268],52,5]
%
% END OF LATTICE L53.5
%
\hbox{}\par\smallskip%
%
% LATTICE L53.6
%
\leavevmode%
\llap{\spacer}%
${L_{53.6}}=\pdualwitharg{5}({L_{53.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{b}{2}{130}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{+}{3}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-151215740\\14271920\\-1414660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14271920\\-1346990\\133515
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1414660\\133515\\-13234
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-3198079\\-63111204\\-294851700
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-29929\\-590622\\-2759351
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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35\\691\\3230
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68\\1339\\6240
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-49\\-968\\-4528
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-139\\-2744\\-12825
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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10\\0\\-20\\-45\\-260\\-650\\-80\\-5
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-260\\-650\\-80\\-5
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-650\\-1820\\-260\\-65
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-80\\-260\\-44\\-20
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-151215740, 14271920, -1414660; 14271920, -1346990, 133515; -1414660, 133515, -13234],[260, 5, 1],[35, 68, -49, -139, -900, -2317, -297, -34; 691, 1339, -968, -2744, -17761, -45721, -5860, -670; 3230, 6240, -4528, -12825, -82980, -213590, -27372, -3125],52,6]
%
% END OF LATTICE L53.6
%
\hbox{}\par\smallskip%
%
% LATTICE L53.7
%
\leavevmode%
\llap{\spacer}%
${L_{53.7}}=\pdualwitharg{13}({L_{53.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{260}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{-}{3}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18460\\2080\\1040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2080\\-234\\-117
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1040\\-117\\-58
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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7\\60\\0
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3\\32\\-13
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-13\\0\\26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-676\\-130\\-1040\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-28\\-260\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1040\\-260\\-2860\\-260
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18460, 2080, 1040; 2080, -234, -117; 1040, -117, -58],[260, 13, 1],[-1, 0, 7, 3, 16, 3, 23, 0; -9, -1, 60, 32, 199, 39, 320, 6; 0, 2, 0, -13, -130, -28, -260, -13],52,7]
%
% END OF LATTICE L53.7
%
\hbox{}\par\smallskip%
%
% LATTICE L53.8
%
\leavevmode%
\llap{\spacer}%
${L_{53.8}}=\pdualwitharg{5.13}(\pfillwitharg{2}({L_{53.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${130}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{-}{3}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62010\\-7087210\\-2797665
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7087210\\810231565\\319837635
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2797665\\319837635\\126255403
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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-13951\\17662932\\-44745090
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\kern3pt%
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1736550\\-2198750149\\5570042010
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685500\\-867952680\\2198764099
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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8\\-10443\\26455
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-151\\192028\\-486460
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-3380\\-650\\-520\\-65
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[62010, -7087210, -2797665; -7087210, 810231565, 319837635; -2797665, 319837635, 126255403],[65, 65, 1],[8, -10, -44, -151, -758, -140, -106, 1; -10443, 12847, 56151, 192028, 960063, 177077, 133773, -2104; 26455, -32545, -142246, -486460, -2432105, -448585, -338884, 5330],52,8]
%
% END OF LATTICE L53.8
%
\hbox{}\par\smallskip%
%
% LATTICE L53.9
%
\leavevmode%
\llap{\spacer}%
${L_{53.9}}=\pdualwitharg{2.5}({L_{53.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{*}{2}{520}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{+}{3}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19163300520\\-329858620\\4713776860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-329858620\\5677880\\-81138420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4713776860\\-81138420\\1159491929
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{$\left.\llap{\phantom{%
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80628438\\5957429\\-327368460
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1527\\112\\-6200
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12231\\910\\-49660
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4269\\316\\-17333
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37338\\2761\\-151600
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19163300520, -329858620, 4713776860; -329858620, 5677880, -81138420; 4713776860, -81138420, 1159491929],[260, 20, 1],[1527, 12231, 4269, 37338, 234643, 601811, 19348, 9896; 112, 910, 316, 2761, 17338, 44460, 1429, 729; -6200, -49660, -17333, -151600, -952700, -2443480, -78557, -40180],52,9]
%
% END OF LATTICE L53.9
%
\hbox{}\par\smallskip%
%
% LATTICE L53.10
%
\leavevmode%
\llap{\spacer}%
${L_{53.10}}=\pdualwitharg{2.13}({L_{53.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${104}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{65}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{-}{3}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43160\\7540\\10660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7540\\2184\\1872
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\kern3pt%
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10660\\1872\\2633
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{$\left.\llap{\phantom{%
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-61751\\-2850\\251940
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446\\21\\-1820
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[43160, 7540, 10660; 7540, 2184, 1872; 10660, 1872, 2633],[260, 52, 1],[51, -1, 207, 446, 3199, 651, 1418, 204; 2, 0, 10, 21, 148, 30, 65, 9; -208, 4, -845, -1820, -13052, -2656, -5785, -832],52,10]
%
% END OF LATTICE L53.10
%
\hbox{}\par\smallskip%
%
% LATTICE L53.11
%
\leavevmode%
\llap{\spacer}%
${L_{53.11}}=\pdualwitharg{5.13}({L_{53.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${130}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{52}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{+}{3}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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20969780\\-20813390\\-198965
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0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21127340, 20969780, 200460; 20969780, -20813390, -198965; 200460, -198965, -1902],[260, 65, 1],[-11, -4, -11, -7, -4, 1, 5, 4; 199, 39, 64, 6, -9, -1, 12, 32; -21970, -4500, -7852, -1365, 520, 210, -728, -2925],52,11]
%
% END OF LATTICE L53.11
%
\hbox{}\par\smallskip%
%
% LATTICE L53.12
%
\leavevmode%
\llap{\spacer}%
${L_{53.12}}=\pdualwitharg{2.5.13}({L_{53.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${520}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{520}\above{1pt}{1pt}{+}{3}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10920\\905060\\222820
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
905060\\105274520\\25912120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
222820\\25912120\\6377973
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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75505\\1170286\\-4757220
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\kern3pt%
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8500386\\131749565\\-535562820
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2092311\\32429241\\-131825071
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-19556\\-303109\\1232140
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-3982\\-61717\\250880
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-1736\\-26905\\109369
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10920, 905060, 222820; 905060, 105274520, 25912120; 222820, 25912120, 6377973],[260, 260, 1],[-19556, -3982, -1736, -1255, -314, 8, -251, -2719; -303109, -61717, -26905, -19444, -4861, 123, -3892, -42150; 1232140, 250880, 109369, 79040, 19760, -500, 15821, 171340],52,12]
%
% END OF LATTICE L53.12
%

\medskip%
%------------------------------WEYL-GROUP-W54--------------------------------
%
\leavevmode\llap{}%
$W_{54\phantom{0}}$%
\qquad\llap{32} lattices, $\chi=48$%
\hfill%
$\infty2222\infty2222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L54.1
%
\leavevmode%
\llap{\spacer}%
${L_{54.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{54},3,2}%
\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114840\\24552\\9504
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24552\\-5190\\-2181
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9504\\-2181\\-410
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{$\left.\llap{\phantom{%
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375935\\1524864\\601920
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\kern3pt%
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-88466\\-358835\\-141645
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1874\\-7601\\-3003
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-8407\\-34100\\-13464
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\HardButStrongLineBreak\kern3pt%
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-1619\\-6567\\-2592
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\HardButStrongLineBreak\kern3pt%
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-1581\\-6413\\-2530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-991\\-4020\\-1584
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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66\\-132\\-99\\-231\\-396
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\0\\18\\0\\-72
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-264\\0\\22\\0
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-396\\-792\\-72\\0\\72
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-429\\-1320\\-198\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\-6072\\-1188\\-1188\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-1188\\-261\\-297\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-1188\\-297\\-385\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-792\\-252\\-396\\-504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-114840, 24552, 9504; 24552, -5190, -2181; 9504, -2181, -410],[792, 3, 1],[-1874, -8407, -1619, -1581, -991, -84, 575, 17, -377, -1145; -7601, -34100, -6567, -6413, -4020, -341, 2332, 69, -1529, -4644; -3003, -13464, -2592, -2530, -1584, -132, 924, 27, -605, -1836],53,1]
%
% END OF LATTICE L54.1
%
\hbox{}\par\smallskip%
%
% LATTICE L54.2
%
\leavevmode%
\llap{\spacer}%
${L_{54.2}}=\pfillwitharg{2.3}({L_{54.1}})$%
{${}={}$}Nikulin $54$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{2,1}{\infty a}{66}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\0\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[66, 0, 0; 0, 0, -1; 0, -1, 0],[66, 1, 1],[-1, 1, 0, -2, -1, -10, -23, -3, -9, -1; 0, 0, -1, -11, -4, -33, -66, -8, -22, -2; -33, 0, 1, -11, -8, -99, -264, -37, -121, -16],53,2]
%
% END OF LATTICE L54.2
%
\hbox{}\par\smallskip%
%
% LATTICE L54.3
%
\leavevmode%
\llap{\spacer}%
${L_{54.3}}=\pfillwitharg{3}({L_{54.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{4,1}{\infty b}{264}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-2461272\\-598752\\-34320
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-2461272, -598752, -34320; -598752, -145658, -8349; -34320, -8349, -478],[264, 1, 1],[-128, 289, 8, -190, -213, -1115, -5261, -347, -1053, -239; 528, -1188, -33, 781, 876, 4587, 21648, 1428, 4334, 984; -33, 0, 2, 0, -8, -66, -396, -29, -99, -28],53,3]
%
% END OF LATTICE L54.3
%
\hbox{}\par\smallskip%
%
% LATTICE L54.4
%
\leavevmode%
\llap{\spacer}%
${L_{54.4}}=\pfillwitharg{2}({L_{54.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{6,1}{\infty a}{66}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-3804174\\-354024\\-132858
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\kern3pt%
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-354024\\-32946\\-12363
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132858\\-12363\\-4634
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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915419\\-10521060\\1824570
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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384\\-4411\\759
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287\\-3300\\576
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-429\\-660\\-198\\-132\\0
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\end{tabular}\endgroup%
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-132\\-594\\-297\\-385\\-198
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3804174, -354024, -132858; -354024, -32946, -12363; -132858, -12363, -4634],[198, 3, 1],[384, -65, -64, 108, 287, 1222, 3277, 1378, 1498, 589; -4411, 748, 735, -1243, -3300, -14047, -37664, -15837, -17215, -6768; 759, -132, -126, 220, 576, 2442, 6534, 2745, 2981, 1170],53,4]
%
% END OF LATTICE L54.4
%
\hbox{}\par\smallskip%
%
% LATTICE L54.5
%
\leavevmode%
\llap{\spacer}%
${L_{54.5}}=\pdualwitharg{2}(\pfillwitharg{2.3}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{4,3}{\infty z}{33}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{44}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1184700\\2178\\-574398
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1184700, 2178, -574398; 2178, -4, 1056; -574398, 1056, -278495],[66, 2, 1],[386, 467, 125, 387, 23, 65, -16, -2, 64, 18; 1023, 1551, 456, 1562, 111, 528, 0, -33, -11, 30; -792, -957, -256, -792, -47, -132, 33, 4, -132, -37],53,5]
%
% END OF LATTICE L54.5
%
\hbox{}\par\smallskip%
%
% LATTICE L54.6
%
\leavevmode%
\llap{\spacer}%
${L_{54.6}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{2,1}{\infty a}{22}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-789690\\2178\\-251262
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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54119\\254364\\-167772
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-150\\-706\\465
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17230\\80981\\-53414
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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0\\0\\0
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92\\341\\-286
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227\\1034\\-704
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92\\456\\-285
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288\\1562\\-891
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35\\222\\-108
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-789690, 2178, -251262; 2178, -6, 693; -251262, 693, -79946],[66, 3, 1],[92, 227, 92, 288, 35, 18, -7, -2, 42, 25; 341, 1034, 456, 1562, 222, 176, 0, -33, -11, 60; -286, -704, -285, -891, -108, -55, 22, 6, -132, -78],53,6]
%
% END OF LATTICE L54.6
%
\hbox{}\par\smallskip%
%
% LATTICE L54.7
%
\leavevmode%
\llap{\spacer}%
${L_{54.7}}=\pdualwitharg{2.3}(\pfillwitharg{2.3}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{4,3}{\infty z}{11}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{132}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\kern3pt%
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10890\\3264\\5346
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80784\\5346\\39647
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-7316002\\-8415\\14908014
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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38435\\44\\-78320
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\HardButStrongLineBreak\kern3pt%
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47844\\55\\-97493
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39070\\45\\-79614
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\HardButStrongLineBreak\kern3pt%
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123888\\143\\-252450
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7766\\9\\-15825
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-66\\-33\\-12\\0\\3\\0\\-33\\-42\\-198\\-21
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{$\left[\!\llap{\phantom{%
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-66\\0\\12\\0\\-12
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-462\\-132\\0\\132\\0
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-66\\-33\\-12\\0\\3
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-286\\-220\\-132\\-264\\0
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-220\\-253\\-198\\-594\\-33
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-264\\-594\\-594\\-2310\\-198
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-33\\-42\\-198\\-21
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[164604, 10890, 80784; 10890, 3264, 5346; 80784, 5346, 39647],[66, 6, 1],[38435, 47844, 39070, 123888, 7766, 9382, -27, 53, 19563, 5275; 44, 55, 45, 143, 9, 11, 0, 0, 22, 6; -78320, -97493, -79614, -252450, -15825, -19118, 55, -108, -39864, -10749],53,7]
%
% END OF LATTICE L54.7
%
\hbox{}\par\smallskip%
%
% LATTICE L54.8
%
\leavevmode%
\llap{\spacer}%
${L_{54.8}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{6,5}{\infty a}{66}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{198}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-842094, -80388, 21780; -80388, -7674, 2079; 21780, 2079, -544],[198, 3, 1],[373, -21, -22, 104, 61, 892, 2551, 367, 3691, 169; -3916, 220, 231, -1089, -640, -9361, -26774, -3852, -38742, -1774; -33, 0, 2, 0, -4, -66, -198, -29, -297, -14],53,8]
%
% END OF LATTICE L54.8
%
\hbox{}\par\smallskip%
%
% LATTICE L54.9
%
\leavevmode%
\llap{\spacer}%
${L_{54.9}}=\pdualwitharg{11}(\pfillwitharg{2.3}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{2,1}{\infty a}{6}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{22}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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541\\-18\\2964
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879\\-29\\4818
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-66, -2178, 0; -2178, -5984, 363; 0, 363, 2],[66, 11, 1],[176, 541, 879, 275, 433, 95, 1, -66, -4, 109; -6, -18, -29, -9, -14, -3, 0, 2, 0, -4; 963, 2964, 4818, 1508, 2376, 522, 6, -363, -23, 594],53,9]
%
% END OF LATTICE L54.9
%
\hbox{}\par\smallskip%
%
% LATTICE L54.10
%
\leavevmode%
\llap{\spacer}%
${L_{54.10}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{4,1}{\infty b}{88}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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182951\\676368\\2422728
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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15\\55\\209
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\HardButStrongLineBreak\kern3pt%
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-107\\-396\\-1408
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-3\\-11\\-42
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211\\781\\2772
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213\\788\\2808
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1687752, 397848, 16368; 397848, -93774, -3861; 16368, -3861, -158],[264, 3, 1],[15, -107, -3, 211, 213, 348, 1559, 300, 878, 183; 55, -396, -11, 781, 788, 1287, 5764, 1109, 3245, 676; 209, -1408, -42, 2772, 2808, 4598, 20636, 3975, 11649, 2436],53,10]
%
% END OF LATTICE L54.10
%
\hbox{}\par\smallskip%
%
% LATTICE L54.11
%
\leavevmode%
\llap{\spacer}%
${L_{54.11}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{54.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5544, -35046, 3762; -35046, 688056, -37308; 3762, -37308, 2945],[198, 6, 1],[22359, 376, 205, 14655, 11888, 87179, 109162, 89417, 94883, 17986; -1309, -22, -12, -858, -696, -5104, -6391, -5235, -5555, -1053; -45144, -759, -414, -29590, -24003, -176022, -220407, -180540, -191576, -36315],53,11]
%
% END OF LATTICE L54.11
%
\hbox{}\par\smallskip%
%
% LATTICE L54.12
%
\leavevmode%
\llap{\spacer}%
${L_{54.12}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{12,11}{\infty z}{33}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{396}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[21245400, -3236310, 10618146; -3236310, 493008, -1617462; 10618146, -1617462, 5306797],[198, 6, 1],[9566, 343, -1, 16355, 1519, 34115, 43338, 11914, 114688, 2452; -649, -22, 0, -1122, -104, -2332, -2959, -813, -7821, -167; -19338, -693, 2, -33066, -3071, -68970, -87615, -24086, -231858, -4957],53,12]
%
% END OF LATTICE L54.12
%
\hbox{}\par\smallskip%
%
% LATTICE L54.13
%
\leavevmode%
\llap{\spacer}%
${L_{54.13}}=\pdualwitharg{2.11}(\pfillwitharg{2.3}({L_{54.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{4,3}{\infty z}{3}\above{1pt}{1pt}{r}{2}{44}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5283696, 15246, 2516052; 15246, 44, 7260; 2516052, 7260, 1198123],[66, 22, 1],[-452, -545, -1605, -453, -299, -83, 10, 0, -82, -236; 195, 243, 728, 210, 145, 48, 0, 1, 33, 98; 948, 1143, 3366, 950, 627, 174, -21, 0, 172, 495],53,13]
%
% END OF LATTICE L54.13
%
\hbox{}\par\smallskip%
%
% LATTICE L54.14
%
\leavevmode%
\llap{\spacer}%
${L_{54.14}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${528}\above{1pt}{1pt}{8,3}{\infty z}{132}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{176}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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617496\\-98064\\19336
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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-3432\\-2640\\-528\\-1056\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3886608, 617496, -121704; 617496, -98064, 19336; -121704, 19336, -3811],[264, 8, 1],[590, -31, -33, 113, 61, 1643, 2264, 642, 2120, 142; -33, 0, 2, 0, -2, -66, -99, -29, -99, -7; -19008, 990, 1064, -3608, -1958, -52800, -72798, -20648, -68200, -4570],53,14]
%
% END OF LATTICE L54.14
%
\hbox{}\par\smallskip%
%
% LATTICE L54.15
%
\leavevmode%
\llap{\spacer}%
${L_{54.15}}=\pdualwitharg{3.11}(\pfillwitharg{2.3}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty a}{2}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{66}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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747\\-766\\2475
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1754082, -10758, -532686; -10758, 66, 3267; -532686, 3267, 161768],[66, 33, 1],[70, 169, 747, 211, 279, 13, -3, 0, 38, 219; -75, -176, -766, -212, -268, -10, 6, 1, -43, -242; 232, 560, 2475, 699, 924, 43, -10, 0, 126, 726],53,15]
%
% END OF LATTICE L54.15
%
\hbox{}\par\smallskip%
%
% LATTICE L54.16
%
\leavevmode%
\llap{\spacer}%
${L_{54.16}}=\pdualwitharg{3}({L_{54.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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-396\\-3564\\-297\\-3465\\-396
\end{tabular}\endgroup%
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0\\-264\\-28\\-396\\-56
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3682008, -327096, 41976; -327096, -29058, 3729; 41976, 3729, -478],[792, 3, 1],[-16, 43, 1, -85, -31, -160, -747, -49, -443, -33; 176, -484, -11, 957, 348, 1793, 8360, 548, 4950, 368; -33, 0, 2, 0, -8, -66, -396, -29, -297, -28],53,16]
%
% END OF LATTICE L54.16
%
\hbox{}\par\smallskip%
%
% LATTICE L54.17
%
\leavevmode%
\llap{\spacer}%
${L_{54.17}}=\pdualwitharg{11}(\pfillwitharg{3}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{4,1}{\infty b}{24}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{88}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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25080\\11880\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11880\\3146\\99
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\99\\-38
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-857\\2640\\18744
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-3317\\10229\\72633
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428\\-1320\\-9373
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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35\\-108\\-768
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\HardButStrongLineBreak\kern3pt%
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1\\-3\\-22
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\HardButStrongLineBreak\kern3pt%
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-23\\71\\504
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\HardButStrongLineBreak\kern3pt%
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-245\\756\\5368
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-12\\24\\0\\-24\\-264
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\22\\0\\-88
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-24\\0\\2\\0
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-132\\-264\\-88\\0\\88
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\kern3pt%
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-39\\-120\\-66\\-12\\0
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\kern3pt%
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-120\\-552\\-396\\-108\\-264
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-396\\-319\\-99\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-108\\-99\\-35\\-132
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-264\\-308\\-132\\-616
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25080, 11880, -528; 11880, 3146, 99; -528, 99, -38],[264, 11, 1],[-1, 35, 1, -23, -245, -106, -463, -322, -84, -183; 3, -108, -3, 71, 756, 327, 1428, 993, 259, 564; 21, -768, -22, 504, 5368, 2322, 10140, 7051, 1839, 4004],53,17]
%
% END OF LATTICE L54.17
%
\hbox{}\par\smallskip%
%
% LATTICE L54.18
%
\leavevmode%
\llap{\spacer}%
${L_{54.18}}=\pdualwitharg{2.3.11}(\pfillwitharg{2.3}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{r}{2}{132}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37620\\20130\\9306
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20130\\-4884\\-2244
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9306\\-2244\\-1031
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-766\\-672945\\1458600
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\kern3pt%
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-981\\-862954\\1870440
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-453\\-398489\\863719
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2\\-1707\\3700
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-2633\\5707
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\HardButStrongLineBreak\kern3pt%
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-29\\-25700\\55704
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-8050\\17448
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-6349\\13761
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-66\\0\\132\\0\\-132
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-24\\-54\\-594\\-210\\-198
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0\\-33\\-462\\-198\\-231
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[37620, 20130, 9306; 20130, -4884, -2244; 9306, -2244, -1031],[66, 66, 1],[-2, -3, -29, -9, -7, -1, 0, 2, 0, -2; -1707, -2633, -25700, -8050, -6349, -932, -6, 1949, 133, -1568; 3700, 5707, 55704, 17448, 13761, 2020, 13, -4224, -288, 3399],53,18]
%
% END OF LATTICE L54.18
%
\hbox{}\par\smallskip%
%
% LATTICE L54.19
%
\leavevmode%
\llap{\spacer}%
${L_{54.19}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{8,3}{\infty z}{44}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{s}{2}{528}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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135015936\\580200\\-1550377
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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2691\\-11\\234344
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-361\\0\\-31438
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-393\\2\\-34224
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3947\\0\\343728
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1371\\-2\\119394
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11757980784, -50527224, 135015936; -50527224, -217104, 580200; 135015936, 580200, -1550377],[264, 24, 1],[2691, -361, -393, 3947, 1371, 10773, 13825, 11409, 36445, 2301; -11, 0, 2, 0, -2, -22, -33, -29, -99, -7; 234344, -31438, -34224, 343728, 119394, 938168, 1203950, 993552, 3173808, 200382],53,19]
%
% END OF LATTICE L54.19
%
\hbox{}\par\smallskip%
%
% LATTICE L54.20
%
\leavevmode%
\llap{\spacer}%
${L_{54.20}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{54.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-208890, 743688, -61182; 743688, -2645742, 217635; -61182, 217635, -17902],[198, 33, 1],[-30, 29, 32, -48, -841, -262, -613, -2660, -244, -911; -61, 58, 66, -96, -1686, -526, -1232, -5349, -491, -1836; -639, 606, 693, -1003, -17622, -5499, -12882, -55935, -5135, -19206],53,20]
%
% END OF LATTICE L54.20
%
\hbox{}\par\smallskip%
%
% LATTICE L54.21
%
\leavevmode%
\llap{\spacer}%
${L_{54.21}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{54.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L54.21
%
\hbox{}\par\smallskip%
%
% LATTICE L54.22
%
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${L_{54.22}}=\pdualwitharg{3.11}(\pfillwitharg{3}({L_{54.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-28248, -3432, 2376; -3432, -198, 165; 2376, 165, -130],[264, 33, 1],[1, -1, -2, 2, 29, 5, 25, 56, 16, 43; 37, -36, -75, 71, 1044, 181, 908, 2037, 583, 1572; 65, -64, -132, 126, 1848, 320, 1604, 3597, 1029, 2772],53,22]
%
% END OF LATTICE L54.22
%
\hbox{}\par\smallskip%
%
% LATTICE L54.23
%
\leavevmode%
\llap{\spacer}%
${L_{54.23}}=\pdualwitharg{2.11}(\pfillwitharg{2}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,7}{\infty z}{3}\above{1pt}{1pt}{r}{2}{396}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{99}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[988434216, -34860078, 488697462; -34860078, 1229448, -17235372; 488697462, -17235372, 241619731],[198, 66, 1],[2367, 79, 98, 1400, 12739, 8618, 10906, 98767, 9585, 20216; 329, 11, 3, 191, 1749, 1187, 1505, 13641, 1325, 2799; -4764, -159, -198, -2818, -25641, -17346, -21951, -198792, -19292, -40689],53,23]
%
% END OF LATTICE L54.23
%
\hbox{}\par\smallskip%
%
% LATTICE L54.24
%
\leavevmode%
\llap{\spacer}%
${L_{54.24}}=\pdualwitharg{2.3.11}(\pfillwitharg{2}({L_{54.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[615558240, 17207190, 304654284; 17207190, 481008, 8516244; 304654284, 8516244, 150780587],[198, 66, 1],[1344, 46, -11, 2287, 2344, 4791, 6089, 18416, 16118, 3791; 43, 1, 4, 90, 87, 172, 214, 641, 555, 128; -2718, -93, 22, -4626, -4741, -9690, -12315, -37246, -32598, -7667],53,24]
%
% END OF LATTICE L54.24
%
\hbox{}\par\smallskip%
%
% LATTICE L54.25
%
\leavevmode%
\llap{\spacer}%
${L_{54.25}}=\pdualwitharg{2}({L_{54.1}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-218422512, -17207784, -912384; -17207784, -1355664, -71880; -912384, -71880, -3811],[792, 24, 1],[238, -31, -35, 113, 119, 939, 1208, 998, 1064, 202; -2629, 341, 387, -1243, -1311, -10351, -13321, -11007, -11737, -2229; -7392, 990, 1080, -3608, -3762, -29568, -37950, -31320, -33352, -6318],53,25]
%
% END OF LATTICE L54.25
%
\hbox{}\par\smallskip%
%
% LATTICE L54.26
%
\leavevmode%
\llap{\spacer}%
${L_{54.26}}=\pdualwitharg{2.3}({L_{54.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${528}\above{1pt}{1pt}{24,11}{\infty z}{132}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{1584}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\kern3pt%
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-126917208\\-10944912\\-1050696
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{$\left.\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1471730832, -126917208, -12184128; -126917208, -10944912, -1050696; -12184128, -1050696, -100811],[792, 24, 1],[-12027, 629, 673, -6877, -1241, -33447, -46103, -13075, -129545, -2893; 144914, -7579, -8109, 82863, 14953, 403007, 555500, 157542, 1560900, 34858; -56760, 2970, 3176, -32472, -5858, -157872, -217602, -61712, -611424, -13654],53,26]
%
% END OF LATTICE L54.26
%
\hbox{}\par\smallskip%
%
% LATTICE L54.27
%
\leavevmode%
\llap{\spacer}%
${L_{54.27}}=\pdualwitharg{2.11}(\pfillwitharg{3}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{8,3}{\infty z}{12}\above{1pt}{1pt}{*}{2}{176}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-9311021808\\9230066472\\77113872
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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121\\0\\14610
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139\\2\\16544
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-1341\\-2\\-161678
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9311021808, 9230066472, 77113872; 9230066472, -9149814960, -76443400; 77113872, -76443400, -638657],[264, 88, 1],[-386, 121, 139, -441, -1341, -2579, -3086, -9034, -2524, -1624; -3, 0, 2, 0, -2, -6, -9, -29, -9, -7; -46248, 14610, 16544, -53248, -161678, -310680, -371538, -1087328, -303680, -195250],53,27]
%
% END OF LATTICE L54.27
%
\hbox{}\par\smallskip%
%
% LATTICE L54.28
%
\leavevmode%
\llap{\spacer}%
${L_{54.28}}=\pdualwitharg{11}({L_{54.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,1}{\infty b}{24}\above{1pt}{1pt}{b}{2}{198}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{792}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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206735\\-1058771\\12106545
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721\\-3692\\42216
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1616\\-8277\\94644
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154\\-789\\9022
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-84744, -415008, -34848; -415008, -1981650, -166221; -34848, -166221, -13942],[792, 33, 1],[144, 721, 1616, 154, 1243, 29, -29, -59, 19, 833; -737, -3692, -8277, -789, -6372, -149, 148, 303, -97, -4260; 8427, 42216, 94644, 9022, 72864, 1704, -1692, -3465, 1109, 48708],53,28]
%
% END OF LATTICE L54.28
%
\hbox{}\par\smallskip%
%
% LATTICE L54.29
%
\leavevmode%
\llap{\spacer}%
${L_{54.29}}=\pdualwitharg{3.11}({L_{54.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,5}{\infty a}{24}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{b}{2}{88}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1288584\\153648\\-7128
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
153648\\-17886\\825
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3\\133\\2322
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\HardButStrongLineBreak\kern3pt%
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13\\580\\10140
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\HardButStrongLineBreak\kern3pt%
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9\\403\\7051
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\HardButStrongLineBreak\kern3pt%
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7\\315\\5517
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\228\\4004
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-120\\-552\\-396\\-324\\-264
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-66\\-396\\-319\\-297\\-308
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\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1288584, 153648, -7128; 153648, -17886, 825; -7128, 825, -38],[792, 33, 1],[3, 13, 9, 7, 5, 0, -1, 0, 2, 7; 133, 580, 403, 315, 228, 1, -44, -1, 87, 308; 2322, 10140, 7051, 5517, 4004, 21, -768, -22, 1512, 5368],53,29]
%
% END OF LATTICE L54.29
%
\hbox{}\par\smallskip%
%
% LATTICE L54.30
%
\leavevmode%
\llap{\spacer}%
${L_{54.30}}=\pdualwitharg{2.3.11}(\pfillwitharg{3}({L_{54.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,3}{\infty z}{4}\above{1pt}{1pt}{*}{2}{528}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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-13626888\\-90595824\\-21860256
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3288120\\-21860256\\-5274755
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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565283\\-21882213\\90334464
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3868968\\-149768227\\618275328
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933664\\-36142248\\149202943
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-589\\22805\\-94144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-683\\26440\\-109150
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-5903\\228494\\-943272
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\HardButStrongLineBreak\kern3pt%
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-1617\\62584\\-258360
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-997\\38578\\-159258
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\-132\\-1848\\-792\\-924
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2044944, -13626888, -3288120; -13626888, -90595824, -21860256; -3288120, -21860256, -5274755],[264, 264, 1],[-589, -683, -5903, -1617, -997, -63, 31, 35, -339, -959; 22805, 26440, 228494, 62584, 38578, 2434, -1201, -1343, 13133, 37139; -94144, -109150, -943272, -258360, -159258, -10048, 4958, 5544, -54216, -153318],53,30]
%
% END OF LATTICE L54.30
%
\hbox{}\par\smallskip%
%
% LATTICE L54.31
%
\leavevmode%
\llap{\spacer}%
${L_{54.31}}=\pdualwitharg{2.11}({L_{54.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,19}{\infty z}{12}\above{1pt}{1pt}{*}{2}{1584}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{396}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4549248\\1423224\\-136265
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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2818595\\174454665\\1916195688
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914104\\56577709\\621444912
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-87368\\-5407570\\-59396305
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-589\\-36457\\-400440
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-151406640, -47522376, 4549248; -47522376, -14864784, 1423224; 4549248, 1423224, -136265],[792, 264, 1],[-589, -683, -5903, -539, -997, -63, 31, 35, -113, -959; -36457, -42274, -365358, -33360, -61704, -3898, 1919, 2163, -6995, -59361; -400440, -464334, -4013064, -366424, -677754, -42816, 21078, 23760, -76832, -652014],53,31]
%
% END OF LATTICE L54.31
%
\hbox{}\par\smallskip%
%
% LATTICE L54.32
%
\leavevmode%
\llap{\spacer}%
${L_{54.32}}=\pdualwitharg{2.3.11}({L_{54.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\kern3pt%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1542234672, -2090830104, 712765152; -2090830104, -2834568528, 966306000; 712765152, 966306000, -329414257],[792, 264, 1],[-187, 61, 65, -667, -671, -1285, -1533, -4481, -3749, -801; -6121, 1982, 2142, -21672, -21832, -41842, -49945, -146031, -122217, -26131; -18360, 5946, 6424, -65016, -65494, -125520, -149826, -438064, -366624, -78386],53,32]
%
% END OF LATTICE L54.32
%

\medskip%
%------------------------------WEYL-GROUP-W55--------------------------------
%
\leavevmode\llap{}%
$W_{55\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=20$%
\hfill%
$622622\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L55.1
%
\leavevmode%
\llap{\spacer}%
${L_{55.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{23\rightarrow N_{55},3,2}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-1\\-196\\-201
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-19\\-3720\\-3780
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1\\196\\200
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-16860888, 101376, -15048; 101376, -606, 87; -15048, 87, -10],[792, 3, 1],[4, -1, -19, -8, -3, 1; 783, -196, -3720, -1566, -587, 196; 792, -201, -3780, -1589, -594, 200],54,1]
%
% END OF LATTICE L55.1
%
\hbox{}\par\smallskip%
%
% LATTICE L55.2
%
\leavevmode%
\llap{\spacer}%
${L_{55.2}}=\pfillwitharg{2.3}({L_{55.1}})$%
{${}={}$}Nikulin $55$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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858\\272\\21
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{$\left.\llap{\phantom{%
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3959\\-12408\\-2112
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[2706, 858, 66; 858, 272, 21; 66, 21, 2],[66, 1, 1],[1, -2, 21, 74, 47, 9; -3, 6, -66, -232, -147, -28; -1, 3, -10, -39, -27, -6],54,2]
%
% END OF LATTICE L55.2
%
\hbox{}\par\smallskip%
%
% LATTICE L55.3
%
\leavevmode%
\llap{\spacer}%
${L_{55.3}}=\pfillwitharg{3}({L_{55.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L55.3
%
\hbox{}\par\smallskip%
%
% LATTICE L55.4
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-922086, 3762, 7722; 3762, -12, -33; 7722, -33, -64],[198, 3, 1],[-1, 3, 11, 7, 1, -1; -42, 131, 474, 299, 40, -44; -99, 294, 1080, 688, 99, -98],54,4]
%
% END OF LATTICE L55.4
%
\hbox{}\par\smallskip%
%
% LATTICE L55.5
%
\leavevmode%
\llap{\spacer}%
${L_{55.5}}=\pdualwitharg{2}(\pfillwitharg{2.3}({L_{55.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L55.5
%
\hbox{}\par\smallskip%
%
% LATTICE L55.6
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-358842, 8316, -118206; 8316, -192, 2739; -118206, 2739, -38938],[66, 3, 1],[19, -27, 997, 3471, 725, 399; -33, 43, -1488, -5197, -1089, -604; -60, 85, -3132, -10905, -2278, -1254],54,6]
%
% END OF LATTICE L55.6
%
\hbox{}\par\smallskip%
%
% LATTICE L55.7
%
\leavevmode%
\llap{\spacer}%
${L_{55.7}}=\pdualwitharg{2.3}(\pfillwitharg{2.3}({L_{55.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7591980, -217206, 3725634; -217206, 6240, -106590; 3725634, -106590, 1828291],[66, 6, 1],[53, 967, -33161, -225873, -46131, -12003; 0, 9, -300, -2045, -418, -109; -108, -1970, 67557, 460158, 93980, 24453],54,7]
%
% END OF LATTICE L55.7
%
\hbox{}\par\smallskip%
%
% LATTICE L55.8
%
\leavevmode%
\llap{\spacer}%
${L_{55.8}}=\pdualwitharg{11}(\pfillwitharg{2.3}({L_{55.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1014882, -13794, 182094; -13794, -176, 2475; 182094, 2475, -32672],[66, 11, 1],[-77, 302, -3493, -12198, -7667, -1417; 2, -3, -10, -27, -12, 0; -429, 1683, -19470, -67991, -42735, -7898],54,8]
%
% END OF LATTICE L55.8
%
\hbox{}\par\smallskip%
%
% LATTICE L55.9
%
\leavevmode%
\llap{\spacer}%
${L_{55.9}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{55.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L55.9
%
\hbox{}\par\smallskip%
%
% LATTICE L55.10
%
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${L_{55.10}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{55.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[97777944, -1350558, 48384864; -1350558, 18660, -668316; 48384864, -668316, 23942977],[198, 6, 1],[196, 1541, 4859, 8273, 3462, 48; 3, 4, 9, 14, 5, 0; -396, -3114, -9819, -16718, -6996, -97],54,10]
%
% END OF LATTICE L55.10
%
\hbox{}\par\smallskip%
%
% LATTICE L55.11
%
\leavevmode%
\llap{\spacer}%
${L_{55.11}}=\pdualwitharg{2.11}(\pfillwitharg{2.3}({L_{55.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14370576, 18150, 6845652; 18150, 44, 8646; 6845652, 8646, 3261035],[66, 22, 1],[0, -3427, 38090, 259645, 159214, 13839; 1, 15, -184, -1251, -765, -66; 0, 7194, -79959, -545050, -334224, -29051],54,11]
%
% END OF LATTICE L55.11
%
\hbox{}\par\smallskip%
%
% LATTICE L55.12
%
\leavevmode%
\llap{\spacer}%
${L_{55.12}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{55.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-23228304, 936408, -11393976; 936408, -37744, 459328; -11393976, 459328, -5588987],[264, 8, 1],[-263, 1507, 1149, 4973, 1319, -207; -2, 3, 5, 27, 12, 0; 536, -3072, -2342, -10136, -2688, 422],54,12]
%
% END OF LATTICE L55.12
%
\hbox{}\par\smallskip%
%
% LATTICE L55.13
%
\leavevmode%
\llap{\spacer}%
${L_{55.13}}=\pdualwitharg{3.11}(\pfillwitharg{2.3}({L_{55.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4915482, -12936, -1484934; -12936, 66, 3927; -1484934, 3927, 448600],[66, 33, 1],[0, 519, -34611, -117965, -24112, -12575; 1, -1033, 68854, 234679, 47969, 25018; 0, 1727, -115170, -392535, -80234, -41844],54,13]
%
% END OF LATTICE L55.13
%
\hbox{}\par\smallskip%
%
% LATTICE L55.14
%
\leavevmode%
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${L_{55.14}}=\pdualwitharg{11}(\pfillwitharg{3}({L_{55.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3989832, 210144, 112728; 210144, -11066, -5929; 112728, -5929, -3154],[264, 11, 1],[494, 209, -15, -44, 61, 375; 10967, 4641, -332, -977, 1353, 8324; -2959, -1254, 88, 264, -363, -2244],54,14]
%
% END OF LATTICE L55.14
%
\hbox{}\par\smallskip%
%
% LATTICE L55.15
%
\leavevmode%
\llap{\spacer}%
${L_{55.15}}=\pdualwitharg{2.3.11}(\pfillwitharg{2.3}({L_{55.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-56240844, -37499154, -17325462; -37499154, -25002912, -11551914; -17325462, -11551914, -5337247],[66, 66, 1],[2, -1, -5, -27, -4, 0; 2345, -3119, 54850, 382825, 80153, 22184; -5082, 6754, -118701, -828498, -173470, -48015],54,15]
%
% END OF LATTICE L55.15
%
\hbox{}\par\smallskip%
%
% LATTICE L55.16
%
\leavevmode%
\llap{\spacer}%
${L_{55.16}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{55.1}}))$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-72336, 40392, -915288; 40392, -17904, 409824; -915288, 409824, -9376313],[264, 24, 1],[6134, 443, -347, -439, 696, 1486; -270181, -19510, 15286, 19336, -30659, -65455; -12408, -896, 702, 888, -1408, -3006],54,16]
%
% END OF LATTICE L55.16
%
\hbox{}\par\smallskip%
%
% LATTICE L55.17
%
\leavevmode%
\llap{\spacer}%
${L_{55.17}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{55.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-56826, 72666, 23562; 72666, -92796, -30063; 23562, -30063, -9734],[198, 33, 1],[-221, -12, 41, 28, -115, -377; -433, -20, 82, 54, -229, -744; 803, 33, -154, -99, 429, 1386],54,17]
%
% END OF LATTICE L55.17
%
\hbox{}\par\smallskip%
%
% LATTICE L55.18
%
\leavevmode%
\llap{\spacer}%
${L_{55.18}}=\pdualwitharg{3.11}(\pfillwitharg{3}({L_{55.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\-33\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\0\\264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-891\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-55\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-132\\-792
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-528792, -509520, 41448; -509520, -490710, 39897; 41448, 39897, -3242],[264, 33, 1],[578, 40, -143, -53, 65, 563; -1161, -80, 288, 106, -131, -1132; -6897, -473, 1716, 627, -781, -6732],54,18]
%
% END OF LATTICE L55.18
%
\hbox{}\par\smallskip%
%
% LATTICE L55.19
%
\leavevmode%
\llap{\spacer}%
${L_{55.19}}=\pdualwitharg{2.11}(\pfillwitharg{2}({L_{55.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{}{6}{132}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{396}\above{1pt}{1pt}{}{6}{132}\above{1pt}{1pt}{l}{2}{99}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50275368\\5823378\\24972156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5823378\\674520\\2892516
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24972156\\2892516\\12403859
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-829\\-579\\1804
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-332\\-247\\726
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-4\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-98\\-3\\198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-185\\-101\\396
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-502\\-336\\1089
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\-66\\-110\\-1782\\-264\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\132\\0\\-792\\-330\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\0\\11\\0\\-66\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1782\\-792\\0\\396\\-198\\-990
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-330\\-66\\-198\\132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-198\\-99\\-990\\0\\99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[50275368, 5823378, 24972156; 5823378, 674520, 2892516; 24972156, 2892516, 12403859],[198, 66, 1],[-829, -332, -5, -98, -185, -502; -579, -247, -4, -3, -101, -336; 1804, 726, 11, 198, 396, 1089],54,19]
%
% END OF LATTICE L55.19
%
\hbox{}\par\smallskip%
%
% LATTICE L55.20
%
\leavevmode%
\llap{\spacer}%
${L_{55.20}}=\pdualwitharg{2}({L_{55.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${144}\above{1pt}{1pt}{}{6}{48}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{}{6}{48}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114971472\\-6002568\\-90010008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6002568\\-313296\\-4698840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90010008\\-4698840\\-70465139
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-931\\-10239\\1872
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4416\\48575\\-8880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7564\\83199\\-15210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9242\\101653\\-18584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
943\\10369\\-1896
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-735\\-8085\\1478
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-72\\-360\\-648\\-288\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\48\\0\\-96\\-120\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\36\\0\\-72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-648\\-96\\0\\16\\-24\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-120\\-72\\-24\\48\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\-36\\-40\\0\\4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-114971472, -6002568, -90010008; -6002568, -313296, -4698840; -90010008, -4698840, -70465139],[792, 24, 1],[-931, 4416, 7564, 9242, 943, -735; -10239, 48575, 83199, 101653, 10369, -8085; 1872, -8880, -15210, -18584, -1896, 1478],54,20]
%
% END OF LATTICE L55.20
%
\hbox{}\par\smallskip%
%
% LATTICE L55.21
%
\leavevmode%
\llap{\spacer}%
${L_{55.21}}=\pdualwitharg{2.11}(\pfillwitharg{3}({L_{55.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{}{6}{528}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19989552\\-6269736\\138600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6269736\\-1966448\\43472
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
138600\\43472\\-961
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75001\\9375\\-10395000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23440\\2929\\-3248784
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
520\\-65\\72071
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\2\\-1496
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\-3\\9240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-5\\6842
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
176\\-264\\-440\\-2376\\-1056\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\528\\0\\-1056\\-1320\\-264
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\0\\44\\0\\-264\\-132
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
176\\-264\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\528\\0
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\kern3pt%
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-440\\0\\44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2376\\-1056\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1056\\-1320\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-264\\-132
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19989552, -6269736, 138600; -6269736, -1966448, 43472; 138600, 43472, -961],[264, 88, 1],[-11, 65, 49, 211, 55, -9; 2, -3, -5, -27, -12, 0; -1496, 9240, 6842, 29216, 7392, -1298],54,21]
%
% END OF LATTICE L55.21
%
\hbox{}\par\smallskip%
%
% LATTICE L55.22
%
\leavevmode%
\llap{\spacer}%
${L_{55.22}}=\pdualwitharg{11}({L_{55.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{}{6}{66}\above{1pt}{1pt}{b}{2}{88}\above{1pt}{1pt}{b}{2}{198}\above{1pt}{1pt}{}{6}{66}\above{1pt}{1pt}{b}{2}{792}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27720\\-26928\\17424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26928\\-6666\\6171
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17424\\6171\\-5018
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-445\\-2652\\-4807
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-983\\-1782
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57\\340\\616
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
220\\1311\\2376
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58\\-346\\-627
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1063\\-6336\\-11484
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\-33\\-220\\-891\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\66\\0\\-396\\-165\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\0\\88\\0\\-132\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-891\\-396\\0\\198\\-99\\-1980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-165\\-132\\-99\\66\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-396\\-792\\-1980\\0\\792
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-27720, -26928, 17424; -26928, -6666, 6171; 17424, 6171, -5018],[792, 33, 1],[-445, -165, 57, 220, -58, -1063; -2652, -983, 340, 1311, -346, -6336; -4807, -1782, 616, 2376, -627, -11484],54,22]
%
% END OF LATTICE L55.22
%
\hbox{}\par\smallskip%
%
% LATTICE L55.23
%
\leavevmode%
\llap{\spacer}%
${L_{55.23}}=\pdualwitharg{2.3.11}(\pfillwitharg{3}({L_{55.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${528}\above{1pt}{1pt}{}{6}{176}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34320\\-152856\\52008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-152856\\-671088\\228360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52008\\228360\\-77707
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1335\\-32752\\-97152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5365\\-131721\\-390720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1827\\44856\\133055
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\3827\\11352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\237\\704
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-245\\-726
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\-264\\-1320\\-7128\\-1056\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\176\\0\\-1056\\-440\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\0\\132\\0\\-264\\-396
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\-264\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\176\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\0\\132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7128\\-1056\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1056\\-440\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-264\\-396
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-34320, -152856, 52008; -152856, -671088, 228360; 52008, 228360, -77707],[264, 264, 1],[156, 11, -9, -11, 18, 38; 3827, 237, -245, -267, 475, 957; 11352, 704, -726, -792, 1408, 2838],54,23]
%
% END OF LATTICE L55.23
%
\hbox{}\par\smallskip%
%
% LATTICE L55.24
%
\leavevmode%
\llap{\spacer}%
${L_{55.24}}=\pdualwitharg{2.11}({L_{55.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{}{6}{528}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{*}{2}{1584}\above{1pt}{1pt}{}{6}{528}\above{1pt}{1pt}{*}{2}{396}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61517808\\-35067384\\243144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35067384\\-19989552\\138600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
243144\\138600\\-961
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\103\\12584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\7\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-1298
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-9\\-792
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\53\\7392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\87\\11286
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
176\\-264\\-440\\-7128\\-1056\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\528\\0\\-3168\\-1320\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\0\\44\\0\\-264\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7128\\-3168\\0\\1584\\-792\\-3960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1056\\-1320\\-264\\-792\\528\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-792\\-396\\-3960\\0\\396
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-61517808, -35067384, 243144; -35067384, -19989552, 138600; 243144, 138600, -961],[792, 264, 1],[-9, -4, 0, 2, -1, -5; 103, 7, -9, -9, 53, 87; 12584, 0, -1298, -792, 7392, 11286],54,24]
%
% END OF LATTICE L55.24
%

\medskip%
%------------------------------WEYL-GROUP-W56--------------------------------
%
\leavevmode\llap{}%
$W_{56\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=12$%
\hfill%
$222222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L56.1
%
\leavevmode%
\llap{\spacer}%
${L_{56.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}23\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{56}}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{92}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{138}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{46}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5244\\276\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
276\\-2\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\18\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\92\\230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-138\\-345
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-23\\-46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-138\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\92\\0\\-552\\-92\\-230
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-4\\-23
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138\\-552\\0\\138\\0\\-207
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-92\\-4\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-230\\-23\\-207\\0\\46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-5244, 276, 0; 276, -2, -5; 0, -5, 2],[276, 1, 1],[1, 5, 0, -7, -1, -1; 18, 92, 0, -138, -20, -23; 48, 230, -1, -345, -48, -46],55,1]
%
% END OF LATTICE L56.1
%
\hbox{}\par\smallskip%
%
% LATTICE L56.2
%
\leavevmode%
\llap{\spacer}%
${L_{56.2}}=\pfillwitharg{2}({L_{56.1}})$%
{${}={}$}Nikulin $56$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{23}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{138}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{46}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1794\\-621\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-621\\215\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\23\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119\\-345\\-69
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-26\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-115\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-69\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\23\\0\\-276\\-23\\-115
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0\\-2\\-23
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\-276\\0\\138\\0\\-207
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-23\\-2\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-115\\-23\\-207\\0\\46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[1794, -621, 0; -621, 215, 0; 0, 0, -1],[69, 1, 1],[1, 8, -1, -119, -9, -40; 3, 23, -3, -345, -26, -115; 0, 0, -1, -69, -5, -23],55,2]
%
% END OF LATTICE L56.2
%
\hbox{}\par\smallskip%
%
% LATTICE L56.3
%
\leavevmode%
\llap{\spacer}%
${L_{56.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{56.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{69}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{46}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{138}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109434\\966\\36777
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
966\\3\\-327
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36777\\-327\\-12359
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-30\\-145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\46\\207
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
74\\45\\219
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-832\\-529\\-2461
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-181\\-852
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2029\\-1265\\-6003
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-23\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\69\\0\\-276\\-69\\-345
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\0\\-6\\-69
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-276\\0\\46\\0\\-207
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-69\\-6\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-345\\-69\\-207\\0\\138
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-109434, 966, 36777; 966, 3, -327; 36777, -327, -12359],[69, 3, 1],[-49, 70, 74, -832, -288, -2029; -30, 46, 45, -529, -181, -1265; -145, 207, 219, -2461, -852, -6003],55,3]
%
% END OF LATTICE L56.3
%
\hbox{}\par\smallskip%
%
% LATTICE L56.4
%
\leavevmode%
\llap{\spacer}%
${L_{56.4}}=\pdualwitharg{3}({L_{56.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{276}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{46}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{138}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126132\\-552\\1380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-552\\6\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1380\\3\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\38\\106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\506\\1380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-115\\-322
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-40\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\69\\207
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-46\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\276\\0\\-552\\-276\\-690
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-12\\-69
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\-552\\0\\46\\0\\-207
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-276\\-12\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-690\\-69\\-207\\0\\138
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-126132, -552, 1380; -552, 6, 3; 1380, 3, -14],[276, 3, 1],[1, 13, 0, -3, -1, 2; 38, 506, 1, -115, -40, 69; 106, 1380, 0, -322, -108, 207],55,4]
%
% END OF LATTICE L56.4
%
\hbox{}\par\smallskip%
%
% LATTICE L56.5
%
\leavevmode%
\llap{\spacer}%
${L_{56.5}}=\pdualwitharg{2}({L_{56.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{92}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{552}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{184}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3802728\\-24012\\935640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24012\\152\\-5908
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
935640\\-5908\\230209
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\0\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
102\\23\\-414
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
340\\69\\-1380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\1\\-65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
249\\0\\-1012
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-276\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\92\\0\\-1104\\-46\\-460
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-4\\-92
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276\\-1104\\0\\552\\0\\-828
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-46\\-4\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-460\\-92\\-828\\0\\184
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3802728, -24012, 935640; -24012, 152, -5908; 935640, -5908, 230209],[276, 4, 1],[31, 102, 2, 340, 16, 249; 0, 23, 5, 69, 1, 0; -126, -414, -8, -1380, -65, -1012],55,5]
%
% END OF LATTICE L56.5
%
\hbox{}\par\smallskip%
%
% LATTICE L56.6
%
\leavevmode%
\llap{\spacer}%
${L_{56.6}}=\pdualwitharg{23}(\pfillwitharg{2}({L_{56.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${69}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{46}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{23}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-563178\\-6693\\195684
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6693\\-23\\2323
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
195684\\2323\\-67993
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-766\\-99\\-2208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\2\\49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
383\\50\\1104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-587\\-72\\-1692
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1540\\-192\\-4439
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-59\\-1349
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69\\0\\-69\\-69\\-69\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-12\\-23\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\0\\46\\0\\-46\\-23
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\-12\\0\\6\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\-23\\-46\\0\\23\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-23\\-9\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-563178, -6693, 195684; -6693, -23, 2323; 195684, 2323, -67993],[69, 23, 1],[-766, 17, 383, -587, -1540, -468; -99, 2, 50, -72, -192, -59; -2208, 49, 1104, -1692, -4439, -1349],55,6]
%
% END OF LATTICE L56.6
%
\hbox{}\par\smallskip%
%
% LATTICE L56.7
%
\leavevmode%
\llap{\spacer}%
${L_{56.7}}=\pdualwitharg{2.3}({L_{56.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{276}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}{184}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{552}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22620408\\-369012\\5569956
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-369012\\6024\\-90864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5569956\\-90864\\1371523
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
97\\-1\\-394
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1733\\0\\-7038
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
133\\2\\-540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
589\\0\\-2392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-1\\-195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1087\\-23\\-4416
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-92\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\276\\0\\-1104\\-138\\-1380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0\\-12\\-276
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\-1104\\0\\184\\0\\-828
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-138\\-12\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1380\\-276\\-828\\0\\552
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22620408, -369012, 5569956; -369012, 6024, -90864; 5569956, -90864, 1371523],[276, 12, 1],[97, 1733, 133, 589, 48, 1087; -1, 0, 2, 0, -1, -23; -394, -7038, -540, -2392, -195, -4416],55,7]
%
% END OF LATTICE L56.7
%
\hbox{}\par\smallskip%
%
% LATTICE L56.8
%
\leavevmode%
\llap{\spacer}%
${L_{56.8}}=\pdualwitharg{3.23}(\pfillwitharg{2}({L_{56.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${23}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{138}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{69}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66102\\59133\\1311
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59133\\52923\\1173
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1311\\1173\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\184
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\-138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-49\\2553
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-12\\639
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\0\\-69\\-23\\-69\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-12\\-69\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\0\\138\\0\\-138\\-69
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-12\\0\\2\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\-69\\-138\\0\\69\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-69\\-9\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[66102, 59133, 1311; 59133, 52923, 1173; 1311, 1173, 26],[69, 69, 1],[-1, 0, 1, -1, -7, -2; -3, 1, 2, -8, -49, -12; 184, -45, -138, 410, 2553, 639],55,8]
%
% END OF LATTICE L56.8
%
\hbox{}\par\smallskip%
%
% LATTICE L56.9
%
\leavevmode%
\llap{\spacer}%
${L_{56.9}}=\pdualwitharg{23}({L_{56.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${276}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{46}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{92}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1380\\276\\276
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
276\\-46\\-23
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
276\\-23\\58
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\34\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\75\\-23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\27\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
276\\0\\-138\\-138\\-276\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-24\\-92\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138\\0\\46\\0\\-92\\-23
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138\\-24\\0\\6\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276\\-92\\-92\\0\\92\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-23\\-9\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1380, 276, 276; 276, -46, -23; 276, -23, 58],[276, 23, 1],[1, 5, 11, 4, -1, -1; 6, 34, 75, 27, -8, -7; 0, -10, -23, -9, 0, 2],55,9]
%
% END OF LATTICE L56.9
%
\hbox{}\par\smallskip%
%
% LATTICE L56.10
%
\leavevmode%
\llap{\spacer}%
${L_{56.10}}=\pdualwitharg{3.23}({L_{56.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${92}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{138}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{276}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
276\\-276\\-276
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276\\138\\69
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276\\69\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\64\\-46
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\34\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\12\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
92\\0\\-138\\-46\\-276\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-276\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138\\0\\138\\0\\-276\\-69
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\-24\\0\\2\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276\\-276\\-276\\0\\276\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-69\\-9\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[276, -276, -276; -276, 138, 69; -276, 69, -34],[276, 69, 1],[21, 11, 0, -1, -1, 4; 64, 34, 1, -3, -4, 12; -46, -24, 0, 2, 0, -9],55,10]
%
% END OF LATTICE L56.10
%
\hbox{}\par\smallskip%
%
% LATTICE L56.11
%
\leavevmode%
\llap{\spacer}%
${L_{56.11}}=\pdualwitharg{2.23}({L_{56.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${276}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{184}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{23}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9384\\22356\\14352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22356\\52072\\33304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14352\\33304\\21287
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
103\\-309\\414
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\76\\-102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-227\\686\\-920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-175\\528\\-708
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74\\223\\-299
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
276\\0\\-276\\-276\\-138\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-48\\-46\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276\\0\\184\\0\\-92\\-92
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276\\-48\\0\\24\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138\\-46\\-92\\0\\23\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-92\\-36\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9384, 22356, 14352; 22356, 52072, 33304; 14352, 33304, 21287],[276, 92, 1],[103, -25, -227, -175, -74, 1; -309, 76, 686, 528, 223, -3; 414, -102, -920, -708, -299, 4],55,11]
%
% END OF LATTICE L56.11
%
\hbox{}\par\smallskip%
%
% LATTICE L56.12
%
\leavevmode%
\llap{\spacer}%
${L_{56.12}}=\pdualwitharg{2.3.23}({L_{56.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${92}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{552}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{69}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
85560\\170844\\42504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
170844\\341688\\85008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42504\\85008\\21149
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-194\\782
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-97\\390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-51\\207
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-98\\396
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
92\\0\\-276\\-92\\-138\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-48\\-138\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276\\0\\552\\0\\-276\\-276
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\-48\\0\\8\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138\\-138\\-276\\0\\69\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-276\\-36\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[85560, 170844, 42504; 170844, 341688, 85008; 42504, 85008, 21149],[276, 276, 1],[-1, 0, 2, 0, -1, -1; -194, -97, -1, -1, -51, -98; 782, 390, 0, 4, 207, 396],55,12]
%
% END OF LATTICE L56.12
%

\medskip%
%------------------------------WEYL-GROUP-W57--------------------------------
%
\leavevmode\llap{}%
$W_{57\phantom{0}}$%
\qquad\llap{16} lattices, $\chi=24$%
\hfill%
$22222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L57.1
%
\leavevmode%
\llap{\spacer}%
${L_{57.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{57}}%
\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3640\\1400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1400\\538\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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8\\-21\\-14
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-13\\-8
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\-168\\-112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\right.$}%
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10\\0\\-10\\-280
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-10\\0\\2\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-168\\0\\56
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-35\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-63\\-21\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-21\\-11\\-168
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0\\-224\\-168\\-3192
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[3640, 1400, 0; 1400, 538, 1; 0, 1, -2],[280, 1, 1],[-2, 8, 5, 65, 2, -8, -5, -65; 5, -21, -13, -168, -5, 21, 13, 168; 0, -14, -8, -112, -5, 7, 5, 56],56,1]
%
% END OF LATTICE L57.1
%
\hbox{}\par\smallskip%
%
% LATTICE L57.2
%
\leavevmode%
\llap{\spacer}%
${L_{57.2}}=\pfillwitharg{2}({L_{57.1}})$%
{${}={}$}Nikulin $57$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1610\\-770\\70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-770\\-368\\35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\35\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2101\\4340\\-840
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1020\\2107\\-408
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\31\\-7
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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17\\-35\\5
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17\\-35\\7
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-15\\31\\-5
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-305\\630\\-112
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-35\\-35\\-5\\0
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-35\\-63\\-21\\-112
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1610, -770, 70; -770, -368, 35; 70, 35, 6],[70, 1, 1],[17, 17, -15, -305, -92, -122, -30, -115; -35, -35, 31, 630, 190, 252, 62, 238; 5, 7, -5, -112, -35, -49, -13, -56],56,2]
%
% END OF LATTICE L57.2
%
\hbox{}\par\smallskip%
%
% LATTICE L57.3
%
\leavevmode%
\llap{\spacer}%
${L_{57.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{57.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3080\\-490\\-1470
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-490\\96\\-244
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1470\\-244\\-701
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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69\\0\\-140
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{$\left[\!\llap{\phantom{%
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110\\-13\\-226
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351\\-42\\-721
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-70\\-70\\-10\\0
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-70\\-126\\-42\\-112
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-10\\-42\\-22\\-84
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0\\-112\\-84\\-399
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3080, -490, -1470; -490, 96, -244; -1470, -244, -701],[70, 2, 1],[-34, 184, 110, 351, 39, -177, -107, -337; 5, -21, -13, -42, -5, 21, 13, 42; 70, -378, -226, -721, -80, 364, 220, 693],56,3]
%
% END OF LATTICE L57.3
%
\hbox{}\par\smallskip%
%
% LATTICE L57.4
%
\leavevmode%
\llap{\spacer}%
${L_{57.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{57.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{70}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2590\\-1120\\-490
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1120\\240\\225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-490\\225\\92
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-29\\0\\-140
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\kern3pt%
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6\\0\\29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\1\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89\\-21\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\-13\\-250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-341\\-84\\-1610
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-10\\-140\\-7\\-35\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-420\\-35\\-315\\-105\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\0\\-5\\-105\\-55\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-420\\0\\70\\0\\-560\\-420\\-3990
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-10\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-420\\0\\70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-35\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-315\\-105\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-105\\-55\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-560\\-420\\-3990
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2590, -1120, -490; -1120, 240, 225; -490, 225, 92],[70, 5, 1],[3, -89, -53, -341, -4, 82, 50, 313; 1, -21, -13, -84, -1, 21, 13, 84; 14, -420, -250, -1610, -19, 385, 235, 1470],56,4]
%
% END OF LATTICE L57.4
%
\hbox{}\par\smallskip%
%
% LATTICE L57.5
%
\leavevmode%
\llap{\spacer}%
${L_{57.5}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{57.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4970\\-1540\\-1400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1540\\336\\441
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1400\\441\\394
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\0\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\41
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\5\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-3\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-79\\-13\\-266
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73\\-12\\-246
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\0\\-70\\-140\\-245\\-35\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-12\\-35\\-9\\-21\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\14\\0\\-35\\-21\\-77\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-12\\0\\2\\0\\-16\\-84\\-114
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\0\\-70\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-12\\0\\2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-245\\-35\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-9\\-21\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-21\\-77\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-16\\-84\\-114
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4970, -1540, -1400; -1540, 336, 441; -1400, 441, 394],[70, 7, 1],[21, -19, -79, -73, -31, 17, 73, 65; 5, -3, -13, -12, -5, 3, 13, 12; 70, -64, -266, -246, -105, 57, 245, 218],56,5]
%
% END OF LATTICE L57.5
%
\hbox{}\par\smallskip%
%
% LATTICE L57.6
%
\leavevmode%
\llap{\spacer}%
${L_{57.6}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{57.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1465660\\-19530\\696430
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19530\\260\\-9280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
696430\\-9280\\330919
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-799\\21\\1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-380\\10\\799
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\6\\-124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
333\\28\\-700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57\\-2\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-49\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-140\\-14\\-70\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\140\\0\\-420\\-70\\-630\\-210\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0\\-10\\-210\\-110\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-420\\0\\35\\0\\-560\\-420\\-1995
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\140\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-420\\0\\35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-70\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-630\\-210\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-210\\-110\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-560\\-420\\-1995
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1465660, -19530, 696430; -19530, 260, -9280; 696430, -9280, 330919],[70, 10, 1],[59, 333, 57, 16, -21, -67, 57, 516; 6, 28, -2, -49, -7, -35, -1, 35; -124, -700, -120, -35, 44, 140, -120, -1085],56,6]
%
% END OF LATTICE L57.6
%
\hbox{}\par\smallskip%
%
% LATTICE L57.7
%
\leavevmode%
\llap{\spacer}%
${L_{57.7}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{57.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
575540\\-14490\\267470
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14490\\364\\-6734
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
267470\\-6734\\124301
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-391\\15\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-182\\7\\391
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
261\\30\\-560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\4\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\-2\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-7\\19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-140\\-140\\-490\\-70\\-70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12\\-70\\-18\\-42\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\28\\0\\-70\\-42\\-154\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-12\\0\\1\\0\\-16\\-84\\-57
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-140\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-12\\0\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-490\\-70\\-70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-18\\-42\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-42\\-154\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-16\\-84\\-57
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[575540, -14490, 267470; -14490, 364, -6734; 267470, -6734, 124301],[70, 14, 1],[261, 41, 39, -9, -131, -15, 39, 61; 30, 4, -2, -7, -35, -5, -1, 5; -560, -88, -84, 19, 280, 32, -84, -131],56,7]
%
% END OF LATTICE L57.7
%
\hbox{}\par\smallskip%
%
% LATTICE L57.8
%
\leavevmode%
\llap{\spacer}%
${L_{57.8}}=\pdualwitharg{5}({L_{57.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{280}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31640\\-560\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\10\\5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\5\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
111\\5040\\2800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-136\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\45\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-182\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-136\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-2576\\-1400
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-10\\-280\\-7\\-35\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-840\\-35\\-315\\-105\\-1120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\0\\-5\\-105\\-55\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-840\\0\\280\\0\\-1120\\-840\\-15960
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-10\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-840\\0\\280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-35\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-315\\-105\\-1120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-105\\-55\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1120\\-840\\-15960
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31640, -560, -280; -560, 10, 5; -280, 5, 2],[280, 5, 1],[0, -4, -3, -57, -1, -3, 0, 1; 0, -182, -136, -2576, -45, -133, 1, 56; -1, -105, -75, -1400, -24, -70, 0, 0],56,8]
%
% END OF LATTICE L57.8
%
\hbox{}\par\smallskip%
%
% LATTICE L57.9
%
\leavevmode%
\llap{\spacer}%
${L_{57.9}}=\pdualwitharg{7}({L_{57.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5320\\-280\\280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\-21\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
119\\2040\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-154\\21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-255\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\270\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\68\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\154\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\224\\-32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\0\\-70\\-280\\-245\\-35\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-24\\-35\\-9\\-21\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\14\\0\\-35\\-21\\-77\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-24\\0\\8\\0\\-32\\-168\\-456
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\0\\-70\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-24\\0\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-245\\-35\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-9\\-21\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-21\\-77\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-32\\-168\\-456
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5320, -280, 280; -280, 14, -21; 280, -21, -38],[280, 7, 1],[16, 4, 9, 13, -1, -1, 0, 11; 270, 68, 154, 224, -15, -17, -1, 184; -35, -9, -21, -32, 0, 2, 0, -24],56,9]
%
% END OF LATTICE L57.9
%
\hbox{}\par\smallskip%
%
% LATTICE L57.10
%
\leavevmode%
\llap{\spacer}%
${L_{57.10}}=\pdualwitharg{2}({L_{57.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\-840\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\176\\64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\64\\23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\105\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\40\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-14\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-24\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-49\\161
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-280\\-280\\-280\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\112\\0\\-168\\-280\\-504\\-168\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\16\\0\\-40\\-168\\-88\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-168\\0\\7\\0\\-224\\-168\\-399
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\112\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-168\\0\\7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-280\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-504\\-168\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-168\\-88\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-224\\-168\\-399
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1680, -840, -280; -840, 176, 64; -280, 64, 23],[280, 8, 1],[-1, 1, 1, 2, 1, -1, -1, -2; 40, -14, -24, -49, -25, 35, 33, 70; -120, 56, 80, 161, 80, -112, -104, -217],56,10]
%
% END OF LATTICE L57.10
%
\hbox{}\par\smallskip%
%
% LATTICE L57.11
%
\leavevmode%
\llap{\spacer}%
${L_{57.11}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{57.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
267330\\-15610\\50960
\end{tabular}\endgroup%
\kern3pt%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[267330, -15610, 50960; -15610, 910, -2975; 50960, -2975, 9714],[70, 35, 1],[-20, -16, -18, 1, 8, 4, -18, -49; 46, 36, 34, -16, -23, -13, 35, 108; 119, 95, 105, -10, -49, -25, 105, 290],56,11]
%
% END OF LATTICE L57.11
%
\hbox{}\par\smallskip%
%
% LATTICE L57.12
%
\leavevmode%
\llap{\spacer}%
${L_{57.12}}=\pdualwitharg{2.5.7}(\pfillwitharg{2}({L_{57.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77980\\-237930\\-116410
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-237930\\709940\\347340
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116410\\347340\\169937
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-1\\3151\\-6440
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
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-3\\-1322\\2700
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-13\\-5484\\11200
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-2529\\5165
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[77980, -237930, -116410; -237930, 709940, 347340; -116410, 347340, 169937],[70, 70, 1],[1, -3, -13, -6, -1, 3, 13, 6; 288, -1322, -5484, -2529, -425, 1185, 5073, 2255; -588, 2700, 11200, 5165, 868, -2420, -10360, -4605],56,12]
%
% END OF LATTICE L57.12
%
\hbox{}\par\smallskip%
%
% LATTICE L57.13
%
\leavevmode%
\llap{\spacer}%
${L_{57.13}}=\pdualwitharg{5.7}({L_{57.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
113960\\-4200\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4200\\70\\35
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\35\\6
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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3\\17\\315
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1\\8\\80
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{$\left.\llap{\phantom{%
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[113960, -4200, -840; -4200, 70, 35; -840, 35, 6],[280, 35, 1],[2, 2, 3, 1, -1, -1, 0, 7; 11, 11, 17, 8, -5, -5, 1, 40; 217, 215, 315, 80, -112, -110, 0, 760],56,13]
%
% END OF LATTICE L57.13
%
\hbox{}\par\smallskip%
%
% LATTICE L57.14
%
\leavevmode%
\llap{\spacer}%
${L_{57.14}}=\pdualwitharg{2.5}({L_{57.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{560}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1081360\\-92680\\5600
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92680\\-7920\\480
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5600\\480\\-29
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-1163\\-1743\\-255640
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{$\left[\!\llap{\phantom{%
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-1\\2\\-160
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\385
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-80\\-280\\-56\\-280\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\560\\0\\-840\\-280\\-2520\\-840\\-1120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\80\\0\\-40\\-840\\-440\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-840\\0\\35\\0\\-1120\\-840\\-1995
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-80\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\560\\0\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-840\\0\\35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-280\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-2520\\-840\\-1120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-840\\-440\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1120\\-840\\-1995
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1081360, -92680, 5600; -92680, -7920, 480; 5600, 480, -29],[280, 40, 1],[-3, -13, -1, 2, 1, -1, -5, -16; -2, 0, 2, 0, -1, -21, -11, -21; -616, -2520, -160, 385, 176, -560, -1160, -3465],56,14]
%
% END OF LATTICE L57.14
%
\hbox{}\par\smallskip%
%
% LATTICE L57.15
%
\leavevmode%
\llap{\spacer}%
${L_{57.15}}=\pdualwitharg{2.7}({L_{57.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${560}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
560\\-13720\\280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13720\\329616\\-6720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\-6720\\137
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-5\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-297\\98\\5544
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-2\\-113
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-5\\-280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\1\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\5\\224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
560\\0\\-560\\-280\\-1960\\-280\\-280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-24\\-280\\-72\\-168\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\0\\112\\0\\-280\\-168\\-616\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-24\\0\\1\\0\\-32\\-168\\-57
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
560\\0\\-560\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\0\\112\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-24\\0\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1960\\-280\\-280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-72\\-168\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-168\\-616\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-32\\-168\\-57
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[560, -13720, 280; -13720, 329616, -6720; 280, -6720, 137],[280, 56, 1],[14, 4, 10, 2, 1, -1, -1, 1; -5, 1, 5, 1, 0, -2, -8, -2; -280, 40, 224, 45, 0, -96, -392, -101],56,15]
%
% END OF LATTICE L57.15
%
\hbox{}\par\smallskip%
%
% LATTICE L57.16
%
\leavevmode%
\llap{\spacer}%
${L_{57.16}}=\pdualwitharg{2.5.7}({L_{57.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{*}{2}{560}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7280\\-3640\\560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3640\\-1680\\280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
560\\280\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-25\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\3\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-2\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\0\\-560\\-280\\-392\\-280\\-280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-120\\-280\\-360\\-840\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\0\\560\\0\\-280\\-840\\-3080\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-120\\0\\5\\0\\-160\\-840\\-285
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\0\\-560\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\0\\560\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-120\\0\\5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-392\\-280\\-280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-360\\-840\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-840\\-3080\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-160\\-840\\-285
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7280, -3640, 560; -3640, -1680, 280; 560, 280, -43],[280, 280, 1],[-3, -3, -1, 2, 5, 5, 7, 0; -2, 0, 2, 0, -1, -3, -11, -3; -56, -40, 0, 25, 56, 40, 0, -25],56,16]
%
% END OF LATTICE L57.16
%

\medskip%
%------------------------------WEYL-GROUP-W58--------------------------------
%
\leavevmode\llap{}%
$W_{58\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=20$%
\hfill%
$2222232$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L58.1
%
\leavevmode%
\llap{\spacer}%
${L_{58.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{58}}%
\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{+}{3}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19740028\\77308\\-33264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77308\\-302\\127
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33264\\127\\-42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\3122\\728
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-8228\\-1914
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\-11634\\-2709
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-137\\-38876\\-9056
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-912\\-258797\\-60291
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-1419\\-331
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\1135\\264
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-42\\-196\\-1386\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\44\\0\\-132\\-1078\\-22\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\14\\0\\-77\\-7\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-196\\-132\\0\\4\\0\\-4\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1386\\-1078\\-77\\0\\154\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-22\\-7\\-4\\0\\2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-22\\-21\\-60\\-385\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-19740028, 77308, -33264; 77308, -302, 127; -33264, 127, -42],[308, 1, 1],[11, -29, -41, -137, -912, -5, 4; 3122, -8228, -11634, -38876, -258797, -1419, 1135; 728, -1914, -2709, -9056, -60291, -331, 264],57,1]
%
% END OF LATTICE L58.1
%
\hbox{}\par\smallskip%
%
% LATTICE L58.2
%
\leavevmode%
\llap{\spacer}%
${L_{58.2}}=\pfillwitharg{2}({L_{58.1}})$%
{${}={}$}Nikulin $58$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{-}{3}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1386\\385\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
385\\107\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\11\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-21\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-82\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
345\\-1232\\-308
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-18\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-11\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-21\\-49\\-693\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\0\\-33\\-539\\-11\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\14\\0\\-77\\-7\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-33\\0\\1\\0\\-2\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693\\-539\\-77\\0\\154\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-11\\-7\\-2\\0\\2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-21\\-30\\-385\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[1386, 385, 0; 385, 107, 0; 0, 0, -1],[77, 1, 1],[-2, -3, 6, 23, 345, 5, 3; 7, 11, -21, -82, -1232, -18, -11; 0, 0, -7, -21, -308, -4, -3],57,2]
%
% END OF LATTICE L58.2
%
\hbox{}\par\smallskip%
%
% LATTICE L58.3
%
\leavevmode%
\llap{\spacer}%
${L_{58.3}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{58.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{77}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{-}{3}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38731\\7392\\-11396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7392\\-1239\\2156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11396\\2156\\-3351
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\-14\\-128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\-22\\-231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\23\\202
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-857\\353\\3143
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1713\\704\\6281
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109\\44\\399
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
86\\-35\\-315
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-49\\-99\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\77\\0\\-231\\-539\\-77\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0\\-11\\-7\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-231\\0\\7\\0\\-14\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-539\\-11\\0\\22\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-77\\-7\\-14\\0\\14\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-77\\-21\\-210\\-385\\-7\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38731, 7392, -11396; 7392, -1239, 2156; -11396, 2156, -3351],[77, 7, 1],[35, 64, -55, -857, -1713, -109, 86; -14, -22, 23, 353, 704, 44, -35; -128, -231, 202, 3143, 6281, 399, -315],57,3]
%
% END OF LATTICE L58.3
%
\hbox{}\par\smallskip%
%
% LATTICE L58.4
%
\leavevmode%
\llap{\spacer}%
${L_{58.4}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{58.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${77}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{+}{3}{22}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-479941\\7623\\-133672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7623\\-55\\2123
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-133672\\2123\\-37230
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
536\\-7\\-1925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-1\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-901\\-7\\3234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1961\\-2\\7040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2486\\0\\8925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-242\\2\\869
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
193\\-1\\-693
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\0\\-231\\-539\\-693\\-77\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-33\\-49\\-11\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\0\\154\\0\\-77\\-77\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-539\\-33\\0\\11\\0\\-22\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693\\-49\\-77\\0\\14\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\-11\\-77\\-22\\0\\22\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-231\\-330\\-385\\-11\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-479941, 7623, -133672; 7623, -55, 2123; -133672, 2123, -37230],[77, 11, 1],[536, 10, -901, -1961, -2486, -242, 193; -7, -1, -7, -2, 0, 2, -1; -1925, -36, 3234, 7040, 8925, 869, -693],57,4]
%
% END OF LATTICE L58.4
%
\hbox{}\par\smallskip%
%
% LATTICE L58.5
%
\leavevmode%
\llap{\spacer}%
${L_{58.5}}=\pdualwitharg{2}({L_{58.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{616}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{+}{3}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1021992664\\-60679388\\251966176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60679388\\3602760\\-14960140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
251966176\\-14960140\\62120753
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107\\0\\-434
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4718\\-77\\19118
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12928\\-203\\52388
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12208\\-189\\49471
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-334119\\-5159\\1353968
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2593\\-39\\10508
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-685\\-10\\2776
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-84\\-98\\-2772\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\44\\0\\-66\\-2156\\-44\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\56\\0\\-308\\-28\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-98\\-66\\0\\1\\0\\-4\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2772\\-2156\\-308\\0\\616\\0\\-1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-44\\-28\\-4\\0\\8\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-44\\-84\\-60\\-1540\\-4\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1021992664, -60679388, 251966176; -60679388, 3602760, -14960140; 251966176, -14960140, 62120753],[308, 4, 1],[107, -4718, -12928, -12208, -334119, -2593, -685; 0, -77, -203, -189, -5159, -39, -10; -434, 19118, 52388, 49471, 1353968, 10508, 2776],57,5]
%
% END OF LATTICE L58.5
%
\hbox{}\par\smallskip%
%
% LATTICE L58.6
%
\leavevmode%
\llap{\spacer}%
${L_{58.6}}=\pdualwitharg{7}({L_{58.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{308}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{+}{3}{14}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1006852\\-46816\\5852
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46816\\-2114\\273
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5852\\273\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\66\\-4466
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\11\\-773
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109\\236\\-16884
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\220\\-15807
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\6\\-469
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-9\\616
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-196\\-198\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\308\\0\\-924\\-1078\\-154\\-154
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-11\\-7\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-196\\-924\\0\\28\\0\\-28\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-1078\\-11\\0\\22\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-154\\-7\\-28\\0\\14\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-154\\-21\\-420\\-385\\-7\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1006852, -46816, 5852; -46816, -2114, 273; 5852, 273, -34],[308, 7, 1],[1, -29, -5, -109, -102, -3, 4; -2, 66, 11, 236, 220, 6, -9; 156, -4466, -773, -16884, -15807, -469, 616],57,6]
%
% END OF LATTICE L58.6
%
\hbox{}\par\smallskip%
%
% LATTICE L58.7
%
\leavevmode%
\llap{\spacer}%
${L_{58.7}}=\pdualwitharg{11}({L_{58.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${308}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{154}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{-}{3}{22}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12012\\-1540\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1540\\-3322\\2079
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\2079\\-1174
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
123\\714\\1232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-168\\-290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-454\\-2632\\-4543
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1521\\-8820\\-15224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-921\\-5341\\-9219
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-325\\-561
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\255\\440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
308\\0\\-462\\-2156\\-1386\\-154\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-132\\-98\\-22\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\0\\154\\0\\-77\\-77\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2156\\-132\\0\\44\\0\\-44\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1386\\-98\\-77\\0\\14\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-154\\-22\\-77\\-44\\0\\22\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-22\\-231\\-660\\-385\\-11\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12012, -1540, -308; -1540, -3322, 2079; -308, 2079, -1174],[308, 11, 1],[123, -29, -454, -1521, -921, -56, 44; 714, -168, -2632, -8820, -5341, -325, 255; 1232, -290, -4543, -15224, -9219, -561, 440],57,7]
%
% END OF LATTICE L58.7
%
\hbox{}\par\smallskip%
%
% LATTICE L58.8
%
\leavevmode%
\llap{\spacer}%
${L_{58.8}}=\pdualwitharg{7.11}(\pfillwitharg{2}({L_{58.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{77}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{154}\above{1pt}{1pt}{+}{3}{154}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
154\\-197351\\-55517
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-197351\\275871211\\77605605
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55517\\77605605\\21831310
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-1999\\7106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-4281\\15218
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-9274\\32967
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-243\\-81467\\289597
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-13126\\46660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\1278\\-4543
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-33\\-539\\-99\\-77\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-231\\-49\\-77\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\22\\0\\-11\\-77\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-539\\-231\\0\\77\\0\\-154\\-2310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-49\\-11\\0\\2\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\-77\\-77\\-154\\0\\154\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-77\\-231\\-2310\\-385\\-77\\154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[154, -197351, -55517; -197351, 275871211, 77605605; -55517, 77605605, 21831310],[77, 77, 1],[-6, -13, -28, -243, -39, 1, 5; -1999, -4281, -9274, -81467, -13126, 0, 1278; 7106, 15218, 32967, 289597, 46660, 0, -4543],57,8]
%
% END OF LATTICE L58.8
%
\hbox{}\par\smallskip%
%
% LATTICE L58.9
%
\leavevmode%
\llap{\spacer}%
${L_{58.9}}=\pdualwitharg{2.7}({L_{58.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{308}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{+}{3}{56}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
660198616\\879956\\162849148
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
879956\\1176\\217056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
162849148\\217056\\40169495
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\0\\-150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3153\\-77\\-12782
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1039\\-29\\-4212
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6439\\-189\\-26103
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24866\\-737\\-100804
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1188\\-39\\-4816
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
297\\-10\\-1204
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-98\\-396\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\308\\0\\-462\\-2156\\-308\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-44\\-28\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-98\\-462\\0\\7\\0\\-28\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-2156\\-44\\0\\88\\0\\-1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-308\\-28\\-28\\0\\56\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-308\\-84\\-420\\-1540\\-28\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[660198616, 879956, 162849148; 879956, 1176, 217056; 162849148, 217056, 40169495],[308, 28, 1],[37, 3153, 1039, 6439, 24866, 1188, 297; 0, -77, -29, -189, -737, -39, -10; -150, -12782, -4212, -26103, -100804, -4816, -1204],57,9]
%
% END OF LATTICE L58.9
%
\hbox{}\par\smallskip%
%
% LATTICE L58.10
%
\leavevmode%
\llap{\spacer}%
${L_{58.10}}=\pdualwitharg{2.11}({L_{58.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${308}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{616}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{-}{3}{88}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
106625288\\-14705460\\26079592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14705460\\2028136\\-3596824
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26079592\\-3596824\\6378835
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
113\\0\\-462
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-7\\86
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1158\\-203\\4620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1261\\-189\\5049
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3215\\-469\\12880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-339\\-39\\1364
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109\\-10\\440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
308\\0\\-924\\-1078\\-2772\\-308\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-66\\-196\\-44\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\0\\616\\0\\-308\\-308\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1078\\-66\\0\\11\\0\\-44\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2772\\-196\\-308\\0\\56\\0\\-1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\-44\\-308\\-44\\0\\88\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-44\\-924\\-660\\-1540\\-44\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[106625288, -14705460, 26079592; -14705460, 2028136, -3596824; 26079592, -3596824, 6378835],[308, 44, 1],[113, -22, -1158, -1261, -3215, -339, -109; 0, -7, -203, -189, -469, -39, -10; -462, 86, 4620, 5049, 12880, 1364, 440],57,10]
%
% END OF LATTICE L58.10
%
\hbox{}\par\smallskip%
%
% LATTICE L58.11
%
\leavevmode%
\llap{\spacer}%
${L_{58.11}}=\pdualwitharg{7.11}({L_{58.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{308}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{154}\above{1pt}{1pt}{-}{3}{154}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-381612\\476168\\137676
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
476168\\-593978\\-171787
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
137676\\-171787\\-49670
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61\\-2\\176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-139\\6\\-406
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276\\11\\-803
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6151\\236\\-17864
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-526\\20\\-1527
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\6\\-539
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
211\\-9\\616
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\-66\\-2156\\-198\\-154\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-924\\-98\\-154\\-154
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\0\\22\\0\\-11\\-77\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2156\\-924\\0\\308\\0\\-308\\-4620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-98\\-11\\0\\2\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-154\\-154\\-77\\-308\\0\\154\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-154\\-231\\-4620\\-385\\-77\\154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-381612, 476168, 137676; 476168, -593978, -171787; 137676, -171787, -49670],[308, 77, 1],[61, -139, -276, -6151, -526, -187, 211; -2, 6, 11, 236, 20, 6, -9; 176, -406, -803, -17864, -1527, -539, 616],57,11]
%
% END OF LATTICE L58.11
%
\hbox{}\par\smallskip%
%
% LATTICE L58.12
%
\leavevmode%
\llap{\spacer}%
${L_{58.12}}=\pdualwitharg{2.7.11}({L_{58.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{l}{2}{77}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{616}\above{1pt}{1pt}{-}{3}{616}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12936\\-345268\\-89320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-345268\\9352728\\2419340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89320\\2419340\\625829
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\17\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\263\\-1022
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145\\1019\\-3960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-926\\6479\\-25179
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-327\\2286\\-8884
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-183\\1268\\-4928
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\317\\-1232
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\-132\\-1078\\-396\\-308\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-462\\-196\\-308\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\88\\0\\-44\\-308\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1078\\-462\\0\\77\\0\\-308\\-4620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-196\\-44\\0\\8\\0\\-1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\-308\\-308\\-308\\0\\616\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-308\\-924\\-4620\\-1540\\-308\\616
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12936, -345268, -89320; -345268, 9352728, 2419340; -89320, 2419340, 625829],[308, 308, 1],[-2, -37, -145, -926, -327, -183, -46; 17, 263, 1019, 6479, 2286, 1268, 317; -66, -1022, -3960, -25179, -8884, -4928, -1232],57,12]
%
% END OF LATTICE L58.12
%

\medskip%
%------------------------------WEYL-GROUP-W59--------------------------------
%
\leavevmode\llap{}%
$W_{59\phantom{0}}$%
\qquad\llap{32} lattices, $\chi=28$%
\hfill%
$\infty222262$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L59.1
%
\leavevmode%
\llap{\spacer}%
${L_{59.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{59},3,2}%
\EasyButWeakLineBreak%
{${78}\above{1pt}{1pt}{12,7}{\infty b}{312}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{r}{2}{234}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{18}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15718248\\-2834208\\-2395224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2834208\\-485598\\-457431
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2395224\\-457431\\-339362
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26015\\78325\\78039
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
222369\\-669500\\-667056
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91435\\-275289\\-274284
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
195783\\-589456\\-587304
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
443503\\-1335282\\-1330407
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13008\\-39164\\-39021
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13102\\39447\\39303
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\-156\\-117\\-312\\-819\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\312\\0\\-312\\-1404\\-156\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\0\\18\\0\\-117\\-27\\-153
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\0\\24\\0\\-24\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-819\\-1404\\-117\\0\\234\\0\\-585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-156\\-27\\-24\\0\\6\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-468\\-153\\-288\\-585\\-9\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-15718248, -2834208, -2395224; -2834208, -485598, -457431; -2395224, -457431, -339362],[936, 3, 1],[-26015, 222369, 91435, 195783, 443503, 13008, -13102; 78325, -669500, -275289, -589456, -1335282, -39164, 39447; 78039, -667056, -274284, -587304, -1330407, -39021, 39303],58,1]
%
% END OF LATTICE L59.1
%
\hbox{}\par\smallskip%
%
% LATTICE L59.2
%
\leavevmode%
\llap{\spacer}%
${L_{59.2}}=\pfillwitharg{2.3}({L_{59.1}})$%
{${}={}$}Nikulin $59$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${78}\above{1pt}{1pt}{2,1}{\infty b}{78}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-13\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\-78\\-39\\-156\\-273\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\78\\0\\-78\\-234\\-78\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\2\\0\\-13\\-9\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-78\\0\\6\\0\\-12\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-273\\-234\\-13\\0\\26\\0\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-78\\-9\\-12\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-78\\-17\\-48\\-65\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[78, 0, 0; 0, 0, -1; 0, -1, 0],[78, 1, 1],[-1, 1, 0, -1, -3, -1, -1; 0, 0, -1, -6, -13, -3, -2; -39, 0, 1, -6, -26, -12, -19],58,2]
%
% END OF LATTICE L59.2
%
\hbox{}\par\smallskip%
%
% LATTICE L59.3
%
\leavevmode%
\llap{\spacer}%
${L_{59.3}}=\pfillwitharg{3}({L_{59.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${78}\above{1pt}{1pt}{4,3}{\infty a}{312}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36514920\\-18104112\\-157872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18104112\\-8976026\\-78273
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-157872\\-78273\\-682
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1102\\2223\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2243\\-4524\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\-117\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1535\\3096\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2591\\5226\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-940\\1896\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-995\\2007\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\-156\\-39\\-312\\-273\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\312\\0\\-312\\-468\\-156\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\2\\0\\-13\\-9\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\0\\24\\0\\-24\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-273\\-468\\-13\\0\\26\\0\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-156\\-9\\-24\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-156\\-17\\-96\\-65\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-36514920, -18104112, -157872; -18104112, -8976026, -78273; -157872, -78273, -682],[312, 1, 1],[-1102, 2243, 58, -1535, -2591, -940, -995; 2223, -4524, -117, 3096, 5226, 1896, 2007; -39, 0, 2, 0, -13, -9, -17],58,3]
%
% END OF LATTICE L59.3
%
\hbox{}\par\smallskip%
%
% LATTICE L59.4
%
\leavevmode%
\llap{\spacer}%
${L_{59.4}}=\pfillwitharg{2}({L_{59.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${78}\above{1pt}{1pt}{6,1}{\infty b}{78}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}{234}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{6}{18}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1921842\\242658\\-401544
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
242658\\-28110\\53187
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-401544\\53187\\-81452
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5729\\16913\\-17199
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24501\\-72332\\73554
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20146\\-59475\\60480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21567\\-63670\\64746
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-97705\\-288444\\293319
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2865\\-8458\\8601
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2887\\8523\\-8667
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\-78\\-117\\-156\\-819\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\78\\0\\-78\\-702\\-78\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\0\\18\\0\\-117\\-27\\-153
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-78\\0\\6\\0\\-12\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-819\\-702\\-117\\0\\234\\0\\-585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-78\\-27\\-12\\0\\6\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-234\\-153\\-144\\-585\\-9\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1921842, 242658, -401544; 242658, -28110, 53187; -401544, 53187, -81452],[234, 3, 1],[5729, -24501, -20146, -21567, -97705, -2865, 2887; 16913, -72332, -59475, -63670, -288444, -8458, 8523; -17199, 73554, 60480, 64746, 293319, 8601, -8667],58,4]
%
% END OF LATTICE L59.4
%
\hbox{}\par\smallskip%
%
% LATTICE L59.5
%
\leavevmode%
\llap{\spacer}%
${L_{59.5}}=\pdualwitharg{2}(\pfillwitharg{2.3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{4,1}{\infty z}{39}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{}{6}{4}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2311764\\3042\\-1126242
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3042\\-4\\1482
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1126242\\1482\\-548681
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\741\\-156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\0\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-39\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-3\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127\\234\\-260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\168\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\321\\-112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\-78\\-78\\-156\\-546\\-78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\39\\0\\-39\\-234\\-78\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\0\\4\\0\\-26\\-18\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-39\\0\\3\\0\\-12\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-546\\-234\\-26\\0\\52\\0\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-78\\-18\\-12\\0\\12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-78\\-34\\-48\\-130\\-6\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2311764, 3042, -1126242; 3042, -4, 1482; -1126242, 1482, -548681],[78, 2, 1],[77, -19, -2, 19, 127, 47, 55; 741, 0, -39, -3, 234, 168, 321; -156, 39, 4, -39, -260, -96, -112],58,5]
%
% END OF LATTICE L59.5
%
\hbox{}\par\smallskip%
%
% LATTICE L59.6
%
\leavevmode%
\llap{\spacer}%
${L_{59.6}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{2,1}{\infty b}{26}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1541046\\3042\\533442
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3042\\-6\\-1053
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
533442\\-1053\\-184654
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\247\\-65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-39\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-2\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-94\\234\\-273
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\56\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\321\\-129
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-26\\-39\\-52\\-273\\-13\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\26\\0\\-26\\-234\\-26\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\6\\0\\-39\\-9\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-26\\0\\2\\0\\-4\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-273\\-234\\-39\\0\\78\\0\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-26\\-9\\-4\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-78\\-51\\-48\\-195\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1541046, 3042, 533442; 3042, -6, -1053; 533442, -1053, -184654],[78, 3, 1],[-22, 9, 2, -9, -94, -12, -44; 247, 0, -39, -2, 234, 56, 321; -65, 26, 6, -26, -273, -35, -129],58,6]
%
% END OF LATTICE L59.6
%
\hbox{}\par\smallskip%
%
% LATTICE L59.7
%
\leavevmode%
\llap{\spacer}%
${L_{59.7}}=\pdualwitharg{2.3}(\pfillwitharg{2.3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{4,1}{\infty z}{13}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{}{6}{12}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9516\\3042\\4368
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3042\\4560\\1404
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4368\\1404\\2005
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2729\\13\\-5954
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
435\\2\\-949
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8402\\39\\-18330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1068\\5\\-2330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4447\\21\\-9702
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\-26\\-78\\-52\\-546\\-26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\13\\0\\-13\\-234\\-26\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\0\\12\\0\\-78\\-18\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-13\\0\\1\\0\\-4\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-546\\-234\\-78\\0\\156\\0\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-26\\-18\\-4\\0\\4\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-78\\-102\\-48\\-390\\-6\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9516, 3042, 4368; 3042, 4560, 1404; 4368, 1404, 2005],[78, 6, 1],[2729, -6, 11, 435, 8402, 1068, 4447; 13, 0, 0, 2, 39, 5, 21; -5954, 13, -24, -949, -18330, -2330, -9702],58,7]
%
% END OF LATTICE L59.7
%
\hbox{}\par\smallskip%
%
% LATTICE L59.8
%
\leavevmode%
\llap{\spacer}%
${L_{59.8}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${78}\above{1pt}{1pt}{6,5}{\infty b}{78}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1999530\\147888\\-39546
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
147888\\-10938\\2925
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39546\\2925\\-760
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
519\\7007\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-338\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-351\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\284\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
231\\3120\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
137\\1850\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
239\\3227\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\-78\\-39\\-156\\-273\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\78\\0\\-78\\-234\\-78\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\2\\0\\-13\\-9\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-78\\0\\6\\0\\-12\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-273\\-234\\-13\\0\\26\\0\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-78\\-9\\-12\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-78\\-17\\-48\\-65\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1999530, 147888, -39546; 147888, -10938, 2925; -39546, 2925, -760],[234, 3, 1],[519, -25, -26, 21, 231, 137, 239; 7007, -338, -351, 284, 3120, 1850, 3227; -39, 0, 2, 0, -13, -9, -17],58,8]
%
% END OF LATTICE L59.8
%
\hbox{}\par\smallskip%
%
% LATTICE L59.9
%
\leavevmode%
\llap{\spacer}%
${L_{59.9}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{4,3}{\infty a}{104}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1864824\\6240\\9360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6240\\-18\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9360\\-33\\-46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-169\\-286
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\1404\\2444
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\581\\1005
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\1248\\2152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\2834\\4875
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\84\\143
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-83\\-144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-52\\-39\\-104\\-273\\-13\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\104\\0\\-104\\-468\\-52\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\6\\0\\-39\\-9\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-104\\0\\8\\0\\-8\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-273\\-468\\-39\\0\\78\\0\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-52\\-9\\-8\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-156\\-51\\-96\\-195\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1864824, 6240, 9360; 6240, -18, -33; 9360, -33, -46],[312, 3, 1],[-2, 17, 7, 15, 34, 1, -1; -169, 1404, 581, 1248, 2834, 84, -83; -286, 2444, 1005, 2152, 4875, 143, -144],58,9]
%
% END OF LATTICE L59.9
%
\hbox{}\par\smallskip%
%
% LATTICE L59.10
%
\leavevmode%
\llap{\spacer}%
${L_{59.10}}=\pdualwitharg{13}(\pfillwitharg{2.3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{2,1}{\infty b}{6}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}{78}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{}{6}{26}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\-3042\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3042\\-9880\\-507
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-507\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-113\\-3\\735
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\2\\-507
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\-156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-1\\216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-323\\-9\\2106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-629\\-17\\4095
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-39\\-156\\-21\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-78\\-18\\-78\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\26\\0\\-13\\-117\\-221
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-78\\0\\78\\0\\-156\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-18\\-13\\0\\2\\0\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-78\\-117\\-156\\0\\78\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-78\\-221\\-624\\-65\\-39\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[78, -3042, 0; -3042, -9880, -507; 0, -507, -2],[78, 13, 1],[-113, -1, 78, 25, -33, -323, -629; -3, 0, 2, 0, -1, -9, -17; 735, 6, -507, -156, 216, 2106, 4095],58,10]
%
% END OF LATTICE L59.10
%
\hbox{}\par\smallskip%
%
% LATTICE L59.11
%
\leavevmode%
\llap{\spacer}%
${L_{59.11}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{12,1}{\infty z}{39}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{468}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{}{6}{36}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
526199238396\\7963029126\\261750015072
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7963029126\\120505368\\3961090860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
261750015072\\3961090860\\130203666959
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2289\\13\\-4602
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20104\\910\\-40443
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26573\\1314\\-53460
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13007\\686\\-26169
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
111068\\6201\\-223470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2710\\199\\-5454
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1171\\129\\-2358
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\-78\\-234\\-156\\-1638\\-78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\39\\0\\-39\\-702\\-78\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\0\\36\\0\\-234\\-54\\-306
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-39\\0\\3\\0\\-12\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1638\\-702\\-234\\0\\468\\0\\-1170
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-78\\-54\\-12\\0\\12\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-234\\-306\\-144\\-1170\\-18\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[526199238396, 7963029126, 261750015072; 7963029126, 120505368, 3961090860; 261750015072, 3961090860, 130203666959],[234, 6, 1],[2289, 20104, 26573, 13007, 111068, 2710, 1171; 13, 910, 1314, 686, 6201, 199, 129; -4602, -40443, -53460, -26169, -223470, -5454, -2358],58,11]
%
% END OF LATTICE L59.11
%
\hbox{}\par\smallskip%
%
% LATTICE L59.12
%
\leavevmode%
\llap{\spacer}%
${L_{59.12}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{12,5}{\infty z}{39}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{}{6}{4}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40770288\\6800742\\20378826
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6800742\\1134444\\3399318
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20378826\\3399318\\10186255
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19301\\-923\\38922
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-522\\-26\\1053
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2611\\-124\\5265
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17226\\-819\\34736
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6784\\-323\\13680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9913\\-473\\19990
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\-78\\-78\\-156\\-546\\-78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\39\\0\\-39\\-234\\-78\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\0\\4\\0\\-26\\-18\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-39\\0\\3\\0\\-12\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-546\\-234\\-26\\0\\52\\0\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-78\\-18\\-12\\0\\12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-78\\-34\\-48\\-130\\-6\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40770288, 6800742, 20378826; 6800742, 1134444, 3399318; 20378826, 3399318, 10186255],[234, 6, 1],[-19301, -522, -1, -2611, -17226, -6784, -9913; -923, -26, 0, -124, -819, -323, -473; 38922, 1053, 2, 5265, 34736, 13680, 19990],58,12]
%
% END OF LATTICE L59.12
%
\hbox{}\par\smallskip%
%
% LATTICE L59.13
%
\leavevmode%
\llap{\spacer}%
${L_{59.13}}=\pdualwitharg{2.13}(\pfillwitharg{2.3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{4,1}{\infty z}{3}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{}{6}{52}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51428208\\51714\\25209912
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51714\\52\\25350
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25209912\\25350\\12357803
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-203\\57\\414
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\-51
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-631\\117\\1287
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-303\\58\\618
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1415\\288\\2886
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1721\\401\\3510
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-78\\-156\\-42\\-78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-39\\-18\\-78\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\0\\52\\0\\-26\\-234\\-442
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-39\\0\\39\\0\\-156\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-18\\-26\\0\\4\\0\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-78\\-234\\-156\\0\\156\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-78\\-442\\-624\\-130\\-78\\52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[51428208, 51714, 25209912; 51714, 52, 25350; 25209912, 25350, 12357803],[78, 26, 1],[-203, 25, 0, -631, -303, -1415, -1721; 57, 0, 1, 117, 58, 288, 401; 414, -51, 0, 1287, 618, 2886, 3510],58,13]
%
% END OF LATTICE L59.13
%
\hbox{}\par\smallskip%
%
% LATTICE L59.14
%
\leavevmode%
\llap{\spacer}%
${L_{59.14}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${624}\above{1pt}{1pt}{8,5}{\infty z}{156}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{208}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{}{6}{16}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\7176\\5616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7176\\-1136\\-744
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5616\\-744\\-469
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\975\\-1248
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-188\\-7371\\9438
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-3449\\4416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-2547\\3261
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-367\\-14378\\18408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-900\\1152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\431\\-552
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\-312\\-312\\-312\\-2184\\-312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\156\\0\\-78\\-936\\-312\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\0\\16\\0\\-104\\-72\\-136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-78\\0\\3\\0\\-24\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2184\\-936\\-104\\0\\208\\0\\-520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\-72\\-24\\0\\48\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-312\\-136\\-96\\-520\\-24\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[624, 7176, 5616; 7176, -1136, -744; 5616, -744, -469],[312, 8, 1],[25, -188, -88, -65, -367, -23, 11; 975, -7371, -3449, -2547, -14378, -900, 431; -1248, 9438, 4416, 3261, 18408, 1152, -552],58,14]
%
% END OF LATTICE L59.14
%
\hbox{}\par\smallskip%
%
% LATTICE L59.15
%
\leavevmode%
\llap{\spacer}%
${L_{59.15}}=\pdualwitharg{3}({L_{59.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${78}\above{1pt}{1pt}{12,11}{\infty a}{312}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5257512\\818064\\59904
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
818064\\-127290\\-9321
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59904\\-9321\\-682
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\247\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89\\-572\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-13\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61\\392\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\650\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\232\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\239\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\-156\\-39\\-312\\-273\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\312\\0\\-312\\-468\\-156\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\2\\0\\-13\\-9\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\0\\24\\0\\-24\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-273\\-468\\-13\\0\\26\\0\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-156\\-9\\-24\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-156\\-17\\-96\\-65\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5257512, 818064, 59904; 818064, -127290, -9321; 59904, -9321, -682],[936, 3, 1],[38, -89, -2, 61, 101, 36, 37; 247, -572, -13, 392, 650, 232, 239; -39, 0, 2, 0, -13, -9, -17],58,15]
%
% END OF LATTICE L59.15
%
\hbox{}\par\smallskip%
%
% LATTICE L59.16
%
\leavevmode%
\llap{\spacer}%
${L_{59.16}}=\pdualwitharg{3.13}(\pfillwitharg{2.3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty b}{2}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{l}{2}{26}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{}{6}{78}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42596658\\-57642\\14611974
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57642\\78\\-19773
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14611974\\-19773\\5012360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\125\\155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-26\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-329\\736\\962
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-237\\532\\693
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-369\\834\\1079
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1347\\3095\\3939
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-39\\-52\\-21\\-13\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-26\\-18\\-26\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\78\\0\\-39\\-117\\-663
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-26\\0\\26\\0\\-52\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-18\\-39\\0\\6\\0\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-26\\-117\\-52\\0\\26\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-78\\-663\\-624\\-195\\-39\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[42596658, -57642, 14611974; -57642, 78, -19773; 14611974, -19773, 5012360],[78, 39, 1],[-53, 13, 0, -329, -237, -369, -1347; 125, -26, 1, 736, 532, 834, 3095; 155, -38, 0, 962, 693, 1079, 3939],58,16]
%
% END OF LATTICE L59.16
%
\hbox{}\par\smallskip%
%
% LATTICE L59.17
%
\leavevmode%
\llap{\spacer}%
${L_{59.17}}=\pdualwitharg{13}(\pfillwitharg{3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{4,3}{\infty a}{24}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{l}{2}{312}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{}{6}{26}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31561608\\-3561480\\38376
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3561480\\-401882\\4329
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38376\\4329\\-46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-372\\-804
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-9\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-365\\3312\\7176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\372\\803
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-166\\1506\\3237
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128\\1161\\2470
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-39\\-312\\-21\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-312\\-36\\-156\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\26\\0\\-13\\-117\\-221
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\0\\312\\0\\-312\\-1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-36\\-13\\0\\2\\0\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-156\\-117\\-312\\0\\78\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-156\\-221\\-1248\\-65\\-39\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-31561608, -3561480, 38376; -3561480, -401882, 4329; 38376, 4329, -46],[312, 13, 1],[-1, 41, 1, -365, -41, -166, -128; 9, -372, -9, 3312, 372, 1506, 1161; 12, -804, -13, 7176, 803, 3237, 2470],58,17]
%
% END OF LATTICE L59.17
%
\hbox{}\par\smallskip%
%
% LATTICE L59.18
%
\leavevmode%
\llap{\spacer}%
${L_{59.18}}=\pdualwitharg{2.3.13}(\pfillwitharg{2.3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{}{6}{156}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-516204\\526890\\259428
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
526890\\175812\\86580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
259428\\86580\\42637
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-23119\\46952
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-97\\197
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\47855\\-97188
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2490\\-5057
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20338\\41304
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-66188\\134420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-386297\\784524
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-78\\-52\\-42\\-26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-13\\-18\\-26\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\0\\156\\0\\-78\\-234\\-1326
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-13\\0\\13\\0\\-52\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-18\\-78\\0\\12\\0\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-26\\-234\\-52\\0\\52\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-78\\-1326\\-624\\-390\\-78\\156
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-516204, 526890, 259428; 526890, 175812, 86580; 259428, 86580, 42637],[78, 78, 1],[-1, 0, 2, 0, -1, -3, -17; -23119, -97, 47855, 2490, -20338, -66188, -386297; 46952, 197, -97188, -5057, 41304, 134420, 784524],58,18]
%
% END OF LATTICE L59.18
%
\hbox{}\par\smallskip%
%
% LATTICE L59.19
%
\leavevmode%
\llap{\spacer}%
${L_{59.19}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{8,5}{\infty z}{52}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{624}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{}{6}{48}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-568464\\9672\\-185952
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9672\\-144\\3216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-185952\\3216\\-60695
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
154\\1053\\-416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-337\\-2314\\910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-551\\-3780\\1488
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-147\\-1008\\397
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2657\\-18213\\7176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\-527\\208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\549\\-216
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\-104\\-312\\-104\\-2184\\-104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\52\\0\\-26\\-936\\-104\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\0\\48\\0\\-312\\-72\\-408
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-26\\0\\1\\0\\-8\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2184\\-936\\-312\\0\\624\\0\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-104\\-72\\-8\\0\\16\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-312\\-408\\-96\\-1560\\-24\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-568464, 9672, -185952; 9672, -144, 3216; -185952, 3216, -60695],[312, 24, 1],[154, -337, -551, -147, -2657, -77, 80; 1053, -2314, -3780, -1008, -18213, -527, 549; -416, 910, 1488, 397, 7176, 208, -216],58,19]
%
% END OF LATTICE L59.19
%
\hbox{}\par\smallskip%
%
% LATTICE L59.20
%
\leavevmode%
\llap{\spacer}%
${L_{59.20}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{6,1}{\infty b}{6}\above{1pt}{1pt}{r}{2}{234}\above{1pt}{1pt}{l}{2}{78}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{}{6}{234}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37674\\-70902\\-6552
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70902\\-132834\\-12285
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6552\\-12285\\-1136
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-7\\93
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\28\\-366
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
122\\303\\-3978
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
133\\326\\-4290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\114\\-1503
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\44\\-585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-45\\585
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-117\\-156\\-63\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-78\\-54\\-78\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\0\\234\\0\\-117\\-351\\-1989
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-78\\0\\78\\0\\-156\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-54\\-117\\0\\18\\0\\-585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-78\\-351\\-156\\0\\78\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-234\\-1989\\-1872\\-585\\-117\\234
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-37674, -70902, -6552; -70902, -132834, -12285; -6552, -12285, -1136],[234, 39, 1],[-3, 11, 122, 133, 47, 19, -17; -7, 28, 303, 326, 114, 44, -45; 93, -366, -3978, -4290, -1503, -585, 585],58,20]
%
% END OF LATTICE L59.20
%
\hbox{}\par\smallskip%
%
% LATTICE L59.21
%
\leavevmode%
\llap{\spacer}%
${L_{59.21}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{6,5}{\infty b}{6}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}{78}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{}{6}{26}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-328770\\-165906\\13104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165906\\-82524\\6513
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13104\\6513\\-514
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-80\\-963
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\38\\456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\41\\494
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-416\\-4992
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-115\\-1381
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-581\\-6981
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-706\\-8489
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-39\\-156\\-21\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-78\\-18\\-78\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\26\\0\\-13\\-117\\-221
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-78\\0\\78\\0\\-156\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-18\\-13\\0\\2\\0\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-78\\-117\\-156\\0\\78\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-78\\-221\\-624\\-65\\-39\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-328770, -165906, 13104; -165906, -82524, 6513; 13104, 6513, -514],[234, 39, 1],[2, -1, -1, 11, 3, 15, 18; -80, 38, 41, -416, -115, -581, -706; -963, 456, 494, -4992, -1381, -6981, -8489],58,21]
%
% END OF LATTICE L59.21
%
\hbox{}\par\smallskip%
%
% LATTICE L59.22
%
\leavevmode%
\llap{\spacer}%
${L_{59.22}}=\pdualwitharg{3.13}(\pfillwitharg{3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty a}{8}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{l}{2}{104}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{}{6}{78}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26520\\2496\\1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2496\\-234\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1248\\-117\\-58
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\59\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\120\\-104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-11\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-39\\-104\\-21\\-13\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-104\\-36\\-52\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\78\\0\\-39\\-117\\-663
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-104\\0\\104\\0\\-104\\-1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-36\\-39\\0\\6\\0\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-52\\-117\\-104\\0\\26\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-156\\-663\\-1248\\-195\\-39\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26520, 2496, 1248; 2496, -234, -117; 1248, -117, -58],[312, 39, 1],[0, 1, 4, 7, 1, 0, -1; -1, 12, 59, 120, 20, 6, -11; 2, -4, -39, -104, -21, -13, 0],58,22]
%
% END OF LATTICE L59.22
%
\hbox{}\par\smallskip%
%
% LATTICE L59.23
%
\leavevmode%
\llap{\spacer}%
${L_{59.23}}=\pdualwitharg{2.13}(\pfillwitharg{2}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,1}{\infty z}{3}\above{1pt}{1pt}{r}{2}{468}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{}{6}{468}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76810500\\1831518\\38096136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1831518\\43680\\908388
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38096136\\908388\\18894755
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\1\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
544\\-38\\-1095
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10463\\-744\\-21060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5561\\-400\\-11193
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3926\\-285\\-7902
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1744\\-131\\-3510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1279\\-99\\-2574
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-234\\-156\\-126\\-78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-39\\-54\\-78\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\0\\468\\0\\-234\\-702\\-3978
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-39\\0\\39\\0\\-156\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-54\\-234\\0\\36\\0\\-1170
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-78\\-702\\-156\\0\\156\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-234\\-3978\\-1872\\-1170\\-234\\468
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[76810500, 1831518, 38096136; 1831518, 43680, 908388; 38096136, 908388, 18894755],[234, 78, 1],[-3, 544, 10463, 5561, 3926, 1744, 1279; 1, -38, -744, -400, -285, -131, -99; 6, -1095, -21060, -11193, -7902, -3510, -2574],58,23]
%
% END OF LATTICE L59.23
%
\hbox{}\par\smallskip%
%
% LATTICE L59.24
%
\leavevmode%
\llap{\spacer}%
${L_{59.24}}=\pdualwitharg{2.3.13}(\pfillwitharg{2}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{12,5}{\infty z}{3}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{}{6}{52}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1190362680\\-32894550\\590116176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32894550\\909012\\-16307304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
590116176\\-16307304\\292547059
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1938\\-49\\-3912
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\-1\\-111
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-4\\26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3342\\-101\\-6747
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1705\\-49\\-3442
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8771\\-241\\-17706
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12880\\-339\\-26000
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-78\\-156\\-42\\-78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-39\\-18\\-78\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\0\\52\\0\\-26\\-234\\-442
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-39\\0\\39\\0\\-156\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-18\\-26\\0\\4\\0\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-78\\-234\\-156\\0\\156\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-78\\-442\\-624\\-130\\-78\\52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1190362680, -32894550, 590116176; -32894550, 909012, -16307304; 590116176, -16307304, 292547059],[234, 78, 1],[1938, 55, -13, 3342, 1705, 8771, 12880; -49, -1, -4, -101, -49, -241, -339; -3912, -111, 26, -6747, -3442, -17706, -26000],58,24]
%
% END OF LATTICE L59.24
%
\hbox{}\par\smallskip%
%
% LATTICE L59.25
%
\leavevmode%
\llap{\spacer}%
${L_{59.25}}=\pdualwitharg{2}({L_{59.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${624}\above{1pt}{1pt}{24,13}{\infty z}{156}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{1872}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{}{6}{144}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1705392\\-102024\\-57096
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102024\\-6000\\-3432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57096\\-3432\\-1909
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67\\247\\1560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
157\\-572\\-3666
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
253\\-924\\-5904
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\-245\\-1563
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1204\\-4407\\-28080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\-125\\-792
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\135\\864
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\-312\\-936\\-312\\-6552\\-312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\156\\0\\-78\\-2808\\-312\\-936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\0\\144\\0\\-936\\-216\\-1224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-78\\0\\3\\0\\-24\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6552\\-2808\\-936\\0\\1872\\0\\-4680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\-216\\-24\\0\\48\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-936\\-1224\\-288\\-4680\\-72\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1705392, -102024, -57096; -102024, -6000, -3432; -57096, -3432, -1909],[936, 24, 1],[-67, 157, 253, 67, 1204, 34, -37; 247, -572, -924, -245, -4407, -125, 135; 1560, -3666, -5904, -1563, -28080, -792, 864],58,25]
%
% END OF LATTICE L59.25
%
\hbox{}\par\smallskip%
%
% LATTICE L59.26
%
\leavevmode%
\llap{\spacer}%
${L_{59.26}}=\pdualwitharg{2.3}({L_{59.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${624}\above{1pt}{1pt}{24,5}{\infty z}{156}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{208}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{}{6}{16}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3944304\\-88920\\2588040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88920\\-1680\\54312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2588040\\54312\\-1648037
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-704\\-23257\\-1872
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5955\\196703\\15834
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2765\\91333\\7352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2032\\67121\\5403
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11421\\377260\\30368
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
695\\22958\\1848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-346\\-11429\\-920
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\-312\\-312\\-312\\-2184\\-312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\156\\0\\-78\\-936\\-312\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\0\\16\\0\\-104\\-72\\-136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-78\\0\\3\\0\\-24\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2184\\-936\\-104\\0\\208\\0\\-520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\-72\\-24\\0\\48\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-312\\-136\\-96\\-520\\-24\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3944304, -88920, 2588040; -88920, -1680, 54312; 2588040, 54312, -1648037],[936, 24, 1],[-704, 5955, 2765, 2032, 11421, 695, -346; -23257, 196703, 91333, 67121, 377260, 22958, -11429; -1872, 15834, 7352, 5403, 30368, 1848, -920],58,26]
%
% END OF LATTICE L59.26
%
\hbox{}\par\smallskip%
%
% LATTICE L59.27
%
\leavevmode%
\llap{\spacer}%
${L_{59.27}}=\pdualwitharg{2.13}(\pfillwitharg{3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{8,5}{\infty z}{12}\above{1pt}{1pt}{*}{2}{208}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{624}\above{1pt}{1pt}{}{6}{208}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1040208\\93912\\3432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93912\\8112\\104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3432\\104\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\24\\-960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-333\\13350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82\\-1951\\78208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\-1404\\56277
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-595\\23848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-405\\16224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\262\\-10504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-312\\-312\\-168\\-312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-78\\-72\\-312\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\0\\208\\0\\-104\\-936\\-1768
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-78\\0\\39\\0\\-312\\-1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-72\\-104\\0\\16\\0\\-520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\-936\\-312\\0\\624\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-312\\-1768\\-1248\\-520\\-312\\208
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1040208, 93912, 3432; 93912, 8112, 104; 3432, 104, -1],[312, 104, 1],[-1, 14, 82, 59, 25, 17, -11; 24, -333, -1951, -1404, -595, -405, 262; -960, 13350, 78208, 56277, 23848, 16224, -10504],58,27]
%
% END OF LATTICE L59.27
%
\hbox{}\par\smallskip%
%
% LATTICE L59.28
%
\leavevmode%
\llap{\spacer}%
${L_{59.28}}=\pdualwitharg{13}({L_{59.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,7}{\infty b}{24}\above{1pt}{1pt}{b}{2}{234}\above{1pt}{1pt}{l}{2}{312}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{}{6}{234}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-498888\\582192\\-57096
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
582192\\-676650\\66417
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57096\\66417\\-6518
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-35\\-453
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\148\\1920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-314\\993\\12870
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-745\\2360\\30576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\444\\5751
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71\\226\\2925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-81\\-1053
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-117\\-312\\-63\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-312\\-108\\-156\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\0\\234\\0\\-117\\-351\\-1989
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\0\\312\\0\\-312\\-3744
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-108\\-117\\0\\18\\0\\-585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-156\\-351\\-312\\0\\78\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-468\\-1989\\-3744\\-585\\-117\\234
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-498888, 582192, -57096; 582192, -676650, 66417; -57096, 66417, -6518],[936, 39, 1],[11, -47, -314, -745, -140, -71, 26; -35, 148, 993, 2360, 444, 226, -81; -453, 1920, 12870, 30576, 5751, 2925, -1053],58,28]
%
% END OF LATTICE L59.28
%
\hbox{}\par\smallskip%
%
% LATTICE L59.29
%
\leavevmode%
\llap{\spacer}%
${L_{59.29}}=\pdualwitharg{3.13}({L_{59.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{12,11}{\infty a}{24}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{l}{2}{312}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{}{6}{26}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2685384\\-144144\\11232
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144144\\-7410\\585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11232\\585\\-46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-44\\-804
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\392\\7176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\44\\803
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\178\\3237
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\137\\2470
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-12\\-39\\-312\\-21\\-39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-312\\-36\\-156\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\26\\0\\-13\\-117\\-221
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\0\\312\\0\\-312\\-1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-36\\-13\\0\\2\\0\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-156\\-117\\-312\\0\\78\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-156\\-221\\-1248\\-65\\-39\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2685384, -144144, 11232; -144144, -7410, 585; 11232, 585, -46],[936, 39, 1],[0, -1, 0, 9, 1, 4, 3; 1, -44, -1, 392, 44, 178, 137; 12, -804, -13, 7176, 803, 3237, 2470],58,29]
%
% END OF LATTICE L59.29
%
\hbox{}\par\smallskip%
%
% LATTICE L59.30
%
\leavevmode%
\llap{\spacer}%
${L_{59.30}}=\pdualwitharg{2.3.13}(\pfillwitharg{3}({L_{59.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,5}{\infty z}{4}\above{1pt}{1pt}{*}{2}{624}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{208}\above{1pt}{1pt}{}{6}{624}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\63336\\1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63336\\-2115984\\-52104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1560\\-52104\\-1283
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\3\\-118
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\71\\-2808
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\20\\-793
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\29\\-1152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\13\\-520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-8\\312
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-312\\-104\\-168\\-104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\4\\0\\-26\\-72\\-104\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\0\\624\\0\\-312\\-936\\-5304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-26\\0\\13\\0\\-104\\-1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-72\\-312\\0\\48\\0\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-104\\-936\\-104\\0\\208\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-312\\-5304\\-1248\\-1560\\-312\\624
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1872, 63336, 1560; 63336, -2115984, -52104; 1560, -52104, -1283],[312, 312, 1],[-1, 3, 59, 15, 20, 6, -11; -2, 3, 71, 20, 29, 13, -8; 80, -118, -2808, -793, -1152, -520, 312],58,30]
%
% END OF LATTICE L59.30
%
\hbox{}\par\smallskip%
%
% LATTICE L59.31
%
\leavevmode%
\llap{\spacer}%
${L_{59.31}}=\pdualwitharg{2.13}({L_{59.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,13}{\infty z}{12}\above{1pt}{1pt}{*}{2}{1872}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{*}{2}{624}\above{1pt}{1pt}{}{6}{1872}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-761904\\-506376\\-3744
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-506376\\19344\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3744\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-26\\3720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\79\\-11298
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\1551\\-221832
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\394\\-56355
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\525\\-75096
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\157\\-22464
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-288\\41184
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-936\\-312\\-504\\-312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\12\\0\\-78\\-216\\-312\\-936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\0\\1872\\0\\-936\\-2808\\-15912
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-78\\0\\39\\0\\-312\\-3744
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-216\\-936\\0\\144\\0\\-4680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\-2808\\-312\\0\\624\\-936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-936\\-15912\\-3744\\-4680\\-936\\1872
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-761904, -506376, -3744; -506376, 19344, 0; -3744, 0, -1],[936, 312, 1],[-1, 3, 59, 15, 20, 6, -11; -26, 79, 1551, 394, 525, 157, -288; 3720, -11298, -221832, -56355, -75096, -22464, 41184],58,31]
%
% END OF LATTICE L59.31
%
\hbox{}\par\smallskip%
%
% LATTICE L59.32
%
\leavevmode%
\llap{\spacer}%
${L_{59.32}}=\pdualwitharg{2.3.13}({L_{59.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{24,5}{\infty z}{12}\above{1pt}{1pt}{*}{2}{208}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{624}\above{1pt}{1pt}{}{6}{208}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6741072\\9390888\\-856440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9390888\\-13080912\\1193088
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-856440\\1193088\\-108809
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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315\\-11\\-2600
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6741072, 9390888, -856440; 9390888, -13080912, 1193088; -856440, 1193088, -108809],[936, 312, 1],[23, -399, -2317, -1657, -698, -452, 315; -1, 14, 82, 59, 25, 17, -11; -192, 3294, 19136, 13689, 5768, 3744, -2600],58,32]
%
% END OF LATTICE L59.32
%

\medskip%
%------------------------------WEYL-GROUP-W60--------------------------------
%
\leavevmode\llap{}%
$W_{60\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=48$%
\hfill%
$222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L60.1
%
\leavevmode%
\llap{\spacer}%
${L_{60.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{60}}%
\EasyButWeakLineBreak%
{${34}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{68}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47940\\19040\\-340
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\kern3pt%
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19040\\7562\\-135
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\kern3pt%
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-340\\-135\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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215\\-544\\-164
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83\\-210\\-65
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\HardButStrongLineBreak\kern3pt%
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0\\0\\-1
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-27\\68\\0
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-2\\5\\0
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{$\left.\llap{\phantom{%
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-153\\-12\\0\\2\\0\\-5
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-1360\\-140\\-65\\-5\\0\\10
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-136\\-20\\-20\\-8\\-204\\-20
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[47940, 19040, -340; 19040, 7562, -135; -340, -135, 2],[340, 1, 1],[2197, 215, 83, 0, -27, -2, 47, 49, 247, 330, 17979, 2642; -5559, -544, -210, 0, 68, 5, -119, -124, -625, -835, -45492, -6685; -1666, -164, -65, -1, 0, 0, -34, -36, -185, -249, -13600, -2000],59,1]
%
% END OF LATTICE L60.1
%
\hbox{}\par\smallskip%
%
% LATTICE L60.2
%
\leavevmode%
\llap{\spacer}%
${L_{60.2}}=\pfillwitharg{2}({L_{60.1}})$%
{${}={}$}Nikulin $60$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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0\\-1\\0
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\kern3pt%
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170\\0\\31
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left.\llap{\phantom{%
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\kern3pt%
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\kern3pt%
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-17\\-4\\-25\\-27\\-680\\-195
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[935, 0, 170; 0, -1, 0; 170, 0, 31],[85, 1, 1],[241, 13, 14, 2, 9, -1, -3, 1, 21, 33, 943, 281; -476, -25, -25, -3, -17, 0, 0, -3, -45, -67, -1887, -560; -1275, -69, -75, -11, -51, 5, 17, -5, -110, -174, -4981, -1485],59,2]
%
% END OF LATTICE L60.2
%
\hbox{}\par\smallskip%
%
% LATTICE L60.3
%
\leavevmode%
\llap{\spacer}%
${L_{60.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{60.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${170}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{85}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420325\\2040\\168810
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-420325, 2040, 168810; 2040, -5, -820; 168810, -820, -67797],[85, 5, 1],[-171, -175, -387, -2643, -72457, -4267, -17815, -883, -142, -2, 513, 35; -68, -63, -137, -931, -25483, -1500, -6256, -309, -49, 1, 187, 12; -425, -435, -962, -6570, -180115, -10607, -44285, -2195, -353, -5, 1275, 87],59,3]
%
% END OF LATTICE L60.3
%
\hbox{}\par\smallskip%
%
% LATTICE L60.4
%
\leavevmode%
\llap{\spacer}%
${L_{60.4}}=\pdualwitharg{2}({L_{60.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4621960, -32980, 1154980; -32980, 232, -8240; 1154980, -8240, 288617],[340, 4, 1],[-11829, -318, -672, -82, -21, 163, 235, -23, -1033, -1623, -23167, -13803; 18632, 503, 1075, 135, 34, -270, -408, 33, 1605, 2545, 36414, 21710; 47872, 1287, 2720, 332, 85, -660, -952, 93, 4180, 6568, 93755, 55860],59,4]
%
% END OF LATTICE L60.4
%
\hbox{}\par\smallskip%
%
% LATTICE L60.5
%
\leavevmode%
\llap{\spacer}%
${L_{60.5}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{60.1}}))$%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1118855, -10455, 460360; -10455, -85, 4301; 460360, 4301, -189418],[85, 17, 1],[-30, -475, -5169, -7027, -11315, -56614, -2778, -2333, -1851, 7, 83, 454; -4, -70, -775, -1059, -1708, -8550, -420, -354, -285, -1, 12, 70; -73, -1156, -12580, -17102, -27538, -137785, -6761, -5678, -4505, 17, 202, 1105],59,5]
%
% END OF LATTICE L60.5
%
\hbox{}\par\smallskip%
%
% LATTICE L60.6
%
\leavevmode%
\llap{\spacer}%
${L_{60.6}}=\pdualwitharg{5}({L_{60.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-45900\\680\\-340
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\2\\5
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-765\\-60\\0\\10\\0\\-5
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-45900, 680, -340; 680, -10, 5; -340, 5, -2],[340, 5, 1],[-133, -13, -1, 0, 1, 0, -3, -3, -3, -20, -1089, -32; -9197, -896, -68, 2, 68, -1, -221, -212, -209, -1387, -75412, -2215; -2125, -200, -13, 5, 0, -3, -85, -60, -52, -330, -17680, -517],59,6]
%
% END OF LATTICE L60.6
%
\hbox{}\par\smallskip%
%
% LATTICE L60.7
%
\leavevmode%
\llap{\spacer}%
${L_{60.7}}=\pdualwitharg{17}({L_{60.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
340\\153\\26
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-7921\\18700\\-7480
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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7\\-16\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1020, -340, 340; -340, -102, 153; 340, 153, 26],[340, 17, 1],[3, 7, -78, -157, -559, -1438, -75, -151, -102, -23, -17, -2; -7, -16, 185, 371, 1320, 3395, 177, 356, 240, 54, 40, 5; 2, 0, -85, -153, -532, -1360, -70, -136, -85, -17, -12, 0],59,7]
%
% END OF LATTICE L60.7
%
\hbox{}\par\smallskip%
%
% LATTICE L60.8
%
\leavevmode%
\llap{\spacer}%
${L_{60.8}}=\pdualwitharg{2.5}({L_{60.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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3656036\\45815\\-14794080
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[59160, -340, 14620; -340, 1400, -80; 14620, -80, 3613],[340, 20, 1],[-145109, -3508, -1034, 84, -21, -85, -4117, -881, -3355, -22029, -298431, -35027; -1819, -44, -13, 1, 0, -1, -51, -11, -42, -276, -3740, -439; 587180, 14195, 4184, -340, 85, 344, 16660, 3565, 13576, 89140, 1207595, 141736],59,8]
%
% END OF LATTICE L60.8
%
\hbox{}\par\smallskip%
%
% LATTICE L60.9
%
\leavevmode%
\llap{\spacer}%
${L_{60.9}}=\pdualwitharg{5.17}(\pfillwitharg{2}({L_{60.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[53210, 915705, -356065; 915705, 15792575, -6140825; -356065, -6140825, 2387814],[85, 85, 1],[1, 7, 14, 94, 151, 151, 37, 31, 5, 1, 1, 1; -490, -4859, -11126, -77776, -126513, -126980, -31374, -27005, -4806, -1884, -943, -476; -1260, -12495, -28611, -200005, -325335, -326536, -80680, -69445, -12359, -4845, -2425, -1224],59,9]
%
% END OF LATTICE L60.9
%
\hbox{}\par\smallskip%
%
% LATTICE L60.10
%
\leavevmode%
\llap{\spacer}%
${L_{60.10}}=\pdualwitharg{2.17}({L_{60.1}})$%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1449080, -202980, 363120; -202980, 28424, -50864; 363120, -50864, 90993],[340, 68, 1],[-1, 122, 3449, 5047, 4158, 41889, 2087, 921, 1766, 168, 14, -169; 0, -11, -315, -463, -382, -3850, -192, -85, -165, -17, -2, 10; 4, -493, -13940, -20400, -16807, -169320, -8436, -3723, -7140, -680, -57, 680],59,10]
%
% END OF LATTICE L60.10
%
\hbox{}\par\smallskip%
%
% LATTICE L60.11
%
\leavevmode%
\llap{\spacer}%
${L_{60.11}}=\pdualwitharg{5.17}({L_{60.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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-1313420\\-170000\\9860
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1313420, -170000, 9860; -170000, -21930, 1275; 9860, 1275, -74],[340, 85, 1],[6, 53, 39, 232, 715, 353, 84, 127, 6, -7, 5, 7; 17, 144, 103, 605, 1856, 915, 217, 324, 14, -20, 16, 21; 1090, 9520, 6953, 41225, 126900, 62628, 14890, 22440, 1037, -1275, 940, 1292],59,11]
%
% END OF LATTICE L60.11
%
\hbox{}\par\smallskip%
%
% LATTICE L60.12
%
\leavevmode%
\llap{\spacer}%
${L_{60.12}}=\pdualwitharg{2.5.17}({L_{60.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26520, -1555500, -384200; -1555500, 92582680, 22867380; -384200, 22867380, 5648109],[340, 340, 1],[-36, -88, -279, -1715, -1337, -2650, -636, -248, -57, 35, -7, -38; 5537, 13414, 42337, 259799, 202420, 401135, 96233, 37471, 8548, -5374, 1120, 5945; -22420, -54315, -171428, -1051960, -819625, -1624248, -389660, -151725, -34612, 21760, -4535, -24072],59,12]
%
% END OF LATTICE L60.12
%

\medskip%
%------------------------------WEYL-GROUP-W61--------------------------------
%
\leavevmode\llap{}%
$W_{61\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=30$%
\hfill%
$22422242\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L61.1
%
\leavevmode%
\llap{\spacer}%
${L_{61.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{61}}%
\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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37932\\-50\\-79
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644\\183425\\193082
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8\\2278\\2399
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-27083796, 37932, 54288; 37932, -50, -79; 54288, -79, -106],[348, 1, 1],[125, 644, 135, 8, 7, -6, -3, 3; 35604, 183425, 38450, 2278, 1992, -1711, -854, 855; 37476, 193082, 40476, 2399, 2100, -1798, -900, 899],60,1]
%
% END OF LATTICE L61.1
%
\hbox{}\par\smallskip%
%
% LATTICE L61.2
%
\leavevmode%
\llap{\spacer}%
${L_{61.2}}=\pfillwitharg{2}({L_{61.1}})$%
{${}={}$}Nikulin $61$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1131, 348, -87; 348, -106, 29; -87, 29, -2],[87, 1, 1],[1, -20, -1, 8, 107, 1064, 109, 10; 3, -58, -3, 23, 309, 3074, 315, 29; 0, 29, -1, -18, -210, -2059, -209, -17],60,2]
%
% END OF LATTICE L61.2
%
\hbox{}\par\smallskip%
%
% LATTICE L61.3
%
\leavevmode%
\llap{\spacer}%
${L_{61.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{61.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-121951554, 1669356, 40578453; 1669356, -22821, -555480; 40578453, -555480, -13502165],[87, 3, 1],[-30233, -935773, -98166, -11743, -1805, 6671, 2052, -4276; -42589, -1318224, -138287, -16543, -2543, 9396, 2891, -6023; -89108, -2758074, -289332, -34611, -5320, 19662, 6048, -12603],60,3]
%
% END OF LATTICE L61.3
%
\hbox{}\par\smallskip%
%
% LATTICE L61.4
%
\leavevmode%
\llap{\spacer}%
${L_{61.4}}=\pdualwitharg{3}({L_{61.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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-696\\-10266\\-2088\\-87
\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3583356, 24012, 853644; 24012, -150, -5883; 853644, -5883, -200930],[348, 3, 1],[161, -412, -207, 206, 2867, 44318, 9291, 551; 7104, -18183, -9134, 9091, 126516, 1955673, 409994, 24314; 476, -1218, -612, 609, 8476, 131022, 27468, 1629],60,4]
%
% END OF LATTICE L61.4
%
\hbox{}\par\smallskip%
%
% LATTICE L61.5
%
\leavevmode%
\llap{\spacer}%
${L_{61.5}}=\pdualwitharg{2}({L_{61.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{232}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{4}{8}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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7169152872\\-79295628\\-1813167696
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-79295628\\877064\\20054848
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1813167696\\20054848\\458572603
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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-2769547735\\16985880\\-10951363032
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30650139\\-187981\\121196972
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700453017\\-4295940\\2769735715
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-28019\\174\\-110793
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-39\\-696\\-66\\0
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-696\\-13688\\-1392\\-116
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7169152872, -79295628, -1813167696; -79295628, 877064, 20054848; -1813167696, 20054848, 458572603],[348, 4, 1],[-28019, -565243, -58479, -6112, -1330, -2171, -219, -3671; 174, 3480, 358, 35, 6, 0, 2, 25; -110793, -2235088, -231238, -24168, -5259, -8584, -866, -14516],60,5]
%
% END OF LATTICE L61.5
%
\hbox{}\par\smallskip%
%
% LATTICE L61.6
%
\leavevmode%
\llap{\spacer}%
${L_{61.6}}=\pdualwitharg{29}(\pfillwitharg{2}({L_{61.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${87}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{29}\above{1pt}{1pt}{}{4}{58}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1406400762\\2264436\\484358319
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2264436\\-3103\\-779868
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
484358319\\-779868\\-166810903
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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{$\left[\!\llap{\phantom{%
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-14810631106\\-480725205\\-43002450045
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18470911\\599530\\53630019
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5100777144\\165561624\\14810031575
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1406400762, 2264436, 484358319; 2264436, -3103, -779868; 484358319, -779868, -166810903],[87, 29, 1],[-1323809, -470975, -1432808, -171404, -79045, 3357, 29954, -62405; -42969, -15287, -46506, -5563, -2565, 109, 972, -2026; -3843660, -1367469, -4160137, -497669, -229506, 9747, 86971, -181192],60,6]
%
% END OF LATTICE L61.6
%
\hbox{}\par\smallskip%
%
% LATTICE L61.7
%
\leavevmode%
\llap{\spacer}%
${L_{61.7}}=\pdualwitharg{2.3}({L_{61.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{696}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{4}{24}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
141166200\\205842348\\-34142280
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
205842348\\300150264\\-49784772
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[141166200, 205842348, -34142280; 205842348, 300150264, -49784772; -34142280, -49784772, 8257609],[348, 12, 1],[7, 505, -45, -347, -614, -35711, -3597, -260; 2, 0, 2, 25, 58, 3480, 358, 35; 41, 2088, -174, -1284, -2189, -126672, -12714, -864],60,7]
%
% END OF LATTICE L61.7
%
\hbox{}\par\smallskip%
%
% LATTICE L61.8
%
\leavevmode%
\llap{\spacer}%
${L_{61.8}}=\pdualwitharg{3.29}(\pfillwitharg{2}({L_{61.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${29}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{87}\above{1pt}{1pt}{}{4}{174}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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95397675\\283712220\\3532896
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
283712220\\843758799\\10506816
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3532896\\10506816\\130835
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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157\\-1572\\122001
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\HardButStrongLineBreak\kern3pt%
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474\\-4703\\364878
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\-475\\36714
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-174\\-87\\-87\\174
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-377\\-348\\-957\\0
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-348\\-354\\-1044\\-87
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[95397675, 283712220, 3532896; 283712220, 843758799, 10506816; 3532896, 10506816, 130835],[87, 87, 1],[149, 157, 474, 53, 9, 1, 0, 26; -1513, -1572, -4703, -475, -59, 0, -13, -311; 117479, 122001, 364878, 36714, 4495, -27, 1044, 24273],60,8]
%
% END OF LATTICE L61.8
%
\hbox{}\par\smallskip%
%
% LATTICE L61.9
%
\leavevmode%
\llap{\spacer}%
${L_{61.9}}=\pdualwitharg{29}({L_{61.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${348}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{116}\above{1pt}{1pt}{*}{4}{58}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108228\\12528\\6264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12528\\-1450\\-725
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6264\\-725\\-362
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2579\\36636\\-29928
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\-4261\\3480
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145\\-2059\\1681
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\792\\-696
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\143\\-118
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\878\\-696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\57\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
348\\0\\-348\\-348\\-4524\\-696\\-3828\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
348\\0\\-348\\-348
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-29
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-348\\0\\116\\-58
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-348\\-29\\-58\\58
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4524\\-696\\-3828\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-696\\-118\\-696\\-29
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3828\\-696\\-4292\\-290
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-29\\-290\\-87
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-108228, 12528, 6264; 12528, -1450, -725; 6264, -725, -362],[348, 29, 1],[53, 10, 63, 5, 7, 0, -3, 0; 792, 143, 878, 57, 60, -1, -26, 14; -696, -118, -696, -29, 0, 2, 0, -29],60,9]
%
% END OF LATTICE L61.9
%
\hbox{}\par\smallskip%
%
% LATTICE L61.10
%
\leavevmode%
\llap{\spacer}%
${L_{61.10}}=\pdualwitharg{3.29}({L_{61.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${116}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{348}\above{1pt}{1pt}{*}{4}{174}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-338427564\\1213476\\-2887356
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1213476\\-4350\\10353
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2887356\\10353\\-24634
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1223479\\1189960\\-142895760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4380\\-4261\\511560
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10439\\10153\\-1219219
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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7\\20\\-812
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-2\\-1\\234
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-9\\-26\\1044
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29\\14\\-3393
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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116\\0\\-348\\-348\\-1508\\-696\\-3828\\0
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0\\6\\0\\-87\\-696\\-354\\-2088\\-87
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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116\\0\\-348\\-348
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-87
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-348\\-87\\-174\\174
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1508\\-696\\-3828\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-696\\-354\\-2088\\-87
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3828\\-2088\\-12876\\-870
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-338427564, 1213476, -2887356; 1213476, -4350, 10353; -2887356, 10353, -24634],[348, 87, 1],[7, -2, -9, 29, 285, 148, 885, 44; 20, -1, -26, 14, 264, 143, 878, 57; -812, 234, 1044, -3393, -33292, -17286, -103356, -5133],60,10]
%
% END OF LATTICE L61.10
%
\hbox{}\par\smallskip%
%
% LATTICE L61.11
%
\leavevmode%
\llap{\spacer}%
${L_{61.11}}=\pdualwitharg{2.29}({L_{61.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${87}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{116}\above{1pt}{1pt}{*}{4}{232}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61944\\24012\\-2436
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24012\\19720\\-8816
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2436\\-8816\\6047
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61802\\-184149\\-243600
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-208953\\622598\\823600
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173637\\-517371\\-684401
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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8542\\-25452\\-33669
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5887\\-17541\\-23204
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17555\\-52307\\-69194
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1707\\-5086\\-6728
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[61944, 24012, -2436; 24012, 19720, -8816; -2436, -8816, 6047],[348, 116, 1],[8542, 5887, 17555, 1707, 287, -1, 103, 1236; -25452, -17541, -52307, -5086, -855, 3, -307, -3683; -33669, -23204, -69194, -6728, -1131, 4, -406, -4872],60,11]
%
% END OF LATTICE L61.11
%
\hbox{}\par\smallskip%
%
% LATTICE L61.12
%
\leavevmode%
\llap{\spacer}%
${L_{61.12}}=\pdualwitharg{2.3.29}({L_{61.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${29}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{348}\above{1pt}{1pt}{*}{4}{696}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59160\\3476172\\-878700
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3476172\\389473944\\-98450592
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-878700\\-98450592\\24886181
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-4261\\27062431\\107059764
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\kern3pt%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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0\\91\\360
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[59160, 3476172, -878700; 3476172, 389473944, -98450592; -878700, -98450592, 24886181],[348, 348, 1],[2, 0, 2, 25, 58, 120, 358, 35; -12440, 91, -13327, -159924, -368721, -762413, -2273639, -221237; -49213, 360, -52722, -632664, -1458671, -3016128, -8994582, -875220],60,12]
%
% END OF LATTICE L61.12
%

\medskip%
%------------------------------WEYL-GROUP-W62--------------------------------
%
\leavevmode\llap{}%
$W_{62\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=42$%
\hfill%
$4222242222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L62.1
%
\leavevmode%
\llap{\spacer}%
${L_{62.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{62}}%
\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22161412\\108836\\217672
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108836\\-534\\-1069
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
217672\\-1069\\-2138
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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586403\\1172808\\59096492
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\kern3pt%
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-2961\\-5923\\-298403
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\kern3pt%
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{$\left.\llap{\phantom{%
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-5\\-2\\-508
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-62\\-105\\-6258
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-767\\-1512\\-77308
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-105\\0\\14\\0\\-56
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-689\\-52\\0\\26\\0
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-1512\\-140\\-56\\0\\28
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-326\\-34\\-21\\-13\\0
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\kern3pt%
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-34\\-8\\-14\\-26\\-28
\end{tabular}\endgroup%
\kern3pt%
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-21\\-14\\-63\\-273\\-532
\end{tabular}\endgroup%
\kern3pt%
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-13\\-26\\-273\\-1547\\-3276
\end{tabular}\endgroup%
\kern3pt%
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0\\-28\\-532\\-3276\\-7084
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-22161412, 108836, 217672; 108836, -534, -1069; 217672, -1069, -2138],[364, 1, 1],[0, -5, -62, -360, -767, -162, -13, -1, 9, 11; 2, -2, -105, -689, -1512, -326, -34, -21, -13, 0; -1, -508, -6258, -36296, -77308, -16325, -1306, -91, 923, 1120],61,1]
%
% END OF LATTICE L62.1
%
\hbox{}\par\smallskip%
%
% LATTICE L62.2
%
\leavevmode%
\llap{\spacer}%
${L_{62.2}}=\pfillwitharg{2}({L_{62.1}})$%
{${}={}$}Nikulin $62$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-17\\-2\\-7\\-13\\-7
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-21\\-7\\-63\\-273\\-266
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-13\\-13\\-273\\-1547\\-1638
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[910, 0, 637; 0, -1, 0; 637, 0, 446],[91, 1, 1],[276, 9, -5, -9, 10, 12, 7, 117, 633, 662; -138, -5, 0, 0, -7, -6, -3, -56, -312, -329; -397, -13, 7, 13, -14, -17, -10, -168, -910, -952],61,2]
%
% END OF LATTICE L62.2
%
\hbox{}\par\smallskip%
%
% LATTICE L62.3
%
\leavevmode%
\llap{\spacer}%
${L_{62.3}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{62.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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-970242\\11375\\139685
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11375\\-133\\-1638
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-970242, 11375, 139685; 11375, -133, -1638; 139685, -1638, -20110],[91, 7, 1],[-12, 13, 117, 5247, 818, 2460, 123, 19, 57, -2; -78, 77, 730, 32838, 5123, 15414, 775, 122, 390, -11; -77, 84, 753, 33761, 5263, 15827, 791, 122, 364, -13],61,3]
%
% END OF LATTICE L62.3
%
\hbox{}\par\smallskip%
%
% LATTICE L62.4
%
\leavevmode%
\llap{\spacer}%
${L_{62.4}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{62.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1727453, -14196, 264173; -14196, -78, 2171; 264173, 2171, -40399],[91, 13, 1],[-169, 167, 11078, 5476, 77741, 33415, 1680, 1851, 65, -167; 2, -1, -105, -53, -756, -326, -17, -21, -1, 0; -1105, 1092, 72436, 35806, 508326, 218491, 10985, 12103, 425, -1092],61,4]
%
% END OF LATTICE L62.4
%
\hbox{}\par\smallskip%
%
% LATTICE L62.5
%
\leavevmode%
\llap{\spacer}%
${L_{62.5}}=\pdualwitharg{2}({L_{62.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[930684664, 364, -235332552; 364, 8, -92; -235332552, -92, 59506095],[364, 4, 1],[0, 1459, 94349, 606681, 331349, 284904, 14369, 16447, 10611, 1039; 1, -30, -1960, -12610, -6888, -5923, -299, -343, -221, -21; 0, 5770, 373128, 2399280, 1310407, 1126728, 56826, 65044, 41964, 4109],61,5]
%
% END OF LATTICE L62.5
%
\hbox{}\par\smallskip%
%
% LATTICE L62.6
%
\leavevmode%
\llap{\spacer}%
${L_{62.6}}=\pdualwitharg{7}({L_{62.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13166972, -314132, 48412; -314132, -7490, 1155; 48412, 1155, -178],[364, 7, 1],[-1, 7, 23, 1009, 313, 469, 45, 3, 5, -1; 2, -2, -15, -689, -216, -326, -34, -3, -13, 0; -259, 1890, 6155, 269815, 83684, 125377, 12012, 796, 1274, -272],61,6]
%
% END OF LATTICE L62.6
%
\hbox{}\par\smallskip%
%
% LATTICE L62.7
%
\leavevmode%
\llap{\spacer}%
${L_{62.7}}=\pdualwitharg{13}({L_{62.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8817172, 247520, -1318408; 247520, -6942, 37011; -1318408, 37011, -197138],[364, 13, 1],[2, 171, 2038, 904, 24993, 5272, 415, 13, -25, -381; 2, -2, -105, -53, -1512, -326, -34, -21, -1, 0; -13, -1144, -13650, -6056, -167440, -35321, -2782, -91, 167, 2548],61,7]
%
% END OF LATTICE L62.7
%
\hbox{}\par\smallskip%
%
% LATTICE L62.8
%
\leavevmode%
\llap{\spacer}%
${L_{62.8}}=\pdualwitharg{7.13}(\pfillwitharg{2}({L_{62.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[182, -7098, -91; -7098, 332605, 4277; -91, 4277, 55],[91, 91, 1],[1, 6, 38, 128, 258, 773, 37, 5, 1, 0; 0, 6, 53, 183, 371, 1116, 56, 9, 3, 1; 0, -455, -4043, -13965, -28314, -85176, -4277, -689, -231, -78],61,8]
%
% END OF LATTICE L62.8
%
\hbox{}\par\smallskip%
%
% LATTICE L62.9
%
\leavevmode%
\llap{\spacer}%
${L_{62.9}}=\pdualwitharg{2.7}({L_{62.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[735453992, 92820, -185914456; 92820, 56, -23464; -185914456, -23464, 46997073],[364, 28, 1],[0, -5213, -37987, -1682217, -130756, -784944, -38395, -5621, -18495, -68; 1, -71, -523, -23179, -1802, -10819, -530, -78, -260, -1; 0, -20622, -150272, -6654648, -517255, -3105144, -151886, -22236, -73164, -269],61,9]
%
% END OF LATTICE L62.9
%
\hbox{}\par\smallskip%
%
% LATTICE L62.10
%
\leavevmode%
\llap{\spacer}%
${L_{62.10}}=\pdualwitharg{2.13}({L_{62.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${104}\above{1pt}{1pt}{*}{4}{52}\above{1pt}{1pt}{*}{2}{728}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{91}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1025160136, 7644, -258052340; 7644, 104, -1924; -258052340, -1924, 64956691],[364, 52, 1],[0, 5491, 355233, 175713, 1247599, 1072728, 54105, 61939, 3075, 3917; 1, -30, -1960, -970, -6888, -5923, -299, -343, -17, -21; 0, 21814, 1411228, 698052, 4956315, 4261608, 214942, 246064, 12216, 15561],61,10]
%
% END OF LATTICE L62.10
%
\hbox{}\par\smallskip%
%
% LATTICE L62.11
%
\leavevmode%
\llap{\spacer}%
${L_{62.11}}=\pdualwitharg{7.13}({L_{62.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-923468, 1223404, 351624; 1223404, -1620710, -465829; 351624, -465829, -133886],[364, 91, 1],[-32, 621, 1757, 5849, 23511, 35150, 3281, 194, 4, -99; 2, -2, -15, -53, -216, -326, -34, -3, -1, 0; -91, 1638, 4667, 15547, 62504, 93457, 8736, 520, 14, -260],61,11]
%
% END OF LATTICE L62.11
%
\hbox{}\par\smallskip%
%
% LATTICE L62.12
%
\leavevmode%
\llap{\spacer}%
${L_{62.12}}=\pdualwitharg{2.7.13}({L_{62.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[728, -13468, 3276; -13468, 11744824, -2874144; 3276, -2874144, 703349],[364, 364, 1],[1, -289, -2115, -7207, -7283, -43723, -2140, -314, -80, -4; 0, -2539, -18543, -63177, -63841, -383256, -18753, -2749, -699, -35; 0, -10374, -75764, -258132, -260845, -1565928, -76622, -11232, -2856, -143],61,12]
%
% END OF LATTICE L62.12
%

\medskip%
%------------------------------WEYL-GROUP-W63--------------------------------
%
\leavevmode\llap{}%
$W_{63\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=32$%
\hfill%
$62226222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L63.1
%
\leavevmode%
\llap{\spacer}%
${L_{63.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}31\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{63}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2765820, 13020, 6324; 13020, -58, -31; 6324, -31, -14],[372, 1, 1],[2, -1, -1, 11, 2, 49, 49, 361; 184, -93, -92, 1023, 185, 4521, 4520, 33294; 495, -246, -248, 2697, 493, 12102, 12104, 89187],62,1]
%
% END OF LATTICE L63.1
%
\hbox{}\par\smallskip%
%
% LATTICE L63.2
%
\leavevmode%
\llap{\spacer}%
${L_{63.2}}=\pfillwitharg{2}({L_{63.1}})$%
{${}={}$}Nikulin $63$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}31\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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5487\\1767\\93
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\kern3pt%
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1767\\569\\30
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93\\30\\2
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{$\left.\llap{\phantom{%
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\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-1
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\6\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-31\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
391\\-1209\\-186
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[5487, 1767, 93; 1767, 569, 30; 93, 30, 2],[93, 1, 1],[1, -2, 10, 391, 23, 290, 134, 1841; -3, 6, -31, -1209, -71, -894, -413, -5673; -1, 3, -4, -186, -12, -159, -74, -1023],62,2]
%
% END OF LATTICE L63.2
%
\hbox{}\par\smallskip%
%
% LATTICE L63.3
%
\leavevmode%
\llap{\spacer}%
${L_{63.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{63.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{62}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3503031\\25482\\-1163523
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25482\\-183\\8463
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1163523\\8463\\-386461
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\right.$}%
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-1100544145\\1098057975\\3337504299
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7548352\\-7531301\\-22891092
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-365388576\\364563150\\1108075445
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
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-367\\368\\1113
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30\\-31\\-91
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\HardButStrongLineBreak\kern3pt%
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5060\\-5052\\-15345
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59289\\-59179\\-179800
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3503031, 25482, -1163523; 25482, -183, 8463; -1163523, 8463, -386461],[93, 3, 1],[-367, 30, 5060, 59289, 9663, 37154, 50716, 228887; 368, -31, -5052, -59179, -9643, -37069, -50598, -228346; 1113, -91, -15345, -179800, -29304, -112673, -153801, -694121],62,3]
%
% END OF LATTICE L63.3
%
\hbox{}\par\smallskip%
%
% LATTICE L63.4
%
\leavevmode%
\llap{\spacer}%
${L_{63.4}}=\pdualwitharg{3}({L_{63.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{62}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1470516\\15996\\8556
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15996\\-174\\-93
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8556\\-93\\-46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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36703\\3399708\\-55056
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-1\\-92\\0
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14\\1302\\-31
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1470516, 15996, 8556; 15996, -174, -93; 8556, -93, -46],[372, 3, 1],[3, -1, -1, 14, 5, 33, 97, 234; 277, -93, -92, 1302, 464, 3057, 8984, 21669; -3, 2, 0, -31, -9, -50, -144, -341],62,4]
%
% END OF LATTICE L63.4
%
\hbox{}\par\smallskip%
%
% LATTICE L63.5
%
\leavevmode%
\llap{\spacer}%
${L_{63.5}}=\pdualwitharg{2}({L_{63.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}31\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
819385800\\-3346884\\202607568
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3346884\\13672\\-827576
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202607568\\-827576\\50098289
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
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8\\-12\\-8\\-372
\end{tabular}\endgroup%
\kern3pt%
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-12\\24\\0\\-372
\end{tabular}\endgroup%
\kern3pt%
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-8\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-372\\-372\\0\\744
\end{tabular}\endgroup%
\kern3pt%
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-12\\-36\\-4\\0
\end{tabular}\endgroup%
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\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[819385800, -3346884, 202607568; -3346884, 13672, -827576; 202607568, -827576, 50098289],[372, 4, 1],[1366, 181, 23, 11959, 1277, 31535, 7906, 233840; 23, 0, 1, 279, 27, 600, 149, 4371; -5524, -732, -93, -48360, -5164, -127524, -31971, -945624],62,5]
%
% END OF LATTICE L63.5
%
\hbox{}\par\smallskip%
%
% LATTICE L63.6
%
\leavevmode%
\llap{\spacer}%
${L_{63.6}}=\pdualwitharg{31}(\pfillwitharg{2}({L_{63.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${62}\above{1pt}{1pt}{}{6}{186}\above{1pt}{1pt}{l}{2}{31}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30049230\\-118203\\-9701016
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-118203\\-434\\-38161
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9701016\\-38161\\-3131851
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{$\left.\llap{\phantom{%
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0\\0\\0
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10184863343\\-665599584\\-31539817632
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45159951\\-2951287\\-139848378
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52225\\-3410\\-161727
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30049230, -118203, -9701016; -118203, -434, -38161; -9701016, -38161, -3131851],[93, 31, 1],[-3804, 961, 52225, 59203, 99685, 1149611, 523061, 228440; 247, -60, -3410, -3867, -6513, -75132, -34186, -14931; 11780, -2976, -161727, -183336, -308698, -3560040, -1619781, -707418],62,6]
%
% END OF LATTICE L63.6
%
\hbox{}\par\smallskip%
%
% LATTICE L63.7
%
\leavevmode%
\llap{\spacer}%
${L_{63.7}}=\pdualwitharg{2.3}({L_{63.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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665880\\343356\\165912
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
343356\\179400\\85560
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[665880, 343356, 165912; 343356, 179400, 85560; 165912, 85560, 41339],[372, 12, 1],[1519, -1, 69, 6224, 1798, 13269, 9882, 96603; 23, 0, 1, 93, 27, 200, 149, 1457; -6144, 4, -279, -25172, -7272, -53668, -39969, -390724],62,7]
%
% END OF LATTICE L63.7
%
\hbox{}\par\smallskip%
%
% LATTICE L63.8
%
\leavevmode%
\llap{\spacer}%
${L_{63.8}}=\pdualwitharg{3.31}(\pfillwitharg{2}({L_{63.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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407526\\-38335251\\-12569043
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[407526, -38335251, -12569043; -38335251, 3606953106, 1182617760; -12569043, 1182617760, 387746867],[93, 93, 1],[14, 1, -248, -95, -485, -1885, -2578, -376; -68607, -1240, 1129393, 430985, 2193899, 8502408, 11622359, 1694307; 209250, 3782, -3444627, -1314496, -6691350, -25932182, -35447973, -5167604],62,8]
%
% END OF LATTICE L63.8
%
\hbox{}\par\smallskip%
%
% LATTICE L63.9
%
\leavevmode%
\llap{\spacer}%
${L_{63.9}}=\pdualwitharg{31}({L_{63.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${62}\above{1pt}{1pt}{}{6}{186}\above{1pt}{1pt}{l}{2}{124}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1594020\\-29016\\-829188
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29016\\-434\\-14911
\end{tabular}\endgroup%
\kern3pt%
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-829188\\-14911\\-430978
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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2885231\\11546880\\-5950512
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48475\\193999\\-99975
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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496\\1986\\-1023
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481\\1924\\-992
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{$\left[\!\llap{\phantom{%
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-93\\-279\\-124\\0
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-279\\-4650\\-4464\\-1023
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1594020, -29016, -829188; -29016, -434, -14911; -829188, -14911, -430978],[372, 31, 1],[-962, 496, 481, -176, -977, -23764, -23749, -5641; -3849, 1986, 1924, -705, -3911, -95106, -95044, -22575; 1984, -1023, -992, 363, 2015, 49011, 48980, 11634],62,9]
%
% END OF LATTICE L63.9
%
\hbox{}\par\smallskip%
%
% LATTICE L63.10
%
\leavevmode%
\llap{\spacer}%
${L_{63.10}}=\pdualwitharg{3.31}({L_{63.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${186}\above{1pt}{1pt}{}{6}{62}\above{1pt}{1pt}{l}{2}{372}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-5499276\\85429056\\-2758752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
85429056\\-1326996354\\42852447
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2758752\\42852447\\-1383826
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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6229723\\12867564\\386045976
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\kern3pt%
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-97055020\\-200468221\\-6014343480
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3134477\\6474297\\194238497
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-2465\\-5093\\-152799
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\HardButStrongLineBreak\kern3pt%
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-16188\\-33437\\-1003160
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-47557\\-98228\\-2946984
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-3699\\-7640\\-229211
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-279\\-1550\\-4464\\-341
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5499276, 85429056, -2758752; 85429056, -1326996354, 42852447; -2758752, 42852447, -1383826],[372, 93, 1],[-2465, -16188, -47557, -3699, -1458, 496, 481, -224; -5093, -33437, -98228, -7640, -3010, 1025, 992, -463; -152799, -1003160, -2946984, -229211, -90303, 30752, 29760, -13891],62,10]
%
% END OF LATTICE L63.10
%
\hbox{}\par\smallskip%
%
% LATTICE L63.11
%
\leavevmode%
\llap{\spacer}%
${L_{63.11}}=\pdualwitharg{2.31}({L_{63.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${248}\above{1pt}{1pt}{}{6}{744}\above{1pt}{1pt}{l}{2}{31}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
449610360\\-291040524\\-329473332
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\kern3pt%
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-291040524\\188395928\\213274048
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-329473332\\213274048\\241437439
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{$\left.\llap{\phantom{%
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16095\\-48271\\64604
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[449610360, -291040524, -329473332; -291040524, 188395928, 213274048; -329473332, 213274048, 241437439],[372, 124, 1],[16095, -92, 749, 6445, 19183, 423428, 105085, 99389; -48271, 279, -2247, -19332, -57538, -1269987, -315180, -298095; 64604, -372, 3007, 25872, 77004, 1699668, 421817, 398952],62,11]
%
% END OF LATTICE L63.11
%
\hbox{}\par\smallskip%
%
% LATTICE L63.12
%
\leavevmode%
\llap{\spacer}%
${L_{63.12}}=\pdualwitharg{2.3.31}({L_{63.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${744}\above{1pt}{1pt}{}{6}{248}\above{1pt}{1pt}{l}{2}{93}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
671487528\\-18845698188\\-4657269252
\end{tabular}\endgroup%
\kern3pt%
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-18845698188\\528915768504\\130709041452
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\kern3pt%
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-4657269252\\130709041452\\32301652805
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[671487528, -18845698188, -4657269252; -18845698188, 528915768504, 130709041452; -4657269252, 130709041452, 32301652805],[372, 372, 1],[-13347, -98950, -73724, -23257, -11393, -10, -496, -1483; -59863, -443293, -330246, -104170, -50968, -31, -2247, -6661; 240312, 1779524, 1325715, 418172, 204600, 124, 9021, 26740],62,12]
%
% END OF LATTICE L63.12
%

\medskip%
%------------------------------WEYL-GROUP-W64--------------------------------
%
\leavevmode\llap{}%
$W_{64\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=40$%
\hfill%
$3222232222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L64.1
%
\leavevmode%
\llap{\spacer}%
${L_{64.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{64}}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-140449\\532000\\-106400
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-36927\\139874\\-27975
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-9\\-7\\-10\\-25\\-152
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-6\\-10\\-24\\-90\\-684
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\kern3pt%
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-5\\-25\\-90\\-415\\-3420
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2676340, -703760, 14820; -703760, -185058, 3897; 14820, 3897, -82],[380, 1, 1],[-16, -30, -77, -301, -2329, -50, -3, 11, 4, -179; 61, 114, 292, 1140, 8816, 189, 11, -42, -15, 684; 7, -5, -42, -235, -2052, -57, -20, -8, 10, 152],63,1]
%
% END OF LATTICE L64.1
%
\hbox{}\par\smallskip%
%
% LATTICE L64.2
%
\leavevmode%
\llap{\spacer}%
${L_{64.2}}=\pfillwitharg{2}({L_{64.1}})$%
{${}={}$}Nikulin $64$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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-4\\-19\\-3
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-2\\-10\\5
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\19\\0
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{$\left.\llap{\phantom{%
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-7\\0\\1\\0\\-19
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-5\\-25\\-45\\-415\\-1710
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0\\-76\\-171\\-1710\\-7239
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3135, 665, 95; 665, -141, -20; 95, -20, -2],[95, 1, 1],[-63, -20, -4, -2, 4, 1, -11, -27, -277, -1182; -295, -94, -19, -10, 19, 5, -51, -126, -1295, -5529; -107, -29, -3, 5, 0, -3, -26, -52, -500, -2090],63,2]
%
% END OF LATTICE L64.2
%
\hbox{}\par\smallskip%
%
% LATTICE L64.3
%
\leavevmode%
\llap{\spacer}%
${L_{64.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{64.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{+}{3}{10}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{95}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187910\\7030\\38095
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\kern3pt%
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7030\\-185\\-1425
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\kern3pt%
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38095\\-1425\\-7723
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-187910, 7030, 38095; 7030, -185, -1425; 38095, -1425, -7723],[95, 5, 1],[230, -10543, -24393, -49231, -1045092, -55170, -16927, -3077, -215, 1232; -3, 160, 369, 744, 15789, 833, 255, 46, 3, -19; 1135, -52030, -120380, -242956, -5157550, -272265, -83535, -15185, -1061, 6080],63,3]
%
% END OF LATTICE L64.3
%
\hbox{}\par\smallskip%
%
% LATTICE L64.4
%
\leavevmode%
\llap{\spacer}%
${L_{64.4}}=\pdualwitharg{2}({L_{64.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24153560, -207100, -6047700; -207100, 1816, 51844; -6047700, 51844, 1514259],[380, 4, 1],[-1033, -3100, -5214, -47162, -96603, -9489, -2161, -47, -187, -3356; -1099, -3297, -5545, -50155, -102733, -10091, -2298, -50, -200, -3572; -4088, -12268, -20634, -186640, -382299, -37552, -8552, -186, -740, -13281],63,4]
%
% END OF LATTICE L64.4
%
\hbox{}\par\smallskip%
%
% LATTICE L64.5
%
\leavevmode%
\llap{\spacer}%
${L_{64.5}}=\pdualwitharg{19}(\pfillwitharg{2}({L_{64.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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-607810\\-9025\\126730
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\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-607810, -9025, 126730; -9025, -133, 1881; 126730, 1881, -26423],[95, 19, 1],[45, -1963, -4547, -45901, -10258, -10291, -3160, -576, -206, 12; 170, -7803, -18053, -182175, -40708, -40830, -12527, -2277, -795, 48; 228, -9975, -23104, -233225, -52121, -52288, -16055, -2926, -1045, 61],63,5]
%
% END OF LATTICE L64.5
%
\hbox{}\par\smallskip%
%
% LATTICE L64.6
%
\leavevmode%
\llap{\spacer}%
${L_{64.6}}=\pdualwitharg{5}({L_{64.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8964580, -74100, 29640; -74100, -610, 245; 29640, 245, -98],[380, 5, 1],[10, 6, 7, 3, 83, 0, -1, 3, 6, 297; -22, -9, -6, -1, 0, 2, -1, -14, -17, -760; 2965, 1790, 2100, 904, 25080, 5, -305, 870, 1769, 87780],63,6]
%
% END OF LATTICE L64.6
%
\hbox{}\par\smallskip%
%
% LATTICE L64.7
%
\leavevmode%
\llap{\spacer}%
${L_{64.7}}=\pdualwitharg{19}({L_{64.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1087940, 81700, -5320; 81700, 6118, -399; -5320, -399, 26],[380, 19, 1],[-5, -2, -1, 1, 1, 1, 0, -3, -19, -9; -79, -29, -14, 5, 8, 9, -6, -56, -320, -148; -2242, -855, -418, 285, 328, 342, -95, -1482, -8835, -4128],63,7]
%
% END OF LATTICE L64.7
%
\hbox{}\par\smallskip%
%
% LATTICE L64.8
%
\leavevmode%
\llap{\spacer}%
${L_{64.8}}=\pdualwitharg{2.5}({L_{64.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[414719080, -1900, -104785760; -1900, 40, 480; -104785760, 480, 26475887],[380, 20, 1],[-33160, -16201, -7911, -6253, -42942, 0, -379, -8669, -23591, -272078; -231, -112, -54, -42, -285, 1, -3, -61, -165, -1900; -131240, -64120, -31310, -24748, -169955, 0, -1500, -34310, -93368, -1076825],63,8]
%
% END OF LATTICE L64.8
%
\hbox{}\par\smallskip%
%
% LATTICE L64.9
%
\leavevmode%
\llap{\spacer}%
${L_{64.9}}=\pdualwitharg{5.19}(\pfillwitharg{2}({L_{64.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[329270, -5415, 5795; -5415, 95, -95; 5795, -95, 102],[95, 95, 1],[7, -479, -1100, -2215, -2473, -2477, -756, -135, -8, 3; 16, -828, -1911, -3854, -4305, -4316, -1322, -239, -16, 5; -380, 26315, 60420, 121657, 135825, 136040, 41515, 7410, 437, -165],63,9]
%
% END OF LATTICE L64.9
%
\hbox{}\par\smallskip%
%
% LATTICE L64.10
%
\leavevmode%
\llap{\spacer}%
${L_{64.10}}=\pdualwitharg{2.19}({L_{64.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${152}\above{1pt}{1pt}{+}{3}{152}\above{1pt}{1pt}{*}{2}{76}\above{1pt}{1pt}{*}{2}{760}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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92354440\\-549860\\-23334280
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32263\\559\\127680
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1096942\\19040\\4341119
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0\\0\\0
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8277\\144\\32756
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845\\17\\3344
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-989\\-15\\-3914
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[92354440, -549860, -23334280; -549860, 3192, 138928; -23334280, 138928, 5895641],[380, 76, 1],[8277, 845, -989, -4897, -212, 941, 3764, 5598, 46378, 4821; 144, 17, -15, -75, -3, 16, 63, 95, 795, 83; 32756, 3344, -3914, -19380, -839, 3724, 14896, 22154, 183540, 19079],63,10]
%
% END OF LATTICE L64.10
%
\hbox{}\par\smallskip%
%
% LATTICE L64.11
%
\leavevmode%
\llap{\spacer}%
${L_{64.11}}=\pdualwitharg{5.19}({L_{64.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${190}\above{1pt}{1pt}{-}{3}{190}\above{1pt}{1pt}{b}{2}{380}\above{1pt}{1pt}{b}{2}{38}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-1\\-9\\-475
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3\\-6\\-190
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3\\0\\100
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26220, -36100, 760; -36100, -49590, 1045; 760, 1045, -22],[380, 95, 1],[-10, -1, 3, 2, 3, 0, -4, -13, -11, -23; -22, -9, -6, -1, 0, 2, -1, -14, -17, -40; -1425, -475, -190, 19, 100, 95, -190, -1140, -1216, -2760],63,11]
%
% END OF LATTICE L64.11
%
\hbox{}\par\smallskip%
%
% LATTICE L64.12
%
\leavevmode%
\llap{\spacer}%
${L_{64.12}}=\pdualwitharg{2.5.19}({L_{64.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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760\\-36100\\9120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36100\\2688120\\-679440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9120\\-679440\\171733
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-391\\-5960\\-23560
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25155\\384419\\1519620
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0\\0\\0
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}}\!\right]$}%
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-147\\-2211\\-8740
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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-6460\\-1520\\0\\152\\0
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-8360\\-3420\\-1140\\-380\\0
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-3420\\-2660\\-1900\\-1900\\-760
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-1140\\-1900\\-2280\\-3420\\-1710
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-380\\-1900\\-3420\\-6308\\-3420
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-760\\-1710\\-3420\\-1905
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[760, -36100, 9120; -36100, 2688120, -679440; 9120, -679440, 171733],[380, 380, 1],[-391, -147, -49, -29, -10, 1, -48, -146, -322, -185; -5960, -2211, -721, -423, -148, 0, -769, -2259, -4941, -2832; -23560, -8740, -2850, -1672, -585, 0, -3040, -8930, -19532, -11195],63,12]
%
% END OF LATTICE L64.12
%

\medskip%
%------------------------------WEYL-GROUP-W65--------------------------------
%
\leavevmode\llap{}%
$W_{65\phantom{0}}$%
\qquad\llap{16} lattices, $\chi=32$%
\hfill%
$22262226\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L65.1
%
\leavevmode%
\llap{\spacer}%
${L_{65.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{65}}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{34}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15547656\\20808\\31824
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20808\\-26\\-45
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31824\\-45\\-62
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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395759\\136022712\\104361912
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\kern3pt%
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-530\\-182162\\-139761
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-810\\-278397\\-213598
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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272\\93486\\71727
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514\\176664\\135541
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115\\39528\\30324
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2\\688\\527
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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-735\\-1377\\-300\\-3
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-1377\\-2635\\-612\\-17
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-300\\-612\\-168\\-12
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-3\\-17\\-12\\-3
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-15547656, 20808, 31824; 20808, -26, -45; 31824, -45, -62],[408, 1, 1],[272, 514, 115, 2, -2, -4, 5, 8; 93486, 176664, 39528, 688, -687, -1377, 1716, 2749; 71727, 135541, 30324, 527, -528, -1054, 1320, 2110],64,1]
%
% END OF LATTICE L65.1
%
\hbox{}\par\smallskip%
%
% LATTICE L65.2
%
\leavevmode%
\llap{\spacer}%
${L_{65.2}}=\pfillwitharg{2}({L_{65.1}})$%
{${}={}$}Nikulin $65$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{34}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3162\\-1530\\306
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1530\\-740\\153
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\kern3pt%
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306\\153\\46
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-207061\\422688\\-27846
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-47\\96\\-6
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-217\\443\\-29
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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-1377\\-2635\\-306\\-17
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-150\\-306\\-42\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-17\\-6\\-3
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3162, -1530, 306; -1530, -740, 153; 306, 153, 46],[102, 1, 1],[25, 125, -47, -217, -7855, -14915, -1693, -73; -51, -255, 96, 443, 16035, 30447, 3456, 149; 3, 17, -6, -29, -1056, -2006, -228, -10],64,2]
%
% END OF LATTICE L65.2
%
\hbox{}\par\smallskip%
%
% LATTICE L65.3
%
\leavevmode%
\llap{\spacer}%
${L_{65.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{65.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{68}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2082840\\10914\\-1033260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10914\\-52\\5412
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-1033260\\5412\\-512581
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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-200\\174\\405
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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-1470\\-2754\\-150\\-6
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-2754\\-5270\\-306\\-34
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-150\\-306\\-21\\-6
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2082840, 10914, -1033260; 10914, -52, 5412; -1033260, 5412, -512581],[102, 2, 1],[320, 638, -200, -1278, -43439, -82085, -4591, -319; -279, -561, 174, 1117, 37998, 71808, 4017, 280; -648, -1292, 405, 2588, 87966, 166226, 9297, 646],64,3]
%
% END OF LATTICE L65.3
%
\hbox{}\par\smallskip%
%
% LATTICE L65.4
%
\leavevmode%
\llap{\spacer}%
${L_{65.4}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{65.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{102}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2345694\\14076\\-774792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14076\\-78\\4647
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-774792\\4647\\-255916
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-11154619\\12396978\\33996294
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66621\\-74042\\-203043
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3684267\\4094607\\11228660
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
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84\\-93\\-256
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
502\\-561\\-1530
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\116\\320
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1006\\1117\\3066
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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2\\0\\-4\\-24\\-245\\-1377\\-50\\-3
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-245\\-1377\\-50\\-3
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1377\\-7905\\-306\\-51
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-306\\-14\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-51\\-6\\-9
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2345694, 14076, -774792; 14076, -78, 4647; -774792, 4647, -255916],[102, 3, 1],[84, 502, -105, -1006, -11397, -64609, -2409, -251; -93, -561, 116, 1117, 12666, 71808, 2678, 280; -256, -1530, 320, 3066, 34735, 196911, 7342, 765],64,4]
%
% END OF LATTICE L65.4
%
\hbox{}\par\smallskip%
%
% LATTICE L65.5
%
\leavevmode%
\llap{\spacer}%
${L_{65.5}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{65.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{204}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
707401824\\-9162558\\349513506
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9162558\\118680\\-4527042
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
349513506\\-4527042\\172687837
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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2356095194\\5716845\\-4768495920
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{$\left[\!\llap{\phantom{%
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503\\2\\-1018
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2671\\0\\-5406
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1129\\2\\-2285
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17405\\40\\-35226
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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4\\0\\-4\\-48\\-490\\-2754\\-50\\-6
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\204\\0\\-204
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-4\\0\\1\\0
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-48\\-204\\0\\12
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-490\\-2754\\-50\\-6
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2754\\-15810\\-306\\-102
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-306\\-7\\-6
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[707401824, -9162558, 349513506; -9162558, 118680, -4527042; 349513506, -4527042, 172687837],[102, 6, 1],[503, 2671, 1129, 17405, 196084, 1111322, 20714, 4438; 2, 0, 2, 40, 475, 2703, 51, 13; -1018, -5406, -2285, -35226, -396854, -2249202, -41923, -8982],64,5]
%
% END OF LATTICE L65.5
%
\hbox{}\par\smallskip%
%
% LATTICE L65.6
%
\leavevmode%
\llap{\spacer}%
${L_{65.6}}=\pdualwitharg{3}({L_{65.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{102}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19120920\\-76296\\60384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76296\\-282\\237
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60384\\237\\-190
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
807295\\57779328\\328598304
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\kern3pt%
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3052\\218435\\1242273
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-2520\\-180360\\-1025731
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19120920, -76296, 60384; -76296, -282, 237; 60384, 237, -190],[408, 3, 1],[254, 1432, 105, 4, -2, -4, 7, 24; 18179, 102493, 7516, 287, -143, -289, 500, 1717; 103387, 582879, 42740, 1629, -814, -1632, 2848, 9768],64,6]
%
% END OF LATTICE L65.6
%
\hbox{}\par\smallskip%
%
% LATTICE L65.7
%
\leavevmode%
\llap{\spacer}%
${L_{65.7}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{65.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${102}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{102}\above{1pt}{1pt}{r}{2}{34}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-6957114\\-89556\\1224408
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89556\\-1054\\15759
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1224408\\15759\\-215488
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{$\left.\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6957114, -89556, 1224408; -89556, -1054, 15759; 1224408, 15759, -215488],[102, 17, 1],[-905, -107, 1129, 3623, 123242, 13700, 26057, 908; -120, -14, 150, 478, 16239, 1805, 3432, 119; -5151, -609, 6426, 20621, 701454, 77976, 148308, 5168],64,7]
%
% END OF LATTICE L65.7
%
\hbox{}\par\smallskip%
%
% LATTICE L65.8
%
\leavevmode%
\llap{\spacer}%
${L_{65.8}}=\pdualwitharg{2}({L_{65.1}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-573648\\11016\\-28152
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-573648, 11016, -28152; 11016, -208, 552; -28152, 552, -1345],[408, 8, 1],[-2095, -3957, -221, -15, 16, 30, -10, -62; -60756, -114750, -6408, -434, 465, 867, -291, -1799; 18936, 35768, 1998, 136, -144, -272, 90, 560],64,8]
%
% END OF LATTICE L65.8
%
\hbox{}\par\smallskip%
%
% LATTICE L65.9
%
\leavevmode%
\llap{\spacer}%
${L_{65.9}}=\pdualwitharg{2.17}(\pfillwitharg{2}({L_{65.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-10328010\\122944\\-5007894
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0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
204\\0\\-204\\-816
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\0\\51\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-816\\-68\\0\\68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24990\\-2754\\-2550\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2754\\-310\\-306\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2550\\-306\\-357\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\-34\\-102\\-102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[867654024, -10328010, 420712158; -10328010, 122944, -5007894; 420712158, -5007894, 203996887],[102, 34, 1],[-2918, -160, -4822, -26888, -925585, -103007, -98345, -7501; 3, 1, 15, 67, 2184, 242, 228, 14; 6018, 330, 9945, 55454, 1908930, 212442, 202827, 15470],64,9]
%
% END OF LATTICE L65.9
%
\hbox{}\par\smallskip%
%
% LATTICE L65.10
%
\leavevmode%
\llap{\spacer}%
${L_{65.10}}=\pdualwitharg{3.17}(\pfillwitharg{2}({L_{65.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${34}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{34}\above{1pt}{1pt}{r}{2}{102}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
338364498\\7899186\\-100834956
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7899186\\184416\\-2354007
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100834956\\-2354007\\30049516
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-413162729\\824252990\\-1321849518
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9640356\\19232354\\-30842811
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
123128500\\-245639375\\393930374
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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457\\-913\\1462
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75\\-149\\240
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1509\\-3008\\4828
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12625\\-25183\\40392
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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34\\0\\-68\\-408\\-4165\\-1377\\-850\\-51
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{$\left[\!\llap{\phantom{%
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34\\0\\-68\\-408
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0\\6\\0\\-102
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-408\\-102\\0\\102
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-4165\\-1377\\-850\\-51
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1377\\-465\\-306\\-51
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-850\\-306\\-238\\-102
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-51\\-102\\-153
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[338364498, 7899186, -100834956; 7899186, 184416, -2354007; -100834956, -2354007, 30049516],[102, 51, 1],[457, 75, 1509, 12625, 144875, 48369, 30787, 3523; -913, -149, -3008, -25183, -289022, -96496, -61422, -7032; 1462, 240, 4828, 40392, 463505, 154749, 98498, 11271],64,10]
%
% END OF LATTICE L65.10
%
\hbox{}\par\smallskip%
%
% LATTICE L65.11
%
\leavevmode%
\llap{\spacer}%
${L_{65.11}}=\pdualwitharg{17}({L_{65.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${102}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{408}\above{1pt}{1pt}{b}{2}{34}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24072\\3264\\1632
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3264\\-442\\-221
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1632\\-221\\-110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1535\\20352\\-19584
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-212\\-2810\\2703
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-1325\\1274
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-2\\-15\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
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5\\36\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\45\\-34
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
102\\0\\-408\\-408\\-12495\\-1377\\-5100\\-51
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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102\\0\\-408\\-408
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0\\2\\0\\-34
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-408\\0\\408\\0
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-408\\-34\\0\\34
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-12495\\-1377\\-5100\\-51
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1377\\-155\\-612\\-17
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-5100\\-612\\-2856\\-204
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24072, 3264, 1632; 3264, -442, -221; 1632, -221, -110],[408, 17, 1],[-2, 0, 5, 4, 110, 12, 43, 0; -15, -1, 36, 45, 1446, 160, 600, 8; 0, 2, 0, -34, -1377, -155, -612, -17],64,11]
%
% END OF LATTICE L65.11
%
\hbox{}\par\smallskip%
%
% LATTICE L65.12
%
\leavevmode%
\llap{\spacer}%
${L_{65.12}}=\pdualwitharg{2.3}({L_{65.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{816}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-860880\\-48552\\-554880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48552\\-2256\\-28968
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-554880\\-28968\\-346427
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-9439676\\-122236443\\25341288
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-5711400\\-73958184\\15332543
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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608\\7871\\-1632
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-260\\-3367\\698
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-3576\\-46307\\9600
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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16\\0\\-16\\-192\\-1960\\-11016\\-200\\-24
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{$\left[\!\llap{\phantom{%
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-1960\\-11016\\-200\\-24
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-11016\\-63240\\-1224\\-408
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-200\\-1224\\-28\\-24
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-860880, -48552, -554880; -48552, -2256, -28968; -554880, -28968, -346427],[408, 24, 1],[298, 608, -260, -3576, -37873, -213533, -3915, -599; 3859, 7871, -3367, -46307, -490426, -2765084, -50696, -7756; -800, -1632, 698, 9600, 101672, 573240, 10510, 1608],64,12]
%
% END OF LATTICE L65.12
%
\hbox{}\par\smallskip%
%
% LATTICE L65.13
%
\leavevmode%
\llap{\spacer}%
${L_{65.13}}=\pdualwitharg{2.3.17}(\pfillwitharg{2}({L_{65.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${68}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{17}\above{1pt}{1pt}{r}{2}{204}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1044961032\\-628251762\\-297695466
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\kern3pt%
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-628251762\\-377717424\\-178980420
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-297695466\\-178980420\\-84809407
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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19232354\\-8078716625\\16981686240
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11562099\\-4856759426\\10209042912
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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25\\-10425\\21913
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478\\-200553\\421566
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{$\left.\llap{\phantom{%
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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-8330\\-2754\\-850\\-102
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1044961032, -628251762, -297695466; -628251762, -377717424, -178980420; -297695466, -178980420, -84809407],[102, 102, 1],[-40, -14, 25, 478, 5413, 1805, 572, 119; 16709, 5917, -10425, -200553, -2273684, -758242, -240350, -50222; -35122, -12438, 21913, 421566, 4779346, 1593846, 505223, 105570],64,13]
%
% END OF LATTICE L65.13
%
\hbox{}\par\smallskip%
%
% LATTICE L65.14
%
\leavevmode%
\llap{\spacer}%
${L_{65.14}}=\pdualwitharg{3.17}({L_{65.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${34}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{136}\above{1pt}{1pt}{b}{2}{102}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32790552\\-1712784\\114240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1712784\\-89454\\5967
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
114240\\5967\\-398
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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215071\\1837888\\89245104
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11154\\95315\\4628403
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-748\\-6392\\-310387
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2\\-15\\-799
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\HardButStrongLineBreak\kern3pt%
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0\\-1\\-15
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7\\52\\2788
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20\\165\\8211
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-4165\\-1377\\-1700\\-51
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-1377\\-465\\-612\\-51
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-1700\\-612\\-952\\-204
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-32790552, -1712784, 114240; -1712784, -89454, 5967; 114240, 5967, -398],[408, 51, 1],[-2, 0, 7, 20, 200, 66, 81, 2; -15, -1, 52, 165, 1707, 565, 700, 23; -799, -15, 2788, 8211, 82960, 27402, 33728, 918],64,14]
%
% END OF LATTICE L65.14
%
\hbox{}\par\smallskip%
%
% LATTICE L65.15
%
\leavevmode%
\llap{\spacer}%
${L_{65.15}}=\pdualwitharg{2.17}({L_{65.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${816}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{204}\above{1pt}{1pt}{*}{2}{272}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14212272\\14437896\\-94248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14437896\\-14667056\\95744
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-94248\\95744\\-625
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-104131\\183963\\43901208
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-91\\160\\38248
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-89\\150\\36414
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\8\\2584
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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816\\0\\-816\\-3264\\-99960\\-11016\\-10200\\-408
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{$\left[\!\llap{\phantom{%
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-3264\\-272\\0\\272
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-99960\\-11016\\-10200\\-408
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-11016\\-1240\\-1224\\-136
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14212272, 14437896, -94248; 14437896, -14667056, 95744; -94248, 95744, -625],[408, 136, 1],[-811, -91, -89, -9, 1, 1, -1, -21; 1446, 160, 150, 8, -15, -1, 9, 45; 343944, 38248, 36414, 2584, -2448, -304, 1530, 10064],64,15]
%
% END OF LATTICE L65.15
%
\hbox{}\par\smallskip%
%
% LATTICE L65.16
%
\leavevmode%
\llap{\spacer}%
${L_{65.16}}=\pdualwitharg{2.3.17}({L_{65.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${272}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{*}{2}{68}\above{1pt}{1pt}{*}{2}{816}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-715632\\-1529592\\-185232
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1529592\\-3264816\\-395352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-185232\\-395352\\-47875
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{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
95315\\-1772979\\14273064
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25192\\-468598\\3772367
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\15\\-120
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-249\\2006
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
165\\-3091\\24888
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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-11016\\-3720\\-1224\\-408
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-3400\\-1224\\-476\\-408
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-715632, -1529592, -185232; -1529592, -3264816, -395352; -185232, -395352, -47875],[408, 408, 1],[-15, -1, 13, 165, 1707, 565, 175, 23; 287, 15, -249, -3091, -31760, -10506, -3248, -406; -2312, -120, 2006, 24888, 255680, 84576, 26146, 3264],64,16]
%
% END OF LATTICE L65.16
%

\medskip%
%------------------------------WEYL-GROUP-W66--------------------------------
%
\leavevmode\llap{}%
$W_{66\phantom{0}}$%
\qquad\llap{48} lattices, $\chi=36$%
\hfill%
$2222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L66.1
%
\leavevmode%
\llap{\spacer}%
${L_{66.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{66},3,2}%
\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21420\\-3780\\13860
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3780\\-642\\2001
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13860\\2001\\-1070
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9409\\67200\\3780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1456\\10399\\585
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2464\\-17600\\-991
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-899\\6420\\360
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\7\\0
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\HardButStrongLineBreak\kern3pt%
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134\\-957\\-54
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
348\\-2485\\-140
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-107\\-6
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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180\\0\\-180\\-1260\\-360
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0\\42\\0\\-210\\-84
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-180\\0\\18\\0\\-18
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\kern3pt%
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-1260\\-210\\0\\70\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-84\\-18\\0\\6
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\kern3pt%
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-8820\\-2520\\-900\\-1260\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-777\\-315\\-525\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\-315\\-153\\-315\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-525\\-315\\-805\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-27\\-105\\-21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-21420, -3780, 13860; -3780, -642, 2001; 13860, 2001, -1070],[1260, 3, 1],[-899, -1, 134, 348, 15, -1789, -783, -470, -1132, -127; 6420, 7, -957, -2485, -107, 12780, 5593, 3357, 8085, 907; 360, 0, -54, -140, -6, 720, 315, 189, 455, 51],65,1]
%
% END OF LATTICE L66.1
%
\hbox{}\par\smallskip%
%
% LATTICE L66.2
%
\leavevmode%
\llap{\spacer}%
${L_{66.2}}=\pfillwitharg{2.3}({L_{66.1}})$%
{${}={}$}Nikulin $66$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3045\\-630\\-105
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-130\\-21
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-21\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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2071\\-11655\\19425
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
408\\-2296\\3825
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-135\\224
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-1\\5\\0
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8\\-42\\21
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2\\-10\\-3
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2\\0\\-175
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\24\\-63
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\42\\0\\-210\\-84
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\kern3pt%
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-10\\0\\2\\0\\-6
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-210\\-210\\0\\70\\0
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-60\\-84\\-6\\0\\6
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-245\\-420\\-50\\-210\\0
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-420\\-777\\-105\\-525\\-21
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-50\\-105\\-17\\-105\\-9
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-525\\-105\\-805\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-9\\-105\\-21
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3045, -630, -105; -630, -130, -21; -105, -21, -2],[105, 1, 1],[-1, 8, 2, 2, -4, -31, -64, -10, -58, -4; 5, -42, -10, 0, 24, 175, 357, 55, 315, 21; 0, 21, -3, -175, -63, -300, -546, -72, -350, -12],65,2]
%
% END OF LATTICE L66.2
%
\hbox{}\par\smallskip%
%
% LATTICE L66.3
%
\leavevmode%
\llap{\spacer}%
${L_{66.3}}=\pfillwitharg{3}({L_{66.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-420\\840
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-38\\81
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\81\\-34
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-5489\\55860\\-2940
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140\\-1425\\74
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-177\\1800\\-100
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-194\\1974\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\336\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-213\\2170\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\204\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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20\\0\\-20\\-420\\-120\\-980\\-840\\-100\\-420\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\42\\0\\-210\\-84
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-20\\0\\2\\0\\-6
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-420\\-210\\0\\70\\0
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-120\\-84\\-6\\0\\6
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-980\\-840\\-100\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-777\\-105\\-525\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-105\\-17\\-105\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-525\\-105\\-805\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-9\\-105\\-21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-4620, -420, 840; -420, -38, 81; 840, 81, -34],[420, 1, 1],[-177, -194, -33, -213, -20, -47, -2, 5, 17, -8; 1800, 1974, 336, 2170, 204, 480, 21, -51, -175, 81; -100, -105, -17, -105, -9, -20, 0, 2, 0, -6],65,3]
%
% END OF LATTICE L66.3
%
\hbox{}\par\smallskip%
%
% LATTICE L66.4
%
\leavevmode%
\llap{\spacer}%
${L_{66.4}}=\pfillwitharg{2}({L_{66.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-147105\\-2205\\-1890
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2205\\-30\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1890\\-36\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2141\\-105315\\-42840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-1181\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-2360\\-961
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\2745\\1125
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\2114\\861
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\HardButStrongLineBreak\kern3pt%
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-23\\1134\\459
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\2520\\1015
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\248\\99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-90\\-630\\-180\\-2205\\-1260\\-450\\-630\\0
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\42\\0\\-210\\-84
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-90\\0\\18\\0\\-18
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-630\\-210\\0\\70\\0
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-180\\-84\\-18\\0\\6
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-2205\\-1260\\-450\\-630\\0
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\kern3pt%
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-1260\\-777\\-315\\-525\\-21
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-450\\-315\\-153\\-315\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-525\\-315\\-805\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-27\\-105\\-21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-147105, -2205, -1890; -2205, -30, -36; -1890, -36, -5],[315, 3, 1],[-56, -43, -23, -51, -5, -16, 1, 5, 9, -1; 2745, 2114, 1134, 2520, 248, 795, -49, -249, -455, 47; 1125, 861, 459, 1015, 99, 315, -21, -99, -175, 21],65,4]
%
% END OF LATTICE L66.4
%
\hbox{}\par\smallskip%
%
% LATTICE L66.5
%
\leavevmode%
\llap{\spacer}%
${L_{66.5}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103215\\-1575\\35700
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1575\\111\\519
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35700\\519\\-12343
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
277451\\154980\\808920
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4404\\-2461\\-12840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-94319\\-52685\\-274991
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
782\\435\\2280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-7\\-35
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-178\\-99\\-519
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-324\\-175\\-945
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
130\\73\\379
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-30\\-630\\-60\\-735\\-420\\-150\\-630\\0
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-60\\-28\\-6\\0\\2\\0\\-7\\-9\\-105\\-7
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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15\\0\\-30\\-630\\-60
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-210\\-28
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\6\\0\\-6
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-210\\0\\210\\0
\end{tabular}\endgroup%
\kern3pt%
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-60\\-28\\-6\\0\\2
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-735\\-420\\-150\\-630\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-259\\-105\\-525\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-150\\-105\\-51\\-315\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-525\\-315\\-2415\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-9\\-105\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-103215, -1575, 35700; -1575, 111, 519; 35700, 519, -12343],[105, 3, 1],[782, -12, -178, -324, 130, 3622, 2581, 1279, 8031, 237; 435, -7, -99, -175, 73, 2025, 1442, 714, 4480, 132; 2280, -35, -519, -945, 379, 10560, 7525, 3729, 23415, 691],65,5]
%
% END OF LATTICE L66.5
%
\hbox{}\par\smallskip%
%
% LATTICE L66.6
%
\leavevmode%
\llap{\spacer}%
${L_{66.6}}=\pdualwitharg{5}(\pfillwitharg{2.3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24045\\-3045\\-3990
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3045\\185\\-615
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3990\\-615\\-641
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-110111\\111930\\578760
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24045, -3045, -3990; -3045, 185, -615; -3990, -615, -641],[105, 5, 1],[-86, 20, 98, 36, -214, -398, -4255, -703, -883, -391; 87, -21, -99, -35, 219, 405, 4326, 714, 896, 396; 452, -105, -515, -189, 1125, 2092, 22365, 3695, 4641, 2055],65,6]
%
% END OF LATTICE L66.6
%
\hbox{}\par\smallskip%
%
% LATTICE L66.7
%
\leavevmode%
\llap{\spacer}%
${L_{66.7}}=\pdualwitharg{7}(\pfillwitharg{2.3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-8799\\12915\\66149
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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2\\-3\\-15
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68\\-99\\-511
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26565, -4515, 4410; -4515, 259, -651; 4410, -651, 713],[105, 7, 1],[-298, 2, 68, 18, -148, -1378, -421, -487, -437, -271; 435, -3, -99, -25, 219, 2025, 618, 714, 640, 396; 2240, -15, -511, -135, 1113, 10360, 3165, 3661, 3285, 2037],65,7]
%
% END OF LATTICE L66.7
%
\hbox{}\par\smallskip%
%
% LATTICE L66.8
%
\leavevmode%
\llap{\spacer}%
${L_{66.8}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{630}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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12\\92\\-9
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7245, -945, 0; -945, 123, -3; 0, -3, -34],[315, 3, 1],[63, 134, 22, 412, 12, 13, -1, -3, -13, 7; 480, 1022, 168, 3150, 92, 100, -7, -23, -105, 53; -50, -105, -17, -315, -9, -10, 0, 2, 0, -6],65,8]
%
% END OF LATTICE L66.8
%
\hbox{}\par\smallskip%
%
% LATTICE L66.9
%
\leavevmode%
\llap{\spacer}%
${L_{66.9}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-275940\\82320\\-1680
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\kern3pt%
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82320\\-24558\\501
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-275940, 82320, -1680; 82320, -24558, 501; -1680, 501, -10],[420, 3, 1],[41, 2, -5, -51, -2, 31, 19, 14, 114, 5; 140, 7, -17, -175, -7, 100, 63, 47, 385, 17; 120, 14, -12, -210, -16, -240, -49, -3, 105, 11],65,9]
%
% END OF LATTICE L66.9
%
\hbox{}\par\smallskip%
%
% LATTICE L66.10
%
\leavevmode%
\llap{\spacer}%
${L_{66.10}}=\pdualwitharg{3.5}(\pfillwitharg{2.3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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300735\\-2461\\886215
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[236996130, -4144140, -80435565; -4144140, 72465, 1406505; -80435565, 1406505, 27299518],[105, 15, 1],[-277, 368, -56, -2658, -1790, -5591, -17483, -7279, -7611, -655; -3, -14, 0, 28, 18, 51, 154, 60, 56, 2; -816, 1085, -165, -7833, -5275, -16476, -51520, -21450, -22428, -1930],65,10]
%
% END OF LATTICE L66.10
%
\hbox{}\par\smallskip%
%
% LATTICE L66.11
%
\leavevmode%
\llap{\spacer}%
${L_{66.11}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7176120, -193620, 1795920; -193620, 5224, -48456; 1795920, -48456, 449453],[420, 4, 1],[1248, 4696, 654, 3428, 178, 58, -125, -1, 1143, 475; -540, -2037, -285, -1505, -81, -30, 42, 0, -490, -204; -5045, -18984, -2644, -13860, -720, -235, 504, 4, -4620, -1920],65,11]
%
% END OF LATTICE L66.11
%
\hbox{}\par\smallskip%
%
% LATTICE L66.12
%
\leavevmode%
\llap{\spacer}%
${L_{66.12}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[380100, 181440, 4200; 181440, 86610, 2005; 4200, 2005, 46],[420, 5, 1],[-23, -10, 13, 17, -4, -57, -340, -63, -87, -46; 48, 21, -27, -35, 9, 120, 714, 132, 182, 96; 8, 0, -10, -28, -30, -32, -105, -5, 7, 15],65,12]
%
% END OF LATTICE L66.12
%
\hbox{}\par\smallskip%
%
% LATTICE L66.13
%
\leavevmode%
\llap{\spacer}%
${L_{66.13}}=\pdualwitharg{3.7}(\pfillwitharg{2.3}({L_{66.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[132197730, -3662190, -44965725; -3662190, 101451, 1245657; -44965725, 1245657, 15294638],[105, 21, 1],[-929, 86, -50, -2076, -1940, -29779, -2645, -7627, -5601, -619; -15, -2, 0, 20, 18, 255, 22, 60, 40, 2; -2730, 253, -147, -6105, -5705, -87570, -7778, -22428, -16470, -1820],65,13]
%
% END OF LATTICE L66.13
%
\hbox{}\par\smallskip%
%
% LATTICE L66.14
%
\leavevmode%
\llap{\spacer}%
${L_{66.14}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{66.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4306260, -1545180, 36540; -1545180, -554442, 13111; 36540, 13111, -310],[420, 7, 1],[-1741, -191, -128, -52, 37, -91, -38, -101, -177, -266; 5820, 639, 429, 175, -123, 300, 126, 336, 590, 888; 40880, 4506, 3052, 1270, -840, 1960, 849, 2303, 4085, 6195],65,14]
%
% END OF LATTICE L66.14
%
\hbox{}\par\smallskip%
%
% LATTICE L66.15
%
\leavevmode%
\llap{\spacer}%
${L_{66.15}}=\pdualwitharg{5.7}(\pfillwitharg{2.3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[140036295, 4866015, -24370290; 4866015, 169085, -846825; -24370290, -846825, 4241122],[105, 35, 1],[-125, 178, -34, -284, -3980, -4071, -5423, -5211, -765, -1265; 372, -540, 102, 856, 11994, 12264, 16335, 15693, 2303, 3801; -644, 915, -175, -1461, -20475, -20944, -27900, -26810, -3936, -6510],65,15]
%
% END OF LATTICE L66.15
%
\hbox{}\par\smallskip%
%
% LATTICE L66.16
%
\leavevmode%
\llap{\spacer}%
${L_{66.16}}=\pdualwitharg{3}({L_{66.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{630}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28980, -12600, -1260; -12600, 5478, 543; -1260, 543, -34],[1260, 3, 1],[597, 648, 109, 2089, 64, 147, 3, -16, -136, 29; 1380, 1498, 252, 4830, 148, 340, 7, -37, -315, 67; -100, -105, -17, -315, -9, -20, 0, 2, 0, -6],65,16]
%
% END OF LATTICE L66.16
%
\hbox{}\par\smallskip%
%
% LATTICE L66.17
%
\leavevmode%
\llap{\spacer}%
${L_{66.17}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[296415, -31185, 11655; -31185, 3225, -1200; 11655, -1200, 446],[315, 15, 1],[2, -1, -11, -13, -17, -38, -104, -34, -8, 2; 93, -49, -525, -623, -817, -1833, -5026, -1650, -392, 92; 198, -105, -1125, -1337, -1755, -3942, -10815, -3555, -847, 195],65,17]
%
% END OF LATTICE L66.17
%
\hbox{}\par\smallskip%
%
% LATTICE L66.18
%
\leavevmode%
\llap{\spacer}%
${L_{66.18}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{66.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[684495, -99540, 5670; -99540, 14475, -825; 5670, -825, 46],[315, 15, 1],[3, -1, -7, -53, -17, -7, -34, 2, 32, 12; 20, -7, -47, -357, -115, -48, -238, 12, 210, 80; -11, 0, 20, 126, 30, -1, -105, -35, -189, -45],65,18]
%
% END OF LATTICE L66.18
%
\hbox{}\par\smallskip%
%
% LATTICE L66.19
%
\leavevmode%
\llap{\spacer}%
${L_{66.19}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[477960, 22260, 120540; 22260, 1800, 5688; 120540, 5688, 30407],[420, 12, 1],[-613, -215, -107, -4993, -637, -5123, -6477, -2761, -15083, -319; -240, -84, -42, -1960, -250, -2010, -2541, -1083, -5915, -125; 2475, 868, 432, 20160, 2572, 20685, 26152, 11148, 60900, 1288],65,19]
%
% END OF LATTICE L66.19
%
\hbox{}\par\smallskip%
%
% LATTICE L66.20
%
\leavevmode%
\llap{\spacer}%
${L_{66.20}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{66.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-69300, -38640, 420; -38640, -21390, 225; 420, 225, -2],[420, 15, 1],[-9, -3, 6, 16, 5, 9, 3, -6, -16, -5; 20, 7, -13, -35, -11, -20, -7, 13, 35, 11; 348, 140, -210, -588, -190, -372, -175, 195, 567, 185],65,20]
%
% END OF LATTICE L66.20
%
\hbox{}\par\smallskip%
%
% LATTICE L66.21
%
\leavevmode%
\llap{\spacer}%
${L_{66.21}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-439110, -77490, 945; -77490, -13629, 168; 945, 168, -2],[315, 21, 1],[136, 12, 34, 8, 2, -4, -1, -1, 3, 9; -495, -44, -126, -30, -8, 15, 4, 6, -10, -32; 22050, 1923, 5355, 1235, 273, -630, -138, 0, 550, 1512],65,21]
%
% END OF LATTICE L66.21
%
\hbox{}\par\smallskip%
%
% LATTICE L66.22
%
\leavevmode%
\llap{\spacer}%
${L_{66.22}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-315, -3780, 315; -3780, -44562, 3717; 315, 3717, -310],[315, 21, 1],[-199, -44, -30, -38, 8, -9, -8, -22, -118, -60; 25, 7, 7, 15, 1, -5, -2, -2, 0, 4; 70, 33, 49, 135, 21, -70, -33, -49, -135, -21],65,22]
%
% END OF LATTICE L66.22
%
\hbox{}\par\smallskip%
%
% LATTICE L66.23
%
\leavevmode%
\llap{\spacer}%
${L_{66.23}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{66.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16582440, -351540, 4139940; -351540, 7400, -87760; 4139940, -87760, 1033569],[420, 20, 1],[-28, -2393, -109, 1623, 3463, 1758, 32668, 4434, 4432, 862; 10, 882, 40, -602, -1284, -652, -12117, -1645, -1645, -321; 113, 9660, 440, -6552, -13980, -7097, -131880, -17900, -17892, -3480],65,23]
%
% END OF LATTICE L66.23
%
\hbox{}\par\smallskip%
%
% LATTICE L66.24
%
\leavevmode%
\llap{\spacer}%
${L_{66.24}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3780, -420, -420; -420, 42, 21; -420, 21, -94],[420, 21, 1],[387, 17, 48, 34, 3, -3, -1, 0, 14, 13; 4240, 186, 524, 370, 32, -40, -11, 1, 155, 143; -840, -37, -105, -75, -7, 0, 2, 0, -30, -28],65,24]
%
% END OF LATTICE L66.24
%
\hbox{}\par\smallskip%
%
% LATTICE L66.25
%
\leavevmode%
\llap{\spacer}%
${L_{66.25}}=\pdualwitharg{3.5.7}(\pfillwitharg{2.3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[23161110, 16137765, -5748960; 16137765, 11239095, -4003860; -5748960, -4003860, 1426351],[105, 105, 1],[87, -1, -99, -5, 73, 405, 206, 714, 128, 132; 14770, -166, -16840, -882, 12264, 68402, 34821, 120805, 21675, 22391; 41811, -470, -47670, -2496, 34720, 193641, 98575, 341985, 61359, 63385],65,25]
%
% END OF LATTICE L66.25
%
\hbox{}\par\smallskip%
%
% LATTICE L66.26
%
\leavevmode%
\llap{\spacer}%
${L_{66.26}}=\pdualwitharg{2.7}(\pfillwitharg{3}({L_{66.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[207178440, -1782900, 51378600; -1782900, 15400, -442148; 51378600, -442148, 12741483],[420, 28, 1],[119152, 53815, 38453, 17971, -7631, 3942, 7732, 23094, 43576, 69178; -25585, -11556, -8258, -3860, 1638, -845, -1659, -4957, -9355, -14853; -481355, -217404, -155344, -72600, 30828, -15925, -31236, -93296, -176040, -279468],65,26]
%
% END OF LATTICE L66.26
%
\hbox{}\par\smallskip%
%
% LATTICE L66.27
%
\leavevmode%
\llap{\spacer}%
${L_{66.27}}=\pdualwitharg{5.7}(\pfillwitharg{3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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32\\60\\2835
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1341060, 374640, 7140; 374640, -104650, -1995; 7140, -1995, -38],[420, 35, 1],[5, 8, 17, 5, 32, 35, 17, 8, 0, -7; 12, 18, 36, 10, 60, 60, 27, 9, -1, -15; 308, 555, 1295, 411, 2835, 3388, 1755, 1015, 51, -525],65,27]
%
% END OF LATTICE L66.27
%
\hbox{}\par\smallskip%
%
% LATTICE L66.28
%
\leavevmode%
\llap{\spacer}%
${L_{66.28}}=\pdualwitharg{2}({L_{66.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6749867880, -104286420, -1687956480; -104286420, 1611240, 26079168; -1687956480, 26079168, 422111533],[1260, 12, 1],[-1073, -1117, -253, 279, 213, 1157, 1215, 295, -181, -199; -255, -238, -60, -70, -4, -135, -217, -135, -385, -61; -4275, -4452, -1008, 1120, 852, 4635, 4872, 1188, -700, -792],65,28]
%
% END OF LATTICE L66.28
%
\hbox{}\par\smallskip%
%
% LATTICE L66.29
%
\leavevmode%
\llap{\spacer}%
${L_{66.29}}=\pdualwitharg{2.3}({L_{66.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[859846680, 4590180, -214982460; 4590180, 24504, -1147656; -214982460, -1147656, 53750813],[1260, 12, 1],[2579, 9668, 1338, 20852, 346, 99, -295, 1, 7267, 993; -1405, -5257, -725, -11235, -181, -45, 182, 0, -3990, -544; 10285, 38556, 5336, 83160, 1380, 395, -1176, 4, 28980, 3960],65,29]
%
% END OF LATTICE L66.29
%
\hbox{}\par\smallskip%
%
% LATTICE L66.30
%
\leavevmode%
\llap{\spacer}%
${L_{66.30}}=\pdualwitharg{5}({L_{66.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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164\\329\\-525
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-772380, 386820, 1260; 386820, -193650, -585; 1260, -585, 26],[1260, 15, 1],[287, 164, -121, -129, -132, -263, -129, 136, 136, 137; 576, 329, -243, -259, -265, -528, -259, 273, 273, 275; -936, -525, 405, 427, 435, 864, 420, -450, -448, -450],65,30]
%
% END OF LATTICE L66.30
%
\hbox{}\par\smallskip%
%
% LATTICE L66.31
%
\leavevmode%
\llap{\spacer}%
${L_{66.31}}=\pdualwitharg{3.5}({L_{66.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{126}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12452580, -1786680, -23940; -1786680, 256350, 3435; -23940, 3435, 46],[1260, 15, 1],[5, 34, 7, 31, 6, 3, 1, -2, -10, -1; 36, 238, 48, 210, 40, 20, 7, -13, -63, -5; -92, -105, 55, 441, 135, 68, 0, -70, -504, -150],65,31]
%
% END OF LATTICE L66.31
%
\hbox{}\par\smallskip%
%
% LATTICE L66.32
%
\leavevmode%
\llap{\spacer}%
${L_{66.32}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[92165640, -705180, 22843380; -705180, 5400, -174780; 22843380, -174780, 5661763],[420, 60, 1],[176, 901, 401, 811, 307, 436, 2669, 1129, 1331, 203; -29, -154, -68, -126, -44, -61, -371, -157, -189, -31; -711, -3640, -1620, -3276, -1240, -1761, -10780, -4560, -5376, -820],65,32]
%
% END OF LATTICE L66.32
%
\hbox{}\par\smallskip%
%
% LATTICE L66.33
%
\leavevmode%
\llap{\spacer}%
${L_{66.33}}=\pdualwitharg{7}({L_{66.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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-161\\522\\5355
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[364140, 112140, 0; 112140, 34314, 21; 0, 21, -2],[1260, 21, 1],[-1333, -58, -161, -37, -8, 37, 4, -1, -17, -46; 4320, 188, 522, 120, 26, -120, -13, 3, 55, 149; 44100, 1923, 5355, 1235, 273, -1260, -138, 0, 550, 1512],65,33]
%
% END OF LATTICE L66.33
%
\hbox{}\par\smallskip%
%
% LATTICE L66.34
%
\leavevmode%
\llap{\spacer}%
${L_{66.34}}=\pdualwitharg{3.7}({L_{66.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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11\\332\\6923
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59\\1770\\37035
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30\\896\\18795
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{$\left.\llap{\phantom{%
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-1260\\-225\\-315\\-1035\\-315
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-39653460, -999180, 110880; -999180, -25158, 2793; 110880, 2793, -310],[1260, 21, 1],[9, 4, 11, 59, 30, 199, 22, 15, 19, -4; 280, 122, 332, 1770, 896, 5920, 653, 443, 555, -121; 5740, 2529, 6923, 37035, 18795, 124460, 13746, 9352, 11790, -2520],65,34]
%
% END OF LATTICE L66.34
%
\hbox{}\par\smallskip%
%
% LATTICE L66.35
%
\leavevmode%
\llap{\spacer}%
${L_{66.35}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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12\\20\\1931
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[630, -30870, 315; -30870, 1318485, -13440; 315, -13440, 137],[315, 105, 1],[-1, -1, 1, 1, 13, 37, 16, 44, 2, 2; -9, -4, 0, 2, 26, 69, 29, 75, 3, -1; -882, -390, 0, 194, 2520, 6678, 2805, 7245, 289, -105],65,35]
%
% END OF LATTICE L66.35
%
\hbox{}\par\smallskip%
%
% LATTICE L66.36
%
\leavevmode%
\llap{\spacer}%
${L_{66.36}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{66.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-375795, -281925, 3780; -281925, -211470, 2835; 3780, 2835, -38],[315, 105, 1],[-1, -4, -10, -10, -24, -15, -16, -10, -2, 4; 2, 10, 28, 30, 76, 50, 55, 37, 9, -11; 49, 345, 1085, 1233, 3255, 2219, 2490, 1750, 468, -420],65,36]
%
% END OF LATTICE L66.36
%
\hbox{}\par\smallskip%
%
% LATTICE L66.37
%
\leavevmode%
\llap{\spacer}%
${L_{66.37}}=\pdualwitharg{2.3.7}(\pfillwitharg{3}({L_{66.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[792120, 290220, 197400; 290220, 106344, 72324; 197400, 72324, 49193],[420, 84, 1],[-3523, -660, -2066, -1684, -278, -443, -1, 83, -299, -383; -25, -5, -17, -15, -3, -5, 0, 2, 0, -2; 14175, 2656, 8316, 6780, 1120, 1785, 4, -336, 1200, 1540],65,37]
%
% END OF LATTICE L66.37
%
\hbox{}\par\smallskip%
%
% LATTICE L66.38
%
\leavevmode%
\llap{\spacer}%
${L_{66.38}}=\pdualwitharg{3.5.7}(\pfillwitharg{3}({L_{66.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2286900, -26880, -8820; -26880, 210, 105; -8820, 105, 34],[420, 105, 1],[51, 11, 30, 4, 1, -3, -1, 0, 2, 9; 176, 38, 104, 14, 4, -8, -3, 1, 7, 31; 12684, 2735, 7455, 993, 245, -756, -250, 0, 498, 2240],65,38]
%
% END OF LATTICE L66.38
%
\hbox{}\par\smallskip%
%
% LATTICE L66.39
%
\leavevmode%
\llap{\spacer}%
${L_{66.39}}=\pdualwitharg{2.5.7}(\pfillwitharg{3}({L_{66.1}}))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[37599240, 472500, -8953560; 472500, 6440, -112140; -8953560, -112140, 2132407],[420, 140, 1],[-636, -5392, -15666, -5900, -46894, -16250, -36823, -26549, -2543, 4679; 1929, 16353, 47511, 17893, 142215, 49281, 111672, 80514, 7712, -14190; -2569, -21780, -63280, -23832, -189420, -65639, -148740, -107240, -10272, 18900],65,39]
%
% END OF LATTICE L66.39
%
\hbox{}\par\smallskip%
%
% LATTICE L66.40
%
\leavevmode%
\llap{\spacer}%
${L_{66.40}}=\pdualwitharg{2.5}({L_{66.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6006709800, 78728580, -1503830160; 78728580, 1031880, -19710360; -1503830160, -19710360, 376496489],[1260, 60, 1],[-2684, -16539, -7043, -2767, -1249, -1036, -5161, -2257, -1573, -1851; 375, 2317, 987, 385, 169, 135, 658, 288, 210, 256; -10701, -65940, -28080, -11032, -4980, -4131, -20580, -9000, -6272, -7380],65,40]
%
% END OF LATTICE L66.40
%
\hbox{}\par\smallskip%
%
% LATTICE L66.41
%
\leavevmode%
\llap{\spacer}%
${L_{66.41}}=\pdualwitharg{2.3.5}({L_{66.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[101619000, -1306620, -25498620; -1306620, 14520, 327840; -25498620, 327840, 6398209],[1260, 60, 1],[9017, 167168, 22582, 67120, 4078, -235, -13483, -627, 25091, 17877; -347, -6433, -869, -2583, -157, 9, 518, 24, -966, -688; 35953, 666540, 90040, 267624, 16260, -937, -53760, -2500, 100044, 71280],65,41]
%
% END OF LATTICE L66.41
%
\hbox{}\par\smallskip%
%
% LATTICE L66.42
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1503363960, -8544060, -376908840; -8544060, 48552, 2142084; -376908840, 2142084, 94494931],[1260, 84, 1],[-59173, -9378, -22184, -4058, 632, 1027, -313, -6889, -5633, -10323; -3810, -605, -1437, -265, 37, 60, -20, -438, -360, -662; -235935, -37392, -88452, -16180, 2520, 4095, -1248, -27468, -22460, -41160],65,42]
%
% END OF LATTICE L66.42
%
\hbox{}\par\smallskip%
%
% LATTICE L66.43
%
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${L_{66.43}}=\pdualwitharg{2.3.7}({L_{66.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11241450360, -36732780, -2819340720; -36732780, 120120, 9212532; -2819340720, 9212532, 707086883],[1260, 84, 1],[-12421, -24372, -72802, -412124, -218090, -375641, -169659, -121229, -169969, 24059; -60, -119, -357, -2025, -1073, -1850, -836, -598, -840, 118; -49525, -97176, -290276, -1643220, -869568, -1497755, -676464, -483364, -677700, 95928],65,43]
%
% END OF LATTICE L66.43
%
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%
% LATTICE L66.44
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L66.44
%
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%
% LATTICE L66.45
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${L_{66.45}}=\pdualwitharg{3.5.7}({L_{66.1}})$%
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% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L66.45
%
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%
% LATTICE L66.46
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[37800, -2994180, -741300; -2994180, 234585960, 58078860; -741300, 58078860, 14379181],[420, 420, 1],[-5, -5, -17, -3, -3, -1, 0, 2, 0, -2; -1721, -2872, -14246, -3176, -4506, -1357, 307, 6551, 1637, 1941; 6951, 11600, 57540, 12828, 18200, 5481, -1240, -26460, -6612, -7840],65,46]
%
% END OF LATTICE L66.46
%
\hbox{}\par\smallskip%
%
% LATTICE L66.47
%
\leavevmode%
\llap{\spacer}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1427800653720, 12698210700, 348468755040; 12698210700, 112932120, 3099122880; 348468755040, 3099122880, 85047217847],[1260, 420, 1],[-485, -2506, -16400, -1354, -6416, -1909, 511, 10415, 955, 4421; 939, 6805, 46011, 3837, 18349, 5457, -1475, -30021, -2771, -13019; 1953, 10020, 65520, 5408, 25620, 7623, -2040, -41580, -3812, -17640],65,47]
%
% END OF LATTICE L66.47
%
\hbox{}\par\smallskip%
%
% LATTICE L66.48
%
\leavevmode%
\llap{\spacer}%
${L_{66.48}}=\pdualwitharg{2.3.5.7}({L_{66.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[365205960, 3080700, 89277300; 3080700, 26040, 753060; 89277300, 753060, 21824527],[1260, 420, 1],[5087, 11499, 8253, 2357, -1475, 223, 1776, 5022, 5608, 14750; -15369, -34742, -24936, -7122, 4456, -673, -5363, -15169, -16941, -44561; -20279, -45840, -32900, -9396, 5880, -889, -7080, -20020, -22356, -58800],65,48]
%
% END OF LATTICE L66.48
%

\medskip%
%------------------------------WEYL-GROUP-W67--------------------------------
%
\leavevmode\llap{}%
$W_{67\phantom{0}}$%
\qquad\llap{88} lattices, $\chi=27$%
\hfill%
$22422222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L67.1
%
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-70\\-630\\-210\\-140\\-210\\0\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-72\\-60\\-120\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-5778780, -1930740, 30240; -1930740, -645076, 10103; 30240, 10103, -158],[420, 1, 1],[-159, 41, 37, -26, -302, -187, -1886, -631; 490, -126, -114, 80, 930, 576, 5810, 1944; 900, -210, -208, 139, 1665, 1040, 10535, 3534],66,1]
%
% END OF LATTICE L67.1
%
\hbox{}\par\smallskip%
%
% LATTICE L67.2
%
\leavevmode%
\llap{\spacer}%
${L_{67.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N'_{33}}%
\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24797640\\31920\\44520
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31920\\-38\\-59
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44520\\-59\\-79
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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4\\875\\1600
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-1\\-219\\-400
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\HardButStrongLineBreak\kern3pt%
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1\\220\\399
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19\\4170\\7590
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11\\2412\\4396
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35\\7668\\13992
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-20\\-84\\-20\\-8\\0\\8\\0\\-24
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-630\\-210\\-140\\-420\\0\\140\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-72\\-60\\-240\\-24\\0\\24
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-24797640, 31920, 44520; 31920, -38, -59; 44520, -59, -79],[840, 1, 1],[4, -2, -1, 1, 19, 11, 107, 35; 875, -441, -219, 220, 4170, 2412, 23450, 7668; 1600, -798, -400, 399, 7590, 4396, 42770, 13992],66,2]
%
% END OF LATTICE L67.2
%
\hbox{}\par\smallskip%
%
% LATTICE L67.3
%
\leavevmode%
\llap{\spacer}%
${L_{67.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32246760\\36960\\22680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36960\\-41\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22680\\-28\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\4925\\3345
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\8001\\5439
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\615\\419
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-616\\-418
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-615\\-420
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\HardButStrongLineBreak\kern3pt%
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3\\1848\\1252
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\HardButStrongLineBreak\kern3pt%
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32\\19705\\13370
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29\\17856\\12120
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-20\\-84\\-20\\-16\\0\\8\\0\\-24
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-35\\-315\\-105\\-140\\-105\\0\\35\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-72\\-120\\-120\\-24\\0\\24
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-32246760, 36960, 22680; 36960, -41, -28; 22680, -28, -13],[840, 1, 1],[8, 13, 1, -1, -1, 3, 32, 29; 4925, 8001, 615, -616, -615, 1848, 19705, 17856; 3345, 5439, 419, -418, -420, 1252, 13370, 12120],66,3]
%
% END OF LATTICE L67.3
%
\hbox{}\par\smallskip%
%
% LATTICE L67.4
%
\leavevmode%
\llap{\spacer}%
${L_{67.4}}=\pfillwitharg{2}({L_{67.1}})$%
{${}={}$}Nikulin $67$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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2730\\525\\0
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\kern3pt%
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525\\101\\0
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\kern3pt%
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0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1\\5\\0
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-4\\21\\0
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1\\-5\\-2
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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5\\-26\\-5
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\HardButStrongLineBreak\kern3pt%
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8\\-42\\-9
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-18\\-30\\-60\\-3\\0\\3
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[2730, 525, 0; 525, 101, 0; 0, 0, -1],[105, 1, 1],[-1, -4, 1, 7, 29, 5, 67, 8; 5, 21, -5, -36, -150, -26, -350, -42; 0, 0, -2, -8, -30, -5, -70, -9],66,4]
%
% END OF LATTICE L67.4
%
\hbox{}\par\smallskip%
%
% LATTICE L67.5
%
\leavevmode%
\llap{\spacer}%
${L_{67.5}}=\pfillwitharg{2}({L_{67.2}})$%
{${}={}$}Nikulin $33'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100590\\52500\\95760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52500\\27401\\49979
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
95760\\49979\\91162
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2408\\-1650\\-1625
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2456\\-1680\\-1659
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\19\\19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1069\\-735\\-720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1151\\-790\\-776
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15257\\-10465\\-10290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5915\\-4056\\-3990
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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-10\\0\\2\\-1\\-15\\-10\\-105\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-105\\-15\\0\\15\\0\\-105\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-42\\-10\\-4\\0\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-315\\-105\\-70\\-105\\0\\35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-36\\-30\\-60\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[100590, 52500, 95760; 52500, 27401, 49979; 95760, 49979, 91162],[210, 1, 1],[2408, 2456, -2, -28, 1069, 1151, 15257, 5915; -1650, -1680, 2, 19, -735, -790, -10465, -4056; -1625, -1659, 1, 19, -720, -776, -10290, -3990],66,5]
%
% END OF LATTICE L67.5
%
\hbox{}\par\smallskip%
%
% LATTICE L67.6
%
\leavevmode%
\llap{\spacer}%
${L_{67.6}}=\main({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9949380\\16380\\23940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16380\\-26\\-41
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23940\\-41\\-55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8\\2570\\1565
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13\\4179\\2541
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\322\\195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-321\\-196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2\\-645\\-390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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3\\962\\588
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64\\20545\\12530
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29\\9312\\5676
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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5\\0\\-5\\-15\\-45\\-10\\-35\\0
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0\\21\\0\\-21\\-105\\-42\\-315\\-84
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-5\\0\\1\\-1\\-15\\-10\\-105\\-36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-105\\-15\\0\\30\\0\\-210\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-42\\-10\\-8\\0\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-315\\-105\\-140\\-210\\0\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-36\\-60\\-120\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-9949380, 16380, 23940; 16380, -26, -41; 23940, -41, -55],[420, 1, 1],[8, 13, 1, -1, -2, 3, 64, 29; 2570, 4179, 322, -321, -645, 962, 20545, 9312; 1565, 2541, 195, -196, -390, 588, 12530, 5676],66,6]
%
% END OF LATTICE L67.6
%
\hbox{}\par\smallskip%
%
% LATTICE L67.7
%
\leavevmode%
\llap{\spacer}%
${L_{67.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{67.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{4}{6}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4974060\\20685\\1667610
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20685\\-78\\-6936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1667610\\-6936\\-559085
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4771\\-2050\\-14205
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1145\\-497\\-3409
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
267\\112\\795
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-268\\-111\\-798
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2482\\-1055\\-7390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3054\\-1304\\-9093
\end{tabular}\endgroup%
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-72718\\-31115\\-216510
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4466\\-1913\\-13297
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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15\\0\\-15\\-45\\-45\\-15\\-105\\0
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\kern3pt%
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0\\7\\0\\-21\\-35\\-21\\-315\\-14
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\kern3pt%
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-15\\0\\3\\-3\\-15\\-15\\-315\\-18
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-21\\-3\\6\\0\\-12\\-420\\-30
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\kern3pt%
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-45\\-35\\-15\\0\\10\\0\\-210\\-20
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\kern3pt%
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-15\\-21\\-15\\-12\\0\\3\\0\\-3
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-315\\-315\\-420\\-210\\0\\210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-18\\-30\\-20\\-3\\0\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4974060, 20685, 1667610; 20685, -78, -6936; 1667610, -6936, -559085],[105, 3, 1],[-4771, -1145, 267, -268, -2482, -3054, -72718, -4466; -2050, -497, 112, -111, -1055, -1304, -31115, -1913; -14205, -3409, 795, -798, -7390, -9093, -216510, -13297],66,7]
%
% END OF LATTICE L67.7
%
\hbox{}\par\smallskip%
%
% LATTICE L67.8
%
\leavevmode%
\llap{\spacer}%
${L_{67.8}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23489970\\44310\\-11714430
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44310\\-76\\22096
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11714430\\22096\\-5841977
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20592\\-7525\\-41320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24238\\-8841\\-48636
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
608\\-219\\-1220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-607\\220\\1218
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5741\\-2115\\-11520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4085\\-1497\\-8197
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
117143\\-42875\\-235060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23462\\-8583\\-47079
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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20\\0\\-20\\-30\\-90\\-10\\-70\\0
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\kern3pt%
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0\\84\\0\\-42\\-210\\-42\\-630\\-84
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-20\\0\\4\\-2\\-30\\-10\\-210\\-36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-42\\-2\\2\\0\\-4\\-140\\-30
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-210\\-30\\0\\30\\0\\-210\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-42\\-10\\-4\\0\\1\\0\\-3
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-630\\-210\\-140\\-210\\0\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-36\\-30\\-60\\-3\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-23489970, 44310, -11714430; 44310, -76, 22096; -11714430, 22096, -5841977],[210, 2, 1],[20592, 24238, 608, -607, 5741, 4085, 117143, 23462; -7525, -8841, -219, 220, -2115, -1497, -42875, -8583; -41320, -48636, -1220, 1218, -11520, -8197, -235060, -47079],66,8]
%
% END OF LATTICE L67.8
%
\hbox{}\par\smallskip%
%
% LATTICE L67.9
%
\leavevmode%
\llap{\spacer}%
${L_{67.9}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{67.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{4}{10}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7001400\\31605\\-1391460
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31605\\-130\\6280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1391460\\6280\\-276539
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
839\\-410\\-4231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3019\\-1491\\-15225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-235\\112\\1185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
236\\-111\\-1190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1310\\-633\\-6606
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2686\\-1304\\-13545
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12790\\-6223\\-64498
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11782\\-5739\\-59415
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-5\\-15\\-9\\-5\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\0\\-105\\-105\\-105\\-315\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\5\\-5\\-15\\-25\\-105\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-105\\-5\\10\\0\\-20\\-140\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-105\\-15\\0\\6\\0\\-42\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-105\\-25\\-20\\0\\5\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-315\\-105\\-140\\-42\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-90\\-150\\-60\\-15\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7001400, 31605, -1391460; 31605, -130, 6280; -1391460, 6280, -276539],[105, 5, 1],[839, 3019, -235, 236, 1310, 2686, 12790, 11782; -410, -1491, 112, -111, -633, -1304, -6223, -5739; -4231, -15225, 1185, -1190, -6606, -13545, -64498, -59415],66,9]
%
% END OF LATTICE L67.9
%
\hbox{}\par\smallskip%
%
% LATTICE L67.10
%
\leavevmode%
\llap{\spacer}%
${L_{67.10}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{4}{3}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29108730\\60270\\-9675120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60270\\-114\\20031
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9675120\\20031\\-3215803
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18212\\-7525\\-54840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7146\\-2947\\-21518
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
538\\-219\\-1620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-537\\220\\1617
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1692\\-705\\-5095
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7225\\-2994\\-21756
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
103598\\-42875\\-311955
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13833\\-5722\\-41654
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\0\\-30\\-45\\-45\\-30\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-21\\-35\\-42\\-315\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\6\\-3\\-15\\-30\\-315\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-21\\-3\\3\\0\\-12\\-210\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-35\\-15\\0\\5\\0\\-105\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-42\\-30\\-12\\0\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-315\\-315\\-210\\-105\\0\\105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-36\\-30\\-20\\-6\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-29108730, 60270, -9675120; 60270, -114, 20031; -9675120, 20031, -3215803],[210, 3, 1],[18212, 7146, 538, -537, 1692, 7225, 103598, 13833; -7525, -2947, -219, 220, -705, -2994, -42875, -5722; -54840, -21518, -1620, 1617, -5095, -21756, -311955, -41654],66,10]
%
% END OF LATTICE L67.10
%
\hbox{}\par\smallskip%
%
% LATTICE L67.11
%
\leavevmode%
\llap{\spacer}%
${L_{67.11}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{67.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{4}{14}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9074940\\42525\\-1287930
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42525\\-182\\6034
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1287930\\6034\\-182785
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3979\\-2050\\-28105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
409\\-213\\-2889
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-223\\112\\1575
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
224\\-111\\-1582
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6214\\-3165\\-43890
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2548\\-1304\\-17997
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8666\\-4445\\-61210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11176\\-5739\\-78939
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-35\\-105\\-315\\-35\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-21\\-105\\-21\\-45\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\7\\-7\\-105\\-35\\-105\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-21\\-7\\14\\0\\-28\\-140\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-105\\-105\\0\\210\\0\\-210\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-21\\-35\\-28\\0\\7\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-45\\-105\\-140\\-210\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-126\\-210\\-420\\-21\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9074940, 42525, -1287930; 42525, -182, 6034; -1287930, 6034, -182785],[105, 7, 1],[3979, 409, -223, 224, 6214, 2548, 8666, 11176; -2050, -213, 112, -111, -3165, -1304, -4445, -5739; -28105, -2889, 1575, -1582, -43890, -17997, -61210, -78939],66,11]
%
% END OF LATTICE L67.11
%
\hbox{}\par\smallskip%
%
% LATTICE L67.12
%
\leavevmode%
\llap{\spacer}%
${L_{67.12}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{}{4}{5}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41043030\\92190\\16442790
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
92190\\-190\\-36935
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16442790\\-36935\\-6587363
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6564\\-1505\\-16376
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38636\\-8841\\-96390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-970\\-219\\-2420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
968\\220\\2415
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1829\\-423\\-4563
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13019\\-2994\\-32480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37337\\-8575\\-93149
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74783\\-17166\\-186570
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-10\\-15\\-9\\-10\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\210\\0\\-105\\-105\\-210\\-315\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\-5\\-15\\-50\\-105\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-105\\-5\\5\\0\\-20\\-70\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-105\\-15\\0\\3\\0\\-21\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-210\\-50\\-20\\0\\10\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-315\\-105\\-70\\-21\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-180\\-150\\-60\\-30\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-41043030, 92190, 16442790; 92190, -190, -36935; 16442790, -36935, -6587363],[210, 5, 1],[-6564, -38636, -970, 968, -1829, -13019, -37337, -74783; -1505, -8841, -219, 220, -423, -2994, -8575, -17166; -16376, -96390, -2420, 2415, -4563, -32480, -93149, -186570],66,12]
%
% END OF LATTICE L67.12
%
\hbox{}\par\smallskip%
%
% LATTICE L67.13
%
\leavevmode%
\llap{\spacer}%
${L_{67.13}}=\pdualwitharg{3}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{4}{6}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-372540\\79380\\-2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79380\\-16914\\537
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\537\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\55\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\28\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\22\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\455\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\68\\-80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\-45\\-45\\-30\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-21\\-35\\-42\\-315\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\3\\-3\\-15\\-30\\-315\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-21\\-3\\6\\0\\-24\\-420\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-35\\-15\\0\\10\\0\\-210\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-42\\-30\\-24\\0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-315\\-315\\-420\\-210\\0\\210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-36\\-60\\-40\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-372540, 79380, -2520; 79380, -16914, 537; -2520, 537, -17],[420, 3, 1],[12, 6, 1, -2, -1, 5, 101, 15; 55, 28, 5, -9, -5, 22, 455, 68; -45, -7, 9, 12, -10, -48, -630, -80],66,13]
%
% END OF LATTICE L67.13
%
\hbox{}\par\smallskip%
%
% LATTICE L67.14
%
\leavevmode%
\llap{\spacer}%
${L_{67.14}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{}{4}{6}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30444894060\\-70747950\\15135332520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70747950\\164406\\-35171538
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15135332520\\-35171538\\7524358273
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1538101\\-1650\\3093900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-523013\\-560\\1052044
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1217\\2\\-2448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17903\\19\\-36012
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-227526\\-245\\457670
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-367535\\-395\\739299
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9744324\\-10465\\19600770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-629650\\-676\\1266545
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-90\\-90\\-30\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-42\\-70\\-42\\-630\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\12\\-6\\-30\\-30\\-630\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-42\\-6\\6\\0\\-12\\-420\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-70\\-30\\0\\10\\0\\-210\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-42\\-30\\-12\\0\\3\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-630\\-630\\-420\\-210\\0\\210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-36\\-30\\-20\\-3\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30444894060, -70747950, 15135332520; -70747950, 164406, -35171538; 15135332520, -35171538, 7524358273],[210, 6, 1],[-1538101, -523013, 1217, 17903, -227526, -367535, -9744324, -629650; -1650, -560, 2, 19, -245, -395, -10465, -676; 3093900, 1052044, -2448, -36012, 457670, 739299, 19600770, 1266545],66,14]
%
% END OF LATTICE L67.14
%
\hbox{}\par\smallskip%
%
% LATTICE L67.15
%
\leavevmode%
\llap{\spacer}%
${L_{67.15}}=\pdualwitharg{3}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{4}{6}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15646260\\47040\\28140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47040\\-132\\-87
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28140\\-87\\-50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\1290\\5070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\98\\392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-298\\-1170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-98\\-393
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\300\\1165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\896\\3504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107\\10640\\41685
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\1292\\5066
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-90\\-90\\-60\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-42\\-70\\-84\\-630\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\12\\-6\\-30\\-60\\-630\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-42\\-6\\6\\0\\-24\\-420\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-70\\-30\\0\\10\\0\\-210\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-84\\-60\\-24\\0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-630\\-630\\-420\\-210\\0\\210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-72\\-60\\-40\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15646260, 47040, 28140; 47040, -132, -87; 28140, -87, -50],[420, 3, 1],[13, 1, -3, -1, 3, 9, 107, 13; 1290, 98, -298, -98, 300, 896, 10640, 1292; 5070, 392, -1170, -393, 1165, 3504, 41685, 5066],66,15]
%
% END OF LATTICE L67.15
%
\hbox{}\par\smallskip%
%
% LATTICE L67.16
%
\leavevmode%
\llap{\spacer}%
${L_{67.16}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{}{4}{7}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53242770\\124110\\22842960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
124110\\-266\\-53249
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22842960\\-53249\\-9800407
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46748\\-7525\\-108920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7862\\-1263\\-18318
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1382\\-219\\-3220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1379\\220\\3213
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13024\\-2115\\-30345
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18543\\-2994\\-43204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37986\\-6125\\-88505
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-106517\\-17166\\-248178
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\0\\-70\\-105\\-315\\-70\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-21\\-105\\-42\\-45\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\14\\-7\\-105\\-70\\-105\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-21\\-7\\7\\0\\-28\\-70\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-105\\-105\\0\\105\\0\\-105\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-42\\-70\\-28\\0\\14\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-45\\-105\\-70\\-105\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-252\\-210\\-420\\-42\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-53242770, 124110, 22842960; 124110, -266, -53249; 22842960, -53249, -9800407],[210, 7, 1],[-46748, -7862, -1382, 1379, -13024, -18543, -37986, -106517; -7525, -1263, -219, 220, -2115, -2994, -6125, -17166; -108920, -18318, -3220, 3213, -30345, -43204, -88505, -248178],66,16]
%
% END OF LATTICE L67.16
%
\hbox{}\par\smallskip%
%
% LATTICE L67.17
%
\leavevmode%
\llap{\spacer}%
${L_{67.17}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{67.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{4}{30}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1579386165\\-13240185\\-535892910
\end{tabular}\endgroup%
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-13240185\\111000\\4492455
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-535892910\\4492455\\181830902
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72921\\-157\\214917
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
113553\\-245\\334670
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6143\\13\\-18105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-626\\2\\-1845
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26146\\-56\\77059
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
190891\\-410\\562605
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
976810\\-2100\\2878911
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
310587\\-668\\915380
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3\\0\\-15\\-45\\-9\\-15\\-21\\0
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0\\35\\0\\-105\\-35\\-105\\-315\\-70
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-15\\0\\15\\-15\\-15\\-75\\-315\\-90
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-45\\-105\\-15\\30\\0\\-60\\-420\\-150
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-9\\-35\\-15\\0\\2\\0\\-42\\-20
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-105\\-75\\-60\\0\\15\\0\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-315\\-315\\-420\\-42\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-90\\-150\\-20\\-15\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1579386165, -13240185, -535892910; -13240185, 111000, 4492455; -535892910, 4492455, 181830902],[105, 15, 1],[72921, 113553, -6143, -626, 26146, 190891, 976810, 310587; -157, -245, 13, 2, -56, -410, -2100, -668; 214917, 334670, -18105, -1845, 77059, 562605, 2878911, 915380],66,17]
%
% END OF LATTICE L67.17
%
\hbox{}\par\smallskip%
%
% LATTICE L67.18
%
\leavevmode%
\llap{\spacer}%
${L_{67.18}}=\pdualwitharg{2}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{4}{8}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1437880080\\-4394460\\357472920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4394460\\13432\\-1092512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
357472920\\-1092512\\88871729
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2493\\-3440\\-10070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
863\\-1197\\-3486
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\-70\\-206
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1834\\-2523\\-7408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11408\\-15705\\-46080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1502\\-2069\\-6067
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58644\\-80815\\-236880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9722\\-13401\\-39270
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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20\\0\\-20\\-60\\-180\\-10\\-140\\0
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\kern3pt%
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0\\84\\0\\-84\\-420\\-42\\-1260\\-168
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-60\\-84\\-4\\8\\0\\-8\\-560\\-120
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-180\\-420\\-60\\0\\120\\0\\-840\\-240
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-10\\-42\\-10\\-8\\0\\1\\0\\-6
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-140\\-1260\\-420\\-560\\-840\\0\\280\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-72\\-120\\-240\\-6\\0\\12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1437880080, -4394460, 357472920; -4394460, 13432, -1092512; 357472920, -1092512, 88871729],[420, 4, 1],[2493, 863, 51, 1834, 11408, 1502, 58644, 9722; -3440, -1197, -70, -2523, -15705, -2069, -80815, -13401; -10070, -3486, -206, -7408, -46080, -6067, -236880, -39270],66,18]
%
% END OF LATTICE L67.18
%
\hbox{}\par\smallskip%
%
% LATTICE L67.19
%
\leavevmode%
\llap{\spacer}%
${L_{67.19}}=\pdualwitharg{2}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{4}{8}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
969298680\\-15464820\\239968680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15464820\\246740\\-3828616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
239968680\\-3828616\\59408899
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7933\\55\\-32040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14330\\105\\-57876
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1651\\13\\-6668
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
207\\2\\-836
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
713\\0\\-2880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1697\\10\\-6854
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64959\\420\\-262360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7234\\48\\-29217
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-60\\-180\\-20\\-140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-84\\-420\\-84\\-1260\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\4\\-4\\-60\\-20\\-420\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-84\\-4\\8\\0\\-16\\-560\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-420\\-60\\0\\120\\0\\-840\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-84\\-20\\-16\\0\\4\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-1260\\-420\\-560\\-840\\0\\280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-36\\-60\\-120\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[969298680, -15464820, 239968680; -15464820, 246740, -3828616; 239968680, -3828616, 59408899],[420, 4, 1],[7933, 14330, 1651, 207, 713, 1697, 64959, 7234; 55, 105, 13, 2, 0, 10, 420, 48; -32040, -57876, -6668, -836, -2880, -6854, -262360, -29217],66,19]
%
% END OF LATTICE L67.19
%
\hbox{}\par\smallskip%
%
% LATTICE L67.20
%
\leavevmode%
\llap{\spacer}%
${L_{67.20}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{420}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{}{4}{10}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
578995620\\-12595590\\275597280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12595590\\274010\\-5995400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
275597280\\-5995400\\131182099
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54335\\-330\\114136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-277121\\-1680\\582120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
219\\2\\-460
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3161\\19\\-6640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24116\\-147\\50658
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64922\\-395\\136375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-344242\\-2093\\723114
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-333655\\-2028\\700875
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-30\\-18\\-10\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-210\\-210\\-210\\-630\\-420
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\-10\\-30\\-50\\-210\\-180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-210\\-10\\10\\0\\-20\\-140\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-210\\-30\\0\\6\\0\\-42\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-210\\-50\\-20\\0\\5\\0\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-630\\-210\\-140\\-42\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-180\\-150\\-60\\-15\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[578995620, -12595590, 275597280; -12595590, 274010, -5995400; 275597280, -5995400, 131182099],[210, 10, 1],[-54335, -277121, 219, 3161, -24116, -64922, -344242, -333655; -330, -1680, 2, 19, -147, -395, -2093, -2028; 114136, 582120, -460, -6640, 50658, 136375, 723114, 700875],66,20]
%
% END OF LATTICE L67.20
%
\hbox{}\par\smallskip%
%
% LATTICE L67.21
%
\leavevmode%
\llap{\spacer}%
${L_{67.21}}=\pdualwitharg{5}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{4}{10}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18612300\\124320\\47040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
124320\\-790\\-325
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47040\\-325\\-116
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\558\\2086
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-1176\\-4410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-682\\-2550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\619\\2320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\1239\\4638
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\3656\\13680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
116\\7189\\26894
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
191\\11838\\44280
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\0\\-20\\-30\\-18\\-20\\-14\\0
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0\\420\\0\\-210\\-210\\-420\\-630\\-840
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\-10\\-30\\-100\\-210\\-360
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-30\\-210\\-10\\10\\0\\-40\\-140\\-300
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-18\\-210\\-30\\0\\6\\0\\-42\\-120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-420\\-100\\-40\\0\\20\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-630\\-210\\-140\\-42\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-360\\-300\\-120\\-60\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18612300, 124320, 47040; 124320, -790, -325; 47040, -325, -116],[420, 5, 1],[9, -19, -11, 10, 20, 59, 116, 191; 558, -1176, -682, 619, 1239, 3656, 7189, 11838; 2086, -4410, -2550, 2320, 4638, 13680, 26894, 44280],66,21]
%
% END OF LATTICE L67.21
%
\hbox{}\par\smallskip%
%
% LATTICE L67.22
%
\leavevmode%
\llap{\spacer}%
${L_{67.22}}=\pdualwitharg{5}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{4}{10}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-818580\\267960\\5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
267960\\-87715\\-1650
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\-1650\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\14\\67
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\105\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\6\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-11\\-65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\30\\190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\119\\651
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
95\\264\\1380
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\0\\-5\\-15\\-9\\-10\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\0\\-105\\-105\\-210\\-315\\-420
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\kern3pt%
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-5\\0\\5\\-5\\-15\\-50\\-105\\-180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-105\\-5\\10\\0\\-40\\-140\\-300
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\kern3pt%
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-9\\-105\\-15\\0\\6\\0\\-42\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-210\\-50\\-40\\0\\20\\0\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-315\\-105\\-140\\-42\\0\\14\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-180\\-300\\-120\\-60\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-818580, 267960, 5040; 267960, -87715, -1650; 5040, -1650, -31],[420, 5, 1],[5, 37, 2, -4, -1, 11, 43, 95; 14, 105, 6, -11, -3, 30, 119, 264; 67, 420, 5, -65, -3, 190, 651, 1380],66,22]
%
% END OF LATTICE L67.22
%
\hbox{}\par\smallskip%
%
% LATTICE L67.23
%
\leavevmode%
\llap{\spacer}%
${L_{67.23}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{67.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{4}{42}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1590863295\\-15722805\\-540921780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15722805\\155400\\5346033
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540921780\\5346033\\183923014
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
441303\\-785\\1297905
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19635\\-35\\57748
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7433\\13\\-21861
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-764\\2\\-2247
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
158216\\-280\\465325
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
231037\\-410\\679497
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
844472\\-1500\\2483655
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
375915\\-668\\1105594
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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105\\0\\-105\\-315\\-315\\-105\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\1\\0\\-21\\-35\\-21\\-45\\-14
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\kern3pt%
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-105\\0\\21\\-21\\-105\\-105\\-315\\-126
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-35\\-105\\0\\70\\0\\-210\\-140
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-21\\-105\\-84\\0\\21\\0\\-21
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-45\\-315\\-420\\-210\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-126\\-210\\-140\\-21\\0\\7
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1590863295, -15722805, -540921780; -15722805, 155400, 5346033; -540921780, 5346033, 183923014],[105, 21, 1],[441303, 19635, -7433, -764, 158216, 231037, 844472, 375915; -785, -35, 13, 2, -280, -410, -1500, -668; 1297905, 57748, -21861, -2247, 465325, 679497, 2483655, 1105594],66,23]
%
% END OF LATTICE L67.23
%
\hbox{}\par\smallskip%
%
% LATTICE L67.24
%
\leavevmode%
\llap{\spacer}%
${L_{67.24}}=\pdualwitharg{3}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{4}{3}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{s}{2}{420}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1803480\\251160\\6720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
251160\\-34977\\-936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6720\\-936\\-25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\45\\195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-13\\-51
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\7\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\60\\170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\112\\372
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
171\\1120\\3990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\124\\460
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\0\\-30\\-45\\-90\\-60\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-21\\-70\\-84\\-630\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\6\\-3\\-30\\-60\\-630\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-21\\-3\\3\\0\\-24\\-420\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-70\\-30\\0\\20\\0\\-420\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-84\\-60\\-24\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-630\\-630\\-420\\-420\\0\\420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-72\\-60\\-80\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1803480, 251160, 6720; 251160, -34977, -936; 6720, -936, -25],[840, 3, 1],[7, -1, -2, 1, 9, 17, 171, 19; 45, -7, -13, 7, 60, 112, 1120, 124; 195, -7, -51, 6, 170, 372, 3990, 460],66,24]
%
% END OF LATTICE L67.24
%
\hbox{}\par\smallskip%
%
% LATTICE L67.25
%
\leavevmode%
\llap{\spacer}%
${L_{67.25}}=\pdualwitharg{3}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{*}{4}{12}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26932920\\33600\\66360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33600\\-39\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66360\\-84\\-163
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\1945\\4695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\1253\\3017
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\419\\1005
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-138\\-336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-140\\-335
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\412\\1008
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\6370\\15435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\2080\\5032
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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30\\0\\-30\\-90\\-45\\-60\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\14\\0\\-42\\-35\\-84\\-315\\-56
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\kern3pt%
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-30\\0\\6\\-6\\-15\\-60\\-315\\-72
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-90\\-42\\-6\\12\\0\\-48\\-420\\-120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-35\\-15\\0\\5\\0\\-105\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-84\\-60\\-48\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-315\\-315\\-420\\-105\\0\\105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-72\\-120\\-40\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26932920, 33600, 66360; 33600, -39, -84; 66360, -84, -163],[840, 3, 1],[14, 9, 3, -1, -1, 3, 46, 15; 1945, 1253, 419, -138, -140, 412, 6370, 2080; 4695, 3017, 1005, -336, -335, 1008, 15435, 5032],66,25]
%
% END OF LATTICE L67.25
%
\hbox{}\par\smallskip%
%
% LATTICE L67.26
%
\leavevmode%
\llap{\spacer}%
${L_{67.26}}=\pdualwitharg{7}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{4}{14}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-370860\\8400\\81480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8400\\-182\\-1925
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81480\\-1925\\-17137
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
383\\8420\\875
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\1935\\201
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\1012\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-1077\\-112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\-2025\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
147\\3230\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
442\\9715\\1010
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1397\\30708\\3192
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-35\\-105\\-315\\-70\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-21\\-105\\-42\\-45\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\7\\-7\\-105\\-70\\-105\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-21\\-7\\14\\0\\-56\\-140\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-105\\-105\\0\\210\\0\\-210\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-42\\-70\\-56\\0\\28\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-45\\-105\\-140\\-210\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-252\\-420\\-840\\-84\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-370860, 8400, 81480; 8400, -182, -1925; 81480, -1925, -17137],[420, 7, 1],[383, 88, 46, -49, -92, 147, 442, 1397; 8420, 1935, 1012, -1077, -2025, 3230, 9715, 30708; 875, 201, 105, -112, -210, 336, 1010, 3192],66,26]
%
% END OF LATTICE L67.26
%
\hbox{}\par\smallskip%
%
% LATTICE L67.27
%
\leavevmode%
\llap{\spacer}%
${L_{67.27}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{}{4}{14}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
373231740\\-11965590\\174065640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11965590\\383614\\-5580442
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
174065640\\-5580442\\81179717
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-257109\\-1650\\551180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37465\\-240\\80316
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
209\\2\\-448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2991\\19\\-6412
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114122\\-735\\244650
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61443\\-395\\131719
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-232708\\-1495\\498870
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315770\\-2028\\676935
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-140\\-210\\-630\\-70\\-70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-42\\-210\\-42\\-90\\-84
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\28\\-14\\-210\\-70\\-210\\-252
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-42\\-14\\14\\0\\-28\\-140\\-210
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-210\\-210\\0\\210\\0\\-210\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-42\\-70\\-28\\0\\7\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-90\\-210\\-140\\-210\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-252\\-210\\-420\\-21\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[373231740, -11965590, 174065640; -11965590, 383614, -5580442; 174065640, -5580442, 81179717],[210, 14, 1],[-257109, -37465, 209, 2991, -114122, -61443, -232708, -315770; -1650, -240, 2, 19, -735, -395, -1495, -2028; 551180, 80316, -448, -6412, 244650, 131719, 498870, 676935],66,27]
%
% END OF LATTICE L67.27
%
\hbox{}\par\smallskip%
%
% LATTICE L67.28
%
\leavevmode%
\llap{\spacer}%
${L_{67.28}}=\pdualwitharg{7}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{4}{14}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1300740\\20160\\13020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20160\\-308\\-203
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13020\\-203\\-130
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\250\\910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\18\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-58\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-54\\-217
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\104\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\220\\755
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\612\\2142
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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140\\0\\-140\\-210\\-630\\-140\\-70\\0
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0\\12\\0\\-42\\-210\\-84\\-90\\-168
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-140\\0\\28\\-14\\-210\\-140\\-210\\-504
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-210\\-42\\-14\\14\\0\\-56\\-140\\-420
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-210\\-210\\0\\210\\0\\-210\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-84\\-140\\-56\\0\\28\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-90\\-210\\-140\\-210\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-504\\-420\\-840\\-84\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1300740, 20160, 13020; 20160, -308, -203; 13020, -203, -130],[420, 7, 1],[13, 1, -3, -3, -1, 5, 11, 31; 250, 18, -58, -54, 0, 104, 220, 612; 910, 72, -210, -217, -105, 336, 755, 2142],66,28]
%
% END OF LATTICE L67.28
%
\hbox{}\par\smallskip%
%
% LATTICE L67.29
%
\leavevmode%
\llap{\spacer}%
${L_{67.29}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{}{4}{15}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
402253530\\12858090\\143544660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12858090\\411015\\4588425
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143544660\\4588425\\51224086
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41320\\-330\\-115761
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70248\\-560\\-196805
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-166\\2\\465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2404\\19\\6735
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6113\\-49\\-17126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
98741\\-790\\-276630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
261783\\-2093\\-733404
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
169155\\-1352\\-473900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-30\\-45\\-9\\-30\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-105\\-35\\-210\\-315\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\-15\\-15\\-150\\-315\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-105\\-15\\15\\0\\-60\\-210\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-35\\-15\\0\\1\\0\\-21\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-210\\-150\\-60\\0\\30\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-315\\-315\\-210\\-21\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-180\\-150\\-20\\-30\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[402253530, 12858090, 143544660; 12858090, 411015, 4588425; 143544660, 4588425, 51224086],[210, 15, 1],[41320, 70248, -166, -2404, 6113, 98741, 261783, 169155; -330, -560, 2, 19, -49, -790, -2093, -1352; -115761, -196805, 465, 6735, -17126, -276630, -733404, -473900],66,29]
%
% END OF LATTICE L67.29
%
\hbox{}\par\smallskip%
%
% LATTICE L67.30
%
\leavevmode%
\llap{\spacer}%
${L_{67.30}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{67.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{4}{70}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1487076675\\19624815\\-277676595
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19624815\\259000\\-3664465
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-277676595\\-3664465\\51849578
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60523\\-181726\\311283
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40393\\-121284\\207750
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5097\\15304\\-26215
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-524\\1574\\-2695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65096\\-195456\\334803
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
158429\\-475697\\814835
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115816\\-347748\\595667
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
773329\\-2321991\\3977400
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-35\\-105\\-63\\-35\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-105\\-105\\-105\\-45\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\35\\-35\\-105\\-175\\-105\\-630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-105\\-35\\70\\0\\-140\\-140\\-1050
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-105\\-105\\0\\42\\0\\-42\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-105\\-175\\-140\\0\\35\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-45\\-105\\-140\\-42\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-630\\-1050\\-420\\-105\\0\\105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1487076675, 19624815, -277676595; 19624815, 259000, -3664465; -277676595, -3664465, 51849578],[105, 35, 1],[60523, 40393, -5097, -524, 65096, 158429, 115816, 773329; -181726, -121284, 15304, 1574, -195456, -475697, -347748, -2321991; 311283, 207750, -26215, -2695, 334803, 814835, 595667, 3977400],66,30]
%
% END OF LATTICE L67.30
%
\hbox{}\par\smallskip%
%
% LATTICE L67.31
%
\leavevmode%
\llap{\spacer}%
${L_{67.31}}=\pdualwitharg{5}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{}{4}{5}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4036200\\28560\\518280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28560\\-190\\-3795
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
518280\\-3795\\-65203
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\2703\\256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128\\-6657\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-3379\\-320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\3328\\315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
245\\12738\\1206
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
711\\36964\\3500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1385\\72002\\6818
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2267\\117852\\11160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-10\\-15\\-18\\-20\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\210\\0\\-105\\-210\\-420\\-630\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\-5\\-30\\-100\\-210\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-105\\-5\\5\\0\\-40\\-140\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-210\\-30\\0\\12\\0\\-84\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-420\\-100\\-40\\0\\40\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-630\\-210\\-140\\-84\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-360\\-300\\-240\\-120\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4036200, 28560, 518280; 28560, -190, -3795; 518280, -3795, -65203],[840, 5, 1],[52, -128, -65, 64, 245, 711, 1385, 2267; 2703, -6657, -3379, 3328, 12738, 36964, 72002, 117852; 256, -630, -320, 315, 1206, 3500, 6818, 11160],66,31]
%
% END OF LATTICE L67.31
%
\hbox{}\par\smallskip%
%
% LATTICE L67.32
%
\leavevmode%
\llap{\spacer}%
${L_{67.32}}=\pdualwitharg{5}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{*}{4}{20}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28581000\\44520\\89040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44520\\-65\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89040\\-140\\-277
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\333\\1117
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\3423\\11445
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\419\\1395
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-82\\-280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-84\\-279
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\244\\840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\994\\3353
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61\\5064\\17040
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2\\0\\-10\\-30\\-9\\-20\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\210\\0\\-210\\-105\\-420\\-315\\-840
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-10\\0\\10\\-10\\-15\\-100\\-105\\-360
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-30\\-210\\-10\\20\\0\\-80\\-140\\-600
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-105\\-15\\0\\3\\0\\-21\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-420\\-100\\-80\\0\\40\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-315\\-105\\-140\\-21\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-360\\-600\\-120\\-120\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-28581000, 44520, 89040; 44520, -65, -140; 89040, -140, -277],[840, 5, 1],[4, 41, 5, -1, -1, 3, 12, 61; 333, 3423, 419, -82, -84, 244, 994, 5064; 1117, 11445, 1395, -280, -279, 840, 3353, 17040],66,32]
%
% END OF LATTICE L67.32
%
\hbox{}\par\smallskip%
%
% LATTICE L67.33
%
\leavevmode%
\llap{\spacer}%
${L_{67.33}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${210}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{}{4}{21}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1026087510\\24298680\\359019780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24298680\\575421\\8501913
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
359019780\\8501913\\125618138
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
389726\\-1650\\-1113735
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18930\\-80\\-54097
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-316\\2\\903
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4534\\19\\12957
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57661\\-245\\-164780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
186269\\-790\\-532308
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
352737\\-1495\\-1008030
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
319095\\-1352\\-911890
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210\\0\\-210\\-315\\-315\\-210\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-21\\-35\\-42\\-45\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\42\\-21\\-105\\-210\\-315\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-21\\-21\\21\\0\\-84\\-210\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-35\\-105\\0\\35\\0\\-105\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-42\\-210\\-84\\0\\42\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-45\\-315\\-210\\-105\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-252\\-210\\-140\\-42\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1026087510, 24298680, 359019780; 24298680, 575421, 8501913; 359019780, 8501913, 125618138],[210, 21, 1],[389726, 18930, -316, -4534, 57661, 186269, 352737, 319095; -1650, -80, 2, 19, -245, -790, -1495, -1352; -1113735, -54097, 903, 12957, -164780, -532308, -1008030, -911890],66,33]
%
% END OF LATTICE L67.33
%
\hbox{}\par\smallskip%
%
% LATTICE L67.34
%
\leavevmode%
\llap{\spacer}%
${L_{67.34}}=\pdualwitharg{2.3}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{4}{24}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5055637440\\-37134300\\1251747420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37134300\\272760\\-9194244
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1251747420\\-9194244\\309925627
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6143\\25\\-24810
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
669\\0\\-2702
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
361\\1\\-1458
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5580\\29\\-22536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10746\\55\\-43400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4055\\20\\-16377
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
154534\\735\\-624120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8418\\39\\-33998
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-180\\-180\\-30\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-84\\-140\\-42\\-1260\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\12\\-12\\-60\\-30\\-1260\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-84\\-12\\24\\0\\-24\\-1680\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-140\\-60\\0\\40\\0\\-840\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-42\\-30\\-24\\0\\3\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-1260\\-1260\\-1680\\-840\\0\\840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-72\\-120\\-80\\-6\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5055637440, -37134300, 1251747420; -37134300, 272760, -9194244; 1251747420, -9194244, 309925627],[420, 12, 1],[6143, 669, 361, 5580, 10746, 4055, 154534, 8418; 25, 0, 1, 29, 55, 20, 735, 39; -24810, -2702, -1458, -22536, -43400, -16377, -624120, -33998],66,34]
%
% END OF LATTICE L67.34
%
\hbox{}\par\smallskip%
%
% LATTICE L67.35
%
\leavevmode%
\llap{\spacer}%
${L_{67.35}}=\pdualwitharg{2.3}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{4}{24}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3023160\\31500\\750960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31500\\492\\7836
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
750960\\7836\\186541
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2244\\-625\\9060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1283\\-357\\5180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-428\\-119\\1728
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-107\\-30\\432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-213\\-60\\860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-587\\-164\\2370
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20077\\-5600\\81060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-721\\-201\\2911
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-180\\-180\\-60\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-84\\-140\\-84\\-1260\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\12\\-12\\-60\\-60\\-1260\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-84\\-12\\24\\0\\-48\\-1680\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-140\\-60\\0\\40\\0\\-840\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-84\\-60\\-48\\0\\12\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-1260\\-1260\\-1680\\-840\\0\\840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-36\\-60\\-40\\-6\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3023160, 31500, 750960; 31500, 492, 7836; 750960, 7836, 186541],[420, 12, 1],[-2244, -1283, -428, -107, -213, -587, -20077, -721; -625, -357, -119, -30, -60, -164, -5600, -201; 9060, 5180, 1728, 432, 860, 2370, 81060, 2911],66,35]
%
% END OF LATTICE L67.35
%
\hbox{}\par\smallskip%
%
% LATTICE L67.36
%
\leavevmode%
\llap{\spacer}%
${L_{67.36}}=\pdualwitharg{7}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{*}{4}{28}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-8704920\\50400\\-927360
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50400\\-287\\5516
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-927360\\5516\\-94315
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-922\\-99355\\3255
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-215\\-23169\\759
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-117\\-12609\\413
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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115\\12392\\-406
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
119\\12825\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-341\\-36744\\1204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-524\\-56465\\1850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3331\\-358944\\11760
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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70\\0\\-70\\-210\\-315\\-140\\-35\\0
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0\\6\\0\\-42\\-105\\-84\\-45\\-168
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-70\\0\\14\\-14\\-105\\-140\\-105\\-504
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-315\\-105\\-105\\0\\105\\0\\-105\\-840
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\kern3pt%
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-140\\-84\\-140\\-112\\0\\56\\0\\-168
\end{tabular}\endgroup%
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-35\\-45\\-105\\-140\\-105\\0\\5\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-504\\-840\\-840\\-168\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8704920, 50400, -927360; 50400, -287, 5516; -927360, 5516, -94315],[840, 7, 1],[-922, -215, -117, 115, 119, -341, -524, -3331; -99355, -23169, -12609, 12392, 12825, -36744, -56465, -358944; 3255, 759, 413, -406, -420, 1204, 1850, 11760],66,36]
%
% END OF LATTICE L67.36
%
\hbox{}\par\smallskip%
%
% LATTICE L67.37
%
\leavevmode%
\llap{\spacer}%
${L_{67.37}}=\pdualwitharg{7}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{}{4}{7}\above{1pt}{1pt}{r}{2}{420}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3916920\\1369200\\-15960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1369200\\-478618\\5579
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15960\\5579\\-65
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\65\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-17\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\9\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\240\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\152\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\220\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
181\\516\\-168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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70\\0\\-70\\-105\\-630\\-140\\-70\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-21\\-210\\-84\\-90\\-168
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-70\\0\\14\\-7\\-210\\-140\\-210\\-504
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-210\\-210\\0\\420\\0\\-420\\-1680
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\kern3pt%
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-140\\-84\\-140\\-56\\0\\56\\0\\-168
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-70\\-90\\-210\\-140\\-420\\0\\20\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-504\\-420\\-1680\\-168\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3916920, 1369200, -15960; 1369200, -478618, 5579; -15960, 5579, -65],[840, 7, 1],[23, -1, -6, 3, 83, 53, 77, 181; 65, -3, -17, 9, 240, 152, 220, 516; -70, -12, 14, 35, 210, 28, -30, -168],66,37]
%
% END OF LATTICE L67.37
%
\hbox{}\par\smallskip%
%
% LATTICE L67.38
%
\leavevmode%
\llap{\spacer}%
${L_{67.38}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{}{4}{30}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36108646770\\6699630\\-17953995780
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6699630\\-1140\\3331200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17953995780\\3331200\\-8927112847
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
593360\\-1505\\-1193352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1164174\\-2947\\-2341360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
87680\\-219\\-176340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87501\\220\\175980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55113\\-141\\-110842
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
588439\\-1497\\-1183455
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3375137\\-8575\\-6787998
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1126686\\-2861\\-2265965
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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12\\0\\-60\\-90\\-18\\-30\\-42\\0
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\kern3pt%
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0\\140\\0\\-210\\-70\\-210\\-630\\-140
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\60\\-30\\-30\\-150\\-630\\-180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-210\\-30\\30\\0\\-60\\-420\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-70\\-30\\0\\2\\0\\-42\\-20
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-210\\-150\\-60\\0\\15\\0\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-630\\-630\\-420\\-42\\0\\42\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-180\\-150\\-20\\-15\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36108646770, 6699630, -17953995780; 6699630, -1140, 3331200; -17953995780, 3331200, -8927112847],[210, 30, 1],[593360, 1164174, 87680, -87501, 55113, 588439, 3375137, 1126686; -1505, -2947, -219, 220, -141, -1497, -8575, -2861; -1193352, -2341360, -176340, 175980, -110842, -1183455, -6787998, -2265965],66,38]
%
% END OF LATTICE L67.38
%
\hbox{}\par\smallskip%
%
% LATTICE L67.39
%
\leavevmode%
\llap{\spacer}%
${L_{67.39}}=\pdualwitharg{3.5}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{4}{30}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4966500\\426720\\78120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
426720\\-36600\\-6705
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78120\\-6705\\-1228
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-160\\1446
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-14\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\196\\-1770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\13\\-135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-55\\491
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-668\\6000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\-1491\\13419
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-878\\7910
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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12\\0\\-60\\-90\\-18\\-60\\-42\\0
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\kern3pt%
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0\\140\\0\\-210\\-70\\-420\\-630\\-280
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\60\\-30\\-30\\-300\\-630\\-360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-210\\-30\\30\\0\\-120\\-420\\-300
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-70\\-30\\0\\2\\0\\-42\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-420\\-300\\-120\\0\\60\\0\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-630\\-630\\-420\\-42\\0\\42\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-360\\-300\\-40\\-60\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4966500, 426720, 78120; 426720, -36600, -6705; 78120, -6705, -1228],[420, 15, 1],[9, 1, -11, -1, 3, 37, 83, 49; -160, -14, 196, 13, -55, -668, -1491, -878; 1446, 140, -1770, -135, 491, 6000, 13419, 7910],66,39]
%
% END OF LATTICE L67.39
%
\hbox{}\par\smallskip%
%
% LATTICE L67.40
%
\leavevmode%
\llap{\spacer}%
${L_{67.40}}=\pdualwitharg{3.5}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{4}{30}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65100\\5040\\2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\-390\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2520\\-195\\-97
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\17\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\42\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-13\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\38\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\147\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\108\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3\\0\\-15\\-45\\-9\\-30\\-21\\0
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\kern3pt%
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0\\35\\0\\-105\\-35\\-210\\-315\\-140
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\kern3pt%
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-15\\0\\15\\-15\\-15\\-150\\-315\\-180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-35\\-15\\0\\2\\0\\-42\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-210\\-150\\-120\\0\\60\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-315\\-315\\-420\\-42\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-180\\-300\\-40\\-60\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-65100, 5040, 2520; 5040, -390, -195; 2520, -195, -97],[420, 15, 1],[1, 2, 0, -1, 0, 3, 10, 7; 17, 42, 7, -13, -1, 38, 147, 108; -9, -35, -15, 0, 2, 0, -42, -40],66,40]
%
% END OF LATTICE L67.40
%
\hbox{}\par\smallskip%
%
% LATTICE L67.41
%
\leavevmode%
\llap{\spacer}%
${L_{67.41}}=\pdualwitharg{2}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{4}{8}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{280}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6112680\\46200\\1485960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46200\\-304\\-11288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1485960\\-11288\\-361157
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
768\\3835\\3040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-382\\-1911\\-1512
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\-959\\-760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
191\\955\\756
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1819\\9090\\7200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
527\\2633\\2086
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10257\\51240\\40600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
839\\4191\\3321
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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80\\0\\-80\\-120\\-360\\-40\\-280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\336\\0\\-168\\-840\\-168\\-2520\\-168
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\kern3pt%
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-80\\0\\16\\-8\\-120\\-40\\-840\\-72
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-168\\-8\\8\\0\\-16\\-560\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-840\\-120\\0\\120\\0\\-840\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-168\\-40\\-16\\0\\4\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-2520\\-840\\-560\\-840\\0\\280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-72\\-60\\-120\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6112680, 46200, 1485960; 46200, -304, -11288; 1485960, -11288, -361157],[840, 8, 1],[768, -382, -192, 191, 1819, 527, 10257, 839; 3835, -1911, -959, 955, 9090, 2633, 51240, 4191; 3040, -1512, -760, 756, 7200, 2086, 40600, 3321],66,41]
%
% END OF LATTICE L67.41
%
\hbox{}\par\smallskip%
%
% LATTICE L67.42
%
\leavevmode%
\llap{\spacer}%
${L_{67.42}}=\pdualwitharg{2}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{s}{2}{336}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{}{4}{8}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12337080\\-5318040\\38640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5318040\\-2292400\\16656
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38640\\16656\\-121
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\105\\1040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-21\\-336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-25\\-248
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\13\\192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71\\180\\2100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\58\\638
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-473\\1190\\12740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\201\\2118
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-120\\-360\\-40\\-280\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\336\\0\\-168\\-840\\-168\\-2520\\-336
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-80\\0\\16\\-8\\-120\\-40\\-840\\-144
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-120\\-168\\-8\\8\\0\\-16\\-560\\-120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-840\\-120\\0\\120\\0\\-840\\-240
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-168\\-40\\-16\\0\\4\\0\\-12
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-2520\\-840\\-560\\-840\\0\\280\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-144\\-120\\-240\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12337080, -5318040, 38640; -5318040, -2292400, 16656; 38640, 16656, -121],[840, 8, 1],[-42, 8, 10, -5, -71, -23, -473, -80; 105, -21, -25, 13, 180, 58, 1190, 201; 1040, -336, -248, 192, 2100, 638, 12740, 2118],66,42]
%
% END OF LATTICE L67.42
%
\hbox{}\par\smallskip%
%
% LATTICE L67.43
%
\leavevmode%
\llap{\spacer}%
${L_{67.43}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{}{4}{35}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9776090730\\96827220\\-3869813010
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96827220\\959035\\-38328535
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3869813010\\-38328535\\1531844696
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-351789\\-330\\-888713
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-256309\\-240\\-647505
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1427\\2\\3605
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20463\\19\\51695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156145\\-147\\-394464
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840687\\-790\\-2123800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-318401\\-299\\-804366
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4320505\\-4056\\-10914750
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-70\\-105\\-63\\-70\\-7\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-105\\-105\\-210\\-45\\-420
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\70\\-35\\-105\\-350\\-105\\-1260
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-105\\-35\\35\\0\\-140\\-70\\-1050
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-105\\-105\\0\\21\\0\\-21\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-210\\-350\\-140\\0\\70\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
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-7\\-45\\-105\\-70\\-21\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-1260\\-1050\\-420\\-210\\0\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9776090730, 96827220, -3869813010; 96827220, 959035, -38328535; -3869813010, -38328535, 1531844696],[210, 35, 1],[-351789, -256309, 1427, 20463, -156145, -840687, -318401, -4320505; -330, -240, 2, 19, -147, -790, -299, -4056; -888713, -647505, 3605, 51695, -394464, -2123800, -804366, -10914750],66,43]
%
% END OF LATTICE L67.43
%
\hbox{}\par\smallskip%
%
% LATTICE L67.44
%
\leavevmode%
\llap{\spacer}%
${L_{67.44}}=\pdualwitharg{2.5}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{4}{40}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15246629160\\-197517180\\3774718500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-197517180\\2558800\\-48900760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3774718500\\-48900760\\934534421
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1803\\11\\-7282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1612\\21\\-6510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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359\\1\\-1450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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8839\\38\\-35700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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10283\\48\\-41532
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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6404\\32\\-25865
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48161\\252\\-194516
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38959\\210\\-157350
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-60\\-36\\-10\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-420\\-420\\-210\\-1260\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\-20\\-60\\-50\\-420\\-360
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-60\\-420\\-20\\40\\0\\-40\\-560\\-600
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-36\\-420\\-60\\0\\24\\0\\-168\\-240
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-210\\-50\\-40\\0\\5\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-1260\\-420\\-560\\-168\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-360\\-600\\-240\\-30\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15246629160, -197517180, 3774718500; -197517180, 2558800, -48900760; 3774718500, -48900760, 934534421],[420, 20, 1],[1803, 1612, 359, 8839, 10283, 6404, 48161, 38959; 11, 21, 1, 38, 48, 32, 252, 210; -7282, -6510, -1450, -35700, -41532, -25865, -194516, -157350],66,44]
%
% END OF LATTICE L67.44
%
\hbox{}\par\smallskip%
%
% LATTICE L67.45
%
\leavevmode%
\llap{\spacer}%
${L_{67.45}}=\pdualwitharg{2.5}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{420}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{4}{40}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9808260\\-101640\\2421300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-101640\\1220\\-25080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2421300\\-25080\\597731
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
913\\248\\-3688
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8422\\2289\\-34020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1015\\276\\-4100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
203\\55\\-820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\27\\-408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
963\\261\\-3890
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7403\\2009\\-29904
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4133\\1122\\-16695
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-60\\-36\\-20\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-420\\-420\\-420\\-1260\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\-20\\-60\\-100\\-420\\-180
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-420\\-20\\40\\0\\-80\\-560\\-300
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\kern3pt%
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-36\\-420\\-60\\0\\24\\0\\-168\\-120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-420\\-100\\-80\\0\\20\\0\\-30
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-1260\\-420\\-560\\-168\\0\\56\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-180\\-300\\-120\\-30\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9808260, -101640, 2421300; -101640, 1220, -25080; 2421300, -25080, 597731],[420, 20, 1],[913, 8422, 1015, 203, 101, 963, 7403, 4133; 248, 2289, 276, 55, 27, 261, 2009, 1122; -3688, -34020, -4100, -820, -408, -3890, -29904, -16695],66,45]
%
% END OF LATTICE L67.45
%
\hbox{}\par\smallskip%
%
% LATTICE L67.46
%
\leavevmode%
\llap{\spacer}%
${L_{67.46}}=\pdualwitharg{3.7}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{4}{42}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-118020\\-43260\\5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43260\\-15855\\1848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\1848\\-215
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\55\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\6\\-19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\17\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-15\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\35\\-195
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\48\\-224
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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105\\0\\-105\\-315\\-315\\-210\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\1\\0\\-21\\-35\\-42\\-45\\-28
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-35\\-105\\0\\70\\0\\-210\\-280
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\kern3pt%
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-210\\-42\\-210\\-168\\0\\84\\0\\-84
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-45\\-315\\-420\\-210\\0\\30\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-252\\-420\\-280\\-84\\0\\28
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-118020, -43260, 5040; -43260, -15855, 1848; 5040, 1848, -215],[420, 21, 1],[-29, -3, -8, 2, 7, -1, -21, -27; 55, 6, 17, -3, -15, -2, 35, 48; -210, -19, -42, 21, 35, -42, -195, -224],66,46]
%
% END OF LATTICE L67.46
%
\hbox{}\par\smallskip%
%
% LATTICE L67.47
%
\leavevmode%
\llap{\spacer}%
${L_{67.47}}=\pdualwitharg{2.3.7}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{}{4}{42}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93204327930\\12721170\\-46341062670
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12721170\\-1596\\6324948
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46341062670\\6324948\\-23040712133
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5559296\\-7525\\-11181240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
311650\\-421\\-626812
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
164344\\-219\\-330540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-163989\\220\\329826
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
516281\\-705\\-1038380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1102577\\-1497\\-2217579
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4517327\\-6125\\-9085560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2111180\\-2861\\-4246151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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420\\0\\-420\\-630\\-630\\-210\\-210\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-252\\-210\\-140\\-21\\0\\7
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-93204327930, 12721170, -46341062670; 12721170, -1596, 6324948; -46341062670, 6324948, -23040712133],[210, 42, 1],[5559296, 311650, 164344, -163989, 516281, 1102577, 4517327, 2111180; -7525, -421, -219, 220, -705, -1497, -6125, -2861; -11181240, -626812, -330540, 329826, -1038380, -2217579, -9085560, -4246151],66,47]
%
% END OF LATTICE L67.47
%
\hbox{}\par\smallskip%
%
% LATTICE L67.48
%
\leavevmode%
\llap{\spacer}%
${L_{67.48}}=\pdualwitharg{3.7}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{*}{4}{42}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17375820\\127260\\83160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127260\\-924\\-609
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83160\\-609\\-398
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-10\\2730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\212
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\20\\-2121
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\30\\-2555
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\32\\-1932
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\30\\-885
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\182
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-420\\-630\\-630\\-420\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\4\\0\\-42\\-70\\-84\\-90\\-56
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\84\\-42\\-210\\-420\\-630\\-504
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-42\\-42\\42\\0\\-168\\-420\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-70\\-210\\0\\70\\0\\-210\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-84\\-420\\-168\\0\\84\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-90\\-630\\-420\\-210\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-504\\-420\\-280\\-84\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17375820, 127260, 83160; 127260, -924, -609; 83160, -609, -398],[420, 21, 1],[13, 1, -3, -10, -12, -9, -4, 1; -10, -2, 2, 20, 30, 32, 30, 16; 2730, 212, -630, -2121, -2555, -1932, -885, 182],66,48]
%
% END OF LATTICE L67.48
%
\hbox{}\par\smallskip%
%
% LATTICE L67.49
%
\leavevmode%
\llap{\spacer}%
${L_{67.49}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{67.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{4}{210}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6641599125\\-4741706865\\1688938965
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4741706865\\-3385296600\\1205801940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1688938965\\1205801940\\-429492151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-410\\-164381\\-463113
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71\\-28189\\-79420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\45957\\129465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111\\-46108\\-129885
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-211\\-85482\\-240821
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1304\\-525996\\-1481865
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-889\\-357866\\-1008207
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1913\\-769276\\-2167270
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-105\\-315\\-63\\-105\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-105\\-35\\-105\\-45\\-70
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\0\\105\\-105\\-105\\-525\\-315\\-630
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-105\\-105\\210\\0\\-420\\-420\\-1050
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-35\\-105\\0\\14\\0\\-42\\-140
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-105\\-525\\-420\\0\\105\\0\\-105
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-45\\-315\\-420\\-42\\0\\6\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-630\\-1050\\-140\\-105\\0\\35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6641599125, -4741706865, 1688938965; -4741706865, -3385296600, 1205801940; 1688938965, 1205801940, -429492151],[105, 105, 1],[-410, -71, 112, -111, -211, -1304, -889, -1913; -164381, -28189, 45957, -46108, -85482, -525996, -357866, -769276; -463113, -79420, 129465, -129885, -240821, -1481865, -1008207, -2167270],66,49]
%
% END OF LATTICE L67.49
%
\hbox{}\par\smallskip%
%
% LATTICE L67.50
%
\leavevmode%
\llap{\spacer}%
${L_{67.50}}=\pdualwitharg{2.7}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{4}{56}\above{1pt}{1pt}{s}{2}{840}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77367360\\-1477980\\19146540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1477980\\28280\\-365764
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19146540\\-365764\\4738303
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2547\\25\\-10290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\0\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
149\\1\\-602
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2800\\29\\-11312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15802\\165\\-63840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1925\\20\\-7777
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10198\\105\\-41200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11446\\117\\-46242
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-140\\-420\\-1260\\-70\\-140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-84\\-420\\-42\\-180\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\28\\-28\\-420\\-70\\-420\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-84\\-28\\56\\0\\-56\\-560\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-420\\-420\\0\\840\\0\\-840\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-42\\-70\\-56\\0\\7\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-180\\-420\\-560\\-840\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-504\\-840\\-1680\\-42\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[77367360, -1477980, 19146540; -1477980, 28280, -365764; 19146540, -365764, 4738303],[420, 28, 1],[2547, 49, 149, 2800, 15802, 1925, 10198, 11446; 25, 0, 1, 29, 165, 20, 105, 117; -10290, -198, -602, -11312, -63840, -7777, -41200, -46242],66,50]
%
% END OF LATTICE L67.50
%
\hbox{}\par\smallskip%
%
% LATTICE L67.51
%
\leavevmode%
\llap{\spacer}%
${L_{67.51}}=\pdualwitharg{2.7}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{4}{56}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25881240\\22119300\\6552420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22119300\\18904172\\5600000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6552420\\5600000\\1658893
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
591\\55\\-2520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
212\\15\\-888
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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237\\13\\-980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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55\\2\\-224
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\HardButStrongLineBreak\kern3pt%
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-319\\0\\1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-5\\10\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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369\\60\\-1660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
379\\48\\-1659
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\-84\\-252\\-420\\-840\\-42\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25881240, 22119300, 6552420; 22119300, 18904172, 5600000; 6552420, 5600000, 1658893],[420, 28, 1],[591, 212, 237, 55, -319, -5, 369, 379; 55, 15, 13, 2, 0, 10, 60, 48; -2520, -888, -980, -224, 1260, -14, -1660, -1659],66,51]
%
% END OF LATTICE L67.51
%
\hbox{}\par\smallskip%
%
% LATTICE L67.52
%
\leavevmode%
\llap{\spacer}%
${L_{67.52}}=\pdualwitharg{3.5}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{4}{15}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2035320\\651840\\23520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
651840\\-208650\\-7515
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23520\\-7515\\-269
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\101\\-636
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\77\\-490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-105\\660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-73\\465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\20\\-122
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
131\\528\\-3300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
347\\1400\\-8778
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
221\\892\\-5600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-30\\-45\\-18\\-60\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\70\\0\\-105\\-70\\-420\\-630\\-280
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\-15\\-30\\-300\\-630\\-360
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-18\\-70\\-30\\0\\4\\0\\-84\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-420\\-300\\-120\\0\\120\\0\\-120
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-42\\-630\\-630\\-420\\-84\\0\\84\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-360\\-300\\-80\\-120\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2035320, 651840, 23520; 651840, -208650, -7515; 23520, -7515, -269],[840, 15, 1],[25, 19, -26, -18, 5, 131, 347, 221; 101, 77, -105, -73, 20, 528, 1400, 892; -636, -490, 660, 465, -122, -3300, -8778, -5600],66,52]
%
% END OF LATTICE L67.52
%
\hbox{}\par\smallskip%
%
% LATTICE L67.53
%
\leavevmode%
\llap{\spacer}%
${L_{67.53}}=\pdualwitharg{3.5}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{*}{4}{60}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-610474200\\357840\\715680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
357840\\-195\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
715680\\-420\\-839
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\277\\8391
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\1029\\31045
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\419\\12585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-26\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-28\\-839
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\76\\2520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\714\\21819
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\1296\\39440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6\\0\\-30\\-90\\-9\\-60\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\70\\0\\-210\\-35\\-420\\-315\\-280
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\-30\\-15\\-300\\-315\\-360
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-90\\-210\\-30\\60\\0\\-240\\-420\\-600
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-9\\-35\\-15\\0\\1\\0\\-21\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-420\\-300\\-240\\0\\120\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-315\\-315\\-420\\-21\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-360\\-600\\-40\\-120\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-610474200, 357840, 715680; 357840, -195, -420; 715680, -420, -839],[840, 15, 1],[10, 37, 15, -1, -1, 3, 26, 47; 277, 1029, 419, -26, -28, 76, 714, 1296; 8391, 31045, 12585, -840, -839, 2520, 21819, 39440],66,53]
%
% END OF LATTICE L67.53
%
\hbox{}\par\smallskip%
%
% LATTICE L67.54
%
\leavevmode%
\llap{\spacer}%
${L_{67.54}}=\pdualwitharg{5.7}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{4}{70}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{420}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-392700\\-73080\\12180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73080\\-13510\\2275
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12180\\2275\\-377
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\20\\-203
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-28\\315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-33\\350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\15\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\118\\-1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\43\\-452
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-287\\564\\-5880
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-35\\-105\\-63\\-70\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\15\\0\\-105\\-105\\-210\\-45\\-420
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\kern3pt%
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-35\\0\\35\\-35\\-105\\-350\\-105\\-1260
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\kern3pt%
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-105\\-105\\-35\\70\\0\\-280\\-140\\-2100
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-105\\-105\\0\\42\\0\\-42\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-210\\-350\\-280\\0\\140\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-45\\-105\\-140\\-42\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-1260\\-2100\\-840\\-420\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-392700, -73080, 12180; -73080, -13510, 2275; 12180, 2275, -377],[420, 35, 1],[-10, -1, 15, 17, -8, -61, -22, -287; 20, 3, -28, -33, 15, 118, 43, 564; -203, -15, 315, 350, -168, -1260, -452, -5880],66,54]
%
% END OF LATTICE L67.54
%
\hbox{}\par\smallskip%
%
% LATTICE L67.55
%
\leavevmode%
\llap{\spacer}%
${L_{67.55}}=\pdualwitharg{2.5.7}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{}{4}{70}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13362465270\\6217890\\-6375780180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6217890\\-2660\\2966810
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6375780180\\2966810\\-3042146197
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
521064\\-1505\\-1092056
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
438158\\-1263\\-918300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77020\\-219\\-161420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76853\\220\\161070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
145169\\-423\\-304248
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
516712\\-1497\\-1082935
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
423401\\-1225\\-887372
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2968159\\-8583\\-6220725
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-140\\-210\\-126\\-70\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\60\\0\\-210\\-210\\-210\\-90\\-420
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\140\\-70\\-210\\-350\\-210\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\-70\\70\\0\\-140\\-140\\-1050
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-210\\-210\\0\\42\\0\\-42\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-210\\-350\\-140\\0\\35\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-90\\-210\\-140\\-42\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-1260\\-1050\\-420\\-105\\0\\105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13362465270, 6217890, -6375780180; 6217890, -2660, 2966810; -6375780180, 2966810, -3042146197],[210, 70, 1],[521064, 438158, 77020, -76853, 145169, 516712, 423401, 2968159; -1505, -1263, -219, 220, -423, -1497, -1225, -8583; -1092056, -918300, -161420, 161070, -304248, -1082935, -887372, -6220725],66,55]
%
% END OF LATTICE L67.55
%
\hbox{}\par\smallskip%
%
% LATTICE L67.56
%
\leavevmode%
\llap{\spacer}%
${L_{67.56}}=\pdualwitharg{5.7}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{*}{4}{70}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7279860\\-33088860\\-4738020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33088860\\-149985500\\-21476945
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4738020\\-21476945\\-3075358
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-621\\-19874\\139748
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71\\-2274\\15990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
759\\24290\\-170800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
214\\6854\\-48195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-476\\-15228\\107079
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2263\\-72412\\509180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-756\\-24192\\170111
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9563\\-306024\\2151870
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-140\\-210\\-126\\-140\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-210\\-210\\-420\\-90\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\140\\-70\\-210\\-700\\-210\\-2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\-70\\70\\0\\-280\\-140\\-2100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-210\\-210\\0\\42\\0\\-42\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-420\\-700\\-280\\0\\140\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-90\\-210\\-140\\-42\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-2520\\-2100\\-840\\-420\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7279860, -33088860, -4738020; -33088860, -149985500, -21476945; -4738020, -21476945, -3075358],[420, 35, 1],[-621, -71, 759, 214, -476, -2263, -756, -9563; -19874, -2274, 24290, 6854, -15228, -72412, -24192, -306024; 139748, 15990, -170800, -48195, 107079, 509180, 170111, 2151870],66,56]
%
% END OF LATTICE L67.56
%
\hbox{}\par\smallskip%
%
% LATTICE L67.57
%
\leavevmode%
\llap{\spacer}%
${L_{67.57}}=\pdualwitharg{3.7}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${210}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{*}{4}{84}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12634440\\61320\\-4872000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61320\\-273\\23856
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4872000\\23856\\-1876909
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3991\\-79835\\9345
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-366\\-7321\\857
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-852\\-17041\\1995
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
287\\5742\\-672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
284\\5680\\-665
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-861\\-17228\\2016
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1877\\-37550\\4395
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4281\\-85640\\10024
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210\\0\\-210\\-630\\-315\\-420\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-42\\-35\\-84\\-45\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\42\\-42\\-105\\-420\\-315\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-42\\-42\\84\\0\\-336\\-420\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-35\\-105\\0\\35\\0\\-105\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-84\\-420\\-336\\0\\168\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-45\\-315\\-420\\-105\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-504\\-840\\-280\\-168\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12634440, 61320, -4872000; 61320, -273, 23856; -4872000, 23856, -1876909],[840, 21, 1],[-3991, -366, -852, 287, 284, -861, -1877, -4281; -79835, -7321, -17041, 5742, 5680, -17228, -37550, -85640; 9345, 857, 1995, -672, -665, 2016, 4395, 10024],66,57]
%
% END OF LATTICE L67.57
%
\hbox{}\par\smallskip%
%
% LATTICE L67.58
%
\leavevmode%
\llap{\spacer}%
${L_{67.58}}=\pdualwitharg{3.7}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${210}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{}{4}{21}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-286440\\15120\\7560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15120\\-798\\-399
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7560\\-399\\-199
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\75\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-19\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\90\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\172\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\250\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\196\\-56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210\\0\\-210\\-315\\-630\\-420\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-21\\-70\\-84\\-90\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\42\\-21\\-210\\-420\\-630\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-21\\-21\\21\\0\\-168\\-420\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-70\\-210\\0\\140\\0\\-420\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-84\\-420\\-168\\0\\168\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-90\\-630\\-420\\-420\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-504\\-420\\-560\\-168\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-286440, 15120, 7560; 15120, -798, -399; 7560, -399, -199],[840, 21, 1],[4, 0, -1, 0, 3, 7, 11, 9; 75, -1, -19, 10, 90, 172, 250, 196; 0, 2, 0, -21, -70, -84, -90, -56],66,58]
%
% END OF LATTICE L67.58
%
\hbox{}\par\smallskip%
%
% LATTICE L67.59
%
\leavevmode%
\llap{\spacer}%
${L_{67.59}}=\pdualwitharg{2.3}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{*}{4}{24}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{840}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74109840\\-5052600\\450240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5052600\\-344472\\30696
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
450240\\30696\\-2735
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-260\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-406\\-112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-44\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\391\\108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\645\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\359\\78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-317\\4725\\840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\89\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\0\\-240\\-360\\-360\\-120\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\112\\0\\-168\\-280\\-168\\-2520\\-56
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\0\\48\\-24\\-120\\-120\\-2520\\-72
\end{tabular}\endgroup%
\kern3pt%
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-360\\-168\\-24\\24\\0\\-48\\-1680\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-280\\-120\\0\\40\\0\\-840\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-168\\-120\\-48\\0\\12\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-2520\\-2520\\-1680\\-840\\0\\840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-72\\-60\\-40\\-6\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-74109840, -5052600, 450240; -5052600, -344472, 30696; 450240, 30696, -2735],[840, 24, 1],[17, 27, 3, -26, -43, -24, -317, -6; -260, -406, -44, 391, 645, 359, 4725, 89; -120, -112, 0, 108, 160, 78, 840, 11],66,59]
%
% END OF LATTICE L67.59
%
\hbox{}\par\smallskip%
%
% LATTICE L67.60
%
\leavevmode%
\llap{\spacer}%
${L_{67.60}}=\pdualwitharg{2.3}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{}{4}{24}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{840}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18890760\\1239840\\147000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1239840\\-80904\\-9600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
147000\\-9600\\-1139
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-1985\\19440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-91\\896
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\473\\-4632
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\90\\-888
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-575\\5620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-761\\7446
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
184\\-17465\\170940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-1043\\10210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\0\\-240\\-360\\-360\\-120\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\112\\0\\-168\\-280\\-168\\-2520\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\0\\48\\-24\\-120\\-120\\-2520\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-168\\-24\\24\\0\\-48\\-1680\\-120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-280\\-120\\0\\40\\0\\-840\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-168\\-120\\-48\\0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-2520\\-2520\\-1680\\-840\\0\\840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-144\\-120\\-80\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18890760, 1239840, 147000; 1239840, -80904, -9600; 147000, -9600, -1139],[840, 24, 1],[21, 1, -5, -1, 6, 8, 184, 11; -1985, -91, 473, 90, -575, -761, -17465, -1043; 19440, 896, -4632, -888, 5620, 7446, 170940, 10210],66,60]
%
% END OF LATTICE L67.60
%
\hbox{}\par\smallskip%
%
% LATTICE L67.61
%
\leavevmode%
\llap{\spacer}%
${L_{67.61}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{}{4}{105}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7196467530\\5589360\\2585575440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5589360\\-3990\\-2007915
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2585575440\\-2007915\\-928955633
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
352676\\-706857\\983136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
98854\\-198129\\275570
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52130\\-104479\\145320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52017\\104254\\-145005
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32752\\-65645\\91301
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
699461\\-1401916\\1949850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
286574\\-574373\\798867
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1339309\\-2684340\\3733520
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-210\\-315\\-63\\-210\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-105\\-35\\-210\\-45\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\210\\-105\\-105\\-1050\\-315\\-1260
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-105\\-105\\105\\0\\-420\\-210\\-1050
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-35\\-105\\0\\7\\0\\-21\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\-1050\\-420\\0\\210\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-45\\-315\\-210\\-21\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-1260\\-1050\\-140\\-210\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7196467530, 5589360, 2585575440; 5589360, -3990, -2007915; 2585575440, -2007915, -928955633],[210, 105, 1],[352676, 98854, 52130, -52017, 32752, 699461, 286574, 1339309; -706857, -198129, -104479, 104254, -65645, -1401916, -574373, -2684340; 983136, 275570, 145320, -145005, 91301, 1949850, 798867, 3733520],66,61]
%
% END OF LATTICE L67.61
%
\hbox{}\par\smallskip%
%
% LATTICE L67.62
%
\leavevmode%
\llap{\spacer}%
${L_{67.62}}=\pdualwitharg{2.3.5}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{4}{120}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2103704400\\34113660\\520789500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34113660\\553200\\8445120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
520789500\\8445120\\128925767
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1412\\35\\-5706
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
433\\14\\-1750
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
438\\11\\-1770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7513\\177\\-30360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2829\\67\\-11432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5193\\124\\-20985
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38705\\931\\-156408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10381\\251\\-41950
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-60\\-180\\-36\\-30\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\140\\0\\-420\\-140\\-210\\-1260\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\60\\-60\\-60\\-150\\-1260\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-420\\-60\\120\\0\\-120\\-1680\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-140\\-60\\0\\8\\0\\-168\\-80
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-210\\-150\\-120\\0\\15\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-360\\-600\\-80\\-30\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2103704400, 34113660, 520789500; 34113660, 553200, 8445120; 520789500, 8445120, 128925767],[420, 60, 1],[1412, 433, 438, 7513, 2829, 5193, 38705, 10381; 35, 14, 11, 177, 67, 124, 931, 251; -5706, -1750, -1770, -30360, -11432, -20985, -156408, -41950],66,62]
%
% END OF LATTICE L67.62
%
\hbox{}\par\smallskip%
%
% LATTICE L67.63
%
\leavevmode%
\llap{\spacer}%
${L_{67.63}}=\pdualwitharg{2.3.5}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{140}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{}{4}{120}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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110040\\16380\\27300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16380\\5100\\4080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27300\\4080\\6773
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
442\\11\\-1788
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1419\\35\\-5740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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534\\13\\-2160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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89\\2\\-360
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\HardButStrongLineBreak\kern3pt%
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-1\\0\\4
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393\\10\\-1590
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3343\\84\\-13524
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639\\16\\-2585
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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-60\\-420\\-300\\-240\\0\\60\\0\\-30
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-84\\-1260\\-1260\\-1680\\-168\\0\\168\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-180\\-300\\-40\\-30\\0\\5
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[110040, 16380, 27300; 16380, 5100, 4080; 27300, 4080, 6773],[420, 60, 1],[442, 1419, 534, 89, -1, 393, 3343, 639; 11, 35, 13, 2, 0, 10, 84, 16; -1788, -5740, -2160, -360, 4, -1590, -13524, -2585],66,63]
%
% END OF LATTICE L67.63
%
\hbox{}\par\smallskip%
%
% LATTICE L67.64
%
\leavevmode%
\llap{\spacer}%
${L_{67.64}}=\pdualwitharg{5.7}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{}{4}{35}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-956760\\-111720\\25200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111720\\-12530\\2905
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25200\\2905\\-661
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\57\\784
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\33\\450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-61\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-95\\-1295
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-72\\-966
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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23\\88\\1260
\end{tabular}\endgroup%
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17\\68\\946
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271\\1092\\15120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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14\\0\\-70\\-105\\-126\\-140\\-14\\0
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\kern3pt%
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\kern3pt%
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-140\\-420\\-700\\-280\\0\\280\\0\\-840
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-14\\-90\\-210\\-140\\-84\\0\\4\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-2520\\-2100\\-1680\\-840\\0\\840
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-956760, -111720, 25200; -111720, -12530, 2905; 25200, 2905, -661],[840, 35, 1],[14, 8, -15, -23, -17, 23, 17, 271; 57, 33, -61, -95, -72, 88, 68, 1092; 784, 450, -840, -1295, -966, 1260, 946, 15120],66,64]
%
% END OF LATTICE L67.64
%
\hbox{}\par\smallskip%
%
% LATTICE L67.65
%
\leavevmode%
\llap{\spacer}%
${L_{67.65}}=\pdualwitharg{5.7}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{*}{4}{140}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{840}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3930360\\43680\\-1447320
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43680\\-455\\16240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1447320\\16240\\-532171
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-467\\-5607\\1099
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-682\\-8187\\1605
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-580\\-6961\\1365
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
119\\1430\\-280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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116\\1392\\-273
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-357\\-4292\\840
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-201\\-2414\\473
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7139\\-85728\\16800
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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14\\0\\-70\\-210\\-63\\-140\\-7\\0
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\kern3pt%
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0\\30\\0\\-210\\-105\\-420\\-45\\-840
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-420\\-700\\-560\\0\\280\\0\\-840
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\kern3pt%
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-7\\-45\\-105\\-140\\-21\\0\\1\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-2520\\-4200\\-840\\-840\\0\\840
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3930360, 43680, -1447320; 43680, -455, 16240; -1447320, 16240, -532171],[840, 35, 1],[-467, -682, -580, 119, 116, -357, -201, -7139; -5607, -8187, -6961, 1430, 1392, -4292, -2414, -85728; 1099, 1605, 1365, -280, -273, 840, 473, 16800],66,65]
%
% END OF LATTICE L67.65
%
\hbox{}\par\smallskip%
%
% LATTICE L67.66
%
\leavevmode%
\llap{\spacer}%
${L_{67.66}}=\pdualwitharg{2.5}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{1680}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{*}{4}{40}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-645960\\32760\\219240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32760\\-1520\\-11440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
219240\\-11440\\-73681
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
160\\799\\352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-382\\-1911\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-999\\-440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
191\\955\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
371\\1854\\816
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
541\\2703\\1190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2113\\10556\\4648
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
866\\4326\\1905
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-80\\-120\\-72\\-40\\-56\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1680\\0\\-840\\-840\\-840\\-2520\\-840
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0\\-840\\-360\\-300\\-120\\-30\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-645960, 32760, 219240; 32760, -1520, -11440; 219240, -11440, -73681],[840, 40, 1],[160, -382, -200, 191, 371, 541, 2113, 866; 799, -1911, -999, 955, 1854, 2703, 10556, 4326; 352, -840, -440, 420, 816, 1190, 4648, 1905],66,66]
%
% END OF LATTICE L67.66
%
\hbox{}\par\smallskip%
%
% LATTICE L67.67
%
\leavevmode%
\llap{\spacer}%
${L_{67.67}}=\pdualwitharg{2.5}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{1680}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{}{4}{40}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-13563480\\5454120\\-75600
\end{tabular}\endgroup%
\kern3pt%
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5454120\\-2193200\\30400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75600\\30400\\-421
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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34\\85\\32
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-76\\-189\\0
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-42\\-105\\-40
\end{tabular}\endgroup%
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39\\97\\0
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77\\192\\36
\end{tabular}\endgroup%
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113\\282\\70
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443\\1106\\308
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364\\909\\270
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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16\\0\\-80\\-120\\-72\\-40\\-56\\0
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-40\\-840\\-200\\-80\\0\\20\\0\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-2520\\-840\\-560\\-168\\0\\56\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1680\\-720\\-600\\-240\\-60\\0\\60
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13563480, 5454120, -75600; 5454120, -2193200, 30400; -75600, 30400, -421],[840, 40, 1],[34, -76, -42, 39, 77, 113, 443, 364; 85, -189, -105, 97, 192, 282, 1106, 909; 32, 0, -40, 0, 36, 70, 308, 270],66,67]
%
% END OF LATTICE L67.67
%
\hbox{}\par\smallskip%
%
% LATTICE L67.68
%
\leavevmode%
\llap{\spacer}%
${L_{67.68}}=\pdualwitharg{2.3.7}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{4}{168}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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673039920\\1047060\\166626600
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1047060\\1848\\259224
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
166626600\\259224\\41252269
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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15961\\25\\-64470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\0\\-206
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1071\\1\\-4326
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19174\\29\\-77448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35492\\55\\-143360
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12774\\20\\-51597
\end{tabular}\endgroup%
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66904\\105\\-270240
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\HardButStrongLineBreak\kern3pt%
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24834\\39\\-100310
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[673039920, 1047060, 166626600; 1047060, 1848, 259224; 166626600, 259224, 41252269],[420, 84, 1],[15961, 51, 1071, 19174, 35492, 12774, 66904, 24834; 25, 0, 1, 29, 55, 20, 105, 39; -64470, -206, -4326, -77448, -143360, -51597, -270240, -100310],66,68]
%
% END OF LATTICE L67.68
%
\hbox{}\par\smallskip%
%
% LATTICE L67.69
%
\leavevmode%
\llap{\spacer}%
${L_{67.69}}=\pdualwitharg{2.3.7}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{}{4}{168}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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193620\\18480\\47460
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18480\\5460\\4452
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47460\\4452\\11635
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3114\\-265\\-12600
\end{tabular}\endgroup%
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259\\-22\\-1048
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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602\\-51\\-2436
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\-7\\-336
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\HardButStrongLineBreak\kern3pt%
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173\\-15\\-700
\end{tabular}\endgroup%
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737\\-63\\-2982
\end{tabular}\endgroup%
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3811\\-325\\-15420
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
974\\-83\\-3941
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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\kern3pt%
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-420\\-84\\-420\\-336\\0\\84\\0\\-42
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\kern3pt%
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0\\-28\\-252\\-420\\-280\\-42\\0\\7
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[193620, 18480, 47460; 18480, 5460, 4452; 47460, 4452, 11635],[420, 84, 1],[3114, 259, 602, 83, 173, 737, 3811, 974; -265, -22, -51, -7, -15, -63, -325, -83; -12600, -1048, -2436, -336, -700, -2982, -15420, -3941],66,69]
%
% END OF LATTICE L67.69
%
\hbox{}\par\smallskip%
%
% LATTICE L67.70
%
\leavevmode%
\llap{\spacer}%
${L_{67.70}}=\pdualwitharg{3.5.7}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{4}{210}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2100\\-76860\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76860\\-2435475\\-53130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-53130\\-1159
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\53\\-2394
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\28\\-1265
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\79\\-3570
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-7\\315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-11\\497
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\39\\-1761
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\-1050\\-840\\0\\420\\0\\-420
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\kern3pt%
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-21\\-45\\-315\\-420\\-42\\0\\6\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-1260\\-2100\\-280\\-420\\0\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2100, -76860, -1680; -76860, -2435475, -53130; -1680, -53130, -1159],[420, 105, 1],[-25, -13, -36, 4, 5, -9, -19, -119; 53, 28, 79, -7, -11, 14, 39, 248; -2394, -1265, -3570, 315, 497, -630, -1761, -11200],66,70]
%
% END OF LATTICE L67.70
%
\hbox{}\par\smallskip%
%
% LATTICE L67.71
%
\leavevmode%
\llap{\spacer}%
${L_{67.71}}=\pdualwitharg{2.3.5.7}(\pfillwitharg{2}({L_{67.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}{420}\above{1pt}{1pt}{}{4}{210}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1306236097110\\513346303890\\248021529000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
513346303890\\201743335920\\97471609830
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
248021529000\\97471609830\\47093078339
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-4743203\\9819054
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-1151961\\2384710
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\19171\\-39690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\275925\\-571200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-49\\-701754\\1452724
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-395\\-5667457\\11732385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-299\\-4292996\\8887068
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-676\\-9708906\\20098715
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-420\\-630\\-126\\-210\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\20\\0\\-210\\-70\\-210\\-90\\-140
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\kern3pt%
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-420\\0\\420\\-210\\-210\\-1050\\-630\\-1260
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-210\\-210\\210\\0\\-420\\-420\\-1050
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-70\\-210\\0\\14\\0\\-42\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\-1050\\-420\\0\\105\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-90\\-630\\-420\\-42\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-1260\\-1050\\-140\\-105\\0\\35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1306236097110, 513346303890, 248021529000; 513346303890, 201743335920, 97471609830; 248021529000, 97471609830, 47093078339],[210, 210, 1],[-330, -80, 2, 19, -49, -395, -299, -676; -4743203, -1151961, 19171, 275925, -701754, -5667457, -4292996, -9708906; 9819054, 2384710, -39690, -571200, 1452724, 11732385, 8887068, 20098715],66,71]
%
% END OF LATTICE L67.71
%
\hbox{}\par\smallskip%
%
% LATTICE L67.72
%
\leavevmode%
\llap{\spacer}%
${L_{67.72}}=\pdualwitharg{3.5.7}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{*}{4}{210}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{420}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6037500\\-187674060\\-4471320
\end{tabular}\endgroup%
\kern3pt%
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-187674060\\-5832758400\\-138965085
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4471320\\-138965085\\-3310834
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-369\\-4430\\186438
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\-518\\21800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
451\\5414\\-227850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
431\\5192\\-218505
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\90\\-3787
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-737\\-8812\\370860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-319\\-3822\\160851
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1489\\-17852\\751310
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-420\\-630\\-126\\-420\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-210\\-70\\-420\\-90\\-280
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\kern3pt%
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-420\\0\\420\\-210\\-210\\-2100\\-630\\-2520
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-210\\-210\\210\\0\\-840\\-420\\-2100
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\kern3pt%
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-126\\-70\\-210\\0\\14\\0\\-42\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-420\\-2100\\-840\\0\\420\\0\\-420
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-90\\-630\\-420\\-42\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-2520\\-2100\\-280\\-420\\0\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6037500, -187674060, -4471320; -187674060, -5832758400, -138965085; -4471320, -138965085, -3310834],[420, 105, 1],[-369, -43, 451, 431, 7, -737, -319, -1489; -4430, -518, 5414, 5192, 90, -8812, -3822, -17852; 186438, 21800, -227850, -218505, -3787, 370860, 160851, 751310],66,72]
%
% END OF LATTICE L67.72
%
\hbox{}\par\smallskip%
%
% LATTICE L67.73
%
\leavevmode%
\llap{\spacer}%
${L_{67.73}}=\pdualwitharg{2.7}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${560}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{*}{4}{56}\above{1pt}{1pt}{l}{2}{840}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36047760\\6280680\\-1195320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6280680\\-1094296\\208264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1195320\\208264\\-39635
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-223\\-1120\\840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\54\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\296\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-157\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-419\\-2085\\1680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-659\\518
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-381\\-1905\\1480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-223\\-1116\\861
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
560\\0\\-560\\-840\\-2520\\-280\\-280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-168\\-840\\-168\\-360\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\0\\112\\-56\\-840\\-280\\-840\\-504
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-168\\-56\\56\\0\\-112\\-560\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-840\\-840\\0\\840\\0\\-840\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-168\\-280\\-112\\0\\28\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-360\\-840\\-560\\-840\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-504\\-420\\-840\\-42\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36047760, 6280680, -1195320; 6280680, -1094296, 208264; -1195320, 208264, -39635],[840, 56, 1],[-223, 11, 59, -32, -419, -132, -381, -223; -1120, 54, 296, -157, -2085, -659, -1905, -1116; 840, -48, -224, 140, 1680, 518, 1480, 861],66,73]
%
% END OF LATTICE L67.73
%
\hbox{}\par\smallskip%
%
% LATTICE L67.74
%
\leavevmode%
\llap{\spacer}%
${L_{67.74}}=\pdualwitharg{2.7}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${560}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{}{4}{56}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9295440\\-1200360\\158760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1200360\\-154952\\20496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
158760\\20496\\-2711
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\430\\4480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\24\\240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-102\\-1064
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-73\\-728
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\105\\1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\111\\1190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\425\\4500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\570\\6006
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
560\\0\\-560\\-840\\-2520\\-280\\-280\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\48\\0\\-168\\-840\\-168\\-360\\-336
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-560\\0\\112\\-56\\-840\\-280\\-840\\-1008
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-840\\-168\\-56\\56\\0\\-112\\-560\\-840
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-840\\-840\\0\\840\\0\\-840\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-168\\-280\\-112\\0\\28\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-360\\-840\\-560\\-840\\0\\40\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-1008\\-840\\-1680\\-84\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9295440, -1200360, 158760; -1200360, -154952, 20496; 158760, 20496, -2711],[840, 56, 1],[21, 1, -5, -3, 8, 6, 22, 29; 430, 24, -102, -73, 105, 111, 425, 570; 4480, 240, -1064, -728, 1260, 1190, 4500, 6006],66,74]
%
% END OF LATTICE L67.74
%
\hbox{}\par\smallskip%
%
% LATTICE L67.75
%
\leavevmode%
\llap{\spacer}%
${L_{67.75}}=\pdualwitharg{2.5.7}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{4}{280}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6897666909840\\-373451250900\\-1985748044700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-373451250900\\20219276840\\107511728380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1985748044700\\107511728380\\571670877787
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9973\\28479\\-39998
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1295\\3678\\-5190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3299\\9415\\-13230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53996\\154309\\-216580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60670\\173373\\-243348
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36990\\105692\\-148365
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39306\\112297\\-157652
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-221108\\631653\\-886830
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-140\\-420\\-252\\-70\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-420\\-420\\-210\\-180\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\140\\-140\\-420\\-350\\-420\\-2520
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-420\\-140\\280\\0\\-280\\-560\\-4200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-420\\-420\\0\\168\\0\\-168\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-210\\-350\\-280\\0\\35\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-180\\-420\\-560\\-168\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-2520\\-4200\\-1680\\-210\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6897666909840, -373451250900, -1985748044700; -373451250900, 20219276840, 107511728380; -1985748044700, 107511728380, 571670877787],[420, 140, 1],[-9973, -1295, -3299, -53996, -60670, -36990, -39306, -221108; 28479, 3678, 9415, 154309, 173373, 105692, 112297, 631653; -39998, -5190, -13230, -216580, -243348, -148365, -157652, -886830],66,75]
%
% END OF LATTICE L67.75
%
\hbox{}\par\smallskip%
%
% LATTICE L67.76
%
\leavevmode%
\llap{\spacer}%
${L_{67.76}}=\pdualwitharg{2.5.7}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{}{2}{140}\above{1pt}{1pt}{}{4}{280}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54001500\\-2340240\\-15075480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2340240\\101500\\653380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15075480\\653380\\4208633
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1537\\4576\\-6216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1988\\5919\\-8040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1627\\4844\\-6580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-277\\823\\-1120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\555\\-756
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1679\\4997\\-6790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1811\\5391\\-7324
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7036\\20946\\-28455
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-140\\-420\\-252\\-140\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-420\\-420\\-420\\-180\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\140\\-140\\-420\\-700\\-420\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-420\\-140\\280\\0\\-560\\-560\\-2100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-420\\-420\\0\\168\\0\\-168\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-420\\-700\\-560\\0\\140\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-180\\-420\\-560\\-168\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-1260\\-2100\\-840\\-210\\0\\105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[54001500, -2340240, -15075480; -2340240, 101500, 653380; -15075480, 653380, 4208633],[420, 140, 1],[-1537, -1988, -1627, -277, -187, -1679, -1811, -7036; 4576, 5919, 4844, 823, 555, 4997, 5391, 20946; -6216, -8040, -6580, -1120, -756, -6790, -7324, -28455],66,76]
%
% END OF LATTICE L67.76
%
\hbox{}\par\smallskip%
%
% LATTICE L67.77
%
\leavevmode%
\llap{\spacer}%
${L_{67.77}}=\pdualwitharg{3.5.7}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{}{4}{105}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}{840}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-233413320\\72762480\\683760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72762480\\-22682415\\-213150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
683760\\-213150\\-2003
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\15\\-231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-19\\315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\94\\-2835
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\74\\-2072
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
119\\508\\-13440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\122\\-3084
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\420\\-10220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-210\\-315\\-126\\-420\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-105\\-70\\-420\\-90\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\210\\-105\\-210\\-2100\\-630\\-2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-105\\-105\\105\\0\\-840\\-420\\-2100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-70\\-210\\0\\28\\0\\-84\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-420\\-2100\\-840\\0\\840\\0\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-90\\-630\\-420\\-84\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-2520\\-2100\\-560\\-840\\0\\280
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-233413320, 72762480, 683760; 72762480, -22682415, -213150; 683760, -213150, -2003],[840, 105, 1],[4, -2, -5, 21, 17, 119, 29, 101; 15, -9, -19, 94, 74, 508, 122, 420; -231, 275, 315, -2835, -2072, -13440, -3084, -10220],66,77]
%
% END OF LATTICE L67.77
%
\hbox{}\par\smallskip%
%
% LATTICE L67.78
%
\leavevmode%
\llap{\spacer}%
${L_{67.78}}=\pdualwitharg{3.5.7}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{*}{4}{420}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{280}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-85986600\\355320\\-11979240
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355320\\-1365\\49560
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-11979240\\49560\\-1668857
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1181\\-4727\\8337
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-624\\-2497\\4405
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-1770\\-7081\\12495
\end{tabular}\endgroup%
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119\\478\\-840
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118\\472\\-833
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-357\\-1436\\2520
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-439\\-1758\\3099
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5553\\-22232\\39200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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42\\0\\-210\\-630\\-63\\-420\\-21\\0
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-21\\-45\\-315\\-420\\-21\\0\\3\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-2520\\-4200\\-280\\-840\\0\\280
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-85986600, 355320, -11979240; 355320, -1365, 49560; -11979240, 49560, -1668857],[840, 105, 1],[-1181, -624, -1770, 119, 118, -357, -439, -5553; -4727, -2497, -7081, 478, 472, -1436, -1758, -22232; 8337, 4405, 12495, -840, -833, 2520, 3099, 39200],66,78]
%
% END OF LATTICE L67.78
%
\hbox{}\par\smallskip%
%
% LATTICE L67.79
%
\leavevmode%
\llap{\spacer}%
${L_{67.79}}=\pdualwitharg{2.3.5}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{560}\above{1pt}{1pt}{s}{2}{240}\above{1pt}{1pt}{*}{4}{120}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-220920\\28560\\-9240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28560\\-3000\\960
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9240\\960\\-307
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\-171\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-189\\-560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\163\\480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\182\\540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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0\\5\\16
\end{tabular}\endgroup%
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-1\\-154\\-450
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-6\\-973\\-2856
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-1\\-165\\-485
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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48\\0\\-240\\-360\\-72\\-120\\-168\\0
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\kern3pt%
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0\\560\\0\\-840\\-280\\-840\\-2520\\-280
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-120\\-840\\-600\\-240\\0\\60\\0\\-30
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-168\\-2520\\-2520\\-1680\\-168\\0\\168\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-360\\-300\\-40\\-30\\0\\5
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-220920, 28560, -9240; 28560, -3000, 960; -9240, 960, -307],[840, 120, 1],[-1, -1, 1, 1, 0, -1, -6, -1; -171, -189, 163, 182, 5, -154, -973, -165; -504, -560, 480, 540, 16, -450, -2856, -485],66,79]
%
% END OF LATTICE L67.79
%
\hbox{}\par\smallskip%
%
% LATTICE L67.80
%
\leavevmode%
\llap{\spacer}%
${L_{67.80}}=\pdualwitharg{2.3.5}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{s}{2}{560}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{}{4}{120}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-203280\\-1228920\\31080
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\kern3pt%
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-1228920\\-7285800\\184320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31080\\184320\\-4663
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\172\\6912
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\14\\560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-212\\-8520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\59\\2372
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\359\\14430
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160\\1603\\64428
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\472\\18970
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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48\\0\\-240\\-360\\-72\\-120\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
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-72\\-280\\-120\\0\\8\\0\\-168\\-80
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-120\\-840\\-600\\-240\\0\\60\\0\\-60
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\kern3pt%
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-168\\-2520\\-2520\\-1680\\-168\\0\\168\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-560\\-720\\-600\\-80\\-60\\0\\20
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-203280, -1228920, 31080; -1228920, -7285800, 184320; 31080, 184320, -4663],[840, 120, 1],[17, 1, -21, -1, 6, 36, 160, 47; 172, 14, -212, -15, 59, 359, 1603, 472; 6912, 560, -8520, -600, 2372, 14430, 64428, 18970],66,80]
%
% END OF LATTICE L67.80
%
\hbox{}\par\smallskip%
%
% LATTICE L67.81
%
\leavevmode%
\llap{\spacer}%
${L_{67.81}}=\pdualwitharg{2.3.7}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1680}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{336}\above{1pt}{1pt}{*}{4}{168}\above{1pt}{1pt}{l}{2}{280}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-17200680\\331800\\-4258800
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\kern3pt%
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331800\\-6384\\82152
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4258800\\82152\\-1054457
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-832\\-5\\3360
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\HardButStrongLineBreak\kern3pt%
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-2\\-1\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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208\\1\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\10\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-277\\15\\1120
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\HardButStrongLineBreak\kern3pt%
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-343\\8\\1386
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-227\\2\\917
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\0\\-1680\\-2520\\-2520\\-840\\-840\\0
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0\\16\\0\\-168\\-280\\-168\\-360\\-56
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\kern3pt%
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0\\-56\\-504\\-420\\-280\\-42\\0\\7
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17200680, 331800, -4258800; 331800, -6384, 82152; -4258800, 82152, -1054457],[840, 168, 1],[-832, -2, 208, 21, -277, -343, -1099, -227; -5, -1, 1, 10, 15, 8, 15, 2; 3360, 8, -840, -84, 1120, 1386, 4440, 917],66,81]
%
% END OF LATTICE L67.81
%
\hbox{}\par\smallskip%
%
% LATTICE L67.82
%
\leavevmode%
\llap{\spacer}%
${L_{67.82}}=\pdualwitharg{2.3.7}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1680}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{}{4}{168}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-233520\\-568680\\-269640
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-568680\\-1380792\\-654696
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-269640\\-654696\\-310421
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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21\\-2000\\4200
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\118\\-248
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-5\\480\\-1008
\end{tabular}\endgroup%
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11\\-1279\\2688
\end{tabular}\endgroup%
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26\\-2865\\6020
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20\\-2139\\4494
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52\\-5455\\11460
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19\\-1966\\4130
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1680\\0\\-1680\\-2520\\-2520\\-840\\-840\\0
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0\\16\\0\\-168\\-280\\-168\\-360\\-112
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\kern3pt%
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-840\\-168\\-840\\-336\\0\\84\\0\\-84
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-840\\-360\\-2520\\-1680\\-840\\0\\120\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-1008\\-840\\-560\\-84\\0\\28
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-233520, -568680, -269640; -568680, -1380792, -654696; -269640, -654696, -310421],[840, 168, 1],[21, -1, -5, 11, 26, 20, 52, 19; -2000, 118, 480, -1279, -2865, -2139, -5455, -1966; 4200, -248, -1008, 2688, 6020, 4494, 11460, 4130],66,82]
%
% END OF LATTICE L67.82
%
\hbox{}\par\smallskip%
%
% LATTICE L67.83
%
\leavevmode%
\llap{\spacer}%
${L_{67.83}}=\pdualwitharg{2.3.5.7}({L_{67.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{4}{840}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3542267400\\150256887060\\37209275880
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\kern3pt%
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150256887060\\6373638566400\\1578353448000
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37209275880\\1578353448000\\390859880249
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5129\\17147\\-68754
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82\\257\\-1030
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1721\\5709\\-22890
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30807\\103174\\-413700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11405\\38204\\-153188
\end{tabular}\endgroup%
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-20524\\68739\\-275625
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21499\\71984\\-288636
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39901\\133568\\-535570
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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84\\0\\-420\\-1260\\-252\\-210\\-84\\0
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\kern3pt%
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0\\20\\0\\-420\\-140\\-210\\-180\\-280
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\kern3pt%
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\kern3pt%
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-84\\-180\\-1260\\-1680\\-168\\0\\24\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-2520\\-4200\\-560\\-210\\0\\140
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3542267400, 150256887060, 37209275880; 150256887060, 6373638566400, 1578353448000; 37209275880, 1578353448000, 390859880249],[420, 420, 1],[-5129, -82, -1721, -30807, -11405, -20524, -21499, -39901; 17147, 257, 5709, 103174, 38204, 68739, 71984, 133568; -68754, -1030, -22890, -413700, -153188, -275625, -288636, -535570],66,83]
%
% END OF LATTICE L67.83
%
\hbox{}\par\smallskip%
%
% LATTICE L67.84
%
\leavevmode%
\llap{\spacer}%
${L_{67.84}}=\pdualwitharg{2.3.5.7}(\main({L_{67.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{420}\above{1pt}{1pt}{}{4}{840}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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388500\\16841160\\4168500
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16841160\\730054500\\180702060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4168500\\180702060\\44727119
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
329\\-964\\3864
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
146\\-439\\1760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
367\\-1152\\4620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\-209\\840
\end{tabular}\endgroup%
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3\\-7\\28
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319\\-891\\3570
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
369\\-1057\\4236
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
487\\-1406\\5635
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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84\\0\\-420\\-1260\\-252\\-420\\-84\\0
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0\\20\\0\\-420\\-140\\-420\\-180\\-140
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-252\\-140\\-420\\0\\56\\0\\-168\\-280
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-420\\-2100\\-1680\\0\\420\\0\\-210
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-180\\-1260\\-1680\\-168\\0\\24\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-1260\\-2100\\-280\\-210\\0\\35
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[388500, 16841160, 4168500; 16841160, 730054500, 180702060; 4168500, 180702060, 44727119],[420, 420, 1],[329, 146, 367, 47, 3, 319, 369, 487; -964, -439, -1152, -209, -7, -891, -1057, -1406; 3864, 1760, 4620, 840, 28, 3570, 4236, 5635],66,84]
%
% END OF LATTICE L67.84
%
\hbox{}\par\smallskip%
%
% LATTICE L67.85
%
\leavevmode%
\llap{\spacer}%
${L_{67.85}}=\pdualwitharg{2.5.7}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{s}{2}{560}\above{1pt}{1pt}{*}{4}{280}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-371280\\183960\\-26040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
183960\\-91000\\12880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26040\\12880\\-1823
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-36\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\105\\700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\99\\672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\91\\630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\35\\248
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\72\\525
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\0\\-560\\-840\\-504\\-280\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-2520\\-2100\\-840\\-210\\0\\105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-371280, 183960, -26040; 183960, -91000, 12880; -26040, 12880, -1823],[840, 280, 1],[-1, -1, 1, 3, 2, 1, 0, -1; -10, -36, 2, 105, 99, 91, 35, 72; -56, -240, 0, 700, 672, 630, 248, 525],66,85]
%
% END OF LATTICE L67.85
%
\hbox{}\par\smallskip%
%
% LATTICE L67.86
%
\leavevmode%
\llap{\spacer}%
${L_{67.86}}=\pdualwitharg{2.5.7}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{s}{2}{240}\above{1pt}{1pt}{*}{2}{560}\above{1pt}{1pt}{}{4}{280}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-212520\\-5822880\\48720
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5822880\\-159223400\\1332240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48720\\1332240\\-11147
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\87\\10472
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\1080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-21\\-107\\-12880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-28\\-3360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\69\\8316
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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34\\161\\19390
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22\\107\\12884
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\HardButStrongLineBreak\kern3pt%
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137\\675\\81270
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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112\\0\\-560\\-840\\-504\\-280\\-56\\0
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\kern3pt%
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-56\\-360\\-840\\-560\\-168\\0\\8\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1680\\-5040\\-4200\\-1680\\-420\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-212520, -5822880, 48720; -5822880, -159223400, 1332240; 48720, 1332240, -11147],[840, 280, 1],[17, 1, -21, -3, 16, 34, 22, 137; 87, 9, -107, -28, 69, 161, 107, 675; 10472, 1080, -12880, -3360, 8316, 19390, 12884, 81270],66,86]
%
% END OF LATTICE L67.86
%
\hbox{}\par\smallskip%
%
% LATTICE L67.87
%
\leavevmode%
\llap{\spacer}%
${L_{67.87}}=\pdualwitharg{2.3.5.7}({L_{67.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{s}{2}{1680}\above{1pt}{1pt}{*}{4}{840}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{r}{2}{420}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-31920\\318360\\15120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
318360\\-3165960\\-150360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15120\\-150360\\-7141
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\-32\\672
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\40\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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10\\21\\-420
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\HardButStrongLineBreak\kern3pt%
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3\\-5\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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8\\-49\\1050
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3\\-35\\744
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\HardButStrongLineBreak\kern3pt%
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2\\-38\\805
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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336\\0\\-1680\\-2520\\-504\\-840\\-168\\0
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0\\80\\0\\-840\\-280\\-840\\-360\\-280
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0\\-280\\-2520\\-2100\\-280\\-210\\0\\35
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-31920, 318360, 15120; 318360, -3165960, -150360; 15120, -150360, -7141],[840, 840, 1],[-1, -1, 1, 10, 3, 8, 3, 2; -32, -2, 40, 21, -5, -49, -35, -38; 672, 40, -840, -420, 112, 1050, 744, 805],66,87]
%
% END OF LATTICE L67.87
%
\hbox{}\par\smallskip%
%
% LATTICE L67.88
%
\leavevmode%
\llap{\spacer}%
${L_{67.88}}=\pdualwitharg{2.3.5.7}({L_{67.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{*}{2}{1680}\above{1pt}{1pt}{}{4}{840}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{s}{2}{420}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-31920\\-13088040\\32760
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\kern3pt%
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-13088040\\-5363079960\\13424040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32760\\13424040\\-33601
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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17\\-32\\-12768
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\800
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-21\\40\\15960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-10\\-21\\-8400
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\HardButStrongLineBreak\kern3pt%
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3\\-19\\-7588
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\HardButStrongLineBreak\kern3pt%
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27\\-91\\-36330
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\HardButStrongLineBreak\kern3pt%
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19\\-53\\-21156
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-104\\-41510
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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336\\0\\-1680\\-2520\\-504\\-840\\-168\\0
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\kern3pt%
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-504\\-280\\-840\\0\\56\\0\\-168\\-560
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\kern3pt%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-560\\-5040\\-4200\\-560\\-420\\0\\140
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-31920, -13088040, 32760; -13088040, -5363079960, 13424040; 32760, 13424040, -33601],[840, 840, 1],[17, 1, -21, -10, 3, 27, 19, 41; -32, 2, 40, -21, -19, -91, -53, -104; -12768, 800, 15960, -8400, -7588, -36330, -21156, -41510],66,88]
%
% END OF LATTICE L67.88
%

\medskip%
%------------------------------WEYL-GROUP-W68--------------------------------
%
\leavevmode\llap{}%
$W_{68\phantom{0}}$%
\qquad\llap{176} lattices, $\chi=72$%
\hfill%
$2222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L68.1
%
\leavevmode%
\llap{\spacer}%
${L_{68.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{68},3,2}%
\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15952860\\39060\\3780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39060\\-48\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3780\\-21\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
145529\\42656460\\172986660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-387\\-113435\\-460014
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-7914\\-32095
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-15952860, 39060, 3780; 39060, -48, -21; 3780, -21, 2],[1260, 3, 1],[325, 363, 27, 47, 31, 26, 2, 11, -1, -3, 0, 7, 32, 109, 43, 1249; 95262, 106400, 7914, 13776, 9086, 7620, 586, 3220, -294, -880, 0, 2052, 9380, 31950, 12604, 366100; 386316, 431490, 32095, 55872, 36855, 30915, 2379, 13090, -1188, -3570, -1, 8316, 38031, 129555, 51111, 1484630],67,1]
%
% END OF LATTICE L68.1
%
\hbox{}\par\smallskip%
%
% LATTICE L68.2
%
\leavevmode%
\llap{\spacer}%
${L_{68.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{3m,3,2}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[6749126720280, -24187683240, 83739600; -24187683240, 86684403, -300108; 83739600, -300108, 1039],[2520, 3, 1],[12529, 27543, 1983, 3181, 1805, 1141, 8, 1, -1, 297, 105, 995, 3067, 9299, 1732, 97439; 3497139, 7687900, 553502, 887892, 503818, 318480, 2233, 280, -279, 82900, 29308, 277728, 856072, 2595570, 483442, 27197520; 332955, 731940, 52696, 84528, 47964, 30330, 216, 280, 9, 7980, 2798, 26460, 81522, 247140, 46029, 2589440],67,2]
%
% END OF LATTICE L68.2
%
\hbox{}\par\smallskip%
%
% LATTICE L68.3
%
\leavevmode%
\llap{\spacer}%
${L_{68.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{32\rightarrow N'_{32},3,m}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[9024120, -22680, 0; -22680, 57, 0; 0, 0, 1],[2520, 3, 1],[-613, -1361, -50, -169, -53, -41, -5, -27, 1, 1, -1, -35, -66, -214, -165, -4733; -244659, -543200, -19956, -67452, -21154, -16365, -1996, -10780, 399, 400, -399, -13968, -26341, -85410, -65854, -1889020; -4221, -9360, -343, -1152, -357, -270, -30, -140, 9, 0, -8, -252, -462, -1485, -1140, -32620],67,3]
%
% END OF LATTICE L68.3
%
\hbox{}\par\smallskip%
%
% LATTICE L68.4
%
\leavevmode%
\llap{\spacer}%
${L_{68.4}}=\pfillwitharg{2.3}({L_{68.1}})$%
{${}={}$}Nikulin $68$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\0\\1
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{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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15469\\-111930\\-16380
\end{tabular}\endgroup%
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2108\\-15253\\-2232
\end{tabular}\endgroup%
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204\\-1476\\-217
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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204\\-1476\\-217
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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682\\-4935\\-720
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\HardButStrongLineBreak\kern3pt%
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101\\-731\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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29\\-210\\-29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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113\\-819\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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31\\-225\\-25
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\HardButStrongLineBreak\kern3pt%
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7\\-51\\-3
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\HardButStrongLineBreak\kern3pt%
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24\\-175\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-840, -105, 0; -105, -13, 0; 0, 0, 1],[105, 1, 1],[204, 682, 101, 29, 113, 31, 7, 24, 0, -2, 1, 5, 125, 139, 163, 2356; -1476, -4935, -731, -210, -819, -225, -51, -175, 0, 15, -7, -36, -903, -1005, -1179, -17045; -217, -720, -105, -29, -105, -25, -3, 0, 1, 0, -3, -7, -147, -155, -177, -2520],67,4]
%
% END OF LATTICE L68.4
%
\hbox{}\par\smallskip%
%
% LATTICE L68.5
%
\leavevmode%
\llap{\spacer}%
${L_{68.5}}=\pfillwitharg{3.2}({L_{68.3}})$%
{${}={}$}Nikulin $32'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{30}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
191730\\18270\\-4200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18270\\1741\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4200\\-400\\93
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9799\\-98490\\19110
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
940\\-9448\\1833
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\1809\\-352
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\-1127\\211
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[191730, 18270, -4200; 18270, 1741, -400; -4200, -400, 93],[210, 1, 1],[112, 319, 16, 1, -2, 16, 35, 1353, 128, 481, 44, 39, 142, 84, 65, 1447; -1127, -3210, -161, -10, 21, -160, -351, -13580, -1285, -4830, -442, -392, -1428, -845, -654, -14560; 211, 600, 30, 2, 0, 35, 72, 2730, 257, 960, 87, 76, 273, 160, 123, 2730],67,5]
%
% END OF LATTICE L68.5
%
\hbox{}\par\smallskip%
%
% LATTICE L68.6
%
\leavevmode%
\llap{\spacer}%
${L_{68.6}}=\main(\pfillwitharg{3}({L_{68.2}}))$%
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{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-964740, -385140, 3360; -385140, -153754, 1341; 3360, 1341, -11],[420, 1, 1],[-366, -2599, -215, -155, -409, -161, -68, -613, -6, 119, 29, 31, -25, -149, -242, -8067; 921, 6540, 541, 390, 1029, 405, 171, 1540, 15, -300, -73, -78, 63, 375, 609, 20300; 475, 3360, 276, 196, 504, 190, 72, 420, -7, -240, -42, -40, 42, 200, 318, 10500],67,6]
%
% END OF LATTICE L68.6
%
\hbox{}\par\smallskip%
%
% LATTICE L68.7
%
\leavevmode%
\llap{\spacer}%
${L_{68.7}}=\pfillwitharg{3}({L_{68.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-10946460, 1680, 32760; 1680, 8, -15; 32760, -15, -86],[420, 1, 1],[71, 257, 22, 17, 50, 23, 13, 209, 13, 23, -1, -3, -1, 12, 22, 771; 26970, 97620, 8356, 6456, 18984, 8730, 4932, 79240, 4926, 8700, -382, -1140, -378, 4560, 8358, 292880; 22336, 80850, 6921, 5348, 15729, 7235, 4089, 65730, 4088, 7230, -315, -944, -315, 3775, 6921, 242550],67,7]
%
% END OF LATTICE L68.7
%
\hbox{}\par\smallskip%
%
% LATTICE L68.8
%
\leavevmode%
\llap{\spacer}%
${L_{68.8}}=\pfillwitharg{3}({L_{68.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-65786280, 13161120, 101640; 13161120, -2632997, -20334; 101640, -20334, -157],[840, 1, 1],[27, 11, -21, -29, 17, 139, 116, 7837, 357, 2549, 213, 157, 431, 181, 44, 1123; 135, 60, -104, -144, 84, 690, 576, 38920, 1773, 12660, 1058, 780, 2142, 900, 219, 5600; -7, -660, -126, -124, 126, 620, 495, 32760, 1483, 10500, 864, 616, 1596, 610, 120, 1680],67,8]
%
% END OF LATTICE L68.8
%
\hbox{}\par\smallskip%
%
% LATTICE L68.9
%
\leavevmode%
\llap{\spacer}%
${L_{68.9}}=\pfillwitharg{3}({L_{68.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-588867720, 1475880, -159600; 1475880, -3699, 400; -159600, 400, -43],[840, 1, 1],[10, 43, 0, -3, -1, 6, 23, 821, 38, 277, 12, 19, 29, 14, 17, 299; 3993, 17160, -1, -1200, -399, 2400, 9198, 328300, 15195, 110760, 4798, 7596, 11592, 5595, 6792, 119420; 23, 0, -10, -28, 0, 55, 192, 6580, 301, 2160, 91, 136, 189, 80, 78, 980],67,9]
%
% END OF LATTICE L68.9
%
\hbox{}\par\smallskip%
%
% LATTICE L68.10
%
\leavevmode%
\llap{\spacer}%
${L_{68.10}}=\pfillwitharg{2}({L_{68.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[1780065, 49455, -3780; 49455, 1374, -105; -3780, -105, 8],[315, 3, 1],[397, 439, 64, 53, 64, 46, 2, 3, -1, 1, 2, 13, 90, 284, 108, 1537; -14112, -15605, -2275, -1884, -2275, -1635, -71, -105, 36, -35, -71, -462, -3199, -10095, -3839, -54635; 2376, 2625, 382, 315, 378, 270, 12, 35, 0, 15, 14, 81, 546, 1710, 648, 9205],67,10]
%
% END OF LATTICE L68.10
%
\hbox{}\par\smallskip%
%
% LATTICE L68.11
%
\leavevmode%
\llap{\spacer}%
${L_{68.11}}=\pfillwitharg{2}({L_{68.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-87570, -41580, 0; -41580, -19662, -9; 0, -9, 1],[630, 3, 1],[10, 19, 0, -37, -166, -562, -221, -6407, -1666, -1859, -138, -239, -156, -128, -9, -33; -21, -40, 0, 78, 350, 1185, 466, 13510, 3513, 3920, 291, 504, 329, 270, 19, 70; -180, -360, -1, 702, 3171, 10755, 4233, 122780, 31932, 35640, 2647, 4590, 3003, 2475, 177, 700],67,11]
%
% END OF LATTICE L68.11
%
\hbox{}\par\smallskip%
%
% LATTICE L68.12
%
\leavevmode%
\llap{\spacer}%
${L_{68.12}}=\main({L_{68.3}})$%
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{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[14888262060, 105590520, -888300; 105590520, 748869, -6300; -888300, -6300, 53],[1260, 3, 1],[469, 1039, 76, 127, 78, 58, 3, 13, -1, 1, 2, 29, 104, 332, 127, 3627; -65877, -145940, -10675, -17838, -10955, -8145, -421, -1820, 141, -140, -281, -4074, -14609, -46635, -17839, -509460; 29916, 66360, 4867, 8190, 5103, 3915, 237, 1540, 0, 120, 119, 1782, 6531, 21015, 8073, 231140],67,12]
%
% END OF LATTICE L68.12
%
\hbox{}\par\smallskip%
%
% LATTICE L68.13
%
\leavevmode%
\llap{\spacer}%
${L_{68.13}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{68.1}}))$%
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{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-422205, 0, 144795; 0, 24, -15; 144795, -15, -49648],[105, 3, 1],[-133441, -141767, -56904, -12293, -8136, 962, 582, -70977, -9371, -16913, -14502, -11509, -47402, -119972, -40252, -1595163; -243300, -258480, -103751, -22413, -14833, 1755, 1061, -129430, -17088, -30840, -26443, -20985, -86429, -218745, -73391, -2908430; -389088, -413365, -165921, -35844, -23723, 2805, 1697, -206955, -27324, -49315, -42285, -33558, -138215, -349815, -117367, -4651185],67,13]
%
% END OF LATTICE L68.13
%
\hbox{}\par\smallskip%
%
% LATTICE L68.14
%
\leavevmode%
\llap{\spacer}%
${L_{68.14}}=\pdualwitharg{2}(\pfillwitharg{3.2}({L_{68.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-3780\\-1818\\-1690
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12390, -3780, 6510; -3780, -1818, -1690; 6510, -1690, 3391],[210, 2, 1],[1329, 1196, -94, -149, -104, 1193, 2192, 38447, 7083, 12824, 2197, 850, 5011, 2312, 1313, 10623; -267, -240, 19, 30, 21, -240, -441, -7735, -1425, -2580, -442, -171, -1008, -465, -264, -2135; -2684, -2415, 190, 301, 210, -2410, -4428, -77665, -14308, -25905, -4438, -1717, -10122, -4670, -2652, -21455],67,14]
%
% END OF LATTICE L68.14
%
\hbox{}\par\smallskip%
%
% LATTICE L68.15
%
\leavevmode%
\llap{\spacer}%
${L_{68.15}}=\pdualwitharg{5}(\pfillwitharg{2.3}({L_{68.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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6773\\-22413\\-35590
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13448\\-44499\\-70665
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-222495, -1050, -41685; -1050, 40, -225; -41685, -225, -7792],[105, 5, 1],[73521, 46865, 31352, 6773, 13448, -106, -962, 7821, 5163, 5591, 7990, 6341, 78350, 13220, 66532, 175775; -243300, -155088, -103751, -22413, -44499, 351, 3183, -25886, -17088, -18504, -26443, -20985, -259287, -43749, -220173, -581686; -386330, -246261, -164745, -35590, -70665, 557, 5055, -41097, -27130, -29379, -41985, -33320, -411705, -69467, -349605, -923643],67,15]
%
% END OF LATTICE L68.15
%
\hbox{}\par\smallskip%
%
% LATTICE L68.16
%
\leavevmode%
\llap{\spacer}%
${L_{68.16}}=\pdualwitharg{3}(\pfillwitharg{3.2}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-73290, 15330, -25620; 15330, -2727, 5310; -25620, 5310, -8951],[210, 3, 1],[874, 529, -59, -193, -23, 773, 474, 49909, 4598, 5551, 1427, 1105, 1087, 1507, 286, 13931; -267, -160, 19, 60, 7, -240, -147, -15470, -1425, -1720, -442, -342, -336, -465, -88, -4270; -2661, -1610, 180, 588, 70, -2355, -1444, -152040, -14007, -16910, -4347, -3366, -3311, -4590, -871, -42420],67,16]
%
% END OF LATTICE L68.16
%
\hbox{}\par\smallskip%
%
% LATTICE L68.17
%
\leavevmode%
\llap{\spacer}%
${L_{68.17}}=\pdualwitharg{7}(\pfillwitharg{2.3}({L_{68.1}}))$%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-97545, -2100, -11655; -2100, 56, -315; -11655, -315, -1352],[105, 7, 1],[51051, 162709, 21770, 4703, 1334, -368, -668, 3879, 3585, 19411, 5548, 4403, 7772, 45898, 46198, 87181; -243300, -775440, -103751, -22413, -6357, 1755, 3183, -18490, -17088, -92520, -26443, -20985, -37041, -218745, -220173, -415490; -383572, -1222515, -163569, -35336, -10023, 2765, 5019, -29145, -26936, -145845, -41685, -33082, -58395, -344855, -347109, -655035],67,17]
%
% END OF LATTICE L68.17
%
\hbox{}\par\smallskip%
%
% LATTICE L68.18
%
\leavevmode%
\llap{\spacer}%
${L_{68.18}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{68.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8190, 315, 1260; 315, -12, -45; 1260, -45, -89],[315, 3, 1],[28, 102, 53, 7, 43, 21, 13, 352, 8, 18, 1, -1, -1, 9, 17, 908; 835, 3040, 1578, 208, 1274, 620, 382, 10290, 233, 520, 24, -30, -28, 270, 508, 27090; -29, -105, -54, -7, -42, -20, -12, -315, -7, -15, 0, 1, 0, -10, -18, -945],67,18]
%
% END OF LATTICE L68.18
%
\hbox{}\par\smallskip%
%
% LATTICE L68.19
%
\leavevmode%
\llap{\spacer}%
${L_{68.19}}=\pdualwitharg{5}(\pfillwitharg{3.2}({L_{68.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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53550\\-4545\\-21220
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247170\\-21220\\-97919
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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462013\\120414\\1140090
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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85\\21\\210
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-623910, 53550, 247170; 53550, -4545, -21220; 247170, -21220, -97919],[210, 5, 1],[-1059, -391, 65, 227, 85, -182, -1678, -11791, -5433, -3937, -1688, -1309, -3872, -359, -1027, -3357; -267, -96, 19, 60, 21, -48, -441, -3094, -1425, -1032, -442, -342, -1008, -93, -264, -854; -2615, -966, 160, 560, 210, -449, -4140, -29092, -13405, -9714, -4165, -3230, -9555, -886, -2535, -8288],67,19]
%
% END OF LATTICE L68.19
%
\hbox{}\par\smallskip%
%
% LATTICE L68.20
%
\leavevmode%
\llap{\spacer}%
${L_{68.20}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{68.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-230580, 2100, 2100; 2100, -18, -21; 2100, -21, -16],[420, 3, 1],[-1, -1, 1, 7, 27, 88, 34, 2929, 253, 281, 62, 35, 22, 17, 1, 11; -68, -70, 67, 476, 1841, 6005, 2321, 199990, 17276, 19190, 4235, 2392, 1505, 1165, 69, 770; -42, -40, 42, 288, 1106, 3600, 1390, 119700, 10338, 11480, 2532, 1428, 896, 690, 40, 420],67,20]
%
% END OF LATTICE L68.20
%
\hbox{}\par\smallskip%
%
% LATTICE L68.21
%
\leavevmode%
\llap{\spacer}%
${L_{68.21}}=\pdualwitharg{2.3}(\pfillwitharg{3.2}({L_{68.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3375407070, -5937960, 1678000170; -5937960, 10446, -2951910; 1678000170, -2951910, 834176297],[210, 6, 1],[-21961, -10453, -3162, -88, 348, -2297, -1938, -116988, -22355, -14167, -7917, -3605, -8965, -16168, -4217, -141522; -1127, -535, -161, -5, 7, -160, -117, -6790, -1285, -805, -442, -196, -476, -845, -218, -7280; 44172, 21025, 6360, 177, -700, 4620, 3898, 235305, 44964, 28495, 15924, 7251, 18032, 32520, 8482, 284655],67,21]
%
% END OF LATTICE L68.21
%
\hbox{}\par\smallskip%
%
% LATTICE L68.22
%
\leavevmode%
\llap{\spacer}%
${L_{68.22}}=\pdualwitharg{3}(\main(\pfillwitharg{3}({L_{68.2}})))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24780, -9660, 2100; -9660, -3753, 831; 2100, 831, -166],[420, 3, 1],[11, 19, -23, -151, -575, -1867, -720, -61961, -2675, -5939, -1309, -737, -461, -353, -20, -199; -23, -40, 48, 316, 1204, 3910, 1508, 129780, 5603, 12440, 2742, 1544, 966, 740, 42, 420; 24, 40, -51, -330, -1253, -4065, -1567, -134820, -5820, -12920, -2847, -1602, -1001, -765, -43, -420],67,22]
%
% END OF LATTICE L68.22
%
\hbox{}\par\smallskip%
%
% LATTICE L68.23
%
\leavevmode%
\llap{\spacer}%
${L_{68.23}}=\pdualwitharg{7}(\pfillwitharg{3.2}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1319010, 91770, 562380; 91770, -6363, -39130; 562380, -39130, -239779],[210, 7, 1],[-1114, -2093, 61, 231, 13, -927, -1714, -8613, -5558, -20147, -1729, -1343, -569, -1853, -1066, -2507; -267, -480, 19, 60, 3, -240, -441, -2210, -1425, -5160, -442, -342, -144, -465, -264, -610; -2569, -4830, 140, 532, 30, -2135, -3948, -19840, -12803, -46410, -3983, -3094, -1311, -4270, -2457, -5780],67,23]
%
% END OF LATTICE L68.23
%
\hbox{}\par\smallskip%
%
% LATTICE L68.24
%
\leavevmode%
\llap{\spacer}%
${L_{68.24}}=\pdualwitharg{3.5}(\pfillwitharg{2.3}({L_{68.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[172794615, -9884700, -58645755; -9884700, 565455, 3354825; -58645755, 3354825, 19904119],[105, 15, 1],[843371, 179073, 358827, 77128, 50221, -1409, -3499, 93909, 61453, 21999, 93585, 73676, 301655, 152213, 254839, 2017347; -960, -203, -403, -84, -49, 3, 3, -133, -84, -29, -119, -90, -357, -177, -293, -2303; 2485080, 527657, 1057320, 227265, 147980, -4152, -10310, 276717, 181080, 64823, 275760, 217095, 888860, 448512, 750910, 5944323],67,24]
%
% END OF LATTICE L68.24
%
\hbox{}\par\smallskip%
%
% LATTICE L68.25
%
\leavevmode%
\llap{\spacer}%
${L_{68.25}}=\pdualwitharg{2}(\main(\pfillwitharg{3}({L_{68.2}})))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[317418360, -1430940, 79167480; -1430940, 6452, -356892; 79167480, -356892, 19745203],[420, 4, 1],[5644, 10187, 3491, 1387, 9047, 4937, 3601, 19524, 2984, 4193, 823, 51, 1019, 2253, 3589, 30806; -9219, -16635, -5698, -2262, -14742, -8040, -5862, -31780, -4857, -6825, -1340, -84, -1680, -3690, -5868, -50330; -22796, -41145, -14100, -5602, -36540, -19940, -14544, -78855, -12052, -16935, -3324, -206, -4116, -9100, -14496, -124425],67,25]
%
% END OF LATTICE L68.25
%
\hbox{}\par\smallskip%
%
% LATTICE L68.26
%
\leavevmode%
\llap{\spacer}%
${L_{68.26}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{68.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9251597040, -107772420, 2290588020; -107772420, 1255448, -26683200; 2290588020, -26683200, 567123001],[420, 4, 1],[19763, 72412, 12722, 2640, 36062, 20376, 15250, 167154, 12841, 36268, 3580, 77, 1976, 6794, 11920, 213226; 0, -15, -7, -3, -63, -45, -39, -455, -36, -105, -11, 0, 21, 15, 9, 35; -79822, -292470, -51384, -10663, -145656, -82300, -61596, -675150, -51866, -146490, -14460, -311, -7980, -27440, -48144, -861210],67,26]
%
% END OF LATTICE L68.26
%
\hbox{}\par\smallskip%
%
% LATTICE L68.27
%
\leavevmode%
\llap{\spacer}%
${L_{68.27}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{68.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-687732570, -7557480, 171990; -7557480, -83049, 1890; 171990, 1890, -43],[630, 3, 1],[9, 21, 1, -1, -1, 4, 8, 871, 27, 99, 26, 7, 22, 11, 7, 389; -817, -1910, -93, 90, 91, -360, -721, -78540, -2435, -8930, -2346, -632, -1988, -995, -634, -35280; 83, 30, -90, -44, 0, 175, 306, 31500, 961, 3450, 873, 218, 609, 260, 129, 5040],67,27]
%
% END OF LATTICE L68.27
%
\hbox{}\par\smallskip%
%
% LATTICE L68.28
%
\leavevmode%
\llap{\spacer}%
${L_{68.28}}=\pdualwitharg{5}(\main(\pfillwitharg{3}({L_{68.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4620, 1260, -840; 1260, 45, -40; -840, -40, 33],[420, 5, 1],[0, -1, -1, -1, 1, 1, 8, 53, 12, 17, 7, 5, 13, 1, 2, 3; -13, -84, -73, -70, 63, 67, 537, 3556, 805, 1140, 469, 334, 861, 65, 123, 140; -20, -132, -115, -110, 105, 107, 855, 5656, 1280, 1812, 745, 530, 1365, 103, 195, 224],67,28]
%
% END OF LATTICE L68.28
%
\hbox{}\par\smallskip%
%
% LATTICE L68.29
%
\leavevmode%
\llap{\spacer}%
${L_{68.29}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{68.1}}))$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-572460, 283920, 420; 283920, -140760, -155; 420, -155, 52],[420, 5, 1],[1359, 1043, 468, 311, 52, -155, -1291, -8465, -3819, -2683, -1083, -721, -1487, -50, 266, 2725; 2744, 2106, 945, 628, 105, -313, -2607, -17094, -7712, -5418, -2187, -1456, -3003, -101, 537, 5502; -2810, -2154, -965, -640, -105, 319, 2655, 17402, 7850, 5514, 2225, 1480, 3045, 101, -555, -5642],67,29]
%
% END OF LATTICE L68.29
%
\hbox{}\par\smallskip%
%
% LATTICE L68.30
%
\leavevmode%
\llap{\spacer}%
${L_{68.30}}=\pdualwitharg{2.5}(\pfillwitharg{3.2}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24030930, -646800, 11410980; -646800, 17410, -307130; 11410980, -307130, 5418453],[210, 10, 1],[-7191, -2050, -1030, -31, 200, -187, -2132, -8343, -7917, -2986, -2747, -1228, -9013, -1072, -4163, -9283; -1127, -321, -161, -5, 21, -32, -351, -1358, -1285, -483, -442, -196, -1428, -169, -654, -1456; 15080, 4299, 2160, 65, -420, 392, 4470, 17493, 16600, 6261, 5760, 2575, 18900, 2248, 8730, 19467],67,30]
%
% END OF LATTICE L68.30
%
\hbox{}\par\smallskip%
%
% LATTICE L68.31
%
\leavevmode%
\llap{\spacer}%
${L_{68.31}}=\pdualwitharg{3.7}(\pfillwitharg{2.3}({L_{68.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[96660165, -8747550, -32877915; -8747550, 791637, 2975385; -32877915, 2975385, 11183069],[105, 21, 1],[809507, 859143, 344067, 73786, 6811, -7213, -3295, 65589, 59893, 106857, 90633, 71114, 41489, 731713, 244795, 1383411; -960, -1015, -403, -84, -7, 15, 3, -95, -84, -145, -119, -90, -51, -885, -293, -1645; 2380182, 2526125, 1011654, 216951, 20026, -21210, -9688, 192855, 176106, 314195, 266490, 209097, 121990, 2151450, 719768, 4067625],67,31]
%
% END OF LATTICE L68.31
%
\hbox{}\par\smallskip%
%
% LATTICE L68.32
%
\leavevmode%
\llap{\spacer}%
${L_{68.32}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{68.2}}))$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[471240, -1680, -840; -1680, 6, 3; -840, 3, 1],[840, 3, 1],[0, -1, -1, -3, -9, -27, -5, -841, -36, -79, -17, -9, -5, -3, 0, 1; 1, -260, -262, -788, -2366, -7100, -1315, -221200, -9469, -20780, -4472, -2368, -1316, -790, 0, 280; 0, -40, -42, -132, -406, -1230, -229, -38640, -1656, -3640, -786, -420, -238, -150, -1, 0],67,32]
%
% END OF LATTICE L68.32
%
\hbox{}\par\smallskip%
%
% LATTICE L68.33
%
\leavevmode%
\llap{\spacer}%
${L_{68.33}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1073804760, 9111480, -487200; 9111480, -77313, 4134; -487200, 4134, -221],[840, 3, 1],[3, 1, -4, -39, -69, -219, -167, -14279, -615, -1361, -149, -165, -50, -36, -3, -1; 355, 120, -473, -4616, -8169, -25930, -19774, -1690780, -72823, -161160, -17644, -19540, -5922, -4265, -356, -140; 27, 40, -30, -372, -700, -2265, -1744, -149940, -6471, -14360, -1581, -1776, -553, -420, -46, -420],67,33]
%
% END OF LATTICE L68.33
%
\hbox{}\par\smallskip%
%
% LATTICE L68.34
%
\leavevmode%
\llap{\spacer}%
${L_{68.34}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{68.1}}))$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13020, -1680, 420; -1680, 168, -35; 420, -35, 6],[420, 7, 1],[-1, -11, -2, -3, -2, -9, -7, -21, -11, -29, -1, 1, 1, 4, 2, 1; -22, -240, -44, -68, -48, -230, -192, -620, -338, -960, -46, 8, 18, 80, 42, 20; -56, -630, -119, -196, -153, -805, -735, -2570, -1456, -4410, -259, -56, 27, 175, 105, 50],67,34]
%
% END OF LATTICE L68.34
%
\hbox{}\par\smallskip%
%
% LATTICE L68.35
%
\leavevmode%
\llap{\spacer}%
${L_{68.35}}=\pdualwitharg{2.7}(\pfillwitharg{3.2}({L_{68.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7501830, -427560, 3479490; -427560, 24374, -198310; 3479490, -198310, 1613853],[210, 14, 1],[-8233, -11731, -1178, -36, 28, -1113, -2494, -6936, -9203, -17329, -3181, -1417, -1481, -6152, -4771, -7594; -1127, -1605, -161, -5, 3, -160, -351, -970, -1285, -2415, -442, -196, -204, -845, -654, -1040; 17612, 25095, 2520, 77, -60, 2380, 5334, 14835, 19684, 37065, 6804, 3031, 3168, 13160, 10206, 16245],67,35]
%
% END OF LATTICE L68.35
%
\hbox{}\par\smallskip%
%
% LATTICE L68.36
%
\leavevmode%
\llap{\spacer}%
${L_{68.36}}=\pdualwitharg{7}(\main(\pfillwitharg{3}({L_{68.2}})))$%
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{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[420, 0, 0; 0, 7, 0; 0, 0, -1],[420, 7, 1],[0, 1, 0, -1, -2, -16, -19, -79, -24, -161, -12, -7, -2, -4, -1, -1; 1, 0, -3, -14, -21, -155, -177, -720, -217, -1440, -105, -58, -15, -25, -3, 0; 0, 0, -7, -42, -69, -525, -609, -2500, -756, -5040, -371, -210, -57, -105, -21, -20],67,36]
%
% END OF LATTICE L68.36
%
\hbox{}\par\smallskip%
%
% LATTICE L68.37
%
\leavevmode%
\llap{\spacer}%
${L_{68.37}}=\pdualwitharg{3.5}(\pfillwitharg{3.2}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20864970, 738150, 7414890; 738150, 26115, 262320; 7414890, 262320, 2635067],[210, 15, 1],[5024, 955, 720, 43, -50, 128, 491, 11559, 5488, 1381, 1908, 1709, 2094, 748, 969, 12969; -1127, -214, -161, -10, 7, -32, -117, -2716, -1285, -322, -442, -392, -476, -169, -218, -2912; -14025, -2666, -2010, -120, 140, -357, -1370, -32256, -15315, -3854, -5325, -4770, -5845, -2088, -2705, -36204],67,37]
%
% END OF LATTICE L68.37
%
\hbox{}\par\smallskip%
%
% LATTICE L68.38
%
\leavevmode%
\llap{\spacer}%
${L_{68.38}}=\pdualwitharg{5.7}(\pfillwitharg{2.3}({L_{68.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[156508590, 14369985, -22031310; 14369985, 1319395, -2022825; -22031310, -2022825, 3101291],[105, 35, 1],[551699, 351313, 234483, 50282, 13921, -985, -6733, 8945, 40837, 43711, 61785, 48474, 84835, 99741, 500513, 188567; -1656057, -1054548, -703852, -150930, -41784, 2958, 20208, -26854, -122595, -131220, -185474, -145512, -254658, -299400, -1502418, -566030; 2839060, 1807869, 1206660, 258755, 71640, -5068, -34650, 46029, 210140, 224931, 317940, 249445, 436560, 513268, 2575650, 970371],67,38]
%
% END OF LATTICE L68.38
%
\hbox{}\par\smallskip%
%
% LATTICE L68.39
%
\leavevmode%
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${L_{68.39}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{68.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[910445760, -871290, 453915630; -871290, 834, -434394; 453915630, -434394, 226306067],[630, 6, 1],[1247, 157, 1, 785, 8143, 28798, 11579, 170064, 88861, 49903, 7508, 6724, 9378, 8747, 936, 5146; -426, -50, 3, -249, -2653, -9435, -3803, -55930, -29238, -16430, -2475, -2223, -3115, -2925, -317, -1750; -2502, -315, -2, -1575, -16338, -57780, -23232, -341215, -178290, -100125, -15064, -13491, -18816, -17550, -1878, -10325],67,39]
%
% END OF LATTICE L68.39
%
\hbox{}\par\smallskip%
%
% LATTICE L68.40
%
\leavevmode%
\llap{\spacer}%
${L_{68.40}}=\pdualwitharg{3}(\main({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3567060, -511560, 6300; -511560, -73362, 903; 6300, 903, -11],[1260, 3, 1],[-34, -243, -61, -15, -41, -17, -8, -283, -2, 3, 7, 3, -1, -13, -22, -2237; 249, 1780, 447, 110, 301, 125, 59, 2100, 15, -20, -51, -22, 7, 95, 161, 16380; 961, 6900, 1746, 436, 1218, 520, 258, 10080, 83, 60, -180, -88, 0, 350, 612, 63000],67,40]
%
% END OF LATTICE L68.40
%
\hbox{}\par\smallskip%
%
% LATTICE L68.41
%
\leavevmode%
\llap{\spacer}%
${L_{68.41}}=\pdualwitharg{3}({L_{68.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-97711740, 45360, 91980; 45360, -12, -45; 91980, -45, -86],[1260, 3, 1],[27, 99, 26, 7, 22, 11, 7, 389, 9, 21, 1, -1, -1, 4, 8, 871; 6470, 23720, 6228, 1676, 5264, 2630, 1672, 92820, 2146, 5000, 234, -240, -238, 960, 1918, 208740; 25486, 93450, 24543, 6608, 20769, 10385, 6609, 367290, 8498, 19830, 945, -944, -945, 3775, 7551, 822150],67,41]
%
% END OF LATTICE L68.41
%
\hbox{}\par\smallskip%
%
% LATTICE L68.42
%
\leavevmode%
\llap{\spacer}%
${L_{68.42}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{68.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[166916610, 3393180, 83180790; 3393180, 76758, 1690920; 83180790, 1690920, 41452099],[630, 6, 1],[426829, 607133, 182344, 2040, -942, 69359, 143740, 8195858, 514703, 962087, 523805, 76421, 550169, 322946, 248565, 8280952; -3139, -4465, -1341, -15, 7, -510, -1057, -60270, -3785, -7075, -3852, -562, -4046, -2375, -1828, -60900; -856378, -1218135, -365850, -4093, 1890, -139160, -288396, -16443945, -1032686, -1930305, -1050948, -153329, -1103844, -647950, -498714, -16614675],67,42]
%
% END OF LATTICE L68.42
%
\hbox{}\par\smallskip%
%
% LATTICE L68.43
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1067640, 10080, -9240; 10080, 90, -85; -9240, -85, 79],[840, 5, 1],[12, 17, 7, 5, 13, 1, 1, 3, 0, -1, -1, -1, 1, 1, 4, 53; 1077, 1524, 626, 444, 1134, 84, 75, 112, -33, -132, -104, -96, 84, 90, 360, 4760; 2560, 3624, 1490, 1060, 2730, 206, 195, 448, -40, -264, -230, -220, 210, 214, 855, 11312],67,43]
%
% END OF LATTICE L68.43
%
\hbox{}\par\smallskip%
%
% LATTICE L68.44
%
\leavevmode%
\llap{\spacer}%
${L_{68.44}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{68.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-401073960, -6366360, 168840; -6366360, -101055, 2680; 168840, 2680, -71],[840, 5, 1],[-83, -119, -25, -37, -50, -4, -17, -29, -1, 7, 4, 9, 1, -3, -53, -363; 5307, 7608, 1598, 2364, 3192, 255, 1080, 1820, 57, -456, -257, -576, -63, 192, 3390, 23212; 2935, 4176, 865, 1240, 1575, 112, 330, -308, -235, -576, -190, -340, 0, 113, 1920, 12908],67,44]
%
% END OF LATTICE L68.44
%
\hbox{}\par\smallskip%
%
% LATTICE L68.45
%
\leavevmode%
\llap{\spacer}%
${L_{68.45}}=\pdualwitharg{3.7}(\pfillwitharg{3.2}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26434170, 983010, 9227400; 983010, 36561, 343140; 9227400, 343140, 3221017],[210, 21, 1],[11822, 11231, 1692, 103, -14, 1582, 1187, 19833, 13162, 16529, 4554, 4061, 708, 8828, 2283, 21807; -1127, -1070, -161, -10, 1, -160, -117, -1940, -1285, -1610, -442, -392, -68, -845, -218, -2080; -33747, -32060, -4830, -294, 40, -4515, -3388, -56610, -37569, -47180, -12999, -11592, -2021, -25200, -6517, -62250],67,45]
%
% END OF LATTICE L68.45
%
\hbox{}\par\smallskip%
%
% LATTICE L68.46
%
\leavevmode%
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${L_{68.46}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{68.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5864355, -977445, 28350; -977445, -162915, 4725; 28350, 4725, -137],[315, 15, 1],[1, 2, 0, -28, -260, -178, -352, -1024, -1333, -298, -222, -194, -258, -44, -18, -12; -6, -14, -1, 189, 1771, 1215, 2405, 7000, 9114, 2038, 1519, 1329, 1771, 303, 125, 84; 0, -69, -35, 720, 7245, 5049, 10065, 29407, 38340, 8589, 6425, 5670, 7665, 1341, 585, 413],67,46]
%
% END OF LATTICE L68.46
%
\hbox{}\par\smallskip%
%
% LATTICE L68.47
%
\leavevmode%
\llap{\spacer}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-484155, -6300, 2520; -6300, -15, 30; 2520, 30, -13],[315, 15, 1],[-2, -2, -7, -1, -1, 1, 9, 92, 14, 10, 25, 3, 15, 1, 1, -8; -25, -22, -69, -9, -7, 9, 79, 798, 121, 86, 213, 25, 119, 7, 1, -126; -460, -447, -1530, -215, -210, 214, 1920, 19593, 2980, 2127, 5310, 635, 3150, 206, 180, -1953],67,47]
%
% END OF LATTICE L68.47
%
\hbox{}\par\smallskip%
%
% LATTICE L68.48
%
\leavevmode%
\llap{\spacer}%
${L_{68.48}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{68.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7406280, -390180, 1832040; -390180, 20592, -96516; 1832040, -96516, 453179],[420, 12, 1],[52, 47, 395, 574, 8809, 28753, 11125, 479737, 41456, 46073, 20359, 2885, 7333, 5837, 405, 4523; 1, 0, 4, 7, 112, 370, 144, 6230, 539, 600, 266, 38, 98, 80, 6, 70; -210, -190, -1596, -2319, -35588, -116160, -44944, -1938090, -167478, -186130, -82248, -11655, -29624, -23580, -1636, -18270],67,48]
%
% END OF LATTICE L68.48
%
\hbox{}\par\smallskip%
%
% LATTICE L68.49
%
\leavevmode%
\llap{\spacer}%
${L_{68.49}}=\pdualwitharg{2.3}(\main(\pfillwitharg{3}({L_{68.2}})))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7979580, -259980, 2017260; -259980, 8472, -65724; 2017260, -65724, 509969],[420, 12, 1],[-3, 21, 154, 330, 2306, 7308, 2786, 59566, 10263, 5679, 4976, 1380, 1684, 1242, 64, 284; 1, -15, -115, -253, -1785, -5675, -2167, -46375, -7993, -4425, -3881, -1079, -1323, -985, -53, -245; 12, -85, -624, -1338, -9352, -29640, -11300, -241605, -41628, -23035, -20184, -5598, -6832, -5040, -260, -1155],67,49]
%
% END OF LATTICE L68.49
%
\hbox{}\par\smallskip%
%
% LATTICE L68.50
%
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${L_{68.50}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{68.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2702598360, 68685960, -371280; 68685960, -1745639, 9436; -371280, 9436, -51],[840, 7, 1],[557, 3721, 275, 157, 43, 79, 7, 1, -5, -41, 1, 27, 49, 381, 223, 1839; 21905, 146340, 10816, 6176, 1692, 3110, 276, 40, -197, -1620, 38, 1060, 1926, 14980, 8769, 72320; -2219, -13860, -882, -308, 6, 280, 105, 120, -49, -1260, -252, -448, -384, -2170, -1050, -7680],67,50]
%
% END OF LATTICE L68.50
%
\hbox{}\par\smallskip%
%
% LATTICE L68.51
%
\leavevmode%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[840, 0, 0; 0, -21, 7; 0, 7, -2],[840, 7, 1],[-24, -161, -6, -7, -1, -2, -1, -1, 0, 1, 0, -1, -1, -8, -19, -79; -434, -2880, -105, -116, -15, -25, -6, 0, 2, 0, -3, -28, -21, -155, -354, -1440; -2275, -15120, -553, -616, -81, -140, -42, -20, 7, 0, -14, -140, -108, -805, -1848, -7540],67,51]
%
% END OF LATTICE L68.51
%
\hbox{}\par\smallskip%
%
% LATTICE L68.52
%
\leavevmode%
\llap{\spacer}%
${L_{68.52}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{68.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5127780, 23100, 23100; 23100, -90, -105; 23100, -105, -104],[420, 15, 1],[-1, -1, 1, 19, 83, 56, 110, 1913, 829, 185, 206, 119, 78, 13, 5, 19; -12, -14, 11, 252, 1113, 753, 1481, 25774, 11172, 2494, 2779, 1608, 1057, 177, 69, 266; -210, -208, 210, 3960, 17290, 11664, 22910, 398412, 172650, 38528, 42900, 24780, 16240, 2706, 1040, 3948],67,52]
%
% END OF LATTICE L68.52
%
\hbox{}\par\smallskip%
%
% LATTICE L68.53
%
\leavevmode%
\llap{\spacer}%
${L_{68.53}}=\pdualwitharg{2.3.5}(\pfillwitharg{3.2}({L_{68.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-480309690, 3645600, -238822920; 3645600, -27270, 1812690; -238822920, 1812690, -118749191],[210, 30, 1],[100403, 6164, -6250, -10807, -2680, 17327, 53232, 560993, 516961, 62432, 160591, 62254, 122723, 34120, 32513, 159265; -267, -16, 19, 30, 7, -48, -147, -1547, -1425, -172, -442, -171, -336, -93, -88, -427; -201930, -12397, 12570, 21735, 5390, -34848, -107060, -1128267, -1039710, -125563, -322980, -125205, -246820, -68622, -65390, -320313],67,53]
%
% END OF LATTICE L68.53
%
\hbox{}\par\smallskip%
%
% LATTICE L68.54
%
\leavevmode%
\llap{\spacer}%
${L_{68.54}}=\pdualwitharg{3.5}(\main(\pfillwitharg{3}({L_{68.2}})))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4498620, 21840, -2227680; 21840, -90, 10815; -2227680, 10815, -1103129],[420, 15, 1],[52, 103, -104, -1961, -8562, -5776, -11345, -197293, -42748, -19079, -21244, -12271, -8042, -1340, -515, -1955; 1, 0, -3, -14, -49, -31, -59, -1008, -217, -96, -105, -58, -35, -5, -1, 0; -105, -208, 210, 3960, 17290, 11664, 22910, 398412, 86325, 38528, 42900, 24780, 16240, 2706, 1040, 3948],67,54]
%
% END OF LATTICE L68.54
%
\hbox{}\par\smallskip%
%
% LATTICE L68.55
%
\leavevmode%
\llap{\spacer}%
${L_{68.55}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{68.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5355, 945, 0; 945, 168, 0; 0, 0, -1],[315, 21, 1],[-1, 1, 1, 4, 3, 59, 21, 41, 73, 79, 11, 8, 1, 1, -1, -1; 6, -5, -6, -27, -22, -450, -164, -325, -582, -635, -90, -69, -10, -30, 4, 5; 0, 0, -7, -63, -69, -1575, -609, -1250, -2268, -2520, -371, -315, -57, -315, -21, -10],67,55]
%
% END OF LATTICE L68.55
%
\hbox{}\par\smallskip%
%
% LATTICE L68.56
%
\leavevmode%
\llap{\spacer}%
${L_{68.56}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{68.1}}))$%
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{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4095, -1260, 0; -1260, 168, 21; 0, 21, -2],[315, 21, 1],[-1, -3, 2, 3, 10, 76, 88, 541, 109, 363, 160, 15, 8, 14, 2, -1; -3, -10, 6, 10, 34, 260, 302, 1860, 375, 1250, 552, 52, 28, 50, 8, 0; -35, -105, 63, 98, 327, 2485, 2877, 17685, 3563, 11865, 5229, 490, 261, 455, 63, -45],67,56]
%
% END OF LATTICE L68.56
%
\hbox{}\par\smallskip%
%
% LATTICE L68.57
%
\leavevmode%
\llap{\spacer}%
${L_{68.57}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{68.3}}))$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14591640, 694680, -36120; 694680, -33072, 1720; -36120, 1720, -89],[840, 8, 1],[-158, -143, -49, -19, -113, -53, -31, -262, -34, -17, 1, 3, 1, -27, -49, -858; -3165, -2865, -982, -381, -2268, -1065, -624, -5285, -687, -345, 19, 60, 21, -540, -981, -17185; 2944, 2655, 904, 346, 2016, 920, 516, 4130, 512, 225, -40, -58, 0, 520, 924, 16030],67,57]
%
% END OF LATTICE L68.57
%
\hbox{}\par\smallskip%
%
% LATTICE L68.58
%
\leavevmode%
\llap{\spacer}%
${L_{68.58}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{68.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-330171240, 16515240, -196560; 16515240, -826096, 9832; -196560, 9832, -117],[840, 8, 1],[-158, -286, -49, -19, -113, -53, -31, -131, -34, -34, 1, 3, 1, -27, -49, -429; -3107, -5625, -964, -374, -2226, -1045, -612, -2590, -673, -675, 19, 59, 21, -530, -963, -8435; 4336, 7770, 1308, 490, 2772, 1220, 648, 2415, 560, 390, -84, -82, 84, 820, 1392, 11865],67,58]
%
% END OF LATTICE L68.58
%
\hbox{}\par\smallskip%
%
% LATTICE L68.59
%
\leavevmode%
\llap{\spacer}%
${L_{68.59}}=\pdualwitharg{5.7}(\pfillwitharg{3.2}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[237532470, 3804150, -94047030; 3804150, 60935, -1506190; -94047030, -1506190, 37236357],[210, 35, 1],[-54013, -30787, -7730, -471, 190, -1449, -16291, -18141, -60191, -45349, -20821, -18563, -9707, -8068, -31294, -19927; -1127, -642, -161, -10, 3, -32, -351, -388, -1285, -966, -442, -392, -204, -169, -654, -416; -136465, -77784, -19530, -1190, 480, -3661, -41160, -45834, -152075, -114576, -52605, -46900, -24525, -20384, -79065, -50346],67,59]
%
% END OF LATTICE L68.59
%
\hbox{}\par\smallskip%
%
% LATTICE L68.60
%
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${L_{68.60}}=\pdualwitharg{3}({L_{68.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-58360680, 26951400, 95760; 26951400, -12446358, -44223; 95760, -44223, -157],[2520, 3, 1],[63, 159, -67, -41, 13, 191, 164, 33659, 513, 3681, 931, 233, 659, 289, 76, 6661; 135, 340, -144, -88, 28, 410, 352, 72240, 1101, 7900, 1998, 500, 1414, 620, 163, 14280; 398, 1200, -306, -220, 42, 1010, 879, 181440, 2770, 19920, 5058, 1276, 3654, 1630, 441, 40320],67,60]
%
% END OF LATTICE L68.60
%
\hbox{}\par\smallskip%
%
% LATTICE L68.61
%
\leavevmode%
\llap{\spacer}%
${L_{68.61}}=\pdualwitharg{3}({L_{68.3}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-43704508680, 367264800, -1370880; 367264800, -3086259, 11520; -1370880, 11520, -43],[2520, 3, 1],[8, 41, 2, -1, -1, 2, 9, 1021, 16, 119, 16, 9, 15, 8, 11, 659; 953, 4880, 237, -120, -119, 240, 1078, 122220, 1915, 14240, 1914, 1076, 1792, 955, 1312, 78540; 263, 240, -270, -268, 0, 535, 1872, 192780, 2941, 21120, 2673, 1336, 1869, 800, 798, 31500],67,61]
%
% END OF LATTICE L68.61
%
\hbox{}\par\smallskip%
%
% LATTICE L68.62
%
\leavevmode%
\llap{\spacer}%
${L_{68.62}}=\pdualwitharg{2.5}(\main(\pfillwitharg{3}({L_{68.2}})))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26895540, -841680, 6657420; -841680, 26340, -208340; 6657420, -208340, 1647903],[420, 20, 1],[36, 13, 35, 47, 935, 197, 1129, 1614, 1392, 463, 679, 191, 731, 41, 61, 52; 43, 12, 11, -1, -21, -1, 9, 28, 29, 12, 25, 13, 105, 13, 51, 56; -140, -51, -140, -190, -3780, -796, -4560, -6517, -5620, -1869, -2740, -770, -2940, -164, -240, -203],67,62]
%
% END OF LATTICE L68.62
%
\hbox{}\par\smallskip%
%
% LATTICE L68.63
%
\leavevmode%
\llap{\spacer}%
${L_{68.63}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{68.1}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5659212720, 89830860, 1410226020; 89830860, 1425920, 22385060; 1410226020, 22385060, 351415917],[420, 20, 1],[9846, 5753, 3111, 130, -1, 413, 4651, 18457, 8790, 6671, 6207, 1423, 21743, 2693, 10879, 25027; 3739, 2187, 1187, 52, 63, 167, 1815, 7133, 3389, 2565, 2377, 542, 8253, 1021, 4125, 9499; -39750, -23226, -12560, -525, 0, -1668, -18780, -74522, -35490, -26934, -25060, -5745, -87780, -10872, -43920, -101038],67,63]
%
% END OF LATTICE L68.63
%
\hbox{}\par\smallskip%
%
% LATTICE L68.64
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[39060, 38220, 4200; 38220, 33054, 3507; 4200, 3507, 368],[420, 21, 1],[1129, 1307, 312, 209, 24, 179, 21, 51, -13, -67, -33, -23, 7, 286, 134, 1809; -5464, -6330, -1513, -1016, -117, -875, -103, -250, 64, 330, 163, 116, -33, -1375, -647, -8750; 39270, 45500, 10878, 7308, 842, 6300, 742, 1800, -462, -2380, -1176, -840, 236, 9870, 4648, 62880],67,64]
%
% END OF LATTICE L68.64
%
\hbox{}\par\smallskip%
%
% LATTICE L68.65
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2184405090, 9150540, -1086086610; 9150540, -38178, 4549650; -1086086610, 4549650, -540002461],[210, 42, 1],[137071, 42826, -7622, -14273, -532, 114539, 70568, 531833, 686357, 414634, 213479, 82892, 23405, 228556, 43797, 154357; -267, -80, 19, 30, 1, -240, -147, -1105, -1425, -860, -442, -171, -48, -465, -88, -305; -275688, -86135, 15330, 28707, 1070, -230370, -141932, -1069665, -1380456, -833945, -429366, -166719, -47074, -459690, -88088, -310455],67,65]
%
% END OF LATTICE L68.65
%
\hbox{}\par\smallskip%
%
% LATTICE L68.66
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2724540, 0, 19320; 0, 21, 0; 19320, 0, 137],[420, 21, 1],[684, 1519, 335, 189, 17, 93, 6, 17, 0, 1, 7, 39, 21, 477, 184, 2263; -217, -480, -105, -58, -5, -25, -1, 0, 1, 0, -3, -14, -7, -155, -59, -720; -96516, -214340, -47271, -26670, -2399, -13125, -847, -2400, 0, -140, -987, -5502, -2963, -67305, -25963, -319320],67,66]
%
% END OF LATTICE L68.66
%
\hbox{}\par\smallskip%
%
% LATTICE L68.67
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18079110, -151830, 34020; -151830, -1245, 285; 34020, 285, -64],[630, 15, 1],[613, 137, 51, 89, 59, 10, 4, 5, -1, -1, 0, 13, 60, 41, 81, 471; 8148, 1822, 679, 1188, 791, 135, 55, 70, -12, -14, -1, 168, 791, 543, 1075, 6258; 361845, 80874, 30110, 52560, 34860, 5913, 2370, 2968, -585, -594, -5, 7650, 35385, 24192, 47805, 278012],67,67]
%
% END OF LATTICE L68.67
%
\hbox{}\par\smallskip%
%
% LATTICE L68.68
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[169660890, 11256210, -116550; 11256210, 746745, -7740; -116550, -7740, 79],[630, 15, 1],[-413, -287, -340, -71, -120, 1, 59, 1339, 209, 151, 187, 37, 1, -18, -144, -2767; 5805, 4034, 4779, 998, 1687, -14, -829, -18816, -2937, -2122, -2628, -520, -14, 253, 2024, 38892; -40565, -28188, -33390, -6970, -11760, 103, 5820, 131922, 20585, 14868, 18405, 3640, 105, -1768, -14145, -271782],67,68]
%
% END OF LATTICE L68.68
%
\hbox{}\par\smallskip%
%
% LATTICE L68.69
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-61273590, -44163945, 15727635; -44163945, -31828755, 11334855; 15727635, 11334855, -4036568],[105, 105, 1],[-243300, -51696, -103751, -22413, -2119, 351, 1061, -3698, -17088, -6168, -26443, -20985, -12347, -43749, -73391, -83098; -18724777, -3978617, -7984890, -1724969, -163090, 27004, 81664, -284577, -1315019, -474671, -2035008, -1614997, -950232, -3366970, -5648274, -6395355; -53527950, -11373551, -22826160, -4931115, -466220, 77196, 233450, -813513, -3759210, -1356929, -5817420, -4616745, -2716400, -9625056, -16146550, -18282207],67,69]
%
% END OF LATTICE L68.69
%
\hbox{}\par\smallskip%
%
% LATTICE L68.70
%
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\llap{\spacer}%
${L_{68.70}}=\pdualwitharg{2.7}(\pfillwitharg{3}({L_{68.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L68.70
%
\hbox{}\par\smallskip%
%
% LATTICE L68.71
%
\leavevmode%
\llap{\spacer}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-42420, 0, -10500; 0, 28, 0; -10500, 0, -2599],[420, 28, 1],[0, -26, -7, 31, 139, 1139, 1367, 1429, 1740, 2926, 877, 259, 151, 311, 83, 21; 1, 0, -3, -7, -21, -155, -177, -180, -217, -360, -105, -29, -15, -25, -3, 0; 0, 105, 28, -126, -564, -4620, -5544, -5795, -7056, -11865, -3556, -1050, -612, -1260, -336, -85],67,71]
%
% END OF LATTICE L68.71
%
\hbox{}\par\smallskip%
%
% LATTICE L68.72
%
\leavevmode%
\llap{\spacer}%
${L_{68.72}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{68.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1595160, -4200, 8400; -4200, 30, 15; 8400, 15, -43],[840, 15, 1],[0, -1, -1, 9, 47, 33, 33, 1159, 252, 113, 127, 75, 51, 9, 2, 17; 1, -36, -38, 316, 1666, 1172, 1173, 41216, 8963, 4020, 4520, 2672, 1820, 322, 72, 616; 0, -208, -210, 1860, 9730, 6834, 6835, 240072, 52200, 23408, 26310, 15540, 10570, 1866, 415, 3528],67,72]
%
% END OF LATTICE L68.72
%
\hbox{}\par\smallskip%
%
% LATTICE L68.73
%
\leavevmode%
\llap{\spacer}%
${L_{68.73}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8997240, 21840, -4455360; 21840, -45, 10815; -4455360, 10815, -2206258],[840, 15, 1],[52, 103, -52, -1961, -4281, -2888, -11345, -197293, -42748, -19079, -10622, -12271, -4021, -670, -515, -1955; 2, 0, -3, -28, -49, -31, -118, -2016, -434, -192, -105, -116, -35, -5, -2, 0; -105, -208, 105, 3960, 8645, 5832, 22910, 398412, 86325, 38528, 21450, 24780, 8120, 1353, 1040, 3948],67,73]
%
% END OF LATTICE L68.73
%
\hbox{}\par\smallskip%
%
% LATTICE L68.74
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[804510, 7560, -5040; 7560, 21, -42; -5040, -42, 31],[630, 21, 1],[-467, -509, -36, -55, -4, -13, 1, 1, -1, -11, -3, -23, -9, -182, -66, -261; -9951, -10850, -768, -1176, -86, -285, 20, 20, -21, -230, -63, -486, -191, -3870, -1405, -5560; -89775, -97860, -6923, -10584, -771, -2520, 189, 190, -189, -2100, -574, -4410, -1728, -34965, -12684, -50170],67,74]
%
% END OF LATTICE L68.74
%
\hbox{}\par\smallskip%
%
% LATTICE L68.75
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8190, 2520, 0; 2520, 777, 0; 0, 0, -1],[630, 21, 1],[-109, -363, -80, -15, -4, -7, -1, 1, 1, 3, -1, -3, -5, -38, -44, -541; 375, 1250, 276, 52, 14, 25, 4, 0, -3, -10, 3, 10, 17, 130, 151, 1860; -749, -2520, -567, -112, -33, -70, -21, -90, -7, 0, 0, -14, -30, -245, -294, -3690],67,75]
%
% END OF LATTICE L68.75
%
\hbox{}\par\smallskip%
%
% LATTICE L68.76
%
\leavevmode%
\llap{\spacer}%
${L_{68.76}}=\pdualwitharg{5.7}(\pfillwitharg{3}({L_{68.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4620, -6720, 1260; -6720, 1610, -595; 1260, -595, 156],[420, 35, 1],[-11, -35, -29, -21, 17, 48, 340, 307, 959, 667, 266, 179, 62, 31, 55, 9; 32, 102, 85, 64, -45, -133, -951, -862, -2696, -1878, -751, -508, -177, -89, -159, -26; 210, 672, 560, 420, -300, -882, -6300, -5708, -17850, -12432, -4970, -3360, -1170, -588, -1050, -172],67,76]
%
% END OF LATTICE L68.76
%
\hbox{}\par\smallskip%
%
% LATTICE L68.77
%
\leavevmode%
\llap{\spacer}%
${L_{68.77}}=\pdualwitharg{2.5.7}(\pfillwitharg{3.2}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-323668590, 4546500, -154439040; 4546500, -63630, 2169370; -154439040, 2169370, -73690861],[210, 70, 1],[62589, 11753, -3440, -6496, -730, 10427, 96388, 48434, 312543, 113291, 97223, 37757, 31991, 20834, 59917, 14088; -267, -48, 19, 30, 3, -48, -441, -221, -1425, -516, -442, -171, -144, -93, -264, -61; -131180, -24633, 7210, 13615, 1530, -21854, -202020, -101513, -655060, -237447, -203770, -79135, -67050, -43666, -125580, -29527],67,77]
%
% END OF LATTICE L68.77
%
\hbox{}\par\smallskip%
%
% LATTICE L68.78
%
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${L_{68.78}}=\pdualwitharg{5.7}(\main(\pfillwitharg{3}({L_{68.2}})))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[571620, 0, 6720; 0, 35, 0; 6720, 0, 79],[420, 35, 1],[0, 1, 7, 37, 59, 89, 514, 421, 636, 847, 311, 175, 47, 17, 16, 3; 1, 0, -3, -14, -21, -31, -177, -144, -217, -288, -105, -58, -15, -5, -3, 0; 0, -84, -595, -3150, -5025, -7581, -43785, -35864, -54180, -72156, -26495, -14910, -4005, -1449, -1365, -256],67,78]
%
% END OF LATTICE L68.78
%
\hbox{}\par\smallskip%
%
% LATTICE L68.79
%
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${L_{68.79}}=\pdualwitharg{2}({L_{68.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[79258656960, -186733260, -19877780160; -186733260, 439944, 46832016; -19877780160, 46832016, 4985274281],[1260, 12, 1],[-434975, -485174, -71992, -31178, -82076, -69746, -5992, -28528, -2677, -1106, -950, -5293, -88122, -294964, -115578, -1673452; 1548, 1715, 251, 105, 259, 195, 11, 35, 0, 5, 7, 24, 343, 1095, 419, 5985; -1734390, -1934550, -287056, -124317, -327264, -278100, -23892, -113750, -10674, -4410, -3788, -21105, -351372, -1176120, -460848, -6672610],67,79]
%
% END OF LATTICE L68.79
%
\hbox{}\par\smallskip%
%
% LATTICE L68.80
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-660140460, -7907760, 165577860; -7907760, -9516, 1983240; 165577860, 1983240, -41530597],[1260, 12, 1],[8484128, 4776686, 1444087, 1304923, 1830147, 1684823, 163017, 293257, 316, -63106, -30013, 109151, 1469359, 5385547, 2193773, 16204273; 79989, 45035, 13615, 12303, 17255, 15885, 1537, 2765, 3, -595, -283, 1029, 13853, 50775, 20683, 152775; 33829092, 19046265, 5758064, 5203170, 7297416, 6717960, 650004, 1169315, 1260, -251625, -119672, 435222, 5858832, 21474000, 8747316, 64611925],67,80]
%
% END OF LATTICE L68.80
%
\hbox{}\par\smallskip%
%
% LATTICE L68.81
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[950047560, -2019780, -238174020; -2019780, 4308, 506352; -238174020, 506352, 59709499],[1260, 12, 1],[-10984, -19378, -19141, -2353, -14115, -7213, -5017, -80503, -4076, -5722, -3449, -157, -3455, -5337, -7533, -183047; -2105, -3715, -3672, -452, -2716, -1390, -968, -15540, -787, -1105, -666, -30, -658, -1020, -1442, -35070; -43796, -77265, -76320, -9382, -56280, -28760, -20004, -320985, -16252, -22815, -13752, -626, -13776, -21280, -30036, -729855],67,81]
%
% END OF LATTICE L68.81
%
\hbox{}\par\smallskip%
%
% LATTICE L68.82
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[92138034240, -223862940, -23107945140; -223862940, 543912, 56144160; -23107945140, 56144160, 5795403961],[1260, 12, 1],[-66893, -248838, -134410, -9679, -137546, -79944, -61058, -2025428, -52087, -147762, -44060, -236, -1264, -19206, -38092, -2138872; 35, 125, 63, 4, 49, 25, 17, 525, 13, 35, 9, 0, 7, 15, 23, 1155; -266722, -992190, -535932, -38593, -548436, -318760, -243456, -8075970, -207686, -589170, -175680, -941, -5040, -76580, -151884, -8528310],67,82]
%
% END OF LATTICE L68.82
%
\hbox{}\par\smallskip%
%
% LATTICE L68.83
%
\leavevmode%
\llap{\spacer}%
${L_{68.83}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{68.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5428920, 965160, -45360; 965160, 171591, -8064; -45360, -8064, 379],[840, 21, 1],[0, 33, 35, 111, 49, 1041, 194, 4679, 1404, 3087, 667, 357, 29, 129, 1, 1; 1, -100, -108, -344, -152, -3230, -602, -14520, -4357, -9580, -2070, -1108, -90, -400, -3, 0; 21, 1820, 1890, 5964, 2630, 55860, 10409, 251040, 75327, 165620, 35784, 19152, 1556, 6930, 56, 120],67,83]
%
% END OF LATTICE L68.83
%
\hbox{}\par\smallskip%
%
% LATTICE L68.84
%
\leavevmode%
\llap{\spacer}%
${L_{68.84}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[21325080, -5880, -76440; -5880, -63, 21; -76440, 21, 274],[840, 21, 1],[-7, 1, 14, 137, 35, 781, 597, 7303, 2203, 4879, 535, 595, 26, 134, 13, 17; 2, 0, -3, -28, -7, -155, -118, -1440, -434, -960, -105, -116, -5, -25, -2, 0; -1953, 280, 3906, 38220, 9764, 217875, 166544, 2037300, 614565, 1361080, 149247, 165984, 7253, 37380, 3626, 4740],67,84]
%
% END OF LATTICE L68.84
%
\hbox{}\par\smallskip%
%
% LATTICE L68.85
%
\leavevmode%
\llap{\spacer}%
${L_{68.85}}=\pdualwitharg{5}({L_{68.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36158220, -303660, 34020; -303660, -2490, 285; 34020, 285, -32],[1260, 15, 1],[613, 137, 51, 89, 59, 10, 4, 5, -1, -1, 0, 13, 60, 41, 81, 471; 8148, 1822, 679, 1188, 791, 135, 55, 70, -12, -14, -1, 168, 791, 543, 1075, 6258; 723690, 161748, 60220, 105120, 69720, 11826, 4740, 5936, -1170, -1188, -10, 15300, 70770, 48384, 95610, 556024],67,85]
%
% END OF LATTICE L68.85
%
\hbox{}\par\smallskip%
%
% LATTICE L68.86
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11344127130, -12667410, 5653996110; -12667410, 14160, -6313530; 5653996110, -6313530, 2817993103],[630, 30, 1],[4785526, 530978, 389749, 328597, 411021, 63499, 19707, 12982, 314, 782, 9071, 70223, 519559, 335321, 644993, 1848178; -6015, -667, -489, -411, -511, -78, -23, -14, 3, -1, -12, -90, -658, -423, -812, -2324; -9601650, -1065351, -781990, -659295, -824670, -127404, -39540, -26047, -630, -1569, -18200, -140895, -1042440, -672786, -1294110, -3708173],67,86]
%
% END OF LATTICE L68.86
%
\hbox{}\par\smallskip%
%
% LATTICE L68.87
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-66058020, 1787940, 7951860; 1787940, -48390, -215265; 7951860, -215265, -956687],[1260, 15, 1],[48631, 21757, 8114, 14221, 9496, 1628, 671, 863, -67, -173, -20, 1967, 9390, 6466, 12819, 74681; 1361451, 609100, 227157, 398130, 265853, 45579, 18787, 24164, -1875, -4844, -561, 55062, 262871, 181017, 358873, 2090732; 97875, 43788, 16330, 28620, 19110, 3276, 1350, 1736, -135, -348, -40, 3960, 18900, 13014, 25800, 150304],67,87]
%
% END OF LATTICE L68.87
%
\hbox{}\par\smallskip%
%
% LATTICE L68.88
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3302460, -1421280, -7560; -1421280, 611670, 3255; -7560, 3255, 17],[1260, 15, 1],[-8, -23, -49, -15, 3, 11, 92, 1849, 140, 199, 247, 59, 151, 11, 18, -1; -18, -52, -111, -34, 7, 25, 209, 4200, 318, 452, 561, 134, 343, 25, 41, 0; -115, -276, -540, -160, 0, 106, 900, 18144, 1375, 1956, 2430, 580, 1470, 104, 150, -504],67,88]
%
% END OF LATTICE L68.88
%
\hbox{}\par\smallskip%
%
% LATTICE L68.89
%
\leavevmode%
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${L_{68.89}}=\pdualwitharg{3.5}({L_{68.1}})$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10827180, 5382720, -18900; 5382720, -2675940, 9465; -18900, 9465, 32],[1260, 15, 1],[1021, 1011, 1756, 499, 248, -249, -2239, -45751, -6961, -4971, -6211, -1489, -3713, -246, -236, 4171; 2046, 2026, 3519, 1000, 497, -499, -4487, -91686, -13950, -9962, -12447, -2984, -7441, -493, -473, 8358; -2180, -2154, -3735, -1060, -525, 529, 4755, 97146, 14780, 10554, 13185, 3160, 7875, 521, 495, -8946],67,89]
%
% END OF LATTICE L68.89
%
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%
% LATTICE L68.90
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L68.90
%
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%
% LATTICE L68.91
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35449680, -578760, 52920; -578760, -9432, 864; 52920, 864, -79],[840, 24, 1],[-1, -1, 1, 5, 41, 136, 53, 1147, 397, 221, 98, 28, 36, 29, 2, 8; 2, 0, -3, -7, -49, -155, -59, -1260, -434, -240, -105, -29, -35, -25, -1, 0; -648, -670, 636, 3270, 26908, 89340, 34832, 754005, 261000, 145310, 64452, 18426, 23716, 19140, 1328, 5355],67,91]
%
% END OF LATTICE L68.91
%
\hbox{}\par\smallskip%
%
% LATTICE L68.92
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-37143120, 56280, 110880; 56280, -72, -168; 110880, -168, -331],[840, 24, 1],[-1, -1, 1, 8, 69, 232, 91, 3953, 685, 191, 170, 49, 64, 53, 4, 37; 2, 0, -3, -7, -49, -155, -59, -2520, -434, -120, -105, -29, -35, -25, -1, 0; -336, -335, 336, 2682, 23128, 77760, 30500, 1324890, 229584, 64015, 56976, 16422, 21448, 17760, 1340, 12390],67,92]
%
% END OF LATTICE L68.92
%
\hbox{}\par\smallskip%
%
% LATTICE L68.93
%
\leavevmode%
\llap{\spacer}%
${L_{68.93}}=\pdualwitharg{3.5.7}(\pfillwitharg{3.2}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-158448990, 3896760, 59638950; 3896760, -95445, -1466430; 59638950, -1466430, -22447429],[210, 105, 1],[42446, 5313, -2335, -8813, -165, 7073, 21794, 65707, 212002, 51231, 65947, 51221, 7233, 14131, 13546, 19109; -85159, -10658, 4689, 17686, 331, -14194, -43735, -131856, -425429, -102806, -132336, -102784, -14514, -28355, -27180, -38340; 118335, 14812, -6510, -24570, -460, 19719, 60760, 183186, 591045, 142828, 183855, 142800, 20165, 39396, 37765, 53274],67,93]
%
% END OF LATTICE L68.93
%
\hbox{}\par\smallskip%
%
% LATTICE L68.94
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[485551080, -4714500, 120209040; -4714500, 45840, -1167180; 120209040, -1167180, 29760439],[420, 60, 1],[52, 51, 5511, 7936, 120653, 78497, 151585, 1305933, 564032, 125301, 276531, 39071, 98713, 15517, 5105, 10263; 1, 0, 4, 7, 112, 74, 144, 1246, 539, 120, 266, 38, 98, 16, 6, 14; -210, -206, -22260, -32055, -487340, -317064, -612280, -5274906, -2278230, -506114, -1116960, -157815, -398720, -62676, -20620, -41454],67,94]
%
% END OF LATTICE L68.94
%
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%
% LATTICE L68.95
%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4484891880, 131460, 1117640580; 131460, 60, 32760; 1117640580, 32760, 278517409],[420, 60, 1],[0, 78, 16223, 46329, 351289, 228369, 440827, 1898473, 1639764, 182118, 803659, 226965, 286397, 44925, 14663, 14585; 1, 0, 4, 14, 112, 74, 144, 623, 539, 60, 266, 76, 98, 16, 6, 7; 0, -313, -65100, -185910, -1409660, -916404, -1768960, -7618233, -6580080, -730807, -3224940, -910770, -1149260, -180276, -58840, -58527],67,95]
%
% END OF LATTICE L68.95
%
\hbox{}\par\smallskip%
%
% LATTICE L68.96
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1609020, 113400, -10080; 113400, 7728, -693; -10080, -693, 62],[1260, 21, 1],[-1, -11, -3, -23, -9, -182, -66, -261, -467, -509, -36, -55, -4, -13, 1, 1; -42, -460, -126, -972, -382, -7740, -2810, -11120, -19902, -21700, -1536, -2352, -172, -570, 40, 40; -630, -6930, -1897, -14616, -5739, -116235, -42189, -166930, -298746, -325710, -23051, -35280, -2577, -8505, 609, 610],67,96]
%
% END OF LATTICE L68.96
%
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%
% LATTICE L68.97
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[591249330, -4267620, 294676200; -4267620, 30786, -2126964; 294676200, -2126964, 146865389],[630, 42, 1],[-751303, -415559, -60627, -50269, -8663, -43016, -1549, 466, 5275, 1099, -1542, -11900, -12060, -267829, -102066, -207696; 1479, 815, 118, 96, 16, 75, 2, 0, -3, 5, 5, 27, 25, 540, 203, 410; 1507464, 833805, 121646, 100863, 17382, 86310, 3108, -935, -10584, -2205, 3094, 23877, 24198, 537390, 204792, 416735],67,97]
%
% END OF LATTICE L68.97
%
\hbox{}\par\smallskip%
%
% LATTICE L68.98
%
\leavevmode%
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${L_{68.98}}=\pdualwitharg{7}(\main({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[210420, 10080, 0; 10080, 483, 0; 0, 0, -1],[1260, 21, 1],[-1, 1, 1, 11, 5, 107, 40, 161, 145, 319, 23, 37, 3, 13, 0, -1; 21, -20, -21, -234, -107, -2295, -859, -3460, -3117, -6860, -495, -798, -65, -285, -1, 20; 0, 0, -7, -126, -69, -1575, -609, -2500, -2268, -5040, -371, -630, -57, -315, -21, -20],67,98]
%
% END OF LATTICE L68.98
%
\hbox{}\par\smallskip%
%
% LATTICE L68.99
%
\leavevmode%
\llap{\spacer}%
${L_{68.99}}=\pdualwitharg{3.7}({L_{68.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16380, 2520, 0; 2520, 168, 21; 0, 21, -2],[1260, 21, 1],[1, 3, -1, -3, -5, -38, -44, -541, -109, -363, -80, -15, -4, -7, -1, 1; -6, -20, 6, 20, 34, 260, 302, 3720, 750, 2500, 552, 104, 28, 50, 8, 0; -70, -210, 63, 196, 327, 2485, 2877, 35370, 7126, 23730, 5229, 980, 261, 455, 63, -90],67,99]
%
% END OF LATTICE L68.99
%
\hbox{}\par\smallskip%
%
% LATTICE L68.100
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-151830, -5040, -75600; -5040, 546, -2520; -75600, -2520, -37643],[630, 42, 1],[-26811, -45377, -20663, -2097, -1289, -2896, -963, -2086, -321, 157, 314, -164, -972, -8409, -10342, -65744; 795, 1345, 612, 62, 38, 85, 28, 60, 9, -5, -9, 5, 29, 250, 307, 1950; 53788, 91035, 41454, 4207, 2586, 5810, 1932, 4185, 644, -315, -630, 329, 1950, 16870, 20748, 131895],67,100]
%
% END OF LATTICE L68.100
%
\hbox{}\par\smallskip%
%
% LATTICE L68.101
%
\leavevmode%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-777420, -272160, 1260; -272160, -95277, 441; 1260, 441, -2],[1260, 21, 1],[1, 7, -2, -7, -12, -92, -107, -1319, -133, -887, -196, -37, -10, -18, -3, -1; -3, -20, 6, 20, 34, 260, 302, 3720, 375, 2500, 552, 104, 28, 50, 8, 0; -35, 0, 63, -14, -93, -875, -1113, -14400, -1477, -10080, -2331, -490, -159, -385, -147, -720],67,101]
%
% END OF LATTICE L68.101
%
\hbox{}\par\smallskip%
%
% LATTICE L68.102
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-207480, -13440, 840; -13440, 70, 35; 840, 35, -3],[840, 35, 1],[-156, -203, -71, -35, -7, -1, 2, 1, 0, -5, -7, -17, -19, -25, -67, -105; -1189, -1548, -542, -268, -54, -8, 15, 8, 1, -36, -52, -128, -144, -190, -510, -800; -57960, -75432, -26390, -13020, -2610, -378, 735, 368, 0, -1848, -2590, -6300, -7050, -9282, -24885, -39008],67,102]
%
% END OF LATTICE L68.102
%
\hbox{}\par\smallskip%
%
% LATTICE L68.103
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17724840, -11760, -52920; -11760, -105, 35; -52920, 35, 158],[840, 35, 1],[2155, 2863, 523, 581, 76, 26, 37, 3, -7, 1, 14, 135, 103, 153, 1753, 1429; -434, -576, -105, -116, -15, -5, -6, 0, 2, 0, -3, -28, -21, -31, -354, -288; 721805, 958944, 175175, 194600, 25455, 8708, 12390, 1004, -2345, 336, 4690, 45220, 34500, 51247, 587160, 478636],67,103]
%
% END OF LATTICE L68.103
%
\hbox{}\par\smallskip%
%
% LATTICE L68.104
%
\leavevmode%
\llap{\spacer}%
${L_{68.104}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{68.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[315, 0, 0; 0, 6195, -315; 0, -315, 16],[315, 105, 1],[1, 0, -1, -7, -7, -31, -59, -24, -217, -48, -35, -29, -5, -5, -1, 0; 0, 1, 7, 48, 49, 219, 419, 171, 1548, 343, 251, 210, 37, 39, 11, 1; 0, 21, 140, 945, 960, 4284, 8190, 3341, 30240, 6699, 4900, 4095, 720, 756, 210, 19],67,104]
%
% END OF LATTICE L68.104
%
\hbox{}\par\smallskip%
%
% LATTICE L68.105
%
\leavevmode%
\llap{\spacer}%
${L_{68.105}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{68.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-561645, -31185, 13545; -31185, -1365, 630; 13545, 630, -286],[315, 105, 1],[-1, -2, 2, 10, 38, 60, 354, 440, 445, 298, 664, 64, 36, 14, 16, 4; -58, -119, 102, 581, 2218, 3506, 20696, 25731, 26026, 17431, 38850, 3747, 2110, 822, 944, 237; -175, -357, 315, 1750, 6675, 10549, 62265, 77409, 78295, 52437, 116865, 11270, 6345, 2471, 2835, 711],67,105]
%
% END OF LATTICE L68.105
%
\hbox{}\par\smallskip%
%
% LATTICE L68.106
%
\leavevmode%
\llap{\spacer}%
${L_{68.106}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{68.3}}))$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[840, 0, 0; 0, -120, 40; 0, 40, -13],[840, 40, 1],[-7, -1, -1, 0, 1, 0, -1, -5, -5, -1, -2, -1, -8, -1, -4, -9; -33, -6, -10, -3, 0, 3, 24, 77, 69, 12, 19, 6, 21, 0, -9, -35; -160, -27, -40, -10, 0, 8, 60, 182, 160, 27, 40, 10, 0, -8, -60, -182],67,106]
%
% END OF LATTICE L68.106
%
\hbox{}\par\smallskip%
%
% LATTICE L68.107
%
\leavevmode%
\llap{\spacer}%
${L_{68.107}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{68.2}}))$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31929240, 618240, -76440; 618240, 11960, -1480; -76440, -1480, 183],[840, 40, 1],[7, 1, -1, -1, 1, 2, 17, 29, 53, 19, 16, 6, 34, 3, 8, 6; -19, -9, -10, -4, 0, 5, 42, 70, 127, 45, 37, 13, 63, 4, 3, -7; 2760, 342, -500, -450, 420, 876, 7440, 12677, 23160, 8298, 6980, 2610, 14700, 1284, 3360, 2443],67,107]
%
% END OF LATTICE L68.107
%
\hbox{}\par\smallskip%
%
% LATTICE L68.108
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[667639560, 20882820, 165117960; 20882820, 653184, 5164656; 165117960, 5164656, 40836317],[420, 84, 1],[-40052, -44137, -19165, -2594, -857, -3863, -35, 439, 344, 17, -689, -715, -1349, -29227, -10995, -66619; 6267, 6910, 3004, 408, 136, 630, 12, -40, -39, 10, 110, 111, 210, 4560, 1718, 10420; 161154, 177590, 77112, 10437, 3448, 15540, 140, -1770, -1386, -70, 2772, 2877, 5428, 117600, 44240, 268050],67,108]
%
% END OF LATTICE L68.108
%
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%
% LATTICE L68.109
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-302535660, 0, -74933460; 0, 84, 0; -74933460, 0, -18559873],[420, 84, 1],[281124, 157437, 141311, 41559, 8023, 49101, 4279, 3236, 0, -1257, -749, 5295, 7595, 185169, 73811, 231034; -217, -120, -105, -29, -5, -25, -1, 0, 1, 0, -3, -7, -7, -155, -59, -180; -1135008, -635635, -570528, -167790, -32392, -198240, -17276, -13065, 0, 5075, 3024, -21378, -30664, -747600, -298004, -932775],67,109]
%
% END OF LATTICE L68.109
%
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%
% LATTICE L68.110
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[69584760, 23158800, -27720; 23158800, 7707570, -9225; -27720, -9225, 11],[2520, 15, 1],[-1, 17, 17, 47, -37, -65, -82, -1065, -719, -337, -137, -287, -243, -55, -18, -55; 3, -52, -52, -144, 112, 198, 250, 3248, 2193, 1028, 418, 876, 742, 168, 55, 168; 0, -768, -770, -2340, 630, 2214, 2985, 39704, 27000, 12768, 5270, 11340, 9870, 2286, 765, 2296],67,110]
%
% END OF LATTICE L68.110
%
\hbox{}\par\smallskip%
%
% LATTICE L68.111
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[148611960, 50195880, -108360; 50195880, 16954395, -36600; -108360, -36600, 79],[2520, 15, 1],[11, 25, 47, 13, -9, -11, -44, -1739, -131, -185, -227, -53, -131, -9, -6, 59; -33, -76, -144, -40, 28, 34, 136, 5376, 405, 572, 702, 164, 406, 28, 19, -168; -205, -924, -2250, -700, 630, 664, 2655, 105336, 7945, 11244, 13860, 3280, 8400, 626, 570, 3024],67,111]
%
% END OF LATTICE L68.111
%
\hbox{}\par\smallskip%
%
% LATTICE L68.112
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1689022440, -20875680, 126884520; -20875680, -258015, 1567965; 126884520, 1567965, -9383494],[2520, 15, 1],[2042, 4979, 227, -58345, -138182, -94987, -376331, -2191499, -1426850, -638227, -118961, -416591, -138864, -23747, -19449, -24869; -167130, -407512, -18579, 4775316, 11309669, 7774323, 30801254, 179365816, 116782218, 52236440, 9736503, 34096380, 11365487, 1943601, 1591826, 2035432; -315, -768, -35, 9000, 21315, 14652, 58050, 338044, 220095, 98448, 18350, 64260, 21420, 3663, 3000, 3836],67,112]
%
% END OF LATTICE L68.112
%
\hbox{}\par\smallskip%
%
% LATTICE L68.113
%
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${L_{68.113}}=\pdualwitharg{3.5}({L_{68.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9273476520, -189254520, 811440; -189254520, -3862335, 16560; 811440, 16560, -71],[2520, 15, 1],[5, 3, -2, -3, -1, 1, 19, 403, 31, 45, 29, 15, 22, 2, 11, 101; -247, -152, 93, 144, 49, -48, -914, -19404, -1493, -2168, -1398, -724, -1064, -97, -536, -4956; -475, -1176, -1170, -700, 0, 233, 3960, 79884, 6055, 8616, 5355, 2560, 3255, 232, 690, -1764],67,113]
%
% END OF LATTICE L68.113
%
\hbox{}\par\smallskip%
%
% LATTICE L68.114
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-352380, -22260, 1680; -22260, -1260, 105; 1680, 105, -8],[420, 105, 1],[-1, -1, -2, 1, 2, 11, 23, 59, 181, 41, 47, 29, 3, 4, 2, 1; 0, -2, -7, -12, -3, -9, -13, -26, -72, -14, -11, 0, 1, 3, 3, 2; -210, -238, -525, 0, 355, 2079, 4445, 11526, 35490, 8078, 9345, 5880, 625, 861, 455, 234],67,114]
%
% END OF LATTICE L68.114
%
\hbox{}\par\smallskip%
%
% LATTICE L68.115
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31903955790, 12526433220, 6052157580; 12526433220, 4918246830, 2376255420; 6052157580, 2376255420, 1148089963],[210, 210, 1],[-1127, -107, -161, -5, 1, -32, -117, -194, -1285, -161, -442, -196, -68, -169, -218, -208; -615843, -58511, -88166, -2676, 756, -16335, -61526, -102920, -683217, -85829, -236599, -105579, -36841, -91920, -118899, -113590; 1280580, 121667, 183330, 5565, -1570, 33978, 127960, 214041, 1420860, 178493, 492030, 219555, 76610, 191142, 247240, 236199],67,115]
%
% END OF LATTICE L68.115
%
\hbox{}\par\smallskip%
%
% LATTICE L68.116
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5460, 5460, -2520; 5460, -5355, 2520; -2520, 2520, -1163],[420, 105, 1],[1, 13, 46, 85, 24, 80, 127, 275, 395, 163, 152, 47, -2, -14, -17, -11; 1, 0, -3, -14, -7, -31, -59, -144, -217, -96, -105, -58, -5, -5, -1, 0; 0, -28, -105, -210, -65, -231, -385, -864, -1260, -532, -525, -210, -5, 21, 35, 24],67,116]
%
% END OF LATTICE L68.116
%
\hbox{}\par\smallskip%
%
% LATTICE L68.117
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-351120, -17640, -4200; -17640, 1288, 392; -4200, 392, 117],[840, 56, 1],[133, 227, 35, 11, 7, 16, 5, 1, -1, -7, -2, 0, 4, 39, 50, 54; -18068, -30870, -4769, -1505, -963, -2215, -699, -140, 140, 990, 287, 11, -531, -5255, -6771, -7330; 65016, 111090, 17164, 5418, 3468, 7980, 2520, 505, -504, -3570, -1036, -42, 1908, 18900, 24360, 26375],67,117]
%
% END OF LATTICE L68.117
%
\hbox{}\par\smallskip%
%
% LATTICE L68.118
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1157520, 318360, 69720; 318360, -87528, -19208; 69720, -19208, -4167],[840, 56, 1],[-2783, -2174, -554, -97, 4, 181, 140, 57, -37, -176, -151, -138, -239, -1356, -1315, -2407; -8386, -6555, -1673, -295, 9, 535, 417, 170, -110, -525, -451, -413, -717, -4075, -3957, -7250; -7952, -6195, -1568, -266, 24, 560, 420, 170, -112, -525, -448, -406, -696, -3920, -3780, -6890],67,118]
%
% END OF LATTICE L68.118
%
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%
% LATTICE L68.119
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12279960, -42840, -47880; -42840, 21, 0; -47880, 0, -23],[2520, 21, 1],[-1, 1, 1, 11, 5, 107, 20, 161, 145, 319, 23, 37, 3, 13, 0, -1; -2037, 2060, 2044, 22464, 10208, 218430, 40826, 328640, 295977, 651140, 46946, 75516, 6122, 26520, -1, -2040; 2079, -2100, -2086, -22932, -10422, -223020, -41685, -335560, -302211, -664860, -47936, -77112, -6252, -27090, 0, 2080],67,119]
%
% END OF LATTICE L68.119
%
\hbox{}\par\smallskip%
%
% LATTICE L68.120
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1847160, 45360, -2520; 45360, -1113, 63; -2520, 63, -2],[2520, 21, 1],[-2, 1, 1, 25, 6, 131, 99, 401, 362, 799, 29, 95, 4, 19, 1, -1; -78, 40, 39, 972, 233, 5085, 3842, 15560, 14046, 31000, 1125, 3684, 155, 735, 38, -40; 63, 0, -35, -1008, -255, -5670, -4326, -17620, -15939, -35280, -1288, -4284, -186, -945, -84, -20],67,120]
%
% END OF LATTICE L68.120
%
\hbox{}\par\smallskip%
%
% LATTICE L68.121
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-77759640, -4445280, -108360; -4445280, -254058, -6195; -108360, -6195, -151],[2520, 21, 1],[-2, -7, 11, 21, 33, 247, 142, 3479, 350, 2327, 511, 95, 25, 43, 3, 1; 7, 20, -42, -76, -118, -880, -505, -12360, -1243, -8260, -1812, -336, -88, -150, -10, 0; 1148, 4200, -6174, -11956, -18846, -141190, -81207, -1990080, -200228, -1331400, -292446, -54404, -14334, -24710, -1743, -720],67,121]
%
% END OF LATTICE L68.121
%
\hbox{}\par\smallskip%
%
% LATTICE L68.122
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1378440, -35280, 0; -35280, 903, 0; 0, 0, -1],[2520, 21, 1],[1, 1, -1, -1, 0, 2, 7, 103, 11, 79, 10, 5, 1, 3, 3, 17; 39, 40, -39, -40, -1, 70, 254, 3780, 405, 2920, 372, 188, 38, 115, 116, 660; -7, 0, 0, -28, -30, -245, -588, -7380, -749, -5040, -567, -224, -33, -70, -42, -180],67,122]
%
% END OF LATTICE L68.122
%
\hbox{}\par\smallskip%
%
% LATTICE L68.123
%
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${L_{68.123}}=\pdualwitharg{2.5.7}(\pfillwitharg{3}({L_{68.1}}))$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L68.123
%
\hbox{}\par\smallskip%
%
% LATTICE L68.124
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L68.124
%
\hbox{}\par\smallskip%
%
% LATTICE L68.125
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17506440, -30240, 15120; -30240, 24, 24; 15120, 24, -13],[2520, 24, 1],[-1, -1, 0, 2, 18, 61, 24, 174, 181, 101, 15, 13, 17, 14, 1, 1; -33, -35, -1, 63, 581, 1980, 781, 5670, 5901, 3295, 490, 426, 560, 465, 34, 35; -1224, -1230, -4, 2430, 21924, 74340, 29256, 212135, 220680, 123150, 18292, 15858, 20748, 17100, 1224, 1225],67,125]
%
% END OF LATTICE L68.125
%
\hbox{}\par\smallskip%
%
% LATTICE L68.126
%
\leavevmode%
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${L_{68.126}}=\pdualwitharg{2}({L_{68.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-138481560, -861840, 115920; -861840, -5304, 720; 115920, 720, -97],[2520, 24, 1],[-1, -1, 0, 5, 46, 157, 62, 901, 469, 131, 39, 34, 45, 38, 3, 9; -33, -35, -1, 168, 1561, 5340, 2111, 30695, 15981, 4465, 1330, 1161, 1540, 1305, 104, 315; -1440, -1455, -8, 7218, 66528, 227160, 89724, 1304030, 678816, 189615, 56456, 49230, 65184, 55080, 4356, 13090],67,126]
%
% END OF LATTICE L68.126
%
\hbox{}\par\smallskip%
%
% LATTICE L68.127
%
\leavevmode%
\llap{\spacer}%
${L_{68.127}}=\pdualwitharg{2.3}({L_{68.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-552517560, 39465720, -221760; 39465720, -2818992, 15840; -221760, 15840, -89],[2520, 24, 1],[56, 51, 53, 7, 43, 21, 13, 352, 16, 9, 1, -1, -1, 9, 17, 908; 835, 760, 789, 104, 637, 310, 191, 5145, 233, 130, 12, -15, -14, 135, 254, 13545; 9064, 8175, 8352, 1066, 6216, 2840, 1596, 38430, 1592, 705, -360, -178, 0, 1600, 2844, 148050],67,127]
%
% END OF LATTICE L68.127
%
\hbox{}\par\smallskip%
%
% LATTICE L68.128
%
\leavevmode%
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${L_{68.128}}=\pdualwitharg{2.3}({L_{68.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-493776360, 32888520, -395640; 32888520, -2190576, 26352; -395640, 26352, -317],[2520, 24, 1],[56, 102, 53, 7, 43, 21, 13, 176, 16, 18, 1, -1, -1, 9, 17, 454; 893, 1625, 843, 111, 679, 330, 203, 2730, 247, 275, 12, -16, -14, 145, 272, 7245; 4336, 7770, 3924, 490, 2772, 1220, 648, 7245, 560, 390, -252, -82, 84, 820, 1392, 35595],67,128]
%
% END OF LATTICE L68.128
%
\hbox{}\par\smallskip%
%
% LATTICE L68.129
%
\leavevmode%
\llap{\spacer}%
${L_{68.129}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{68.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[197190, 57330, -12600; 57330, 12705, -2835; -12600, -2835, 632],[630, 105, 1],[109, 27, 12, 29, 4, 7, 5, 1, -1, -3, -3, -11, -1, 2, 10, 11; 5538, 1418, 661, 1704, 247, 447, 329, 66, -66, -202, -205, -792, -95, 9, 431, 542; 26775, 6846, 3185, 8190, 1185, 2142, 1575, 316, -315, -966, -980, -3780, -450, 63, 2100, 2624],67,129]
%
% END OF LATTICE L68.129
%
\hbox{}\par\smallskip%
%
% LATTICE L68.130
%
\leavevmode%
\llap{\spacer}%
${L_{68.130}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{68.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8190, 1260, 630; 1260, 105, 0; 630, 0, -17],[630, 105, 1],[205, 137, 152, 29, 8, 3, 3, 1, -1, -1, 1, 5, 9, 14, 82, 203; -1967, -1318, -1470, -284, -80, -31, -34, -12, 11, 14, 3, -42, -83, -132, -781, -1944; 6335, 4242, 4725, 910, 255, 98, 105, 36, -35, -42, 0, 140, 270, 427, 2520, 6264],67,130]
%
% END OF LATTICE L68.130
%
\hbox{}\par\smallskip%
%
% LATTICE L68.131
%
\leavevmode%
\llap{\spacer}%
${L_{68.131}}=\pdualwitharg{2.5}({L_{68.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[45376508520, -25334820, -11380262040; -25334820, 14160, 6353880; -11380262040, 6353880, 2854128013],[1260, 60, 1],[-2408062, -534373, -392241, -165349, -413649, -63905, -19833, -13065, -158, -787, -9129, -35336, -522881, -337465, -649117, -1859995; -6015, -1334, -978, -411, -1022, -156, -46, -28, 3, -2, -24, -90, -1316, -846, -1624, -4648; -9601650, -2130702, -1563980, -659295, -1649340, -254808, -79080, -52094, -630, -3138, -36400, -140895, -2084880, -1345572, -2588220, -7416346],67,131]
%
% END OF LATTICE L68.131
%
\hbox{}\par\smallskip%
%
% LATTICE L68.132
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3730585320, 2501645580, -929650680; 2501645580, 1677546660, -623402580; -929650680, -623402580, 231666163],[1260, 60, 1],[1781632, 197632, 289985, 244169, 304645, 46829, 14231, 4541, -316, 292, 6901, 52717, 388089, 250057, 480579, 688193; -24375, -2704, -3968, -3342, -4172, -642, -196, -63, 3, -4, -94, -720, -5306, -3420, -6574, -9415; 7083900, 785799, 1153000, 970830, 1211280, 186192, 56580, 18053, -1260, 1161, 27440, 209610, 1543080, 994248, 1910820, 2736307],67,132]
%
% END OF LATTICE L68.132
%
\hbox{}\par\smallskip%
%
% LATTICE L68.133
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[166881960, -2846340, -41936580; -2846340, 48180, 715260; -41936580, 715260, 10538447],[1260, 60, 1],[8151, 2273, 5959, -469, -10323, -1563, -6245, -18313, -4359, -1009, -271, 1101, 14747, 2195, 9405, 31585; -943, -263, -690, 54, 1190, 180, 718, 2100, 499, 115, 24, -128, -1708, -254, -1088, -3654; 32500, 9063, 23760, -1870, -41160, -6232, -24900, -73017, -17380, -4023, -1080, 4390, 58800, 8752, 37500, 125937],67,133]
%
% END OF LATTICE L68.133
%
\hbox{}\par\smallskip%
%
% LATTICE L68.134
%
\leavevmode%
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${L_{68.134}}=\pdualwitharg{2.3.5}({L_{68.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[138000250080, 2150392860, -34538050260; 2150392860, 33508560, -538190160; -34538050260, -538190160, 8644019957],[1260, 60, 1],[-38713, -22647, -36925, -558, -1475, -1903, -19787, -230071, -36311, -27369, -75611, -5694, -86053, -10601, -42733, -295097; 5006, 2929, 4779, 73, 217, 251, 2585, 29967, 4726, 3559, 9819, 738, 11137, 1371, 5525, 38157; -154370, -90306, -147240, -2225, -5880, -7588, -78900, -917406, -144790, -109134, -301500, -22705, -343140, -42272, -170400, -1176714],67,134]
%
% END OF LATTICE L68.134
%
\hbox{}\par\smallskip%
%
% LATTICE L68.135
%
\leavevmode%
\llap{\spacer}%
${L_{68.135}}=\pdualwitharg{3.5.7}(\pfillwitharg{3}({L_{68.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22680, -15960, 0; -15960, 210, 105; 0, 105, -1],[840, 105, 1],[0, -3, -7, 3, 7, 39, 41, 211, 324, 147, 169, 105, 11, 15, 4, 5; 1, -4, -10, 4, 10, 56, 59, 304, 467, 212, 244, 152, 16, 22, 6, 8; 0, -448, -1050, 420, 1030, 5754, 6055, 31176, 47880, 21728, 24990, 15540, 1630, 2226, 595, 744],67,135]
%
% END OF LATTICE L68.135
%
\hbox{}\par\smallskip%
%
% LATTICE L68.136
%
\leavevmode%
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${L_{68.136}}=\pdualwitharg{3.5.7}(\pfillwitharg{3}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1329720, -1353240, -5880; -1353240, 1376445, 5985; -5880, 5985, 26],[840, 105, 1],[9, -1, -17, -153, -38, -168, -639, -1559, -2349, -1039, -568, -627, -27, -27, -11, -1; 2, 0, -3, -28, -7, -31, -118, -288, -434, -192, -105, -116, -5, -5, -2, 0; 1575, -224, -3150, -28140, -6980, -30849, -117320, -286212, -431235, -190736, -104265, -115080, -4955, -4956, -2030, -228],67,136]
%
% END OF LATTICE L68.136
%
\hbox{}\par\smallskip%
%
% LATTICE L68.137
%
\leavevmode%
\llap{\spacer}%
${L_{68.137}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{68.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1680, 49560, -840; 49560, -1458360, 24720; -840, 24720, -419],[840, 120, 1],[-1, -1, 1, 17, 153, 104, 205, 893, 1549, 173, 386, 112, 148, 25, 10, 10; 2, 0, -3, -7, -49, -31, -59, -252, -434, -48, -105, -29, -35, -5, -1, 0; 120, 2, -180, -450, -3220, -2052, -3920, -16779, -28920, -3202, -7020, -1950, -2380, -348, -80, -21],67,137]
%
% END OF LATTICE L68.137
%
\hbox{}\par\smallskip%
%
% LATTICE L68.138
%
\leavevmode%
\llap{\spacer}%
${L_{68.138}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{68.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1680, -840, -1680; -840, -360, -840; -1680, -840, -1679],[840, 120, 1],[-1, -1, 1, 32, 293, 200, 395, 3445, 2989, 167, 746, 217, 288, 49, 20, 41; 2, 0, -3, -7, -49, -31, -59, -504, -434, -24, -105, -29, -35, -5, -1, 0; 0, 1, 0, -30, -280, -192, -380, -3318, -2880, -161, -720, -210, -280, -48, -20, -42],67,138]
%
% END OF LATTICE L68.138
%
\hbox{}\par\smallskip%
%
% LATTICE L68.139
%
\leavevmode%
\llap{\spacer}%
${L_{68.139}}=\pdualwitharg{2.7}({L_{68.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6293513520, -27839700, -1578405780; -27839700, 123144, 6982164; -1578405780, 6982164, 395862311],[1260, 84, 1],[-2623, -1106, 1524, 5894, 11952, 265466, 101172, 205884, 372379, 411946, 60102, 24919, 8590, 42664, 1538, -464; -33, 5, 33, 108, 211, 4635, 1757, 3565, 6441, 7115, 1035, 426, 145, 705, 23, -5; -10458, -4410, 6076, 23499, 47652, 1058400, 403368, 820850, 1484658, 1642410, 239624, 99351, 34248, 170100, 6132, -1850],67,139]
%
% END OF LATTICE L68.139
%
\hbox{}\par\smallskip%
%
% LATTICE L68.140
%
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${L_{68.140}}=\pdualwitharg{2.7}(\main({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L68.140
%
\hbox{}\par\smallskip%
%
% LATTICE L68.141
%
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\llap{\spacer}%
${L_{68.141}}=\pdualwitharg{2.3.7}({L_{68.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[60515280, -364140, -15174180; -364140, 2184, 91308; -15174180, 91308, 3804919],[1260, 84, 1],[165, -158, -316, 86, 1008, 8706, 10700, 67996, 13863, 46918, 21358, 1083, 1330, 2984, 990, 2144; 1, -5, -9, -1, -1, 5, 13, 105, 23, 85, 45, 3, 5, 15, 7, 15; 658, -630, -1260, 343, 4020, 34720, 42672, 271170, 55286, 187110, 85176, 4319, 5304, 11900, 3948, 8550],67,141]
%
% END OF LATTICE L68.141
%
\hbox{}\par\smallskip%
%
% LATTICE L68.142
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16421580, -124740, -4116420; -124740, 2184, 31248; -4116420, 31248, 1031869],[1260, 84, 1],[-1257, 79, 316, -2356, -10368, -85146, -102428, -321818, -130707, -220019, -198262, -19638, -11626, -24824, -7542, -8092; -84, 5, 21, -157, -691, -5675, -6827, -21450, -8712, -14665, -13215, -1309, -775, -1655, -503, -540; -5012, 315, 1260, -9394, -41340, -339500, -408408, -1283175, -521164, -877275, -790524, -78302, -46356, -98980, -30072, -32265],67,142]
%
% END OF LATTICE L68.142
%
\hbox{}\par\smallskip%
%
% LATTICE L68.143
%
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${L_{68.143}}=\pdualwitharg{5.7}({L_{68.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[394380, -467460, 23940; -467460, 485310, -24885; 23940, -24885, 1276],[1260, 105, 1],[109, 27, 12, 29, 4, 7, 5, 1, -1, -3, -3, -11, -1, 2, 10, 11; -7368, -1906, -901, -2364, -347, -633, -469, -94, 96, 290, 295, 1152, 145, 27, -541, -714; -145530, -37632, -17780, -46620, -6840, -12474, -9240, -1852, 1890, 5712, 5810, 22680, 2850, 504, -10710, -14108],67,143]
%
% END OF LATTICE L68.143
%
\hbox{}\par\smallskip%
%
% LATTICE L68.144
%
\leavevmode%
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${L_{68.144}}=\pdualwitharg{2.5.7}(\pfillwitharg{2}({L_{68.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2315011230, -162301230, 1153807830; -162301230, 11378640, -80891370; 1153807830, -80891370, 575060929],[630, 210, 1],[-656720, -72554, -52783, -43457, -7381, -7091, -941, 156, 5336, 178, -1499, -10825, -10713, -47191, -89529, -36344; 6687, 739, 538, 444, 76, 75, 14, 0, -27, 1, 17, 111, 109, 480, 911, 370; 1318590, 145677, 105980, 87255, 14820, 14238, 1890, -313, -10710, -357, 3010, 21735, 21510, 94752, 179760, 72973],67,144]
%
% END OF LATTICE L68.144
%
\hbox{}\par\smallskip%
%
% LATTICE L68.145
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5149620, 1716120, 20160; 1716120, 571935, 6720; 20160, 6720, 79],[1260, 105, 1],[-2125, -943, -346, -583, -52, -56, -17, -3, 1, -1, -8, -125, -66, -298, -573, -469; 8283, 3676, 1349, 2274, 203, 219, 67, 12, -3, 4, 31, 486, 257, 1161, 2233, 1828; -162540, -72156, -26495, -44730, -4005, -4347, -1365, -256, 0, -84, -595, -9450, -5025, -22743, -43785, -35864],67,145]
%
% END OF LATTICE L68.145
%
\hbox{}\par\smallskip%
%
% LATTICE L68.146
%
\leavevmode%
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${L_{68.146}}=\pdualwitharg{3.5.7}({L_{68.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-492660, 1260, 0; 1260, 3360, -1365; 0, -1365, 554],[1260, 105, 1],[-1, -1, 1, 5, 9, 14, 82, 203, 205, 137, 152, 29, 8, 3, 3, 1; -398, -392, 384, 1932, 3476, 5406, 31660, 78372, 79142, 52888, 58674, 11192, 3086, 1156, 1150, 372; -980, -966, 945, 4760, 8565, 13321, 78015, 193122, 195020, 130326, 144585, 27580, 7605, 2849, 2835, 918],67,146]
%
% END OF LATTICE L68.146
%
\hbox{}\par\smallskip%
%
% LATTICE L68.147
%
\leavevmode%
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${L_{68.147}}=\pdualwitharg{2.3.5.7}(\pfillwitharg{2}({L_{68.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6570270, 750960, 3274110; 750960, 85890, 374220; 3274110, 374220, 1631561],[630, 210, 1],[61441, 20377, 44539, 4095, 2121, 718, 523, 148, 35, 115, 1568, 1028, 3002, 4405, 25092, 30590; 493, 163, 354, 32, 16, 5, 2, 0, -1, 1, 15, 9, 25, 36, 203, 246; -123410, -40929, -89460, -8225, -4260, -1442, -1050, -297, -70, -231, -3150, -2065, -6030, -8848, -50400, -61443],67,147]
%
% END OF LATTICE L68.147
%
\hbox{}\par\smallskip%
%
% LATTICE L68.148
%
\leavevmode%
\llap{\spacer}%
${L_{68.148}}=\pdualwitharg{3.5.7}(\main({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16380, 20160, -5040; 20160, -12705, 1155; -5040, 1155, 554],[1260, 105, 1],[-36, -69, 65, 331, 595, 925, 5416, 13405, 6768, 9045, 10033, 1913, 527, 197, 194, 59; -73, -140, 132, 672, 1208, 1878, 10996, 27216, 13741, 18364, 20370, 3884, 1070, 400, 394, 120; -175, -336, 315, 1610, 2895, 4501, 26355, 65232, 32935, 44016, 48825, 9310, 2565, 959, 945, 288],67,148]
%
% END OF LATTICE L68.148
%
\hbox{}\par\smallskip%
%
% LATTICE L68.149
%
\leavevmode%
\llap{\spacer}%
${L_{68.149}}=\pdualwitharg{2.3.7}(\pfillwitharg{3}({L_{68.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38640, -19320, 1680; -19320, -9576, 840; 1680, 840, -73],[840, 168, 1],[-1, -3, -2, 2, 4, 103, 42, 133, 325, 183, 83, 25, 5, 32, 3, 2; 2, 0, -3, -7, -7, -155, -59, -180, -434, -240, -105, -29, -5, -25, -1, 0; 0, -70, -84, -42, 4, 420, 224, 795, 2016, 1190, 588, 210, 52, 420, 56, 45],67,149]
%
% END OF LATTICE L68.149
%
\hbox{}\par\smallskip%
%
% LATTICE L68.150
%
\leavevmode%
\llap{\spacer}%
${L_{68.150}}=\pdualwitharg{2.3.7}(\pfillwitharg{3}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2659440, -835800, 5880; -835800, -262584, 1848; 5880, 1848, -13],[840, 168, 1],[-1, -1, -2, -1, 0, 7, 4, 29, 37, 11, 11, 4, 1, 8, 1, 1; 2, 0, -3, -7, -7, -155, -59, -360, -434, -120, -105, -29, -5, -25, -1, 0; -168, -455, -1344, -1470, -1016, -19320, -6748, -39090, -46200, -12425, -10248, -2394, -272, 0, 308, 450],67,150]
%
% END OF LATTICE L68.150
%
\hbox{}\par\smallskip%
%
% LATTICE L68.151
%
\leavevmode%
\llap{\spacer}%
${L_{68.151}}=\pdualwitharg{2.3.5.7}(\pfillwitharg{3}({L_{68.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3360, -390180, -96600; -390180, 49759920, 12319440; -96600, 12319440, 3050017],[420, 420, 1],[1, -1, -17, -17, -33, -145, -275, -335, -1009, -223, -487, -67, -23, -23, -5, -1; 52, 357, 5199, 5173, 10091, 44441, 84397, 102897, 310016, 68547, 149835, 20666, 7135, 7237, 1733, 459; -210, -1442, -21000, -20895, -40760, -179508, -340900, -415626, -1252230, -276878, -605220, -83475, -28820, -29232, -7000, -1854],67,151]
%
% END OF LATTICE L68.151
%
\hbox{}\par\smallskip%
%
% LATTICE L68.152
%
\leavevmode%
\llap{\spacer}%
${L_{68.152}}=\pdualwitharg{2.3.5.7}(\main(\pfillwitharg{3}({L_{68.2}})))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[420, 1680, 420; 1680, 4650660, 1158780; 420, 1158780, 288727],[420, 420, 1],[1, 1, 25, 47, 45, 197, 373, 227, 1367, 151, 659, 181, 31, 31, 7, 1; 0, 75, 2093, 4029, 3877, 16995, 32197, 19600, 118044, 13041, 56929, 15645, 2681, 2679, 593, 74; 0, -301, -8400, -16170, -15560, -68208, -129220, -78663, -473760, -52339, -228480, -62790, -10760, -10752, -2380, -297],67,152]
%
% END OF LATTICE L68.152
%
\hbox{}\par\smallskip%
%
% LATTICE L68.153
%
\leavevmode%
\llap{\spacer}%
${L_{68.153}}=\pdualwitharg{2.5.7}(\pfillwitharg{3}({L_{68.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10653720, 13915440, -3564960; 13915440, -18175640, 4656400; -3564960, 4656400, -1192911],[840, 280, 1],[-95, 235, 419, -13, -977, -1900, -12187, -2635, -32461, -11087, -8558, -2700, -1736, -813, -1378, -78; -1, -3, -2, 8, 36, 59, 354, 74, 901, 303, 227, 67, 39, 16, 21, 1; 280, -714, -1260, 70, 3060, 5908, 37800, 8163, 100520, 34314, 26460, 8330, 5340, 2492, 4200, 237],67,153]
%
% END OF LATTICE L68.153
%
\hbox{}\par\smallskip%
%
% LATTICE L68.154
%
\leavevmode%
\llap{\spacer}%
${L_{68.154}}=\pdualwitharg{2.5.7}(\pfillwitharg{3}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9376080, -4688040, 23520; -4688040, -2343880, 11760; 23520, 11760, -59],[840, 280, 1],[-1, -1, -2, 3, 16, 27, 164, 69, 421, 71, 107, 32, 19, 8, 11, 1; 2, 0, -3, -7, -21, -31, -177, -72, -434, -72, -105, -29, -15, -5, -3, 0; 0, -399, -1400, -210, 2160, 4536, 29820, 13042, 80640, 13839, 21560, 6930, 4560, 2184, 3780, 398],67,154]
%
% END OF LATTICE L68.154
%
\hbox{}\par\smallskip%
%
% LATTICE L68.155
%
\leavevmode%
\llap{\spacer}%
${L_{68.155}}=\pdualwitharg{5.7}({L_{68.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[172476360, -57609720, -153720; -57609720, 19242510, 51345; -153720, 51345, 137],[2520, 105, 1],[1, -11, -15, -101, -35, -133, -116, -179, -793, -341, -117, -163, -9, 1, 6, 3; 3, -28, -38, -252, -86, -324, -281, -432, -1911, -820, -280, -384, -20, 6, 16, 8; 0, -1848, -2590, -18900, -7050, -27846, -24885, -39008, -173880, -75432, -26390, -39060, -2610, -1134, 735, 368],67,155]
%
% END OF LATTICE L68.155
%
\hbox{}\par\smallskip%
%
% LATTICE L68.156
%
\leavevmode%
\llap{\spacer}%
${L_{68.156}}=\pdualwitharg{5.7}({L_{68.3}})$%
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{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1419846120, -157646160, 473760; -157646160, 17503395, -52605; 473760, -52605, 158],[2520, 105, 1],[65, -1, -36, -1021, -257, -1141, -4349, -3541, -16013, -7087, -1293, -4295, -186, -188, -81, -3; 522, -8, -289, -8196, -2063, -9159, -34910, -28424, -128538, -56888, -10379, -34476, -1493, -1509, -650, -24; -21105, 336, 11725, 332640, 83745, 371826, 1417290, 1153996, 5218605, 2309664, 421400, 1399860, 60630, 61299, 26460, 1004],67,156]
%
% END OF LATTICE L68.156
%
\hbox{}\par\smallskip%
%
% LATTICE L68.157
%
\leavevmode%
\llap{\spacer}%
${L_{68.157}}=\pdualwitharg{3.5.7}({L_{68.2}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-139257720, -4974480, 141120; -4974480, -177555, 5040; 141120, 5040, -143],[2520, 105, 1],[0, -3, -7, 1, 7, 13, 41, 211, 108, 147, 169, 35, 11, 5, 4, 5; 1, -28, -72, 0, 52, 102, 328, 1704, 875, 1196, 1386, 292, 94, 44, 37, 48; 35, -3948, -9450, 980, 8730, 16408, 51975, 268056, 137305, 187068, 215460, 44800, 14160, 6482, 5250, 6624],67,157]
%
% END OF LATTICE L68.157
%
\hbox{}\par\smallskip%
%
% LATTICE L68.158
%
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${L_{68.158}}=\pdualwitharg{3.5.7}({L_{68.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1434704040, 628052040, -1428840; 628052040, 274934310, -625485; -1428840, -625485, 1423],[2520, 105, 1],[8, 3, -1, 13, 16, 27, 329, 833, 424, 573, 325, 131, 20, 9, 31, 31; -17, -8, 3, -16, -23, -40, -494, -1260, -643, -872, -498, -204, -32, -15, -56, -60; 560, -504, 315, 6020, 5955, 9527, 113190, 282528, 143080, 192024, 107415, 41860, 6015, 2443, 6510, 4752],67,158]
%
% END OF LATTICE L68.158
%
\hbox{}\par\smallskip%
%
% LATTICE L68.159
%
\leavevmode%
\llap{\spacer}%
${L_{68.159}}=\pdualwitharg{2.5}({L_{68.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-93829680, 11221560, -113400; 11221560, -1341960, 13560; -113400, 13560, -137],[2520, 120, 1],[-1, -1, 0, 14, 130, 89, 176, 256, 1333, 149, 111, 97, 129, 22, 9, 3; -12, -14, -1, 189, 1771, 1215, 2405, 3500, 18228, 2038, 1519, 1329, 1771, 303, 125, 42; -360, -558, -100, 7110, 67620, 46548, 92280, 134407, 700200, 78318, 58420, 51210, 68460, 11772, 4920, 1673],67,159]
%
% END OF LATTICE L68.159
%
\hbox{}\par\smallskip%
%
% LATTICE L68.160
%
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${L_{68.160}}=\pdualwitharg{2.5}({L_{68.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-398648880, -37464840, 471240; -37464840, -3519960, 44280; 471240, 44280, -557],[2520, 120, 1],[-1, -1, 0, 29, 270, 185, 366, 1065, 2773, 155, 231, 202, 269, 46, 19, 13; -12, -14, -1, 399, 3731, 2559, 5065, 14742, 38388, 2146, 3199, 2799, 3731, 639, 265, 182; -1800, -1959, -80, 56250, 525000, 359928, 712260, 2072854, 5397480, 301719, 449720, 393390, 524160, 89712, 37140, 25466],67,160]
%
% END OF LATTICE L68.160
%
\hbox{}\par\smallskip%
%
% LATTICE L68.161
%
\leavevmode%
\llap{\spacer}%
${L_{68.161}}=\pdualwitharg{2.3.5}({L_{68.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-39536280, 5647320, -22680; 5647320, -806640, 3240; -22680, 3240, -13],[2520, 120, 1],[-28, -5, -25, -3, -15, -1, -1, 8, 4, 1, 7, 1, 1, -1, -9, -92; -173, -31, -156, -19, -98, -7, -10, 21, 17, 5, 39, 6, 7, -6, -55, -567; 5680, 987, 4680, 490, 1680, -8, -780, -8946, -2800, -507, -2520, -250, 0, 248, 1980, 19026],67,161]
%
% END OF LATTICE L68.161
%
\hbox{}\par\smallskip%
%
% LATTICE L68.162
%
\leavevmode%
\llap{\spacer}%
${L_{68.162}}=\pdualwitharg{2.3.5}({L_{68.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2308524120, -364021560, 574560; -364021560, 57401040, -90600; 574560, -90600, 143],[2520, 120, 1],[-28, -10, -25, -3, -15, -1, -1, 4, 4, 2, 7, 1, 1, -1, -9, -46; -115, -41, -102, -12, -56, -3, 2, 42, 31, 13, 39, 5, 7, -4, -37, -189; 39640, 14202, 35820, 4450, 24780, 2116, 5280, 10521, 3560, 198, -3420, -850, 420, 1484, 12720, 65079],67,162]
%
% END OF LATTICE L68.162
%
\hbox{}\par\smallskip%
%
% LATTICE L68.163
%
\leavevmode%
\llap{\spacer}%
${L_{68.163}}=\pdualwitharg{2.7}({L_{68.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[803880, -126000, -35280; -126000, 12264, 3528; -35280, 3528, 1013],[2520, 168, 1],[-1, -7, -3, -7, -3, -38, -9, -6, -35, -13, 0, 4, 2, 23, 4, 1; 135, 955, 412, 978, 434, 5745, 1444, 1040, 6381, 2665, 131, -435, -253, -3030, -539, -135; -504, -3570, -1540, -3654, -1620, -21420, -5376, -3865, -23688, -9870, -476, 1638, 948, 11340, 2016, 505],67,163]
%
% END OF LATTICE L68.163
%
\hbox{}\par\smallskip%
%
% LATTICE L68.164
%
\leavevmode%
\llap{\spacer}%
${L_{68.164}}=\pdualwitharg{2.7}({L_{68.3}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3966480, -395640, 136080; -395640, -33096, 11424; 136080, 11424, -3943],[2520, 168, 1],[-1, -3, -3, -10, -7, -134, -47, -91, -323, -87, -24, -17, -2, -1, 2, 1; -510, -1595, -1607, -5403, -3809, -73185, -25729, -49890, -177186, -47765, -13201, -9405, -1127, -855, 1049, 530; -1512, -4725, -4760, -16002, -11280, -216720, -76188, -147730, -524664, -141435, -39088, -27846, -3336, -2520, 3108, 1570],67,164]
%
% END OF LATTICE L68.164
%
\hbox{}\par\smallskip%
%
% LATTICE L68.165
%
\leavevmode%
\llap{\spacer}%
${L_{68.165}}=\pdualwitharg{2.3.7}({L_{68.2}})$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L68.165
%
\hbox{}\par\smallskip%
%
% LATTICE L68.166
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7978320, -607320, -196560; -607320, -45192, -14616; -196560, -14616, -4727],[2520, 168, 1],[-1, -1, 1, 2, 7, 54, 63, 389, 157, 131, 116, 11, 6, 11, 2, 1; 430, 455, -339, -819, -2911, -22545, -26351, -162870, -65758, -54895, -48657, -4625, -2533, -4675, -869, -450; -1288, -1365, 1008, 2450, 8712, 67480, 78876, 487530, 196840, 164325, 145656, 13846, 7584, 14000, 2604, 1350],67,166]
%
% END OF LATTICE L68.166
%
\hbox{}\par\smallskip%
%
% LATTICE L68.167
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[90201676320, 1449669060, 21531159720; 1449669060, 23298240, 346036320; 21531159720, 346036320, 5139492499],[1260, 420, 1],[-539780, -119281, -86797, -35759, -12179, -11809, -1789, 169, 3632, 137, -2561, -8920, -17607, -77549, -147141, -59741; 1594563, 352367, 256405, 105633, 35975, 34875, 5269, -505, -10779, -415, 7559, 26349, 52013, 229089, 434671, 176481; 2153970, 475986, 346360, 142695, 48600, 47124, 7140, -674, -14490, -546, 10220, 35595, 70260, 309456, 587160, 238394],67,167]
%
% END OF LATTICE L68.167
%
\hbox{}\par\smallskip%
%
% LATTICE L68.168
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-443290680, 7679700, -116970840; 7679700, -131460, 2025240; -116970840, 2025240, -30864119],[1260, 420, 1],[1304704, 146296, 219329, 194489, 37843, 46829, 20855, 1043, -316, -1364, -1931, 22909, 34623, 170569, 341475, 71423; -3930171, -440689, -660688, -585864, -113996, -141066, -62824, -3142, 951, 4109, 5818, -69006, -104294, -513804, -1028626, -215148; -5202540, -583359, -874580, -775530, -150900, -186732, -83160, -4159, 1260, 5439, 7700, -91350, -138060, -680148, -1361640, -284801],67,168]
%
% END OF LATTICE L68.168
%
\hbox{}\par\smallskip%
%
% LATTICE L68.169
%
\leavevmode%
\llap{\spacer}%
${L_{68.169}}=\pdualwitharg{2.3.5.7}({L_{68.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9766517040, 942622380, 1740192300; 942622380, 90977880, 167955900; 1740192300, 167955900, 310066451],[1260, 420, 1],[931, 200, 2842, 1852, 13318, 20844, 122622, 152254, 153929, 103040, 229448, 11053, 12492, 4966, 6428, 2606; -2797, -599, -8511, -5555, -39959, -62545, -367957, -456885, -461915, -309209, -688557, -33171, -37493, -14907, -19303, -7827; -3710, -798, -11340, -7385, -53100, -83104, -488880, -607014, -613690, -410802, -914760, -44065, -49800, -19796, -25620, -10386],67,169]
%
% END OF LATTICE L68.169
%
\hbox{}\par\smallskip%
%
% LATTICE L68.170
%
\leavevmode%
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${L_{68.170}}=\pdualwitharg{2.3.5.7}(\main({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[210673260, 49204260, 16046100; 49204260, 11492040, 3747660; 16046100, 3747660, 1222181],[1260, 420, 1],[-175, -58, -1582, -1352, -4378, -6648, -38538, -23729, -47837, -15946, -70436, -6650, -3624, -1354, -1472, -277; 521, 173, 4719, 4031, 13051, 19817, 114875, 70731, 142591, 47531, 209949, 19821, 10801, 4035, 4385, 825; 700, 231, 6300, 5390, 17460, 26516, 153720, 94653, 190820, 63609, 280980, 26530, 14460, 5404, 5880, 1107],67,170]
%
% END OF LATTICE L68.170
%
\hbox{}\par\smallskip%
%
% LATTICE L68.171
%
\leavevmode%
\llap{\spacer}%
${L_{68.171}}=\pdualwitharg{2.3.5.7}(\pfillwitharg{3}({L_{68.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2995440, -6789720, 12600; -6789720, -15389640, 28560; 12600, 28560, -53],[840, 840, 1],[901, 101, 227, 67, 13, 16, 7, 1, -1, -1, -2, 8, 12, 59, 118, 74; -434, -48, -105, -29, -5, -5, -1, 0, 2, 0, -3, -7, -7, -31, -59, -36; -21000, -2002, -2940, 210, 380, 1092, 1120, 237, 840, -238, -2100, -1890, -940, -2772, -3920, -1917],67,171]
%
% END OF LATTICE L68.171
%
\hbox{}\par\smallskip%
%
% LATTICE L68.172
%
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${L_{68.172}}=\pdualwitharg{2.3.5.7}(\pfillwitharg{3}({L_{68.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3297840, -13291320, 27720; -13291320, -53567640, 111720; 27720, 111720, -233],[840, 840, 1],[2341, 131, 587, 172, 33, 40, 17, 5, -1, -1, -2, 23, 32, 155, 308, 385; -434, -24, -105, -29, -5, -5, -1, 0, 2, 0, -3, -7, -7, -31, -59, -72; 69720, 4039, 19320, 6510, 1520, 2352, 1540, 594, 840, -119, -1680, -630, 440, 3528, 8260, 11166],67,172]
%
% END OF LATTICE L68.172
%
\hbox{}\par\smallskip%
%
% LATTICE L68.173
%
\leavevmode%
\llap{\spacer}%
${L_{68.173}}=\pdualwitharg{2.5.7}({L_{68.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5040, -7560, 2520; -7560, -10643640, 3561600; 2520, 3561600, -1191791],[2520, 840, 1],[2269, 255, 192, 172, 34, 43, 20, 1, -1, -3, -3, 17, 29, 146, 295, 62; 354168, 39872, 30123, 27195, 5441, 6999, 3373, 173, 0, -520, -609, 2319, 4397, 22515, 45817, 9665; 1058400, 119154, 90020, 81270, 16260, 20916, 10080, 517, 0, -1554, -1820, 6930, 13140, 67284, 136920, 28883],67,173]
%
% END OF LATTICE L68.173
%
\hbox{}\par\smallskip%
%
% LATTICE L68.174
%
\leavevmode%
\llap{\spacer}%
${L_{68.174}}=\pdualwitharg{2.5.7}({L_{68.3}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5040, -2520, 0; -2520, -2342760, 11760; 0, 11760, -59],[2520, 840, 1],[829, 47, 72, 67, 14, 19, 10, 1, -1, -1, -3, 2, 9, 50, 105, 45; 1224, 70, 109, 105, 23, 33, 19, 2, 0, -2, -7, -3, 11, 69, 151, 66; 241920, 13839, 21560, 20790, 4560, 6552, 3780, 398, 0, -399, -1400, -630, 2160, 13608, 29820, 13042],67,174]
%
% END OF LATTICE L68.174
%
\hbox{}\par\smallskip%
%
% LATTICE L68.175
%
\leavevmode%
\llap{\spacer}%
${L_{68.175}}=\pdualwitharg{2.3.5.7}({L_{68.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8986320, -8187480, -2716560; -8187480, -7458360, -2474640; -2716560, -2474640, -821071],[2520, 840, 1],[-1, -1, 1, 5, 19, 30, 177, 110, 445, 149, 332, 32, 18, 7, 8, 1; 94, 238, 417, -679, -2987, -4873, -29179, -18279, -74158, -24926, -55965, -5493, -3185, -1299, -1681, -237; -280, -714, -1260, 2030, 8940, 14588, 87360, 54729, 222040, 74634, 167580, 16450, 9540, 3892, 5040, 711],67,175]
%
% END OF LATTICE L68.175
%
\hbox{}\par\smallskip%
%
% LATTICE L68.176
%
\leavevmode%
\llap{\spacer}%
${L_{68.176}}=\pdualwitharg{2.3.5.7}({L_{68.3}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9893520, -3197880, -1038240; -3197880, -1032360, -335160; -1038240, -335160, -108811],[2520, 840, 1],[-1, -1, 1, 10, 39, 62, 367, 457, 925, 155, 692, 67, 38, 15, 18, 5; 94, 119, -3, -1099, -4367, -6973, -41359, -51558, -104398, -17503, -78225, -7593, -4325, -1719, -2101, -594; -280, -357, 0, 3290, 13080, 20888, 123900, 154458, 312760, 52437, 234360, 22750, 12960, 5152, 6300, 1782],67,176]
%
% END OF LATTICE L68.176
%

\medskip%
%------------------------------WEYL-GROUP-W69--------------------------------
%
\leavevmode\llap{}%
$W_{69\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=6$%
\hfill%
$22222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L69.1
%
\leavevmode%
\llap{\spacer}%
${L_{69.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{69}}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3898860\\1115100\\10080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1115100\\-318926\\-2883
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10080\\-2883\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-66\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\45\\50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
121\\420\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\7\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-101\\-350\\-350
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-84\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-5\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\84\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-420\\0\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3898860, 1115100, 10080; 1115100, -318926, -2883; 10080, -2883, -26],[420, 1, 1],[-19, 13, 121, 2, -101; -66, 45, 420, 7, -350; -48, 50, 336, -1, -350],68,1]
%
% END OF LATTICE L69.1
%
\hbox{}\par\smallskip%
%
% LATTICE L69.2
%
\leavevmode%
\llap{\spacer}%
${L_{69.2}}=\pfillwitharg{2}({L_{69.1}})$%
{${}={}$}Nikulin $69$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3990\\-1365\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1365\\467\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-35\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\84\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\32\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\210\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-21\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-5\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\21\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-105\\0\\35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[3990, -1365, 0; -1365, 467, 0; 0, 0, -1],[105, 1, 1],[1, -12, 29, 11, 72; 3, -35, 84, 32, 210; 0, 5, -21, -6, -35],68,2]
%
% END OF LATTICE L69.2
%
\hbox{}\par\smallskip%
%
% LATTICE L69.3
%
\leavevmode%
\llap{\spacer}%
${L_{69.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{69.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396795\\3885\\132825
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3885\\-33\\-1302
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132825\\-1302\\-44462
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-23\\-77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76\\-70\\-225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
104\\91\\308
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\69\\228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\70\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-7\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-15\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\7\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-15\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-105\\0\\105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-396795, 3885, 132825; 3885, -33, -1302; 132825, -1302, -44462],[105, 3, 1],[-26, -76, 104, 77, 71; -23, -70, 91, 69, 70; -77, -225, 308, 228, 210],68,3]
%
% END OF LATTICE L69.3
%
\hbox{}\par\smallskip%
%
% LATTICE L69.4
%
\leavevmode%
\llap{\spacer}%
${L_{69.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{69.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-583905\\5985\\-116235
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5985\\-55\\1190
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116235\\1190\\-23138
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62\\-69\\-315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-14\\-61
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-248\\273\\1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\69\\310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\14\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-105\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-5\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\0\\105\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-5\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-105\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-583905, 5985, -116235; 5985, -55, 1190; -116235, 1190, -23138],[105, 5, 1],[62, 12, -248, -61, -11; -69, -14, 273, 69, 14; -315, -61, 1260, 310, 56],68,4]
%
% END OF LATTICE L69.4
%
\hbox{}\par\smallskip%
%
% LATTICE L69.5
%
\leavevmode%
\llap{\spacer}%
${L_{69.5}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{69.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-771855\\8085\\-109725
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8085\\-77\\1148
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109725\\1148\\-15598
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\-69\\-399
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\-70\\-385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\39\\228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\69\\392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\10\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-21\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-35\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\3\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-35\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-15\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-771855, 8085, -109725; 8085, -77, 1148; -109725, 1148, -15598],[105, 7, 1],[56, 54, -32, -55, -7; -69, -70, 39, 69, 10; -399, -385, 228, 392, 50],68,5]
%
% END OF LATTICE L69.5
%
\hbox{}\par\smallskip%
%
% LATTICE L69.6
%
\leavevmode%
\llap{\spacer}%
${L_{69.6}}=\pdualwitharg{3}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3968580\\-1698900\\18480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1698900\\-727278\\7911
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18480\\7911\\-86
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-26\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\5\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71\\168\\196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\33\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-70\\-210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-28\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-15\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\28\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-15\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-420\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3968580, -1698900, 18480; -1698900, -727278, 7911; 18480, 7911, -86],[420, 3, 1],[11, -2, -71, -14, 29; -26, 5, 168, 33, -70; -28, 30, 196, 27, -210],68,6]
%
% END OF LATTICE L69.6
%
\hbox{}\par\smallskip%
%
% LATTICE L69.7
%
\leavevmode%
\llap{\spacer}%
${L_{69.7}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{69.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65606730\\-1303365\\-22261785
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1303365\\25905\\442260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22261785\\442260\\7553906
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190\\1\\-560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-170\\0\\-501
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2482\\7\\-7315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-911\\4\\-2685
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1311\\7\\-3864
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-35\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-15\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\35\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-105\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[65606730, -1303365, -22261785; -1303365, 25905, 442260; -22261785, 442260, 7553906],[105, 15, 1],[-190, -170, -2482, -911, -1311; 1, 0, 7, 4, 7; -560, -501, -7315, -2685, -3864],68,7]
%
% END OF LATTICE L69.7
%
\hbox{}\par\smallskip%
%
% LATTICE L69.8
%
\leavevmode%
\llap{\spacer}%
${L_{69.8}}=\pdualwitharg{2}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2697240\\66780\\687120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66780\\2248\\17184
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
687120\\17184\\175093
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
257\\300\\-1038
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
411\\480\\-1660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2033\\2373\\-8211
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
515\\601\\-2080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3449\\4025\\-13930
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-42\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\40\\0\\-20\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\21\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-20\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-210\\0\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2697240, 66780, 687120; 66780, 2248, 17184; 687120, 17184, 175093],[420, 4, 1],[257, 411, 2033, 515, 3449; 300, 480, 2373, 601, 4025; -1038, -1660, -8211, -2080, -13930],68,8]
%
% END OF LATTICE L69.8
%
\hbox{}\par\smallskip%
%
% LATTICE L69.9
%
\leavevmode%
\llap{\spacer}%
${L_{69.9}}=\pdualwitharg{5}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{420}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-513660\\172620\\5460
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
172620\\-58010\\-1835
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5460\\-1835\\-58
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-18\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61\\168\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-14\\-28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-420\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-5\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\420\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-5\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-420\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-513660, 172620, 5460; 172620, -58010, -1835; 5460, -1835, -58],[420, 5, 1],[-7, 1, 61, 2, -5; -18, 3, 168, 5, -14; -90, -1, 420, 30, -28],68,9]
%
% END OF LATTICE L69.9
%
\hbox{}\par\smallskip%
%
% LATTICE L69.10
%
\leavevmode%
\llap{\spacer}%
${L_{69.10}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{69.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59913210\\-1473675\\-20372625
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1473675\\36267\\501102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20372625\\501102\\6927418
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-238\\1\\-700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-964\\0\\-2835
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-424\\1\\-1247
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1121\\4\\-3297
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1173\\5\\-3450
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-7\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\210\\0\\-105\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\1\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-105\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-15\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[59913210, -1473675, -20372625; -1473675, 36267, 501102; -20372625, 501102, 6927418],[105, 21, 1],[-238, -964, -424, -1121, -1173; 1, 0, 1, 4, 5; -700, -2835, -1247, -3297, -3450],68,10]
%
% END OF LATTICE L69.10
%
\hbox{}\par\smallskip%
%
% LATTICE L69.11
%
\leavevmode%
\llap{\spacer}%
${L_{69.11}}=\pdualwitharg{7}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4620\\-420\\420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\14\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\-7\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-54\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\45\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\48\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-50\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-84\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-35\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\12\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-35\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-60\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4620, -420, 420; -420, 14, -7; 420, -7, -2],[420, 7, 1],[-1, 1, 1, 0, -1; -54, 45, 48, 1, -50; -42, 35, 36, 0, -40],68,11]
%
% END OF LATTICE L69.11
%
\hbox{}\par\smallskip%
%
% LATTICE L69.12
%
\leavevmode%
\llap{\spacer}%
${L_{69.12}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{69.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62682585\\1946070\\-11050725
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1946070\\60445\\-343070
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11050725\\-343070\\1948214
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-490\\1473\\-2520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\396\\-679
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-872\\2619\\-4485
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-769\\2311\\-3955
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-161\\484\\-828
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\0\\-105\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-35\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\0\\15\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-35\\0\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-15\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[62682585, 1946070, -11050725; 1946070, 60445, -343070; -11050725, -343070, 1948214],[105, 35, 1],[-490, -132, -872, -769, -161; 1473, 396, 2619, 2311, 484; -2520, -679, -4485, -3955, -828],68,12]
%
% END OF LATTICE L69.12
%
\hbox{}\par\smallskip%
%
% LATTICE L69.13
%
\leavevmode%
\llap{\spacer}%
${L_{69.13}}=\pdualwitharg{2.3}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59125080\\-595140\\14724360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-595140\\6072\\-148224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14724360\\-148224\\3666919
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\-30\\-206
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
713\\-420\\-2880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1454\\-854\\-5873
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1123\\-659\\-4536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5667\\-3325\\-22890
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-14\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\120\\0\\-60\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\7\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-60\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-210\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[59125080, -595140, 14724360; -595140, 6072, -148224; 14724360, -148224, 3666919],[420, 12, 1],[51, 713, 1454, 1123, 5667; -30, -420, -854, -659, -3325; -206, -2880, -5873, -4536, -22890],68,13]
%
% END OF LATTICE L69.13
%
\hbox{}\par\smallskip%
%
% LATTICE L69.14
%
\leavevmode%
\llap{\spacer}%
${L_{69.14}}=\pdualwitharg{3.5}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45780\\-11760\\1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11760\\-2910\\315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1260\\315\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\56\\700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\-168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-140\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-15\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\140\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-420\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-45780, -11760, 1260; -11760, -2910, 315; 1260, 315, -34],[420, 15, 1],[-1, 0, 5, 1, -1; -10, 1, 56, 9, -14; -130, 9, 700, 120, -168],68,14]
%
% END OF LATTICE L69.14
%
\hbox{}\par\smallskip%
%
% LATTICE L69.15
%
\leavevmode%
\llap{\spacer}%
${L_{69.15}}=\pdualwitharg{2.5}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2211720\\-32340\\545160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32340\\920\\-7940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
545160\\-7940\\134377
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
349\\99\\-1410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
299\\85\\-1208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2365\\672\\-9555
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99\\28\\-400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
149\\42\\-602
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-210\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-20\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\105\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-20\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-210\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2211720, -32340, 545160; -32340, 920, -7940; 545160, -7940, 134377],[420, 20, 1],[349, 299, 2365, 99, 149; 99, 85, 672, 28, 42; -1410, -1208, -9555, -400, -602],68,15]
%
% END OF LATTICE L69.15
%
\hbox{}\par\smallskip%
%
% LATTICE L69.16
%
\leavevmode%
\llap{\spacer}%
${L_{69.16}}=\pdualwitharg{3.7}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30660\\-5460\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5460\\966\\-147
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\-147\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-28\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\210\\0\\-105\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\4\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-105\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-60\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30660, -5460, 840; -5460, 966, -147; 840, -147, 22],[420, 21, 1],[-1, 1, 1, 1, -1; -10, 5, 8, 9, -10; -28, 0, 16, 21, -30],68,16]
%
% END OF LATTICE L69.16
%
\hbox{}\par\smallskip%
%
% LATTICE L69.17
%
\leavevmode%
\llap{\spacer}%
${L_{69.17}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{69.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-566627565\\-404185110\\143966130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-404185110\\-288312045\\102693465
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143966130\\102693465\\-36578242
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-4356\\-12320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-2546\\-7203
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\2484\\7025
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69\\12881\\36435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\337\\954
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-35\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-105\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\5\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-105\\0\\210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-15\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-566627565, -404185110, 143966130; -404185110, -288312045, 102693465; 143966130, 102693465, -36578242],[105, 105, 1],[-23, -14, 13, 69, 2; -4356, -2546, 2484, 12881, 337; -12320, -7203, 7025, 36435, 954],68,17]
%
% END OF LATTICE L69.17
%
\hbox{}\par\smallskip%
%
% LATTICE L69.18
%
\leavevmode%
\llap{\spacer}%
${L_{69.18}}=\pdualwitharg{2.7}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1341480\\-128940\\331800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128940\\12376\\-31892
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
331800\\-31892\\82067
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73\\-3\\294
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-139\\-5\\560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\0\\-93
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
111\\2\\-448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\0\\-190
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-42\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\280\\0\\-140\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\3\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-140\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-30\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1341480, -128940, 331800; -128940, 12376, -31892; 331800, -31892, 82067],[420, 28, 1],[-73, -139, 23, 111, 47; -3, -5, 0, 2, 0; 294, 560, -93, -448, -190],68,18]
%
% END OF LATTICE L69.18
%
\hbox{}\par\smallskip%
%
% LATTICE L69.19
%
\leavevmode%
\llap{\spacer}%
${L_{69.19}}=\pdualwitharg{5.7}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19149060\\6404580\\-93240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6404580\\-2142070\\31185
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93240\\31185\\-454
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-18\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-9\\203
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-24\\1020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\7\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\-68
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-420\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-35\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\60\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-35\\0\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-60\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19149060, 6404580, -93240; 6404580, -2142070, 31185; -93240, 31185, -454],[420, 35, 1],[-5, -4, -13, 2, 1; -18, -9, -24, 7, 2; -210, 203, 1020, 70, -68],68,19]
%
% END OF LATTICE L69.19
%
\hbox{}\par\smallskip%
%
% LATTICE L69.20
%
\leavevmode%
\llap{\spacer}%
${L_{69.20}}=\pdualwitharg{2.3.5}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33459720\\-385140\\8282820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385140\\4440\\-95340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8282820\\-95340\\2050379
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\-3\\-190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
95\\-5\\-384
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
303\\-21\\-1225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89\\-10\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
135\\-14\\-546
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-70\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-60\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\35\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-210\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[33459720, -385140, 8282820; -385140, 4440, -95340; 8282820, -95340, 2050379],[420, 60, 1],[47, 95, 303, 89, 135; -3, -5, -21, -10, -14; -190, -384, -1225, -360, -546],68,20]
%
% END OF LATTICE L69.20
%
\hbox{}\par\smallskip%
%
% LATTICE L69.21
%
\leavevmode%
\llap{\spacer}%
${L_{69.21}}=\pdualwitharg{2.3.7}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
645960\\-65940\\159600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65940\\6216\\-16296
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
159600\\-16296\\39433
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59\\-2\\238
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-937\\-30\\3780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-1\\121
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
229\\7\\-924
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
171\\5\\-690
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-14\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\840\\0\\-420\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\1\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-420\\0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-30\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[645960, -65940, 159600; -65940, 6216, -16296; 159600, -16296, 39433],[420, 84, 1],[-59, -937, -30, 229, 171; -2, -30, -1, 7, 5; 238, 3780, 121, -924, -690],68,21]
%
% END OF LATTICE L69.21
%
\hbox{}\par\smallskip%
%
% LATTICE L69.22
%
\leavevmode%
\llap{\spacer}%
${L_{69.22}}=\pdualwitharg{3.5.7}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-559020\\-18235140\\-521220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18235140\\-594213270\\-16984695
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-521220\\-16984695\\-485482
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69\\276\\-9730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-794\\27993
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-335\\-1332\\46960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\-277\\9765
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\266\\-9378
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-140\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-105\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\20\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-105\\0\\210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-60\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-559020, -18235140, -521220; -18235140, -594213270, -16984695; -521220, -16984695, -485482],[420, 105, 1],[69, -200, -335, -69, 67; 276, -794, -1332, -277, 266; -9730, 27993, 46960, 9765, -9378],68,22]
%
% END OF LATTICE L69.22
%
\hbox{}\par\smallskip%
%
% LATTICE L69.23
%
\leavevmode%
\llap{\spacer}%
${L_{69.23}}=\pdualwitharg{2.5.7}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57021720\\-412860\\-14432880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-412860\\8120\\108360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14432880\\108360\\3656039
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5461\\-16590\\22050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6893\\-20940\\27832
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7768\\-23598\\31365
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1803\\-5477\\7280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\-161\\214
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-210\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-140\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\15\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-140\\0\\280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-30\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[57021720, -412860, -14432880; -412860, 8120, 108360; -14432880, 108360, 3656039],[420, 140, 1],[5461, 6893, 7768, 1803, 53; -16590, -20940, -23598, -5477, -161; 22050, 27832, 31365, 7280, 214],68,23]
%
% END OF LATTICE L69.23
%
\hbox{}\par\smallskip%
%
% LATTICE L69.24
%
\leavevmode%
\llap{\spacer}%
${L_{69.24}}=\pdualwitharg{2.3.5.7}({L_{69.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
276140760\\10013510220\\2480197860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10013510220\\363113317560\\89937779400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2480197860\\89937779400\\22276253093
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\122\\-490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1931\\7356\\-29484
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1023\\3876\\-15535
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2345\\8803\\-35280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\259\\-1038
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-70\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-420\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\5\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-420\\0\\840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-30\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[276140760, 10013510220, 2480197860; 10013510220, 363113317560, 89937779400; 2480197860, 89937779400, 22276253093],[420, 420, 1],[-23, -1931, -1023, -2345, -69; 122, 7356, 3876, 8803, 259; -490, -29484, -15535, -35280, -1038],68,24]
%
% END OF LATTICE L69.24
%

\medskip%
%------------------------------WEYL-GROUP-W70--------------------------------
%
\leavevmode\llap{}%
$W_{70\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=12$%
\hfill%
$222222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L70.1
%
\leavevmode%
\llap{\spacer}%
${L_{70.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{70}}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-713580\\-355320\\420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-355320\\-176926\\207
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\207\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
191\\-384\\-366
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
383\\-770\\-728
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
97\\-195\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-94\\189\\182
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-189\\380\\360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-30\\-42\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-56\\-140\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\0\\-60\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-56\\0\\14\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-140\\-60\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-15\\-7\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-713580, -355320, 420; -355320, -176926, 207; 420, 207, 2],[420, 1, 1],[191, 383, 97, -94, -189, 0; -384, -770, -195, 189, 380, 0; -366, -728, -180, 182, 360, -1],69,1]
%
% END OF LATTICE L70.1
%
\hbox{}\par\smallskip%
%
% LATTICE L70.2
%
\leavevmode%
\llap{\spacer}%
${L_{70.2}}=\pfillwitharg{2}({L_{70.1}})$%
{${}={}$}Nikulin $70$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3045\\630\\315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
630\\130\\63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
315\\63\\19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-21\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-42\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\90\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\280\\-49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\280\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\37\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-15\\-21\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-28\\-35\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\0\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-28\\0\\14\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-35\\-30\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-15\\-7\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[3045, 630, 315; 630, 130, 63; 315, 63, 19],[105, 1, 1],[4, 8, -17, -53, -53, -7; -21, -42, 90, 280, 280, 37; 3, 7, -15, -49, -50, -7],69,2]
%
% END OF LATTICE L70.2
%
\hbox{}\par\smallskip%
%
% LATTICE L70.3
%
\leavevmode%
\llap{\spacer}%
${L_{70.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{70.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-211785\\1365\\71190
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1365\\-3\\-459
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71190\\-459\\-23930
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\-3\\-113
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-226\\-21\\-672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\-5\\-110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
113\\7\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
111\\10\\330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-5\\-21\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-84\\-105\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\10\\0\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-84\\0\\42\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-105\\-30\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-15\\-21\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-211785, 1365, 71190; 1365, -3, -459; 71190, -459, -23930],[105, 3, 1],[-38, -226, -37, 113, 111, -1; -3, -21, -5, 7, 10, 1; -113, -672, -110, 336, 330, -3],69,3]
%
% END OF LATTICE L70.3
%
\hbox{}\par\smallskip%
%
% LATTICE L70.4
%
\leavevmode%
\llap{\spacer}%
${L_{70.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{70.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-274575\\1995\\-54390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1995\\-5\\395
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54390\\395\\-10774
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
98\\-9\\-495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
194\\-21\\-980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-3\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-97\\7\\490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\2\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\-105\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\35\\0\\-140\\-35\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\6\\0\\-6\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-140\\0\\70\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-35\\-6\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-15\\-35\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-274575, 1995, -54390; 1995, -5, 395; -54390, 395, -10774],[105, 5, 1],[98, 194, 19, -97, -19, 1; -9, -21, -3, 7, 2, 1; -495, -980, -96, 490, 96, -5],69,4]
%
% END OF LATTICE L70.4
%
\hbox{}\par\smallskip%
%
% LATTICE L70.5
%
\leavevmode%
\llap{\spacer}%
${L_{70.5}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{70.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-341565\\2625\\-48300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2625\\-7\\371
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48300\\371\\-6830
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
92\\-9\\-651
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-3\\-184
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89\\-15\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\1\\92
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89\\10\\630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-105\\-21\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4\\-35\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\0\\210\\0\\-210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-4\\0\\2\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-35\\-210\\0\\35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-105\\-7\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-341565, 2625, -48300; 2625, -7, 371; -48300, 371, -6830],[105, 7, 1],[92, 26, 89, -13, -89, 1; -9, -3, -15, 1, 10, 1; -651, -184, -630, 92, 630, -7],69,5]
%
% END OF LATTICE L70.5
%
\hbox{}\par\smallskip%
%
% LATTICE L70.6
%
\leavevmode%
\llap{\spacer}%
${L_{70.6}}=\pdualwitharg{3}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
154140\\43680\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43680\\12378\\-357
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-357\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-18\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\-154\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-25\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\60\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-10\\-42\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-168\\-420\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\0\\-60\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-168\\0\\42\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-420\\-60\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-15\\-21\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[154140, 43680, -1260; 43680, 12378, -357; -1260, -357, 10],[420, 3, 1],[5, 43, 7, -2, -17, -2; -18, -154, -25, 7, 60, 7; -14, -84, -10, 0, 0, -3],69,6]
%
% END OF LATTICE L70.6
%
\hbox{}\par\smallskip%
%
% LATTICE L70.7
%
\leavevmode%
\llap{\spacer}%
${L_{70.7}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{70.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46360965\\-81165\\-15730680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81165\\150\\27540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15730680\\27540\\5337557
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-587\\2\\-1730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3919\\7\\-11550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-265\\0\\-781
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1817\\0\\-5355
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-284\\1\\-837
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-341\\3\\-1005
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-5\\-105\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\0\\-420\\-105\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\0\\-6\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-420\\0\\210\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-105\\-6\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-15\\-105\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[46360965, -81165, -15730680; -81165, 150, 27540; -15730680, 27540, 5337557],[105, 15, 1],[-587, -3919, -265, -1817, -284, -341; 2, 7, 0, 0, 1, 3; -1730, -11550, -781, -5355, -837, -1005],69,7]
%
% END OF LATTICE L70.7
%
\hbox{}\par\smallskip%
%
% LATTICE L70.8
%
\leavevmode%
\llap{\spacer}%
${L_{70.8}}=\pdualwitharg{2}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1454877480\\-4713660\\362599860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4713660\\15272\\-1174788
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
362599860\\-1174788\\90370949
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
572\\-1149\\-2310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1293\\-2618\\-5222
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1337\\-2730\\-5400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
617\\-1260\\-2492
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
234\\-470\\-945
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
106\\-207\\-428
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-60\\-84\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-112\\-70\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\0\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-112\\0\\56\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-70\\-60\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-60\\-28\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1454877480, -4713660, 362599860; -4713660, 15272, -1174788; 362599860, -1174788, 90370949],[420, 4, 1],[572, 1293, 1337, 617, 234, 106; -1149, -2618, -2730, -1260, -470, -207; -2310, -5222, -5400, -2492, -945, -428],69,8]
%
% END OF LATTICE L70.8
%
\hbox{}\par\smallskip%
%
% LATTICE L70.9
%
\leavevmode%
\llap{\spacer}%
${L_{70.9}}=\pdualwitharg{5}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
312900\\-134820\\-2100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-134820\\58090\\905
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2100\\905\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\12\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-42\\-140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-9\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\7\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\16\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\16\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-30\\-210\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\140\\0\\-280\\-140\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\6\\0\\-12\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-280\\0\\70\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-140\\-12\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-15\\-35\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[312900, -134820, -2100; -134820, 58090, 905; -2100, 905, 14],[420, 5, 1],[5, -19, -4, 3, 7, 7; 12, -42, -9, 7, 16, 16; -30, -140, -18, 0, 16, 15],69,9]
%
% END OF LATTICE L70.9
%
\hbox{}\par\smallskip%
%
% LATTICE L70.10
%
\leavevmode%
\llap{\spacer}%
${L_{70.10}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{70.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45613995\\-95235\\-15509760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95235\\210\\32382
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15509760\\32382\\5273659
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-695\\2\\-2044
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-661\\1\\-1944
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1559\\0\\-4585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-305\\0\\-897
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1678\\5\\-4935
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-407\\3\\-1197
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-35\\-21\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-12\\-105\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\70\\0\\-210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-12\\0\\6\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-105\\-210\\0\\105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-105\\-21\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[45613995, -95235, -15509760; -95235, 210, 32382; -15509760, 32382, 5273659],[105, 21, 1],[-695, -661, -1559, -305, -1678, -407; 2, 1, 0, 0, 5, 3; -2044, -1944, -4585, -897, -4935, -1197],69,10]
%
% END OF LATTICE L70.10
%
\hbox{}\par\smallskip%
%
% LATTICE L70.11
%
\leavevmode%
\llap{\spacer}%
${L_{70.11}}=\pdualwitharg{7}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5045460\\-191100\\-13860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-191100\\-7238\\-525
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13860\\-525\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-126\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-26\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\45\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\25\\19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\280\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-210\\-42\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-140\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\210\\0\\-420\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-8\\0\\2\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-140\\-420\\0\\140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-105\\-7\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5045460, -191100, -13860; -191100, -7238, -525; -13860, -525, -38],[420, 7, 1],[5, 1, -2, -1, -11, 0; -126, -26, 45, 25, 280, 1; -84, -6, 105, 19, 140, -14],69,11]
%
% END OF LATTICE L70.11
%
\hbox{}\par\smallskip%
%
% LATTICE L70.12
%
\leavevmode%
\llap{\spacer}%
${L_{70.12}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{70.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39997545\\115185\\-7709100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115185\\350\\-22190
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7709100\\-22190\\1485853
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1429\\4293\\-7350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-453\\1360\\-2330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-641\\1923\\-3297
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-209\\627\\-1075
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-230\\691\\-1183
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-279\\840\\-1435
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\0\\-105\\-105\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-20\\-35\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\0\\42\\0\\-42\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-20\\0\\10\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-35\\-42\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-105\\-35\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[39997545, 115185, -7709100; 115185, 350, -22190; -7709100, -22190, 1485853],[105, 35, 1],[-1429, -453, -641, -209, -230, -279; 4293, 1360, 1923, 627, 691, 840; -7350, -2330, -3297, -1075, -1183, -1435],69,12]
%
% END OF LATTICE L70.12
%
\hbox{}\par\smallskip%
%
% LATTICE L70.13
%
\leavevmode%
\llap{\spacer}%
${L_{70.13}}=\pdualwitharg{2.3}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2672040\\-60900\\667380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60900\\1320\\-15204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
667380\\-15204\\166687
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-253\\99\\1022
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-863\\336\\3486
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99\\-40\\-400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
603\\-238\\-2436
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\10\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-303\\119\\1224
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-84\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-336\\-210\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\40\\0\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-336\\0\\168\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-210\\-60\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-60\\-84\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2672040, -60900, 667380; -60900, 1320, -15204; 667380, -15204, 166687],[420, 12, 1],[-253, -863, 99, 603, -26, -303; 99, 336, -40, -238, 10, 119; 1022, 3486, -400, -2436, 105, 1224],69,13]
%
% END OF LATTICE L70.13
%
\hbox{}\par\smallskip%
%
% LATTICE L70.14
%
\leavevmode%
\llap{\spacer}%
${L_{70.14}}=\pdualwitharg{3.5}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2100\\-840\\420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-330\\195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\195\\38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\34\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-42\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-9\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\7\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\52\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\61\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-10\\-210\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-840\\-420\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\2\\0\\-12\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-840\\0\\210\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-420\\-12\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-15\\-105\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2100, -840, 420; -840, -330, 195; 420, 195, 38],[420, 15, 1],[-15, 19, 4, -3, -23, -27; 34, -42, -9, 7, 52, 61; -10, 0, 2, 0, -12, -15],69,14]
%
% END OF LATTICE L70.14
%
\hbox{}\par\smallskip%
%
% LATTICE L70.15
%
\leavevmode%
\llap{\spacer}%
${L_{70.15}}=\pdualwitharg{2.5}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-444360\\-420\\-110040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\1400\\20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110040\\20\\-27239
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1166\\-417\\4710
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1369\\-490\\5530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\36\\-408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1213\\434\\-4900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\9\\-101
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-406\\-145\\1640
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-420\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\140\\0\\-560\\-70\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\24\\0\\-12\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-560\\0\\280\\0\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-70\\-12\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-60\\-140\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-444360, -420, -110040; -420, 1400, 20; -110040, 20, -27239],[420, 20, 1],[-1166, -1369, 101, 1213, 25, -406; -417, -490, 36, 434, 9, -145; 4710, 5530, -408, -4900, -101, 1640],69,15]
%
% END OF LATTICE L70.15
%
\hbox{}\par\smallskip%
%
% LATTICE L70.16
%
\leavevmode%
\llap{\spacer}%
${L_{70.16}}=\pdualwitharg{3.7}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{420}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12700380\\2762340\\-25200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2762340\\-600810\\5481
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25200\\5481\\-50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-154
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\20\\175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\153
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\1680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-70\\-42\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-420\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\70\\0\\-420\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-24\\0\\6\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-420\\-420\\0\\420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-105\\-21\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12700380, 2762340, -25200; 2762340, -600810, 5481; -25200, 5481, -50],[420, 21, 1],[-1, 1, 4, 1, 1, -2; -6, 4, 20, 6, 20, -9; -154, -66, 175, 153, 1680, 21],69,16]
%
% END OF LATTICE L70.16
%
\hbox{}\par\smallskip%
%
% LATTICE L70.17
%
\leavevmode%
\llap{\spacer}%
${L_{70.17}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{70.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-249420885\\-178013535\\63406035
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-178013535\\-127049475\\45253320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63406035\\45253320\\-16118626
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5880\\-16520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5002\\-14055
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1153\\-3241
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2483\\6975
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3423\\9618
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-151\\-420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-35\\-105\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-60\\-105\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\14\\0\\-42\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-60\\0\\30\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-105\\-42\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-105\\-105\\0\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-249420885, -178013535, 63406035; -178013535, -127049475, 45253320; 63406035, 45253320, -16118626],[105, 105, 1],[-3, -3, -1, 1, 2, 1; -5880, -5002, -1153, 2483, 3423, -151; -16520, -14055, -3241, 6975, 9618, -420],69,17]
%
% END OF LATTICE L70.17
%
\hbox{}\par\smallskip%
%
% LATTICE L70.18
%
\leavevmode%
\llap{\spacer}%
${L_{70.18}}=\pdualwitharg{2.7}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3755640\\-279300\\931560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-279300\\24808\\-69300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
931560\\-69300\\231067
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1945\\42\\7854
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-789\\17\\3186
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9049\\195\\36540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1157\\25\\4672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3233\\70\\13055
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-735\\16\\2968
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-420\\-84\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-16\\-70\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\840\\0\\-420\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-16\\0\\8\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-70\\-420\\0\\35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-420\\-28\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3755640, -279300, 931560; -279300, 24808, -69300; 931560, -69300, 231067],[420, 28, 1],[-1945, -789, -9049, -1157, -3233, -735; 42, 17, 195, 25, 70, 16; 7854, 3186, 36540, 4672, 13055, 2968],69,18]
%
% END OF LATTICE L70.18
%
\hbox{}\par\smallskip%
%
% LATTICE L70.19
%
\leavevmode%
\llap{\spacer}%
${L_{70.19}}=\pdualwitharg{5.7}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128100\\-50820\\1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50820\\-20090\\665
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\665\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-210\\-210\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-40\\-140\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\42\\0\\-84\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-40\\0\\10\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-140\\-84\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-105\\-35\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-128100, -50820, 1680; -50820, -20090, 665; 1680, 665, -22],[420, 35, 1],[-1, -1, -1, 0, 1, 1; -18, -6, -3, 1, 4, 1; -630, -260, -168, 30, 196, 105],69,19]
%
% END OF LATTICE L70.19
%
\hbox{}\par\smallskip%
%
% LATTICE L70.20
%
\leavevmode%
\llap{\spacer}%
${L_{70.20}}=\pdualwitharg{2.3.5}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4340280\\-398580\\1086540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-398580\\36600\\-99780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1086540\\-99780\\272003
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
181\\-19\\-730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
677\\-70\\-2730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-209\\14\\840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-3\\-81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
223\\-25\\-900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-420\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-1680\\-210\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\8\\0\\-12\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-1680\\0\\840\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-210\\-12\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-60\\-420\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4340280, -398580, 1086540; -398580, 36600, -99780; 1086540, -99780, 272003],[420, 60, 1],[181, 677, -1, -209, 20, 223; -19, -70, 0, 14, -3, -25; -730, -2730, 4, 840, -81, -900],69,20]
%
% END OF LATTICE L70.20
%
\hbox{}\par\smallskip%
%
% LATTICE L70.21
%
\leavevmode%
\llap{\spacer}%
${L_{70.21}}=\pdualwitharg{2.3.7}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33849480\\-168420\\8380680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168420\\4200\\-41664
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8380680\\-41664\\2074945
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-891\\-37\\3598
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1520\\-63\\6138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8806\\-365\\35560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4508\\-187\\18204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13131\\-545\\53025
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2621\\-109\\10584
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-140\\-84\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-48\\-210\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\280\\0\\-420\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-48\\0\\24\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\-420\\0\\105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-420\\-84\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[33849480, -168420, 8380680; -168420, 4200, -41664; 8380680, -41664, 2074945],[420, 84, 1],[-891, -1520, -8806, -4508, -13131, -2621; -37, -63, -365, -187, -545, -109; 3598, 6138, 35560, 18204, 53025, 10584],69,21]
%
% END OF LATTICE L70.21
%
\hbox{}\par\smallskip%
%
% LATTICE L70.22
%
\leavevmode%
\llap{\spacer}%
${L_{70.22}}=\pdualwitharg{3.5.7}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\2940\\1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2940\\5250\\2205
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1260\\2205\\926
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\118\\-280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\114\\-270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-95\\225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-284\\672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-89\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-70\\-210\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-120\\-420\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\14\\0\\-84\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-120\\0\\30\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-420\\-84\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-105\\-105\\0\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4620, 2940, 1260; 2940, 5250, 2205; 1260, 2205, 926],[420, 105, 1],[-1, -1, 0, 1, 3, 1; 118, 114, 3, -95, -284, -89; -280, -270, -7, 225, 672, 210],69,22]
%
% END OF LATTICE L70.22
%
\hbox{}\par\smallskip%
%
% LATTICE L70.23
%
\leavevmode%
\llap{\spacer}%
${L_{70.23}}=\pdualwitharg{2.5.7}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27522600\\191940\\-6671280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
191940\\1400\\-46480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6671280\\-46480\\1617103
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1819\\5364\\-7350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-443\\1307\\-1790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-395\\1167\\-1596
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\293\\-400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-123\\363\\-497
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-589\\1736\\-2380
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-420\\-420\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-80\\-70\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\168\\0\\-84\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-80\\0\\40\\0\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-70\\-84\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-420\\-140\\0\\280
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[27522600, 191940, -6671280; 191940, 1400, -46480; -6671280, -46480, 1617103],[420, 140, 1],[-1819, -443, -395, -99, -123, -589; 5364, 1307, 1167, 293, 363, 1736; -7350, -1790, -1596, -400, -497, -2380],69,23]
%
% END OF LATTICE L70.23
%
\hbox{}\par\smallskip%
%
% LATTICE L70.24
%
\leavevmode%
\llap{\spacer}%
${L_{70.24}}=\pdualwitharg{2.3.5.7}({L_{70.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5865720\\13668060\\3383940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13668060\\31848600\\7885080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3383940\\7885080\\1952189
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\331\\-1330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\7\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-133\\532
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-31\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\172\\-693
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\941\\-3780
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-140\\-420\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-240\\-210\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\56\\0\\-84\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-240\\0\\120\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\-84\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-420\\-420\\0\\840
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5865720, 13668060, 3383940; 13668060, 31848600, 7885080; 3383940, 7885080, 1952189],[420, 420, 1],[-4, 1, 3, 3, -1, -12; 331, 7, -133, -31, 172, 941; -1330, -30, 532, 120, -693, -3780],69,24]
%
% END OF LATTICE L70.24
%

\medskip%
%------------------------------WEYL-GROUP-W71--------------------------------
%
\leavevmode\llap{}%
$W_{71\phantom{0}}$%
\qquad\llap{48} lattices, $\chi=16$%
\hfill%
$222262$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L71.1
%
\leavevmode%
\llap{\spacer}%
${L_{71.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{23\rightarrow N_{71},3,2}%
\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{126}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-7158060\\-2882880\\10080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2882880\\-1161066\\4059
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10080\\4059\\-14
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\294\\994
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\470\\1620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-142\\357\\1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-93\\-324
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\5\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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28\\0\\-126\\-70\\-126\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-60\\-180\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\0\\126\\0\\-126\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-60\\0\\10\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-180\\-126\\0\\18\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-63\\-15\\-9\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-7158060, -2882880, 10080; -2882880, -1161066, 4059; 10080, 4059, -14],[1260, 3, 1],[-117, -187, -142, 2, 37, -2; 294, 470, 357, -5, -93, 5; 994, 1620, 1260, -10, -324, 9],70,1]
%
% END OF LATTICE L71.1
%
\hbox{}\par\smallskip%
%
% LATTICE L71.2
%
\leavevmode%
\llap{\spacer}%
${L_{71.2}}=\pfillwitharg{2.3}({L_{71.1}})$%
{${}={}$}Nikulin $71$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2310\\315\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
315\\43\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\7\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\15\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-21\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-65\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-51\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-15\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-21\\-35\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-30\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\14\\0\\-14\\-21
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-30\\0\\10\\0\\-15
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\kern3pt%
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-21\\-30\\-14\\0\\2\\-3
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-21\\-15\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[2310, 315, 0; 315, 43, 0; 0, 0, -1],[105, 1, 1],[-1, -2, 3, 9, 7, 2; 7, 15, -21, -65, -51, -15; 0, 0, -7, -15, -11, -3],70,2]
%
% END OF LATTICE L71.2
%
\hbox{}\par\smallskip%
%
% LATTICE L71.3
%
\leavevmode%
\llap{\spacer}%
${L_{71.3}}=\pfillwitharg{3}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3371340\\-674100\\7980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-674100\\-134786\\1595
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7980\\1595\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-411\\2072\\1386
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-601\\3030\\2040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-143\\721\\490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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24\\-121\\-82
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\126\\81
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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28\\0\\-42\\-70\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\60\\0\\-60\\-60\\-30
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\kern3pt%
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-42\\0\\14\\0\\-14\\-21
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-60\\0\\10\\0\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-60\\-14\\0\\2\\-3
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-21\\-15\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3371340, -674100, 7980; -674100, -134786, 1595; 7980, 1595, -18],[420, 1, 1],[-411, -601, -143, 1, 24, -25; 2072, 3030, 721, -5, -121, 126; 1386, 2040, 490, 0, -82, 81],70,3]
%
% END OF LATTICE L71.3
%
\hbox{}\par\smallskip%
%
% LATTICE L71.4
%
\leavevmode%
\llap{\spacer}%
${L_{71.4}}=\pfillwitharg{2}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{126}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3465\\-630\\630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\339\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
630\\-99\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\147\\308
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\235\\495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\357\\756
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-93\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-63\\-35\\-63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-30\\-90\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\0\\126\\0\\-126\\-63
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-30\\0\\10\\0\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-90\\-126\\0\\18\\-9
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-63\\-15\\-9\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3465, -630, 630; -630, 339, -99; 630, -99, -14],[315, 3, 1],[30, 48, 73, -1, -19, 1; 147, 235, 357, -5, -93, 5; 308, 495, 756, -10, -198, 9],70,4]
%
% END OF LATTICE L71.4
%
\hbox{}\par\smallskip%
%
% LATTICE L71.5
%
\leavevmode%
\llap{\spacer}%
${L_{71.5}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1095255\\328020\\367710
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
328020\\-97674\\-110139
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
367710\\-110139\\-123451
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10236\\679\\29883
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4977\\330\\14530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7078\\469\\20664
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-5\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1192\\-79\\-3480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
421\\28\\1229
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-63\\-105\\-63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-30\\-30\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\0\\42\\0\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-30\\0\\30\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-30\\-42\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-21\\-15\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1095255, 328020, 367710; 328020, -97674, -110139; 367710, -110139, -123451],[105, 3, 1],[10236, 4977, 7078, -72, -1192, 421; 679, 330, 469, -5, -79, 28; 29883, 14530, 20664, -210, -3480, 1229],70,5]
%
% END OF LATTICE L71.5
%
\hbox{}\par\smallskip%
%
% LATTICE L71.6
%
\leavevmode%
\llap{\spacer}%
${L_{71.6}}=\pdualwitharg{5}(\pfillwitharg{2.3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1475565\\490770\\-292740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
490770\\-162790\\97355
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-292740\\97355\\-58077
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5836\\679\\30555
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1703\\198\\8916
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4038\\469\\21140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-1\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
680\\-79\\-3560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-719\\84\\3765
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-105\\-35\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-6\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\0\\70\\0\\-70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-6\\0\\2\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-30\\-70\\0\\10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-105\\-15\\-15\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1475565, 490770, -292740; 490770, -162790, 97355; -292740, 97355, -58077],[105, 5, 1],[-5836, -1703, -4038, 8, 680, -719; 679, 198, 469, -1, -79, 84; 30555, 8916, 21140, -42, -3560, 3765],70,6]
%
% END OF LATTICE L71.6
%
\hbox{}\par\smallskip%
%
% LATTICE L71.7
%
\leavevmode%
\llap{\spacer}%
${L_{71.7}}=\pdualwitharg{7}(\pfillwitharg{2.3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1870995\\653520\\-265020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
653520\\-227906\\92561
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-265020\\92561\\-37539
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-598\\97\\4461
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6109\\990\\45570
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-414\\67\\3088
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-5\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
488\\-79\\-3640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-515\\84\\3843
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-35\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\0\\-210\\-210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0\\-14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-210\\0\\70\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-210\\-14\\0\\14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-21\\-105\\-21\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1870995, 653520, -265020; 653520, -227906, 92561; -265020, 92561, -37539],[105, 7, 1],[-598, -6109, -414, 28, 488, -515; 97, 990, 67, -5, -79, 84; 4461, 45570, 3088, -210, -3640, 3843],70,7]
%
% END OF LATTICE L71.7
%
\hbox{}\par\smallskip%
%
% LATTICE L71.8
%
\leavevmode%
\llap{\spacer}%
${L_{71.8}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${63}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9909585\\-51345\\40635
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51345\\-246\\207
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40635\\207\\-166
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\3612\\20412
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\1610\\9105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\665\\3766
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-225\\-1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-111\\-628
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\223\\1257
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\0\\-63\\-315\\-63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-90\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\0\\14\\0\\-14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-90\\0\\90\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-30\\-14\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-21\\-45\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9909585, -51345, 40635; -51345, -246, 207; 40635, 207, -166],[315, 3, 1],[65, 29, 12, -4, -2, 4; 3612, 1610, 665, -225, -111, 223; 20412, 9105, 3766, -1260, -628, 1257],70,8]
%
% END OF LATTICE L71.8
%
\hbox{}\par\smallskip%
%
% LATTICE L71.9
%
\leavevmode%
\llap{\spacer}%
${L_{71.9}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3834180\\15540\\9240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15540\\-54\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9240\\-39\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\1274\\7392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\500\\2890
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\224\\1281
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-110\\-645
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-56\\-321
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\55\\322
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-126\\-210\\-126\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-60\\-60\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\0\\42\\0\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-60\\0\\30\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-60\\-42\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-21\\-15\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3834180, 15540, 9240; 15540, -54, -39; 9240, -39, -22],[420, 3, 1],[23, 9, 4, -2, -1, 1; 1274, 500, 224, -110, -56, 55; 7392, 2890, 1281, -645, -321, 322],70,9]
%
% END OF LATTICE L71.9
%
\hbox{}\par\smallskip%
%
% LATTICE L71.10
%
\leavevmode%
\llap{\spacer}%
${L_{71.10}}=\pdualwitharg{3.5}(\pfillwitharg{2.3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11223448635\\-22618995\\-3809101065
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22618995\\45585\\7676610
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3809101065\\7676610\\1292762269
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21946\\2779\\-64680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1838\\230\\-5417
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11365\\1386\\-33495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-113\\8\\-333
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-341\\42\\-1005
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1839\\239\\-5420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\0\\-315\\-105\\-315\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-6\\-30\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\0\\210\\0\\-210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-6\\0\\6\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-30\\-210\\0\\30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-105\\-15\\-15\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11223448635, -22618995, -3809101065; -22618995, 45585, 7676610; -3809101065, 7676610, 1292762269],[105, 15, 1],[-21946, -1838, -11365, -113, -341, -1839; 2779, 230, 1386, 8, 42, 239; -64680, -5417, -33495, -333, -1005, -5420],70,10]
%
% END OF LATTICE L71.10
%
\hbox{}\par\smallskip%
%
% LATTICE L71.11
%
\leavevmode%
\llap{\spacer}%
${L_{71.11}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
680928360\\-4407900\\169485120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4407900\\28536\\-1097140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
169485120\\-1097140\\42185357
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6652\\-5411\\-26866
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8401\\-6840\\-33930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3501\\-2856\\-14140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
203\\-170\\-820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
103\\-84\\-416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1652\\-1341\\-6672
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-84\\-140\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-120\\-120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\56\\0\\-56\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-120\\0\\40\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-120\\-56\\0\\8\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-84\\-60\\-12\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[680928360, -4407900, 169485120; -4407900, 28536, -1097140; 169485120, -1097140, 42185357],[420, 4, 1],[6652, 8401, 3501, 203, 103, 1652; -5411, -6840, -2856, -170, -84, -1341; -26866, -33930, -14140, -820, -416, -6672],70,11]
%
% END OF LATTICE L71.11
%
\hbox{}\par\smallskip%
%
% LATTICE L71.12
%
\leavevmode%
\llap{\spacer}%
${L_{71.12}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-688380\\16800\\8400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16800\\-410\\-205
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8400\\-205\\-102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\728\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\210\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\245\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-41\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\48\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-210\\-70\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12\\-60\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\70\\0\\-70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-12\\0\\2\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-60\\-70\\0\\10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-105\\-15\\-15\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-688380, 16800, 8400; 16800, -410, -205; 8400, -205, -102],[420, 5, 1],[17, 5, 6, 0, -1, 1; 728, 210, 245, -1, -41, 48; -70, -12, 0, 2, 0, -15],70,12]
%
% END OF LATTICE L71.12
%
\hbox{}\par\smallskip%
%
% LATTICE L71.13
%
\leavevmode%
\llap{\spacer}%
${L_{71.13}}=\pdualwitharg{3.7}(\pfillwitharg{2.3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{}{6}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6659410065\\-20615385\\-2265034485
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20615385\\63819\\7011816
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2265034485\\7011816\\770395751
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3766\\397\\-11076
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11008\\1150\\-32375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1933\\198\\-5685
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-607\\40\\-1785
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-407\\42\\-1197
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2223\\239\\-6538
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-9\\-105\\-63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\35\\0\\-210\\-210\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\6\\0\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-210\\0\\210\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-210\\-42\\0\\42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-21\\-105\\-21\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6659410065, -20615385, -2265034485; -20615385, 63819, 7011816; -2265034485, 7011816, 770395751],[105, 21, 1],[-3766, -11008, -1933, -607, -407, -2223; 397, 1150, 198, 40, 42, 239; -11076, -32375, -5685, -1785, -1197, -6538],70,13]
%
% END OF LATTICE L71.13
%
\hbox{}\par\smallskip%
%
% LATTICE L71.14
%
\leavevmode%
\llap{\spacer}%
${L_{71.14}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-464226420\\16415700\\-693420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16415700\\-580482\\24521
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693420\\24521\\-1034
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-179\\-4980\\1942
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1817\\-50550\\19740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\-1697\\664
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\195\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\2003\\-784
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-2226\\861
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-70\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-420\\-420\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-420\\0\\70\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-420\\-14\\0\\14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-21\\-105\\-21\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-464226420, 16415700, -693420; 16415700, -580482, 24521; -693420, 24521, -1034],[420, 7, 1],[-179, -1817, -61, 7, 72, -80; -4980, -50550, -1697, 195, 2003, -2226; 1942, 19740, 664, -70, -784, 861],70,14]
%
% END OF LATTICE L71.14
%
\hbox{}\par\smallskip%
%
% LATTICE L71.15
%
\leavevmode%
\llap{\spacer}%
${L_{71.15}}=\pdualwitharg{5.7}(\pfillwitharg{2.3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{}{6}{210}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5818166130\\24876600\\-1115959005
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24876600\\106365\\-4771480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1115959005\\-4771480\\214047601
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2582\\8143\\-13280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4528\\14274\\-23289
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1325\\4173\\-6815
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-83\\257\\-427
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-279\\879\\-1435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4573\\14436\\-23520
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-15\\-35\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-42\\-210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\10\\0\\-70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-42\\0\\14\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-210\\-70\\0\\70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-105\\-105\\-105\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5818166130, 24876600, -1115959005; 24876600, 106365, -4771480; -1115959005, -4771480, 214047601],[105, 35, 1],[-2582, -4528, -1325, -83, -279, -4573; 8143, 14274, 4173, 257, 879, 14436; -13280, -23289, -6815, -427, -1435, -23520],70,15]
%
% END OF LATTICE L71.15
%
\hbox{}\par\smallskip%
%
% LATTICE L71.16
%
\leavevmode%
\llap{\spacer}%
${L_{71.16}}=\pdualwitharg{3}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${252}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120870540\\-181440\\142380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-181440\\-246\\207
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
142380\\207\\-166
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\20874\\81774
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\9310\\36480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\1925\\7546
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-645\\-2520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-321\\-1258
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\643\\2517
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252\\0\\-126\\-630\\-126\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-180\\-60\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\0\\14\\0\\-14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-180\\0\\90\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-60\\-14\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-21\\-45\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-120870540, -181440, 142380; -181440, -246, 207; 142380, 207, -166],[1260, 3, 1],[65, 29, 6, -2, -1, 2; 20874, 9310, 1925, -645, -321, 643; 81774, 36480, 7546, -2520, -1258, 2517],70,16]
%
% END OF LATTICE L71.16
%
\hbox{}\par\smallskip%
%
% LATTICE L71.17
%
\leavevmode%
\llap{\spacer}%
${L_{71.17}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{630}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58905\\4410\\2205
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4410\\-330\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2205\\-165\\-82
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\56\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\105\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-27\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-315\\-35\\-315\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-6\\-90\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\0\\630\\0\\-630\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-6\\0\\2\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-90\\-630\\0\\90\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-315\\-15\\-45\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-58905, 4410, 2205; 4410, -330, -165; 2205, -165, -82],[315, 15, 1],[3, 1, 8, 0, -2, 0; 56, 16, 105, -1, -27, 7; -35, -6, 0, 2, 0, -15],70,17]
%
% END OF LATTICE L71.17
%
\hbox{}\par\smallskip%
%
% LATTICE L71.18
%
\leavevmode%
\llap{\spacer}%
${L_{71.18}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${315}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38481030\\6230070\\-61425
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6230070\\-1008645\\9945
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61425\\9945\\-98
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-292\\-1701\\10395
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-163\\1008
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-378\\2380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\64\\-400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-76\\435
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
315\\0\\-315\\-315\\-315\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-18\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\0\\70\\0\\-70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-18\\0\\18\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-30\\-70\\0\\10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-105\\-45\\-15\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38481030, 6230070, -61425; 6230070, -1008645, 9945; -61425, 9945, -98],[315, 15, 1],[-292, -28, -65, 1, 11, -13; -1701, -163, -378, 6, 64, -76; 10395, 1008, 2380, -18, -400, 435],70,18]
%
% END OF LATTICE L71.18
%
\hbox{}\par\smallskip%
%
% LATTICE L71.19
%
\leavevmode%
\llap{\spacer}%
${L_{71.19}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1175804280\\-5869500\\291116280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5869500\\29304\\-1453224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
291116280\\-1453224\\72077207
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10555\\56\\-42630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4172\\25\\-16850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4742\\35\\-19152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\5\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
514\\1\\-2076
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1029\\4\\-4156
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-252\\-420\\-252\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-120\\-120\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\0\\168\\0\\-168\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-120\\0\\120\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-120\\-168\\0\\24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-84\\-60\\-12\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1175804280, -5869500, 291116280; -5869500, 29304, -1453224; 291116280, -1453224, 72077207],[420, 12, 1],[10555, 4172, 4742, 208, 514, 1029; 56, 25, 35, 5, 1, 4; -42630, -16850, -19152, -840, -2076, -4156],70,19]
%
% END OF LATTICE L71.19
%
\hbox{}\par\smallskip%
%
% LATTICE L71.20
%
\leavevmode%
\llap{\spacer}%
${L_{71.20}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2448180\\-23100\\7980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23100\\-210\\75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7980\\75\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\140\\3150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\14\\346
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\49\\1365
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-13\\-345
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-630\\-210\\-630\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12\\-60\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\0\\210\\0\\-210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-12\\0\\6\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-60\\-210\\0\\30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-105\\-15\\-15\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2448180, -23100, 7980; -23100, -210, 75; 7980, 75, -26],[420, 15, 1],[9, 1, 4, 0, -1, 0; 140, 14, 49, -1, -13, 2; 3150, 346, 1365, -3, -345, 5],70,20]
%
% END OF LATTICE L71.20
%
\hbox{}\par\smallskip%
%
% LATTICE L71.21
%
\leavevmode%
\llap{\spacer}%
${L_{71.21}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{s}{2}{126}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1191330\\76230\\9135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76230\\-4557\\-525
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9135\\-525\\-59
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\133\\-718
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\1460\\-7875
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\309\\-1665
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-45\\245
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-573\\3087
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\46\\-252
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-9\\-35\\-63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\0\\-210\\-630\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\18\\0\\-126\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-210\\0\\70\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-630\\-126\\0\\126\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-63\\-105\\-63\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1191330, 76230, 9135; 76230, -4557, -525; 9135, -525, -59],[315, 21, 1],[3, 33, 7, -1, -13, 1; 133, 1460, 309, -45, -573, 46; -718, -7875, -1665, 245, 3087, -252],70,21]
%
% END OF LATTICE L71.21
%
\hbox{}\par\smallskip%
%
% LATTICE L71.22
%
\leavevmode%
\llap{\spacer}%
${L_{71.22}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{630}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1100295\\-45675\\371700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45675\\-1722\\15225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
371700\\15225\\-125326
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
827\\3369\\2862
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2579\\10505\\8925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
152\\619\\526
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-364\\-1485\\-1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-178\\-725\\-616
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
358\\1459\\1239
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-9\\-315\\-63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\0\\-630\\-210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\2\\0\\-14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-630\\0\\630\\0\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-210\\-14\\0\\14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-21\\-315\\-21\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1100295, -45675, 371700; -45675, -1722, 15225; 371700, 15225, -125326],[315, 21, 1],[827, 2579, 152, -364, -178, 358; 3369, 10505, 619, -1485, -725, 1459; 2862, 8925, 526, -1260, -616, 1239],70,22]
%
% END OF LATTICE L71.22
%
\hbox{}\par\smallskip%
%
% LATTICE L71.23
%
\leavevmode%
\llap{\spacer}%
${L_{71.23}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{}{6}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
157080\\35700\\39060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35700\\16440\\8920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39060\\8920\\9713
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7086\\147\\-28630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1733\\36\\-7002
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3361\\70\\-13580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99\\2\\-400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1886\\39\\-7620
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-420\\-140\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\280\\0\\-280\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-24\\0\\8\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-120\\-280\\0\\40\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-420\\-60\\-60\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[157080, 35700, 39060; 35700, 16440, 8920; 39060, 8920, 9713],[420, 20, 1],[7086, 1733, 3361, -1, 99, 1886; 147, 36, 70, 0, 2, 39; -28630, -7002, -13580, 4, -400, -7620],70,23]
%
% END OF LATTICE L71.23
%
\hbox{}\par\smallskip%
%
% LATTICE L71.24
%
\leavevmode%
\llap{\spacer}%
${L_{71.24}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{}{6}{14}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-345660\\11760\\-21420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11760\\-378\\777
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21420\\777\\-1222
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67\\-1074\\492
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-181\\-2900\\1330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-176\\81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\690\\-315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\320\\-147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-337\\154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-18\\-210\\-126\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\140\\0\\-420\\-420\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\6\\0\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-420\\0\\210\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-420\\-42\\0\\42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-21\\-105\\-21\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-345660, 11760, -21420; 11760, -378, 777; -21420, 777, -1222],[420, 21, 1],[-67, -181, -11, 43, 20, -21; -1074, -2900, -176, 690, 320, -337; 492, 1330, 81, -315, -147, 154],70,24]
%
% END OF LATTICE L71.24
%
\hbox{}\par\smallskip%
%
% LATTICE L71.25
%
\leavevmode%
\llap{\spacer}%
${L_{71.25}}=\pdualwitharg{3.5.7}(\pfillwitharg{2.3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{}{6}{70}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1505364105\\-1024233000\\364818720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1024233000\\-696874185\\248217690
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
364818720\\248217690\\-88411973
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
97\\119718\\336510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\81529\\229166
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\82870\\232935
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1128\\-3171
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-79\\-97684\\-274575
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\34379\\96635
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-45\\-105\\-315\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-42\\-210\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\30\\0\\-210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-42\\0\\42\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-210\\-210\\0\\210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-105\\-105\\-105\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1505364105, -1024233000, 364818720; -1024233000, -696874185, 248217690; 364818720, 248217690, -88411973],[105, 105, 1],[97, 66, 67, -1, -79, 28; 119718, 81529, 82870, -1128, -97684, 34379; 336510, 229166, 232935, -3171, -274575, 96635],70,25]
%
% END OF LATTICE L71.25
%
\hbox{}\par\smallskip%
%
% LATTICE L71.26
%
\leavevmode%
\llap{\spacer}%
${L_{71.26}}=\pdualwitharg{2.7}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{}{6}{168}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3527137320\\439422900\\875642880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
439422900\\54745208\\109090604
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
875642880\\109090604\\217386051
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34690\\703\\-140086
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
291581\\5910\\-1177470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15485\\314\\-62532
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3987\\-80\\16100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3453\\70\\-13944
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67084\\1359\\-270900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-140\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-840\\-840\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-56\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-840\\0\\280\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-840\\-56\\0\\56\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-84\\-420\\-84\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3527137320, 439422900, 875642880; 439422900, 54745208, 109090604; 875642880, 109090604, 217386051],[420, 28, 1],[34690, 291581, 15485, -3987, 3453, 67084; 703, 5910, 314, -80, 70, 1359; -140086, -1177470, -62532, 16100, -13944, -270900],70,26]
%
% END OF LATTICE L71.26
%
\hbox{}\par\smallskip%
%
% LATTICE L71.27
%
\leavevmode%
\llap{\spacer}%
${L_{71.27}}=\pdualwitharg{5.7}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{}{6}{210}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70298340\\20821500\\607740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20821500\\-6167070\\-180005
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
607740\\-180005\\-5254
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\144\\-1810
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\258\\-3402
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\35\\-505
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-41\\595
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\108\\-1155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-30\\-70\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-84\\-420\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\0\\-70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-84\\0\\14\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-420\\-70\\0\\70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-105\\-105\\-105\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-70298340, 20821500, 607740; 20821500, -6167070, -180005; 607740, -180005, -5254],[420, 35, 1],[27, 47, 6, -2, -7, 22; 144, 258, 35, -9, -41, 108; -1810, -3402, -505, 77, 595, -1155],70,27]
%
% END OF LATTICE L71.27
%
\hbox{}\par\smallskip%
%
% LATTICE L71.28
%
\leavevmode%
\llap{\spacer}%
${L_{71.28}}=\pdualwitharg{2}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{504}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3781494360\\-7082460\\-947435580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7082460\\13272\\1774476
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-947435580\\1774476\\237375517
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6759\\-3633\\-26950
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8329\\-4480\\-33210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10681\\-5754\\-42588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-627\\-340\\-2500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1571\\-846\\-6264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1887\\-1013\\-7524
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-252\\-140\\-252\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-120\\-360\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\0\\504\\0\\-504\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-120\\0\\40\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-360\\-504\\0\\72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-252\\-60\\-36\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3781494360, -7082460, -947435580; -7082460, 13272, 1774476; -947435580, 1774476, 237375517],[1260, 12, 1],[-6759, -8329, -10681, -627, -1571, -1887; -3633, -4480, -5754, -340, -846, -1013; -26950, -33210, -42588, -2500, -6264, -7524],70,28]
%
% END OF LATTICE L71.28
%
\hbox{}\par\smallskip%
%
% LATTICE L71.29
%
\leavevmode%
\llap{\spacer}%
${L_{71.29}}=\pdualwitharg{2.3}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${252}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{s}{2}{360}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
332439677640\\-810639900\\-83374601940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-810639900\\1976712\\203305392
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-83374601940\\203305392\\20910031853
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-122515\\-231\\-488502
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53623\\-110\\-213810
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23897\\-56\\-95284
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6907\\-30\\-27540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\0\\-1256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9125\\-13\\-36384
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252\\0\\-252\\-1260\\-252\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-360\\-120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\0\\56\\0\\-56\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-360\\0\\360\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-120\\-56\\0\\8\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-84\\-180\\-12\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[332439677640, -810639900, -83374601940; -810639900, 1976712, 203305392; -83374601940, 203305392, 20910031853],[1260, 12, 1],[-122515, -53623, -23897, -6907, -315, -9125; -231, -110, -56, -30, 0, -13; -488502, -213810, -95284, -27540, -1256, -36384],70,29]
%
% END OF LATTICE L71.29
%
\hbox{}\par\smallskip%
%
% LATTICE L71.30
%
\leavevmode%
\llap{\spacer}%
${L_{71.30}}=\pdualwitharg{5}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{630}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1558620\\22680\\11340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22680\\-330\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11340\\-165\\-82
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\238\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\74\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\273\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-69\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-630\\-70\\-630\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12\\-180\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\0\\630\\0\\-630\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-12\\0\\2\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-180\\-630\\0\\90\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-315\\-15\\-45\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1558620, 22680, 11340; 22680, -330, -165; 11340, -165, -82],[1260, 15, 1],[3, 1, 4, 0, -1, 0; 238, 74, 273, -1, -69, 7; -70, -12, 0, 2, 0, -15],70,30]
%
% END OF LATTICE L71.30
%
\hbox{}\par\smallskip%
%
% LATTICE L71.31
%
\leavevmode%
\llap{\spacer}%
${L_{71.31}}=\pdualwitharg{3.5}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1260}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2462056380\\90861120\\-491400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90861120\\-3353190\\18135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-491400\\18135\\-98
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-635\\-17094\\20790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\-1642\\2016
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71\\-1911\\2380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\27\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\323\\-400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-377\\435
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1260\\0\\-630\\-630\\-630\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-36\\-60\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\0\\70\\0\\-70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-36\\0\\18\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-60\\-70\\0\\10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-105\\-45\\-15\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2462056380, 90861120, -491400; 90861120, -3353190, 18135; -491400, 18135, -98],[1260, 15, 1],[-635, -61, -71, 1, 12, -14; -17094, -1642, -1911, 27, 323, -377; 20790, 2016, 2380, -18, -400, 435],70,31]
%
% END OF LATTICE L71.31
%
\hbox{}\par\smallskip%
%
% LATTICE L71.32
%
\leavevmode%
\llap{\spacer}%
${L_{71.32}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{}{6}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
133288680\\-716100\\33001080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-716100\\3960\\-177300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33001080\\-177300\\8170771
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15962\\-35\\-64470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1283\\-3\\-5182
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7799\\-21\\-31500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
205\\-1\\-828
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1129\\-3\\-4560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1540\\-3\\-6220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-1260\\-420\\-1260\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-24\\-120\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\0\\840\\0\\-840\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-24\\0\\24\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-120\\-840\\0\\120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-420\\-60\\-60\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[133288680, -716100, 33001080; -716100, 3960, -177300; 33001080, -177300, 8170771],[420, 60, 1],[15962, 1283, 7799, 205, 1129, 1540; -35, -3, -21, -1, -3, -3; -64470, -5182, -31500, -828, -4560, -6220],70,32]
%
% END OF LATTICE L71.32
%
\hbox{}\par\smallskip%
%
% LATTICE L71.33
%
\leavevmode%
\llap{\spacer}%
${L_{71.33}}=\pdualwitharg{7}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}{126}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
122220\\-84420\\6300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84420\\-23898\\5187
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6300\\5187\\-782
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103\\-418\\-3602
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1141\\-4630\\-39900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-122\\-495\\-4266
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\65\\560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
227\\921\\7938
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-61\\-525
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-18\\-70\\-126\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-420\\-1260\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\18\\0\\-126\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-420\\0\\70\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-1260\\-126\\0\\126\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-63\\-105\\-63\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[122220, -84420, 6300; -84420, -23898, 5187; 6300, 5187, -782],[1260, 21, 1],[-103, -1141, -122, 16, 227, -15; -418, -4630, -495, 65, 921, -61; -3602, -39900, -4266, 560, 7938, -525],70,33]
%
% END OF LATTICE L71.33
%
\hbox{}\par\smallskip%
%
% LATTICE L71.34
%
\leavevmode%
\llap{\spacer}%
${L_{71.34}}=\pdualwitharg{3.7}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{630}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26740980\\-221760\\-12195540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-221760\\-1722\\-100737
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12195540\\-100737\\-5560558
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4951\\-39486\\11574
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15451\\-123230\\36120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-456\\-3637\\1066
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1078\\8595\\-2520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
533\\4251\\-1246
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1069\\-8525\\2499
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-18\\-630\\-126\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-1260\\-420\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\2\\0\\-14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-1260\\0\\630\\0\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-420\\-14\\0\\14\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-21\\-315\\-21\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26740980, -221760, -12195540; -221760, -1722, -100737; -12195540, -100737, -5560558],[1260, 21, 1],[-4951, -15451, -456, 1078, 533, -1069; -39486, -123230, -3637, 8595, 4251, -8525; 11574, 36120, 1066, -2520, -1246, 2499],70,34]
%
% END OF LATTICE L71.34
%
\hbox{}\par\smallskip%
%
% LATTICE L71.35
%
\leavevmode%
\llap{\spacer}%
${L_{71.35}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{s}{2}{630}\above{1pt}{1pt}{}{6}{210}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-125055\\-88515\\6930
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88515\\-61530\\4830
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6930\\4830\\-379
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\43\\640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\76\\1134
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\57\\855
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\-133
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-105\\-1575
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\64\\945
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-45\\-35\\-315\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-42\\-630\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\90\\0\\-630\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-42\\0\\14\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-630\\-630\\0\\630\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-315\\-105\\-315\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-125055, -88515, 6930; -88515, -61530, 4830; 6930, 4830, -379],[315, 105, 1],[5, 9, 7, -1, -13, 7; 43, 76, 57, -9, -105, 64; 640, 1134, 855, -133, -1575, 945],70,35]
%
% END OF LATTICE L71.35
%
\hbox{}\par\smallskip%
%
% LATTICE L71.36
%
\leavevmode%
\llap{\spacer}%
${L_{71.36}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{126}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{}{6}{210}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-219870\\-290430\\84735
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-290430\\-375585\\109620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84735\\109620\\-31994
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\2103\\7335
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\1421\\4956
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\476\\1660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-36\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-562\\-1960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\632\\2205
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-45\\-315\\-315\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-126\\-210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\10\\0\\-70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-126\\0\\126\\0\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-210\\-70\\0\\70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-105\\-315\\-105\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-219870, -290430, 84735; -290430, -375585, 109620; 84735, 109620, -31994],[315, 105, 1],[49, 33, 11, -1, -13, 15; 2103, 1421, 476, -36, -562, 632; 7335, 4956, 1660, -126, -1960, 2205],70,36]
%
% END OF LATTICE L71.36
%
\hbox{}\par\smallskip%
%
% LATTICE L71.37
%
\leavevmode%
\llap{\spacer}%
${L_{71.37}}=\pdualwitharg{2.3.7}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{840}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{}{6}{56}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91166040\\25769100\\22651020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25769100\\7283976\\6402564
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22651020\\6402564\\5627849
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1487\\8\\-5994
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4254\\25\\-17150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
738\\5\\-2976
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
416\\5\\-1680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
438\\1\\-1764
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
945\\4\\-3808
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-36\\-420\\-252\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\140\\0\\-840\\-840\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\24\\0\\-168\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-840\\0\\840\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-840\\-168\\0\\168\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-84\\-420\\-84\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[91166040, 25769100, 22651020; 25769100, 7283976, 6402564; 22651020, 6402564, 5627849],[420, 84, 1],[1487, 4254, 738, 416, 438, 945; 8, 25, 5, 5, 1, 4; -5994, -17150, -2976, -1680, -1764, -3808],70,37]
%
% END OF LATTICE L71.37
%
\hbox{}\par\smallskip%
%
% LATTICE L71.38
%
\leavevmode%
\llap{\spacer}%
${L_{71.38}}=\pdualwitharg{3.5.7}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{}{6}{70}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-807660\\843360\\344400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
843360\\-880530\\-359625
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
344400\\-359625\\-146858
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
443\\20\\990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
331\\14\\742
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\7\\405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-1\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-327\\-13\\-735
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\2\\35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-90\\-210\\-630\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-84\\-420\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\30\\0\\-210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-84\\0\\42\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-420\\-210\\0\\210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-105\\-105\\-105\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-807660, 843360, 344400; 843360, -880530, -359625; 344400, -359625, -146858],[420, 105, 1],[443, 331, 180, -10, -327, 17; 20, 14, 7, -1, -13, 2; 990, 742, 405, -21, -735, 35],70,38]
%
% END OF LATTICE L71.38
%
\hbox{}\par\smallskip%
%
% LATTICE L71.39
%
\leavevmode%
\llap{\spacer}%
${L_{71.39}}=\pdualwitharg{2.5.7}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{}{6}{840}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
225898680\\-5599860\\-60125100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5599860\\143080\\1493660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60125100\\1493660\\16005279
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27652\\-83935\\111710
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47023\\-142734\\189966
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12847\\-38996\\51900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-201\\610\\-812
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2911\\-8836\\11760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52294\\-158733\\211260
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-60\\-140\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-168\\-840\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\0\\-280\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-168\\0\\56\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-840\\-280\\0\\280\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-420\\-420\\-420\\840
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[225898680, -5599860, -60125100; -5599860, 143080, 1493660; -60125100, 1493660, 16005279],[420, 140, 1],[27652, 47023, 12847, -201, 2911, 52294; -83935, -142734, -38996, 610, -8836, -158733; 111710, 189966, 51900, -812, 11760, 211260],70,39]
%
% END OF LATTICE L71.39
%
\hbox{}\par\smallskip%
%
% LATTICE L71.40
%
\leavevmode%
\llap{\spacer}%
${L_{71.40}}=\pdualwitharg{2.5}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{2520}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{360}\above{1pt}{1pt}{}{6}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
849240\\64260\\-212940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64260\\37560\\-16080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-212940\\-16080\\53393
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8761\\35\\-34930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1997\\8\\-7962
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10429\\42\\-41580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1535\\6\\-6120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2769\\11\\-11040
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-1260\\-140\\-1260\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-360\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\0\\2520\\0\\-2520\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-24\\0\\8\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-360\\-2520\\0\\360\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-1260\\-60\\-180\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[849240, 64260, -212940; 64260, 37560, -16080; -212940, -16080, 53393],[1260, 60, 1],[-8761, -1997, -10429, 1, -1535, -2769; 35, 8, 42, 0, 6, 11; -34930, -7962, -41580, 4, -6120, -11040],70,40]
%
% END OF LATTICE L71.40
%
\hbox{}\par\smallskip%
%
% LATTICE L71.41
%
\leavevmode%
\llap{\spacer}%
${L_{71.41}}=\pdualwitharg{2.3.5}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1260}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{}{6}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7766083080\\266680260\\-1947023820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
266680260\\9163560\\-66859020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1947023820\\-66859020\\488135617
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1206011\\-11823\\4808790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
102623\\-1006\\409194
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
218531\\-2142\\871360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9805\\-96\\39096
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6325\\-62\\25220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
97403\\-955\\388380
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1260\\0\\-1260\\-1260\\-1260\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-72\\-120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\0\\280\\0\\-280\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-72\\0\\72\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-120\\-280\\0\\40\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-420\\-180\\-60\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7766083080, 266680260, -1947023820; 266680260, 9163560, -66859020; -1947023820, -66859020, 488135617],[1260, 60, 1],[1206011, 102623, 218531, 9805, 6325, 97403; -11823, -1006, -2142, -96, -62, -955; 4808790, 409194, 871360, 39096, 25220, 388380],70,41]
%
% END OF LATTICE L71.41
%
\hbox{}\par\smallskip%
%
% LATTICE L71.42
%
\leavevmode%
\llap{\spacer}%
${L_{71.42}}=\pdualwitharg{2.7}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{*}{2}{504}\above{1pt}{1pt}{}{6}{168}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
118205768520\\-1032447780\\-29657373480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1032447780\\9017736\\259037184
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29657373480\\259037184\\7440921139
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
639\\25\\2546
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4059\\230\\16170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
163\\48\\648
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-913\\30\\-3640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
443\\48\\1764
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1855\\43\\7392
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-36\\-140\\-252\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-840\\-2520\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\72\\0\\-504\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-840\\0\\280\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-2520\\-504\\0\\504\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-252\\-420\\-252\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[118205768520, -1032447780, -29657373480; -1032447780, 9017736, 259037184; -29657373480, 259037184, 7440921139],[1260, 84, 1],[639, 4059, 163, -913, 443, 1855; 25, 230, 48, 30, 48, 43; 2546, 16170, 648, -3640, 1764, 7392],70,42]
%
% END OF LATTICE L71.42
%
\hbox{}\par\smallskip%
%
% LATTICE L71.43
%
\leavevmode%
\llap{\spacer}%
${L_{71.43}}=\pdualwitharg{2.3.7}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{2520}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{}{6}{168}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
345027969720\\28945764540\\-86357997180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28945764540\\2428374984\\-7244914824
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-86357997180\\-7244914824\\21614780051
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60931\\-33\\-243450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190421\\-110\\-760830
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12543\\-8\\-50116
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31219\\-30\\-124740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-855\\0\\-3416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30001\\-13\\-119868
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-36\\-1260\\-252\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-2520\\-840\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\8\\0\\-56\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-2520\\0\\2520\\0\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-840\\-56\\0\\56\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-84\\-1260\\-84\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[345027969720, 28945764540, -86357997180; 28945764540, 2428374984, -7244914824; -86357997180, -7244914824, 21614780051],[1260, 84, 1],[-60931, -190421, -12543, -31219, -855, -30001; -33, -110, -8, -30, 0, -13; -243450, -760830, -50116, -124740, -3416, -119868],70,43]
%
% END OF LATTICE L71.43
%
\hbox{}\par\smallskip%
%
% LATTICE L71.44
%
\leavevmode%
\llap{\spacer}%
${L_{71.44}}=\pdualwitharg{5.7}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{630}\above{1pt}{1pt}{}{6}{210}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1171854180\\-228808440\\2163420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-228808440\\-44675610\\422415
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2163420\\422415\\-3994
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-42\\2600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-58\\4158
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-9\\1215
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\-343
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\15\\-2205
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-53\\2520
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-90\\-70\\-630\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-84\\-1260\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\90\\0\\-630\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-84\\0\\14\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-1260\\-630\\0\\630\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-315\\-105\\-315\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1171854180, -228808440, 2163420; -228808440, -44675610, 422415; 2163420, 422415, -3994],[1260, 105, 1],[13, 19, 4, -2, -7, 15; -42, -58, -9, 7, 15, -53; 2600, 4158, 1215, -343, -2205, 2520],70,44]
%
% END OF LATTICE L71.44
%
\hbox{}\par\smallskip%
%
% LATTICE L71.45
%
\leavevmode%
\llap{\spacer}%
${L_{71.45}}=\pdualwitharg{3.5.7}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{126}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{}{6}{210}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30979620\\-9452520\\996660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9452520\\-2883930\\303975
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
996660\\303975\\-31994
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2005\\8118\\14670
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1355\\5486\\9912
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-227\\919\\1660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-69\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
268\\-1085\\-1960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-301\\1219\\2205
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\0\\-90\\-630\\-630\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-252\\-420\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\10\\0\\-70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-252\\0\\126\\0\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-420\\-70\\0\\70\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-105\\-315\\-105\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30979620, -9452520, 996660; -9452520, -2883930, 303975; 996660, 303975, -31994],[1260, 105, 1],[-2005, -1355, -227, 17, 268, -301; 8118, 5486, 919, -69, -1085, 1219; 14670, 9912, 1660, -126, -1960, 2205],70,45]
%
% END OF LATTICE L71.45
%
\hbox{}\par\smallskip%
%
% LATTICE L71.46
%
\leavevmode%
\llap{\spacer}%
${L_{71.46}}=\pdualwitharg{2.3.5.7}(\pfillwitharg{3}({L_{71.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{}{6}{280}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27720\\1204980\\304920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1204980\\53019960\\13416900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
304920\\13416900\\3395197
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-147\\2668\\-10530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81\\1465\\-5782
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67\\1201\\-4740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\149\\-588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-71\\1277\\-5040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103\\1880\\-7420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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60\\0\\-180\\-420\\-1260\\0
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0\\28\\0\\-168\\-840\\-140
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-180\\0\\120\\0\\-840\\-420
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-420\\-168\\0\\168\\0\\-420
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-1260\\-840\\-840\\0\\840\\-420
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0\\-140\\-420\\-420\\-420\\280
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[27720, 1204980, 304920; 1204980, 53019960, 13416900; 304920, 13416900, 3395197],[420, 420, 1],[-147, -81, -67, -9, -71, -103; 2668, 1465, 1201, 149, 1277, 1880; -10530, -5782, -4740, -588, -5040, -7420],70,46]
%
% END OF LATTICE L71.46
%
\hbox{}\par\smallskip%
%
% LATTICE L71.47
%
\leavevmode%
\llap{\spacer}%
${L_{71.47}}=\pdualwitharg{2.5.7}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{360}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{2}{2520}\above{1pt}{1pt}{}{6}{840}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5300363880\\-27078660\\1349753580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27078660\\167160\\-6917400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1349753580\\-6917400\\343735199
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-92377\\277821\\368330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-144447\\434420\\575946
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103199\\310368\\411480
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941\\-2830\\-3752
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-127351\\383004\\507780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-212357\\638657\\846720
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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20\\0\\-180\\-140\\-1260\\0
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0\\84\\0\\-168\\-2520\\-420
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\kern3pt%
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-180\\0\\360\\0\\-2520\\-1260
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-140\\-168\\0\\56\\0\\-420
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-1260\\-2520\\-2520\\0\\2520\\-1260
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-1260\\-420\\-1260\\840
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5300363880, -27078660, 1349753580; -27078660, 167160, -6917400; 1349753580, -6917400, 343735199],[1260, 420, 1],[-92377, -144447, -103199, 941, -127351, -212357; 277821, 434420, 310368, -2830, 383004, 638657; 368330, 575946, 411480, -3752, 507780, 846720],70,47]
%
% END OF LATTICE L71.47
%
\hbox{}\par\smallskip%
%
% LATTICE L71.48
%
\leavevmode%
\llap{\spacer}%
${L_{71.48}}=\pdualwitharg{2.3.5.7}({L_{71.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{504}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{}{6}{840}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93697125480\\8230981500\\17286447780
\end{tabular}\endgroup%
\kern3pt%
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8230981500\\723064440\\1518557040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17286447780\\1518557040\\3189225679
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45413\\-136848\\-180990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26799\\-80755\\-106806
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7999\\-24103\\-31880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2023\\-6093\\-8064
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1651\\-4975\\-6580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26241\\-79078\\-104580
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\0\\-180\\-1260\\-1260\\0
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0\\84\\0\\-504\\-840\\-420
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\kern3pt%
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-180\\0\\40\\0\\-280\\-420
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-1260\\-504\\0\\504\\0\\-1260
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-1260\\-840\\-280\\0\\280\\-420
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0\\-420\\-420\\-1260\\-420\\840
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[93697125480, 8230981500, 17286447780; 8230981500, 723064440, 1518557040; 17286447780, 1518557040, 3189225679],[1260, 420, 1],[45413, 26799, 7999, 2023, 1651, 26241; -136848, -80755, -24103, -6093, -4975, -79078; -180990, -106806, -31880, -8064, -6580, -104580],70,48]
%
% END OF LATTICE L71.48
%

\medskip%
%------------------------------WEYL-GROUP-W72--------------------------------
%
\leavevmode\llap{}%
$W_{72\phantom{0}}$%
\qquad\llap{88} lattices, $\chi=24$%
\hfill%
$22222222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L72.1
%
\leavevmode%
\llap{\spacer}%
${L_{72.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{72}}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-133966140\\1288140\\-32760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1288140\\-12386\\315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32760\\315\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\1146\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\1043\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\209\\38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\1155\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-104\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-105\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\518\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\312\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-420\\-36\\-60\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-210\\-28\\-70\\-140\\-14
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\kern3pt%
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-12\\0\\2\\0\\-4\\-20\\-56\\-10
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-420\\-210\\0\\210\\0\\-210\\-840\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-28\\-4\\0\\4\\0\\-28\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-70\\-20\\-210\\0\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-140\\-56\\-840\\-28\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-10\\-210\\-12\\-10\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-133966140, 1288140, -32760; 1288140, -12386, 315; -32760, 315, -8],[420, 1, 1],[11, 10, 2, 11, -1, -1, 5, 3; 1146, 1043, 209, 1155, -104, -105, 518, 312; 72, 112, 38, 420, 0, -40, -84, -2],71,1]
%
% END OF LATTICE L72.1
%
\hbox{}\par\smallskip%
%
% LATTICE L72.2
%
\leavevmode%
\llap{\spacer}%
${L_{72.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{34}}%
\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34131720\\52920\\26040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52920\\-79\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26040\\-42\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\5700\\10692
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\5698\\10696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\670\\1259
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\5355\\10080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-336\\-628
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-335\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\1680\\3136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\671\\1257
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-12\\-420\\-72\\-60\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-210\\-56\\-70\\-280\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\1\\0\\-4\\-10\\-56\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-210\\0\\105\\0\\-105\\-840\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-56\\-4\\0\\8\\0\\-56\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-70\\-10\\-105\\0\\5\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-280\\-56\\-840\\-56\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-5\\-105\\-12\\-5\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-34131720, 52920, 26040; 52920, -79, -42; 26040, -42, -19],[840, 1, 1],[17, 17, 2, 16, -1, -1, 5, 2; 5700, 5698, 670, 5355, -336, -335, 1680, 671; 10692, 10696, 1259, 10080, -628, -630, 3136, 1257],71,2]
%
% END OF LATTICE L72.2
%
\hbox{}\par\smallskip%
%
% LATTICE L72.3
%
\leavevmode%
\llap{\spacer}%
${L_{72.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-434280\\-43680\\2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43680\\-4393\\253
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2520\\253\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\240\\192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\63\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-10\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61\\-630\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-52\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\280\\196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\52\\39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-840\\-72\\-120\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-210\\-28\\-70\\-140\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\4\\0\\-8\\-40\\-112\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-210\\0\\420\\0\\-420\\-1680\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-28\\-8\\0\\8\\0\\-56\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-70\\-40\\-420\\0\\20\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-140\\-112\\-1680\\-56\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-10\\-210\\-12\\-10\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-434280, -43680, 2520; -43680, -4393, 253; 2520, 253, -14],[840, 1, 1],[-23, -6, 1, 61, 5, -1, -27, -5; 240, 63, -10, -630, -52, 10, 280, 52; 192, 56, -2, -420, -40, 0, 196, 39],71,3]
%
% END OF LATTICE L72.3
%
\hbox{}\par\smallskip%
%
% LATTICE L72.4
%
\leavevmode%
\llap{\spacer}%
${L_{72.4}}=\pfillwitharg{2}({L_{72.1}})$%
{${}={}$}Nikulin $72$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3360\\-1155\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\397\\35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\35\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-210\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\134\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3026\\8820\\-315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
165\\481\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
734\\2140\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
689\\2009\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\175\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-6\\-210\\-9\\-30\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-210\\-14\\-70\\-70\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-2\\-20\\-28\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\0\\210\\0\\-210\\-420\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-14\\-2\\0\\1\\0\\-7\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-70\\-20\\-210\\0\\10\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-70\\-28\\-420\\-7\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-5\\-105\\-3\\-5\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[3360, -1155, -105; -1155, 397, 35; -105, 35, -34],[105, 1, 1],[1, -72, 46, 3026, 165, 734, 689, 60; 3, -210, 134, 8820, 481, 2140, 2009, 175; 0, 7, -5, -315, -17, -75, -70, -6],71,4]
%
% END OF LATTICE L72.4
%
\hbox{}\par\smallskip%
%
% LATTICE L72.5
%
\leavevmode%
\llap{\spacer}%
${L_{72.5}}=\pfillwitharg{2}({L_{72.2}})$%
{${}={}$}Nikulin $34'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28770\\14280\\630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14280\\7088\\315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
630\\315\\59
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5741\\11592\\-606
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2451\\4949\\-259
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\-212\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
988\\-1995\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-419\\846\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4128\\8335\\-435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14207\\28686\\-1498
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3185\\6431\\-336
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-6\\-210\\-18\\-30\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-105\\-14\\-35\\-70\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\1\\0\\-2\\-10\\-28\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-105\\0\\105\\0\\-105\\-420\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-14\\-2\\0\\2\\0\\-14\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-35\\-10\\-105\\0\\5\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-70\\-28\\-420\\-14\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-5\\-105\\-6\\-5\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[28770, 14280, 630; 14280, 7088, 315; 630, 315, 59],[210, 1, 1],[-5741, -2451, 105, 988, -419, -4128, -14207, -3185; 11592, 4949, -212, -1995, 846, 8335, 28686, 6431; -606, -259, 11, 105, -44, -435, -1498, -336],71,5]
%
% END OF LATTICE L72.5
%
\hbox{}\par\smallskip%
%
% LATTICE L72.6
%
\leavevmode%
\llap{\spacer}%
${L_{72.6}}=\main({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-217140\\-43680\\1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43680\\-8786\\253
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1260\\253\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\120\\192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\63\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61\\-315\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-26\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\140\\196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\26\\39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-70\\-20\\-210\\0\\10\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-140\\-56\\-840\\-28\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-5\\-105\\-6\\-5\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-217140, -43680, 1260; -43680, -8786, 253; 1260, 253, -7],[420, 1, 1],[-23, -12, 1, 61, 5, -1, -27, -5; 120, 63, -5, -315, -26, 5, 140, 26; 192, 112, -2, -420, -40, 0, 196, 39],71,6]
%
% END OF LATTICE L72.6
%
\hbox{}\par\smallskip%
%
% LATTICE L72.7
%
\leavevmode%
\llap{\spacer}%
${L_{72.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{72.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132720\\4725\\44835
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4725\\-123\\-1596
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44835\\-1596\\-15146
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
258\\-3\\764
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1312\\-14\\3885
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
230\\-2\\681
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
272\\0\\805
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\1\\-231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76\\0\\-225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
468\\-7\\1386
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
233\\-3\\690
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-70\\-9\\-30\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-210\\-42\\-210\\-210\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-6\\-60\\-84\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-210\\0\\70\\0\\-210\\-420\\-105
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-42\\-6\\0\\3\\0\\-21\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-210\\-60\\-210\\0\\30\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-210\\-84\\-420\\-21\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-15\\-105\\-9\\-15\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-132720, 4725, 44835; 4725, -123, -1596; 44835, -1596, -15146],[105, 3, 1],[258, 1312, 230, 272, -78, -76, 468, 233; -3, -14, -2, 0, 1, 0, -7, -3; 764, 3885, 681, 805, -231, -225, 1386, 690],71,7]
%
% END OF LATTICE L72.7
%
\hbox{}\par\smallskip%
%
% LATTICE L72.8
%
\leavevmode%
\llap{\spacer}%
${L_{72.8}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120559740\\1221990\\-60310950
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1221990\\-12386\\611310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60310950\\611310\\-30171023
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4660\\-4137\\-9399
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3804\\-3353\\-7672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-227\\209\\458
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1354\\1155\\2730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
402\\-366\\-811
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7124\\-6385\\-14370
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12011\\-10731\\-24227
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5289\\-4712\\-10668
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-6\\-210\\-9\\-30\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-210\\-14\\-70\\-70\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-2\\-20\\-28\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\0\\210\\0\\-210\\-420\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-14\\-2\\0\\1\\0\\-7\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-70\\-20\\-210\\0\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-70\\-28\\-420\\-7\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-10\\-210\\-6\\-10\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-120559740, 1221990, -60310950; 1221990, -12386, 611310; -60310950, 611310, -30171023],[210, 2, 1],[4660, 3804, -227, -1354, 402, 7124, 12011, 5289; -4137, -3353, 209, 1155, -366, -6385, -10731, -4712; -9399, -7672, 458, 2730, -811, -14370, -24227, -10668],71,8]
%
% END OF LATTICE L72.8
%
\hbox{}\par\smallskip%
%
% LATTICE L72.9
%
\leavevmode%
\llap{\spacer}%
${L_{72.9}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{72.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145740\\6405\\-28665
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6405\\-205\\1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28665\\1260\\-5638
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-614\\-9\\3120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1040\\-14\\5285
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-182\\-2\\925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128\\0\\651
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62\\1\\-315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-61
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-372\\-7\\1890
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-185\\-3\\940
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-30\\-210\\-45\\-30\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-210\\-70\\-70\\-350\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\0\\-10\\-20\\-140\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\0\\42\\0\\-42\\-420\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-70\\-10\\0\\5\\0\\-35\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-70\\-20\\-42\\0\\2\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-350\\-140\\-420\\-35\\0\\35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-25\\-105\\-15\\-5\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-145740, 6405, -28665; 6405, -205, 1260; -28665, 1260, -5638],[105, 5, 1],[-614, -1040, -182, -128, 62, 12, -372, -185; -9, -14, -2, 0, 1, 0, -7, -3; 3120, 5285, 925, 651, -315, -61, 1890, 940],71,9]
%
% END OF LATTICE L72.9
%
\hbox{}\par\smallskip%
%
% LATTICE L72.10
%
\leavevmode%
\llap{\spacer}%
${L_{72.10}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187425210\\1866060\\-62506710
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1866060\\-18579\\622335
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62506710\\622335\\-20846122
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2185\\-2758\\-6634
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2684\\-3353\\-8148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-157\\209\\477
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-323\\385\\980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
559\\-732\\-1698
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4989\\-6385\\-15150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16847\\-21462\\-51156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3714\\-4712\\-11277
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2\\0\\-6\\-70\\-18\\-30\\-42\\0
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0\\21\\0\\-105\\-42\\-105\\-210\\-21
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-6\\0\\3\\0\\-6\\-30\\-84\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-105\\-30\\-105\\0\\15\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-210\\-84\\-420\\-42\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-15\\-105\\-18\\-15\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-187425210, 1866060, -62506710; 1866060, -18579, 622335; -62506710, 622335, -20846122],[210, 3, 1],[2185, 2684, -157, -323, 559, 4989, 16847, 3714; -2758, -3353, 209, 385, -732, -6385, -21462, -4712; -6634, -8148, 477, 980, -1698, -15150, -51156, -11277],71,10]
%
% END OF LATTICE L72.10
%
\hbox{}\par\smallskip%
%
% LATTICE L72.11
%
\leavevmode%
\llap{\spacer}%
${L_{72.11}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{72.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165480\\8085\\-23205
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8085\\-287\\1134
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23205\\1134\\-3254
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-554\\-9\\3948
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-134\\-2\\955
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-164\\-2\\1169
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82\\0\\585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\1\\-399
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\0\\-385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-1\\342
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-167\\-3\\1190
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-42\\-210\\-63\\-210\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-30\\-14\\-70\\-10\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\14\\0\\-14\\-140\\-28\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-30\\0\\30\\0\\-210\\-60\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-14\\-14\\0\\7\\0\\-7\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-70\\-140\\-210\\0\\70\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-10\\-28\\-60\\-7\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-35\\-105\\-21\\-35\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-165480, 8085, -23205; 8085, -287, 1134; -23205, 1134, -3254],[105, 7, 1],[-554, -134, -164, -82, 56, 54, -48, -167; -9, -2, -2, 0, 1, 0, -1, -3; 3948, 955, 1169, 585, -399, -385, 342, 1190],71,11]
%
% END OF LATTICE L72.11
%
\hbox{}\par\smallskip%
%
% LATTICE L72.12
%
\leavevmode%
\llap{\spacer}%
${L_{72.12}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-321297270\\3154200\\128486820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3154200\\-30965\\-1261365
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
128486820\\-1261365\\-51381896
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8923\\-8274\\-22110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3672\\-3353\\-9100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\209\\515
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
271\\231\\672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-747\\-732\\-1850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1349\\-1277\\-3342
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22829\\-21462\\-56560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5043\\-4712\\-12495
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\0\\-30\\-210\\-90\\-30\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\35\\0\\-105\\-70\\-35\\-350\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\5\\0\\-10\\-10\\-140\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-105\\0\\21\\0\\-21\\-420\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-70\\-10\\0\\10\\0\\-70\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-35\\-10\\-21\\0\\1\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-350\\-140\\-420\\-70\\0\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-25\\-105\\-30\\-5\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-321297270, 3154200, 128486820; 3154200, -30965, -1261365; 128486820, -1261365, -51381896],[210, 5, 1],[-8923, -3672, 208, 271, -747, -1349, -22829, -5043; -8274, -3353, 209, 231, -732, -1277, -21462, -4712; -22110, -9100, 515, 672, -1850, -3342, -56560, -12495],71,12]
%
% END OF LATTICE L72.12
%
\hbox{}\par\smallskip%
%
% LATTICE L72.13
%
\leavevmode%
\llap{\spacer}%
${L_{72.13}}=\pdualwitharg{3}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-1260\\2940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-318\\603
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2940\\603\\-329
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\404\\52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-124\\651\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-47\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
207\\-1085\\-140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\-278\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-251\\1316\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\257\\33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-140\\-36\\-60\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-210\\-84\\-210\\-420\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\6\\0\\-12\\-60\\-168\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-210\\0\\70\\0\\-210\\-840\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-84\\-12\\0\\12\\0\\-84\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-210\\-60\\-210\\0\\30\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-420\\-168\\-840\\-84\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-15\\-105\\-18\\-15\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4620, -1260, 2940; -1260, -318, 603; 2940, 603, -329],[420, 3, 1],[-77, -124, 9, 207, 53, -1, -251, -49; 404, 651, -47, -1085, -278, 5, 1316, 257; 52, 84, -6, -140, -36, 0, 168, 33],71,13]
%
% END OF LATTICE L72.13
%
\hbox{}\par\smallskip%
%
% LATTICE L72.14
%
\leavevmode%
\llap{\spacer}%
${L_{72.14}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
668268930\\-5300190\\332214960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5300190\\42528\\-2634870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
332214960\\-2634870\\165153241
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
481994\\1932\\-969527
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1234615\\4949\\-2483418
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52907\\-212\\106422
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165857\\-665\\333620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105553\\423\\-212319
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2079561\\8335\\-4183020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3578450\\14343\\-7198023
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1604442\\6431\\-3227322
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-70\\-9\\-30\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-210\\-42\\-210\\-210\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-6\\-60\\-84\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-210\\0\\70\\0\\-210\\-420\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-42\\-6\\0\\3\\0\\-21\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-210\\-60\\-210\\0\\30\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-210\\-84\\-420\\-21\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-30\\-210\\-18\\-30\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[668268930, -5300190, 332214960; -5300190, 42528, -2634870; 332214960, -2634870, 165153241],[210, 6, 1],[481994, 1234615, -52907, -165857, 105553, 2079561, 3578450, 1604442; 1932, 4949, -212, -665, 423, 8335, 14343, 6431; -969527, -2483418, 106422, 333620, -212319, -4183020, -7198023, -3227322],71,14]
%
% END OF LATTICE L72.14
%
\hbox{}\par\smallskip%
%
% LATTICE L72.15
%
\leavevmode%
\llap{\spacer}%
${L_{72.15}}=\pdualwitharg{3}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-529620\\-14700\\-3780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14700\\-408\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3780\\-105\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-108\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-112\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\34\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\280\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\36\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-140\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-672\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-178\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-140\\-36\\-60\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-210\\-84\\-210\\-420\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\6\\0\\-12\\-60\\-168\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-210\\0\\70\\0\\-210\\-840\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-84\\-12\\0\\12\\0\\-84\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-210\\-60\\-210\\0\\30\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-420\\-168\\-840\\-84\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-30\\-210\\-36\\-30\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-529620, -14700, -3780; -14700, -408, -105; -3780, -105, -26],[420, 3, 1],[3, 3, -1, -8, -1, 4, 19, 5; -108, -112, 34, 280, 36, -140, -672, -178; 2, 21, 9, 35, 0, -15, -42, -6],71,15]
%
% END OF LATTICE L72.15
%
\hbox{}\par\smallskip%
%
% LATTICE L72.16
%
\leavevmode%
\llap{\spacer}%
${L_{72.16}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-455223090\\4442340\\195063330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4442340\\-43351\\-1903545
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
195063330\\-1903545\\-83584738
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10501\\-8274\\-24318
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-620\\-479\\-1436
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
239\\209\\553
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
233\\165\\540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-865\\-732\\-2002
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7891\\-6385\\-18270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3823\\-3066\\-8852
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5922\\-4712\\-13713
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-42\\-210\\-126\\-210\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-15\\-14\\-35\\-10\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\7\\0\\-14\\-70\\-28\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-15\\0\\15\\0\\-105\\-60\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-14\\-14\\0\\14\\0\\-14\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-35\\-70\\-105\\0\\35\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-10\\-28\\-60\\-14\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-35\\-105\\-42\\-35\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-455223090, 4442340, 195063330; 4442340, -43351, -1903545; 195063330, -1903545, -83584738],[210, 7, 1],[-10501, -620, 239, 233, -865, -7891, -3823, -5922; -8274, -479, 209, 165, -732, -6385, -3066, -4712; -24318, -1436, 553, 540, -2002, -18270, -8852, -13713],71,16]
%
% END OF LATTICE L72.16
%
\hbox{}\par\smallskip%
%
% LATTICE L72.17
%
\leavevmode%
\llap{\spacer}%
${L_{72.17}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{72.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
197382255\\52605\\-66976035
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52605\\15\\-17850
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66976035\\-17850\\22726406
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1454\\25\\-4285
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7874\\154\\-23205
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1756\\40\\-5175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1114\\28\\-3283
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-170\\0\\-501
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3456\\35\\-10185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1303\\18\\-3840
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-30\\-70\\-45\\-30\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\210\\0\\-210\\-210\\-210\\-1050\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\0\\-30\\-60\\-420\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-210\\0\\14\\0\\-42\\-420\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-210\\-30\\0\\15\\0\\-105\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-210\\-60\\-42\\0\\6\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-1050\\-420\\-420\\-105\\0\\105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-75\\-105\\-45\\-15\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[197382255, 52605, -66976035; 52605, 15, -17850; -66976035, -17850, 22726406],[105, 15, 1],[-1454, -7874, -1756, -1114, 0, -170, -3456, -1303; 25, 154, 40, 28, 1, 0, 35, 18; -4285, -23205, -5175, -3283, 0, -501, -10185, -3840],71,17]
%
% END OF LATTICE L72.17
%
\hbox{}\par\smallskip%
%
% LATTICE L72.18
%
\leavevmode%
\llap{\spacer}%
${L_{72.18}}=\pdualwitharg{2}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4375560\\-75180\\1082760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75180\\14176\\-18480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1082760\\-18480\\267937
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27535\\-1068\\111198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49449\\-1918\\199696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10931\\-424\\44144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102857\\-3990\\415380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\1\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5413\\-210\\21860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22017\\-854\\88914
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8276\\-321\\33422
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-24\\-840\\-18\\-120\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-840\\-28\\-280\\-280\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-4\\-80\\-112\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-840\\0\\840\\0\\-840\\-1680\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-28\\-4\\0\\1\\0\\-14\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-280\\-80\\-840\\0\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-280\\-112\\-1680\\-14\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-20\\-420\\-6\\-20\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4375560, -75180, 1082760; -75180, 14176, -18480; 1082760, -18480, 267937],[420, 4, 1],[-27535, -49449, -10931, -102857, 26, -5413, -22017, -8276; -1068, -1918, -424, -3990, 1, -210, -854, -321; 111198, 199696, 44144, 415380, -105, 21860, 88914, 33422],71,18]
%
% END OF LATTICE L72.18
%
\hbox{}\par\smallskip%
%
% LATTICE L72.19
%
\leavevmode%
\llap{\spacer}%
${L_{72.19}}=\pdualwitharg{2}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{840}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36228360\\-152460\\9000180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-152460\\644\\-37876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9000180\\-37876\\2235907
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
329\\-276\\-1329
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1213\\-1022\\-4900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
303\\-256\\-1224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3743\\-3150\\-15120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\-42\\-206
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
307\\-250\\-1240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
331\\-273\\-1337
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
203\\-169\\-820
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-420\\-18\\-60\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-840\\-56\\-280\\-140\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-8\\-80\\-56\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-840\\0\\840\\0\\-840\\-840\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-56\\-8\\0\\4\\0\\-14\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-280\\-80\\-840\\0\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-140\\-56\\-840\\-14\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-20\\-420\\-12\\-20\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[36228360, -152460, 9000180; -152460, 644, -37876; 9000180, -37876, 2235907],[420, 4, 1],[329, 1213, 303, 3743, 51, 307, 331, 203; -276, -1022, -256, -3150, -42, -250, -273, -169; -1329, -4900, -1224, -15120, -206, -1240, -1337, -820],71,19]
%
% END OF LATTICE L72.19
%
\hbox{}\par\smallskip%
%
% LATTICE L72.20
%
\leavevmode%
\llap{\spacer}%
${L_{72.20}}=\pdualwitharg{5}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-327180\\110460\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110460\\-37280\\-285
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\-285\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-36\\-330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\14\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\36\\325
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\210\\1953
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\80\\760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\22\\217
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\112\\1190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-14\\-110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-420\\-180\\-60\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-210\\-140\\-70\\-700\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\10\\0\\-20\\-20\\-280\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-210\\0\\42\\0\\-42\\-840\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-140\\-20\\0\\20\\0\\-140\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-70\\-20\\-42\\0\\2\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-700\\-280\\-840\\-140\\0\\140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-50\\-210\\-60\\-10\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-327180, 110460, 840; 110460, -37280, -285; 840, -285, -2],[420, 5, 1],[-13, 5, 13, 76, 29, 8, 41, -5; -36, 14, 36, 210, 80, 22, 112, -14; -330, 105, 325, 1953, 760, 217, 1190, -110],71,20]
%
% END OF LATTICE L72.20
%
\hbox{}\par\smallskip%
%
% LATTICE L72.21
%
\leavevmode%
\llap{\spacer}%
${L_{72.21}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5921790\\-637350\\2813790
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-637350\\70880\\-302820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2813790\\-302820\\1336997
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
281293\\5796\\-590685
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240179\\4949\\-504350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10291\\-212\\21610
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19361\\-399\\40656
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20532\\423\\-43115
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80907\\1667\\-169896
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
696119\\14343\\-1461775
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
312116\\6431\\-655410
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-30\\-210\\-45\\-30\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-210\\-70\\-70\\-350\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\0\\-10\\-20\\-140\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\0\\42\\0\\-42\\-420\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-70\\-10\\0\\5\\0\\-35\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-70\\-20\\-42\\0\\2\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-350\\-140\\-420\\-35\\0\\35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-50\\-210\\-30\\-10\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5921790, -637350, 2813790; -637350, 70880, -302820; 2813790, -302820, 1336997],[210, 10, 1],[281293, 240179, -10291, -19361, 20532, 80907, 696119, 312116; 5796, 4949, -212, -399, 423, 1667, 14343, 6431; -590685, -504350, 21610, 40656, -43115, -169896, -1461775, -655410],71,21]
%
% END OF LATTICE L72.21
%
\hbox{}\par\smallskip%
%
% LATTICE L72.22
%
\leavevmode%
\llap{\spacer}%
${L_{72.22}}=\pdualwitharg{5}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12180\\420\\1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\85\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\-160\\-127
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\432\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\35\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-77\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-147\\-147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\54\\50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\101\\97
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
269\\1596\\1540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\160\\155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-420\\-180\\-60\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-210\\-140\\-70\\-700\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\10\\0\\-20\\-20\\-280\\-25
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-210\\0\\42\\0\\-42\\-840\\-105
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-140\\-20\\0\\20\\0\\-140\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-70\\-20\\-42\\0\\2\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-700\\-280\\-840\\-140\\0\\140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-25\\-105\\-30\\-5\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12180, 420, 1680; 420, 85, -160; 1680, -160, -127],[420, 5, 1],[73, 6, -13, -25, 9, 17, 269, 27; 432, 35, -77, -147, 54, 101, 1596, 160; 420, 35, -75, -147, 50, 97, 1540, 155],71,22]
%
% END OF LATTICE L72.22
%
\hbox{}\par\smallskip%
%
% LATTICE L72.23
%
\leavevmode%
\llap{\spacer}%
${L_{72.23}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{72.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180471585\\59535\\-61366725
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59535\\21\\-20244
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61366725\\-20244\\20866858
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1640\\25\\-4823
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1268\\22\\-3729
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1978\\40\\-5817
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-896\\20\\-2635
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-964\\0\\-2835
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-558\\5\\-1641
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1471\\18\\-4326
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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7\\0\\-42\\-70\\-63\\-210\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-30\\-42\\-210\\-30\\-21
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\kern3pt%
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-42\\0\\42\\0\\-42\\-420\\-84\\-105
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\kern3pt%
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-70\\-30\\0\\10\\0\\-210\\-60\\-105
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-42\\-42\\0\\21\\0\\-21\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\-420\\-210\\0\\210\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-30\\-84\\-60\\-21\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-105\\-105\\-63\\-105\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[180471585, 59535, -61366725; 59535, 21, -20244; -61366725, -20244, 20866858],[105, 21, 1],[-1640, -1268, -1978, -896, 0, -964, -558, -1471; 25, 22, 40, 20, 1, 0, 5, 18; -4823, -3729, -5817, -2635, 0, -2835, -1641, -4326],71,23]
%
% END OF LATTICE L72.23
%
\hbox{}\par\smallskip%
%
% LATTICE L72.24
%
\leavevmode%
\llap{\spacer}%
${L_{72.24}}=\pdualwitharg{3}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-150360\\0\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\39\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\-12\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\164\\532
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\490\\1596
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\54\\177
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\105\\350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-56\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-55\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\56\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\55\\177
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-12\\-140\\-72\\-60\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-210\\-168\\-210\\-840\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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-140\\-210\\0\\35\\0\\-105\\-840\\-105
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-168\\-12\\0\\24\\0\\-168\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-210\\-30\\-105\\0\\15\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-840\\-168\\-840\\-168\\0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-15\\-105\\-36\\-15\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-150360, 0, 840; 0, 39, -12; 840, -12, -1],[840, 3, 1],[3, 9, 1, 2, -1, -1, 1, 1; 164, 490, 54, 105, -56, -55, 56, 55; 532, 1596, 177, 350, -180, -180, 168, 177],71,24]
%
% END OF LATTICE L72.24
%
\hbox{}\par\smallskip%
%
% LATTICE L72.25
%
\leavevmode%
\llap{\spacer}%
${L_{72.25}}=\pdualwitharg{3}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2787960\\383040\\15960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
383040\\-52626\\-2193
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15960\\-2193\\-91
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\112\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\134\\102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99\\700\\490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\64\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-120\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-532\\-336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-57\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-24\\-280\\-72\\-120\\-168\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-210\\-84\\-210\\-420\\-21
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\12\\0\\-24\\-120\\-336\\-30
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-210\\0\\140\\0\\-420\\-1680\\-210
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-84\\-24\\0\\24\\0\\-168\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-210\\-120\\-420\\0\\60\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-420\\-336\\-1680\\-168\\0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-30\\-210\\-36\\-30\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2787960, 383040, 15960; 383040, -52626, -2193; 15960, -2193, -91],[840, 3, 1],[-1, 16, 19, 99, 9, -17, -75, -8; -8, 112, 134, 700, 64, -120, -532, -57; 16, 105, 102, 490, 36, -90, -336, -30],71,25]
%
% END OF LATTICE L72.25
%
\hbox{}\par\smallskip%
%
% LATTICE L72.26
%
\leavevmode%
\llap{\spacer}%
${L_{72.26}}=\pdualwitharg{7}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1297380\\516600\\-3780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
516600\\205702\\-1505
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3780\\-1505\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\12\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-68
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-17\\-266
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-45\\-660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-14\\-196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-84\\-420\\-252\\-420\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\2\\0\\-30\\-28\\-70\\-20\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\14\\0\\-28\\-140\\-56\\-35
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-28\\-28\\0\\28\\0\\-28\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-70\\-140\\-210\\0\\70\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-20\\-56\\-120\\-28\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-35\\-105\\-42\\-35\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1297380, 516600, -3780; 516600, 205702, -1505; -3780, -1505, 11],[420, 7, 1],[-5, 1, 6, 16, 5, -2, -3, -3; 12, -3, -17, -45, -14, 5, 8, 8; -84, -68, -266, -660, -196, 0, 64, 63],71,26]
%
% END OF LATTICE L72.26
%
\hbox{}\par\smallskip%
%
% LATTICE L72.27
%
\leavevmode%
\llap{\spacer}%
${L_{72.27}}=\pdualwitharg{7}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-101220\\6300\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6300\\-392\\105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\105\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-46\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-60\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\120\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\44\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\42\\-28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-84\\-420\\-252\\-420\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-30\\-28\\-70\\-20\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\14\\0\\-28\\-140\\-56\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-30\\0\\30\\0\\-210\\-120\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-28\\-28\\0\\28\\0\\-28\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-70\\-140\\-210\\0\\70\\0\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-20\\-56\\-120\\-28\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-70\\-210\\-84\\-70\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-101220, 6300, -1680; 6300, -392, 105; -1680, 105, -26],[420, 7, 1],[-1, -1, -3, -4, 1, 8, 3, 3; -24, -16, -46, -60, 16, 120, 44, 42; -42, -1, 7, 15, 0, -35, -18, -28],71,27]
%
% END OF LATTICE L72.27
%
\hbox{}\par\smallskip%
%
% LATTICE L72.28
%
\leavevmode%
\llap{\spacer}%
${L_{72.28}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2326170\\-465990\\1081080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-465990\\99232\\-216510
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1081080\\-216510\\502429
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
277370\\5796\\-594321
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33833\\707\\-72494
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10147\\-212\\21742
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13637\\-285\\29220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20245\\423\\-43379
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
398889\\8335\\-854700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
98058\\2049\\-210109
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
307762\\6431\\-659442
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-42\\-210\\-63\\-210\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-30\\-14\\-70\\-10\\-14
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-42\\0\\14\\0\\-14\\-140\\-28\\-70
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\kern3pt%
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-210\\-30\\0\\30\\0\\-210\\-60\\-210
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-14\\-14\\0\\7\\0\\-7\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-70\\-140\\-210\\0\\70\\0\\-70
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-10\\-28\\-60\\-7\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-70\\-210\\-42\\-70\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2326170, -465990, 1081080; -465990, 99232, -216510; 1081080, -216510, 502429],[210, 14, 1],[277370, 33833, -10147, -13637, 20245, 398889, 98058, 307762; 5796, 707, -212, -285, 423, 8335, 2049, 6431; -594321, -72494, 21742, 29220, -43379, -854700, -210109, -659442],71,28]
%
% END OF LATTICE L72.28
%
\hbox{}\par\smallskip%
%
% LATTICE L72.29
%
\leavevmode%
\llap{\spacer}%
${L_{72.29}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4851210\\708750\\1725780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
708750\\106320\\252105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1725780\\252105\\613933
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140293\\3864\\392780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-179681\\4949\\503055
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7699\\-212\\-21555
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4828\\-133\\-13517
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30721\\846\\86010
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60528\\1667\\169461
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1041557\\28686\\2916060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-233499\\6431\\653730
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-30\\-70\\-90\\-30\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\0\\-105\\-210\\-105\\-1050\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\15\\0\\-30\\-30\\-420\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-105\\0\\7\\0\\-21\\-420\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-210\\-30\\0\\30\\0\\-210\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-105\\-30\\-21\\0\\3\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-1050\\-420\\-420\\-210\\0\\210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-75\\-105\\-90\\-15\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4851210, 708750, 1725780; 708750, 106320, 252105; 1725780, 252105, 613933],[210, 15, 1],[-140293, -179681, 7699, 4828, -30721, -60528, -1041557, -233499; 3864, 4949, -212, -133, 846, 1667, 28686, 6431; 392780, 503055, -21555, -13517, 86010, 169461, 2916060, 653730],71,29]
%
% END OF LATTICE L72.29
%
\hbox{}\par\smallskip%
%
% LATTICE L72.30
%
\leavevmode%
\llap{\spacer}%
${L_{72.30}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{72.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
154856940\\-71190\\-30146235
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71190\\35\\13860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30146235\\13860\\5868614
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3368\\10179\\-17325
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-868\\2626\\-4465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1354\\4102\\-6965
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-368\\1116\\-1893
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\396\\-679
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-382\\1151\\-1965
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1007\\3039\\-5180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\0\\-210\\-210\\-315\\-210\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-30\\-70\\-70\\-50\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\70\\0\\-70\\-140\\-140\\-175
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-30\\0\\6\\0\\-42\\-60\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-70\\-70\\0\\35\\0\\-35\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-70\\-140\\-42\\0\\14\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-50\\-140\\-60\\-35\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-175\\-105\\-105\\-35\\0\\35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[154856940, -71190, -30146235; -71190, 35, 13860; -30146235, 13860, 5868614],[105, 35, 1],[-3368, -868, -1354, -368, 0, -132, -382, -1007; 10179, 2626, 4102, 1116, 1, 396, 1151, 3039; -17325, -4465, -6965, -1893, 0, -679, -1965, -5180],71,30]
%
% END OF LATTICE L72.30
%
\hbox{}\par\smallskip%
%
% LATTICE L72.31
%
\leavevmode%
\llap{\spacer}%
${L_{72.31}}=\pdualwitharg{5}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{280}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1455720\\11760\\26040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11760\\-95\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26040\\-210\\-419
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\8172\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\9184\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\1259\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\2520\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-124\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-126\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\616\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\753\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\-60\\-420\\-360\\-60\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\140\\0\\-210\\-280\\-70\\-1400\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\5\\0\\-20\\-10\\-280\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-210\\0\\21\\0\\-21\\-840\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-280\\-20\\0\\40\\0\\-280\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-70\\-10\\-21\\0\\1\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-1400\\-280\\-840\\-280\\0\\280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-25\\-105\\-60\\-5\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1455720, 11760, 26040; 11760, -95, -210; 26040, -210, -419],[840, 5, 1],[65, 73, 10, 20, -1, -1, 5, 6; 8172, 9184, 1259, 2520, -124, -126, 616, 753; -60, -70, -10, -21, 0, 1, 0, -5],71,31]
%
% END OF LATTICE L72.31
%
\hbox{}\par\smallskip%
%
% LATTICE L72.32
%
\leavevmode%
\llap{\spacer}%
${L_{72.32}}=\pdualwitharg{5}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15860040\\-3967320\\38640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3967320\\-992405\\9665
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38640\\9665\\-94
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-281\\1176\\5400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-169\\707\\3220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\414\\1870
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-211\\882\\3948
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-46\\-208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-112\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\88\\415
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\-120\\-840\\-360\\-120\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\35\\0\\-210\\-140\\-70\\-700\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\20\\0\\-40\\-40\\-560\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-210\\0\\84\\0\\-84\\-1680\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-140\\-40\\0\\40\\0\\-280\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-70\\-40\\-84\\0\\4\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-700\\-560\\-1680\\-280\\0\\280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-50\\-210\\-60\\-10\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15860040, -3967320, 38640; -3967320, -992405, 9665; 38640, 9665, -94],[840, 5, 1],[-281, -169, -99, -211, -1, 11, 27, -21; 1176, 707, 414, 882, 4, -46, -112, 88; 5400, 3220, 1870, 3948, 0, -208, -420, 415],71,32]
%
% END OF LATTICE L72.32
%
\hbox{}\par\smallskip%
%
% LATTICE L72.33
%
\leavevmode%
\llap{\spacer}%
${L_{72.33}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7520730\\1031940\\2627310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1031940\\148848\\360465
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2627310\\360465\\917831
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276865\\3864\\791014
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50657\\707\\144729
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15193\\-212\\-43407
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6806\\-95\\-19445
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60625\\846\\173208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-597246\\8335\\1706355
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-293639\\4098\\838938
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-460803\\6431\\1316532
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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14\\0\\-42\\-70\\-126\\-210\\-42\\0
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0\\3\\0\\-15\\-42\\-105\\-30\\-21
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-42\\0\\21\\0\\-42\\-210\\-84\\-105
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-70\\-15\\0\\5\\0\\-105\\-60\\-105
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-42\\-42\\0\\42\\0\\-42\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-105\\-210\\-105\\0\\105\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-30\\-84\\-60\\-42\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-105\\-105\\-126\\-105\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7520730, 1031940, 2627310; 1031940, 148848, 360465; 2627310, 360465, 917831],[210, 21, 1],[-276865, -50657, 15193, 6806, -60625, -597246, -293639, -460803; 3864, 707, -212, -95, 846, 8335, 4098, 6431; 791014, 144729, -43407, -19445, 173208, 1706355, 838938, 1316532],71,33]
%
% END OF LATTICE L72.33
%
\hbox{}\par\smallskip%
%
% LATTICE L72.34
%
\leavevmode%
\llap{\spacer}%
${L_{72.34}}=\pdualwitharg{2.3}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
762720\\-102900\\189840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102900\\14808\\-25620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
189840\\-25620\\47251
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1118\\-41\\-4514
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5534\\-203\\-22344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1064\\-39\\-4296
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2878\\-105\\-11620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\1\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
966\\-35\\-3900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3256\\-119\\-13146
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1119\\-41\\-4518
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-24\\-280\\-18\\-120\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-840\\-84\\-840\\-840\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-12\\-240\\-336\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-840\\0\\280\\0\\-840\\-1680\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-84\\-12\\0\\3\\0\\-42\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-840\\-240\\-840\\0\\120\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-840\\-336\\-1680\\-42\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-60\\-420\\-18\\-60\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[762720, -102900, 189840; -102900, 14808, -25620; 189840, -25620, 47251],[420, 12, 1],[1118, 5534, 1064, 2878, -26, 966, 3256, 1119; -41, -203, -39, -105, 1, -35, -119, -41; -4514, -22344, -4296, -11620, 105, -3900, -13146, -4518],71,34]
%
% END OF LATTICE L72.34
%
\hbox{}\par\smallskip%
%
% LATTICE L72.35
%
\leavevmode%
\llap{\spacer}%
${L_{72.35}}=\pdualwitharg{2.3}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
198217320\\-6351660\\49486080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6351660\\203532\\-1585728
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49486080\\-1585728\\12354481
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-8\\19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-253\\-42\\1008
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93\\-4\\372
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-177\\-70\\700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-14\\-150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
341\\-110\\-1380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
202\\-77\\-819
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\-29\\-204
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-12\\-140\\-18\\-60\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-840\\-168\\-840\\-420\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0\\-24\\-240\\-168\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-840\\0\\280\\0\\-840\\-840\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-168\\-24\\0\\12\\0\\-42\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-840\\-240\\-840\\0\\120\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-420\\-168\\-840\\-42\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-60\\-420\\-36\\-60\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[198217320, -6351660, 49486080; -6351660, 203532, -1585728; 49486080, -1585728, 12354481],[420, 12, 1],[-5, -253, -93, -177, 37, 341, 202, 50; -8, -42, -4, -70, -14, -110, -77, -29; 19, 1008, 372, 700, -150, -1380, -819, -204],71,35]
%
% END OF LATTICE L72.35
%
\hbox{}\par\smallskip%
%
% LATTICE L72.36
%
\leavevmode%
\llap{\spacer}%
${L_{72.36}}=\pdualwitharg{7}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2252040\\26880\\-5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26880\\259\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5040\\-56\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\228\\3444
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\46\\692
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\46\\693
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\45\\690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-48\\-700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-95\\-1400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-48\\-704
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-84\\-420\\-504\\-420\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-30\\-56\\-70\\-40\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\7\\0\\-28\\-70\\-56\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-30\\0\\15\\0\\-105\\-120\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-56\\-28\\0\\56\\0\\-56\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-70\\-70\\-105\\0\\35\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-40\\-56\\-120\\-56\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-35\\-105\\-84\\-35\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2252040, 26880, -5040; 26880, 259, -56; -5040, -56, 11],[840, 7, 1],[5, 1, 1, 1, -1, -2, -1, 0; 228, 46, 46, 45, -48, -95, -48, -1; 3444, 692, 693, 690, -700, -1400, -704, -7],71,36]
%
% END OF LATTICE L72.36
%
\hbox{}\par\smallskip%
%
% LATTICE L72.37
%
\leavevmode%
\llap{\spacer}%
${L_{72.37}}=\pdualwitharg{7}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12825960\\-3853080\\-16800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3853080\\1157513\\5047
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16800\\5047\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\24\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-76
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-34\\-602
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-90\\-1500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-28\\-448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\16\\148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\16\\147
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-168\\-840\\-504\\-840\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-30\\-28\\-70\\-20\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\28\\0\\-56\\-280\\-112\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-30\\0\\60\\0\\-420\\-240\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-28\\-56\\0\\56\\0\\-56\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-70\\-280\\-420\\0\\140\\0\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-20\\-112\\-240\\-56\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-70\\-210\\-84\\-70\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12825960, -3853080, -16800; -3853080, 1157513, 5047; -16800, 5047, 22],[840, 7, 1],[7, -1, -11, -29, -9, 3, 5, 5; 24, -3, -34, -90, -28, 10, 16, 16; -168, -76, -602, -1500, -448, 0, 148, 147],71,37]
%
% END OF LATTICE L72.37
%
\hbox{}\par\smallskip%
%
% LATTICE L72.38
%
\leavevmode%
\llap{\spacer}%
${L_{72.38}}=\pdualwitharg{3.5}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171780\\11340\\2100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11340\\-330\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2100\\-75\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\70\\-460
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-63\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-205\\1350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-413\\2716
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-484\\3180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-139\\912
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-770\\5040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\68\\-450
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-60\\-140\\-180\\-60\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\210\\0\\-210\\-420\\-210\\-2100\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\30\\0\\-60\\-60\\-840\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-210\\0\\14\\0\\-42\\-840\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-420\\-60\\0\\60\\0\\-420\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-210\\-60\\-42\\0\\6\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-2100\\-840\\-840\\-420\\0\\420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-150\\-210\\-180\\-30\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-171780, 11340, 2100; 11340, -330, -75; 2100, -75, -16],[420, 15, 1],[-1, 1, 3, 6, 7, 2, 11, -1; 70, -63, -205, -413, -484, -139, -770, 68; -460, 420, 1350, 2716, 3180, 912, 5040, -450],71,38]
%
% END OF LATTICE L72.38
%
\hbox{}\par\smallskip%
%
% LATTICE L72.39
%
\leavevmode%
\llap{\spacer}%
${L_{72.39}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276792811680\\226771650\\-137627568960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
226771650\\-185790\\112755930
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-137627568960\\112755930\\-68431501609
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
138796\\-1379\\-279145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
342484\\-3353\\-688800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19481\\209\\39180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8402\\77\\16898
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34942\\-366\\-70275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
126012\\-1277\\-253434
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1065929\\-10731\\-2143785
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
470811\\-4712\\-946890
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-30\\-70\\-45\\-30\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\210\\0\\-210\\-210\\-210\\-1050\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\0\\-30\\-60\\-420\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-210\\0\\14\\0\\-42\\-420\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-210\\-30\\0\\15\\0\\-105\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-210\\-60\\-42\\0\\6\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-1050\\-420\\-420\\-105\\0\\105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-150\\-210\\-90\\-30\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-276792811680, 226771650, -137627568960; 226771650, -185790, 112755930; -137627568960, 112755930, -68431501609],[210, 30, 1],[138796, 342484, -19481, -8402, 34942, 126012, 1065929, 470811; -1379, -3353, 209, 77, -366, -1277, -10731, -4712; -279145, -688800, 39180, 16898, -70275, -253434, -2143785, -946890],71,39]
%
% END OF LATTICE L72.39
%
\hbox{}\par\smallskip%
%
% LATTICE L72.40
%
\leavevmode%
\llap{\spacer}%
${L_{72.40}}=\pdualwitharg{3.5}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{420}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7980\\-2520\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-465\\-690
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-690\\59
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87\\176\\200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\91\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\-93\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\-105\\-119
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-26\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\77\\87
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-733\\1484\\1680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85\\172\\195
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-60\\-140\\-180\\-60\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\210\\0\\-210\\-420\\-210\\-2100\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\30\\0\\-60\\-60\\-840\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-210\\0\\14\\0\\-42\\-840\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-420\\-60\\0\\60\\0\\-420\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-210\\-60\\-42\\0\\6\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-2100\\-840\\-840\\-420\\0\\420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-75\\-105\\-90\\-15\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7980, -2520, -1260; -2520, -465, -690; -1260, -690, 59],[420, 15, 1],[-87, -45, 46, 52, 13, -38, -733, -85; 176, 91, -93, -105, -26, 77, 1484, 172; 200, 105, -105, -119, -30, 87, 1680, 195],71,40]
%
% END OF LATTICE L72.40
%
\hbox{}\par\smallskip%
%
% LATTICE L72.41
%
\leavevmode%
\llap{\spacer}%
${L_{72.41}}=\pdualwitharg{2}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{840}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-24\\-840\\-36\\-120\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-840\\-56\\-280\\-140\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-8\\-80\\-56\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-840\\0\\840\\0\\-840\\-840\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-56\\-8\\0\\4\\0\\-14\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-280\\-80\\-840\\0\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-140\\-56\\-840\\-14\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-40\\-840\\-24\\-40\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[840, 0, 0; 0, 8, 0; 0, 0, -1],[840, 8, 1],[-1, -1, 0, 1, 0, -1, -1, -1; -3, 0, 1, 0, -1, -10, -7, -5; -30, -28, 0, 0, -2, -40, -35, -32],71,41]
%
% END OF LATTICE L72.41
%
\hbox{}\par\smallskip%
%
% LATTICE L72.42
%
\leavevmode%
\llap{\spacer}%
${L_{72.42}}=\pdualwitharg{2}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{840}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-175560\\10920\\5880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10920\\272\\112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5880\\112\\43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-222\\441
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\224\\-448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\223\\-444
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-210\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-222\\442
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-2000\\3980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-2667\\5306
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-889\\1768
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-420\\-18\\-60\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-840\\-56\\-280\\-280\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-8\\-80\\-112\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-840\\0\\840\\0\\-840\\-1680\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-56\\-8\\0\\4\\0\\-28\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-280\\-80\\-840\\0\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-280\\-112\\-1680\\-28\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-40\\-840\\-24\\-40\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-175560, 10920, 5880; 10920, 272, 112; 5880, 112, 43],[840, 8, 1],[1, -1, -1, 1, 1, 9, 12, 4; -222, 224, 223, -210, -222, -2000, -2667, -889; 441, -448, -444, 420, 442, 3980, 5306, 1768],71,42]
%
% END OF LATTICE L72.42
%
\hbox{}\par\smallskip%
%
% LATTICE L72.43
%
\leavevmode%
\llap{\spacer}%
${L_{72.43}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${210}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68915910\\4027800\\-27284040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4027800\\248080\\-1594635
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27284040\\-1594635\\10801843
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3758983\\11592\\9496410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
229256\\707\\579175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68758\\-212\\-173705
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18481\\-57\\-46689
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
274367\\846\\693140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
540585\\1667\\1365693
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1328907\\4098\\3357250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2085433\\6431\\5268480
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210\\0\\-210\\-210\\-630\\-210\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-15\\-70\\-35\\-50\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\35\\0\\-70\\-70\\-140\\-175
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-15\\0\\3\\0\\-21\\-60\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-70\\-70\\0\\70\\0\\-70\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-35\\-70\\-21\\0\\7\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-50\\-140\\-60\\-70\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-175\\-105\\-210\\-35\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[68915910, 4027800, -27284040; 4027800, 248080, -1594635; -27284040, -1594635, 10801843],[210, 35, 1],[3758983, 229256, -68758, -18481, 274367, 540585, 1328907, 2085433; 11592, 707, -212, -57, 846, 1667, 4098, 6431; 9496410, 579175, -173705, -46689, 693140, 1365693, 3357250, 5268480],71,43]
%
% END OF LATTICE L72.43
%
\hbox{}\par\smallskip%
%
% LATTICE L72.44
%
\leavevmode%
\llap{\spacer}%
${L_{72.44}}=\pdualwitharg{2.5}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61333440\\-385980\\15159480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385980\\2440\\-95400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15159480\\-95400\\3746893
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\15\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
312\\91\\-1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
926\\255\\-3740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6988\\1911\\-28224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
796\\217\\-3215
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1232\\335\\-4976
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5286\\1435\\-21350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
359\\97\\-1450
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-120\\-840\\-90\\-120\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\280\\0\\-840\\-140\\-280\\-1400\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\40\\0\\-20\\-80\\-560\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-840\\0\\168\\0\\-168\\-1680\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-140\\-20\\0\\5\\0\\-70\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-280\\-80\\-168\\0\\8\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-1400\\-560\\-1680\\-70\\0\\140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-100\\-420\\-30\\-20\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[61333440, -385980, 15159480; -385980, 2440, -95400; 15159480, -95400, 3746893],[420, 20, 1],[52, 312, 926, 6988, 796, 1232, 5286, 359; 15, 91, 255, 1911, 217, 335, 1435, 97; -210, -1260, -3740, -28224, -3215, -4976, -21350, -1450],71,44]
%
% END OF LATTICE L72.44
%
\hbox{}\par\smallskip%
%
% LATTICE L72.45
%
\leavevmode%
\llap{\spacer}%
${L_{72.45}}=\pdualwitharg{2.5}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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2394420\\-381360\\580020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-381360\\60740\\-92380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
580020\\-92380\\140503
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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85\\9\\-345
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-5\\-1\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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851\\105\\-3444
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299\\37\\-1210
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335\\41\\-1356
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925\\112\\-3745
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237\\28\\-960
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-100\\-420\\-60\\-20\\0\\20
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2394420, -381360, 580020; -381360, 60740, -92380; 580020, -92380, 140503],[420, 20, 1],[85, -139, -5, 851, 299, 335, 925, 237; 9, -21, -1, 105, 37, 41, 112, 28; -345, 560, 20, -3444, -1210, -1356, -3745, -960],71,45]
%
% END OF LATTICE L72.45
%
\hbox{}\par\smallskip%
%
% LATTICE L72.46
%
\leavevmode%
\llap{\spacer}%
${L_{72.46}}=\pdualwitharg{3.7}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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9660\\-840\\0
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\140
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-115
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-126
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\HardButStrongLineBreak\kern3pt%
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1\\10\\105
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1\\12\\120
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1\\13\\126
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9660, -840, 0; -840, -147, 21; 0, 21, -2],[420, 21, 1],[1, 0, -1, -1, -1, 1, 1, 1; 16, 2, -8, -10, -12, 10, 12, 13; 140, 9, -105, -115, -126, 105, 120, 126],71,46]
%
% END OF LATTICE L72.46
%
\hbox{}\par\smallskip%
%
% LATTICE L72.47
%
\leavevmode%
\llap{\spacer}%
${L_{72.47}}=\pdualwitharg{3.7}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-548940\\137340\\-2100
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
137340\\-34356\\525
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2100\\525\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3\\-18\\-406
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0\\-1\\-69
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\9\\63
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\HardButStrongLineBreak\kern3pt%
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1\\5\\65
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\HardButStrongLineBreak\kern3pt%
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-1\\-4\\0
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\HardButStrongLineBreak\kern3pt%
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-11\\-55\\-735
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-5\\-26\\-402
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-210\\-210\\-252\\-210\\0\\84
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-548940, 137340, -2100; 137340, -34356, 525; -2100, 525, -8],[420, 21, 1],[-3, 0, 2, 1, -1, -11, -5, -7; -18, -1, 9, 5, -4, -55, -26, -38; -406, -69, 63, 65, 0, -735, -402, -672],71,47]
%
% END OF LATTICE L72.47
%
\hbox{}\par\smallskip%
%
% LATTICE L72.48
%
\leavevmode%
\llap{\spacer}%
${L_{72.48}}=\pdualwitharg{2.3.7}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-391894606530\\225125670\\-194849575307
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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212418\\-1379\\-427231
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75212\\-479\\-151272
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52592\\-366\\-105777
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958392\\-6385\\-1927590
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232105\\-1533\\-466827
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718935\\-4712\\-1445976
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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-210\\-210\\-420\\-210\\0\\210\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
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-21\\-30\\-84\\-60\\-21\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-210\\-210\\-126\\-210\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-788204862060, 452787930, -391894606530; 452787930, -260106, 225125670; -391894606530, 225125670, -194849575307],[210, 42, 1],[212418, 75212, -29089, -9402, 52592, 958392, 232105, 718935; -1379, -479, 209, 55, -366, -6385, -1533, -4712; -427231, -151272, 58506, 18910, -105777, -1927590, -466827, -1445976],71,48]
%
% END OF LATTICE L72.48
%
\hbox{}\par\smallskip%
%
% LATTICE L72.49
%
\leavevmode%
\llap{\spacer}%
${L_{72.49}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{72.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-121405935\\-86159430\\30688560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-86159430\\-61144860\\21778785
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30688560\\21778785\\-7757242
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3\\34798\\97685
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-2\\25258\\70905
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-2\\30932\\86835
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0\\1034\\2903
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\-10548\\-29610
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0\\-2042\\-5733
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-1\\9036\\25365
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-3\\31457\\88305
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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-210\\-210\\-420\\-42\\0\\42\\0\\-105
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-105\\-150\\-420\\-60\\-105\\0\\15\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-525\\-105\\-315\\-105\\0\\105
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-121405935, -86159430, 30688560; -86159430, -61144860, 21778785; 30688560, 21778785, -7757242],[105, 105, 1],[-3, -2, -2, 0, 1, 0, -1, -3; 34798, 25258, 30932, 1034, -10548, -2042, 9036, 31457; 97685, 70905, 86835, 2903, -29610, -5733, 25365, 88305],71,49]
%
% END OF LATTICE L72.49
%
\hbox{}\par\smallskip%
%
% LATTICE L72.50
%
\leavevmode%
\llap{\spacer}%
${L_{72.50}}=\pdualwitharg{2.7}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
729120\\94500\\181440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
94500\\12712\\23520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
181440\\23520\\45151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2674\\-93\\10794
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-546\\-19\\2204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-548\\-19\\2212
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-446\\-15\\1800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\1\\-105
\end{tabular}\endgroup%
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-1318\\-45\\5320
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-492\\-17\\1986
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\HardButStrongLineBreak\kern3pt%
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-1037\\-36\\4186
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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84\\0\\-168\\-840\\-126\\-840\\-84\\0
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\kern3pt%
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0\\8\\0\\-120\\-28\\-280\\-40\\-28
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\kern3pt%
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-840\\-280\\-560\\-840\\0\\280\\0\\-140
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\kern3pt%
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-84\\-40\\-112\\-240\\-14\\0\\4\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-140\\-420\\-42\\-140\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[729120, 94500, 181440; 94500, 12712, 23520; 181440, 23520, 45151],[420, 28, 1],[-2674, -546, -548, -446, 26, -1318, -492, -1037; -93, -19, -19, -15, 1, -45, -17, -36; 10794, 2204, 2212, 1800, -105, 5320, 1986, 4186],71,50]
%
% END OF LATTICE L72.50
%
\hbox{}\par\smallskip%
%
% LATTICE L72.51
%
\leavevmode%
\llap{\spacer}%
${L_{72.51}}=\pdualwitharg{2.7}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37508520\\-216300\\9299220
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216300\\1148\\-53620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9299220\\-53620\\2305489
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1825\\423\\7371
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1131\\262\\4568
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1851\\428\\7476
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1441\\330\\5820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
565\\-132\\-2282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2877\\-670\\-11620
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\-18\\-311
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-617\\143\\2492
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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21\\0\\-84\\-420\\-126\\-420\\-21\\0
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-126\\-56\\-56\\0\\28\\0\\-14\\-84
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\kern3pt%
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-420\\-280\\-560\\-840\\0\\280\\0\\-140
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-21\\-20\\-56\\-120\\-14\\0\\1\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-140\\-420\\-84\\-140\\0\\28
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[37508520, -216300, 9299220; -216300, 1148, -53620; 9299220, -53620, 2305489],[420, 28, 1],[-1825, -1131, -1851, -1441, 565, 2877, 77, -617; 423, 262, 428, 330, -132, -670, -18, 143; 7371, 4568, 7476, 5820, -2282, -11620, -311, 2492],71,51]
%
% END OF LATTICE L72.51
%
\hbox{}\par\smallskip%
%
% LATTICE L72.52
%
\leavevmode%
\llap{\spacer}%
${L_{72.52}}=\pdualwitharg{3.5}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{840}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\-4200\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4200\\-2730\\615
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\615\\-137
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\260\\1280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\896\\4410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\125\\615
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\84\\413
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\-69
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\67\\330
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-60\\-140\\-360\\-60\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-210\\-840\\-210\\-4200\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\15\\0\\-60\\-30\\-840\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-210\\0\\7\\0\\-21\\-840\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-840\\-60\\0\\120\\0\\-840\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-210\\-30\\-21\\0\\3\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-4200\\-840\\-840\\-840\\0\\840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-75\\-105\\-180\\-15\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[840, -4200, 840; -4200, -2730, 615; 840, 615, -137],[840, 15, 1],[19, 65, 9, 6, -1, -1, 1, 5; 260, 896, 125, 84, -12, -14, 0, 67; 1280, 4410, 615, 413, -60, -69, 0, 330],71,52]
%
% END OF LATTICE L72.52
%
\hbox{}\par\smallskip%
%
% LATTICE L72.53
%
\leavevmode%
\llap{\spacer}%
${L_{72.53}}=\pdualwitharg{3.5}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190680\\36960\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36960\\-6555\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\-195\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\112\\2080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\217\\3990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\134\\2430
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\98\\1736
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-18\\-348
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-112\\-2100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\18\\345
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-120\\-280\\-360\\-120\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\0\\-210\\-420\\-210\\-2100\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\60\\0\\-120\\-120\\-1680\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-210\\0\\28\\0\\-84\\-1680\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-420\\-120\\0\\120\\0\\-840\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-210\\-120\\-84\\0\\12\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-2100\\-1680\\-1680\\-840\\0\\840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-150\\-210\\-180\\-30\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-190680, 36960, 840; 36960, -6555, -195; 840, -195, -2],[840, 15, 1],[31, 60, 37, 27, 1, -5, -31, 5; 112, 217, 134, 98, 4, -18, -112, 18; 2080, 3990, 2430, 1736, 0, -348, -2100, 345],71,53]
%
% END OF LATTICE L72.53
%
\hbox{}\par\smallskip%
%
% LATTICE L72.54
%
\leavevmode%
\llap{\spacer}%
${L_{72.54}}=\pdualwitharg{5.7}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
330540\\38220\\-3360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38220\\4340\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3360\\-385\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-108\\-2310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-20\\-425
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-42\\-875
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-237
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\189
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-420\\-420\\-1260\\-420\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-30\\-140\\-70\\-100\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\70\\0\\-140\\-140\\-280\\-350
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-30\\0\\6\\0\\-42\\-120\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-140\\-140\\0\\140\\0\\-140\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-70\\-140\\-42\\0\\14\\0\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-100\\-280\\-120\\-140\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-350\\-210\\-420\\-70\\0\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[330540, 38220, -3360; 38220, 4340, -385; -3360, -385, 34],[420, 35, 1],[-11, -2, -4, -1, 1, 1, 1, -1; -108, -20, -42, -12, 4, 8, 8, -10; -2310, -425, -875, -237, 140, 189, 190, -210],71,54]
%
% END OF LATTICE L72.54
%
\hbox{}\par\smallskip%
%
% LATTICE L72.55
%
\leavevmode%
\llap{\spacer}%
${L_{72.55}}=\pdualwitharg{2.5.7}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-113737994580\\222050850\\-54269303970
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
222050850\\-433510\\105950040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54269303970\\105950040\\-25894226149
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
294029\\-4137\\-616245
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34716\\-479\\-72760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13393\\209\\28070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2608\\33\\5466
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24231\\-366\\-50785
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88394\\-1277\\-185262
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107056\\-1533\\-224375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
331653\\-4712\\-695100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\0\\-210\\-210\\-315\\-210\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-30\\-70\\-70\\-50\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\70\\0\\-70\\-140\\-140\\-350
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-30\\0\\6\\0\\-42\\-60\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-70\\-70\\0\\35\\0\\-35\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-70\\-140\\-42\\0\\14\\0\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-50\\-140\\-60\\-35\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-350\\-210\\-210\\-70\\0\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-113737994580, 222050850, -54269303970; 222050850, -433510, 105950040; -54269303970, 105950040, -25894226149],[210, 70, 1],[294029, 34716, -13393, -2608, 24231, 88394, 107056, 331653; -4137, -479, 209, 33, -366, -1277, -1533, -4712; -616245, -72760, 28070, 5466, -50785, -185262, -224375, -695100],71,55]
%
% END OF LATTICE L72.55
%
\hbox{}\par\smallskip%
%
% LATTICE L72.56
%
\leavevmode%
\llap{\spacer}%
${L_{72.56}}=\pdualwitharg{5.7}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
141540\\68460\\-2100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68460\\33110\\-1015
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2100\\-1015\\31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\48\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\11\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\27\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\9\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-8\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-420\\-420\\-1260\\-420\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-30\\-140\\-70\\-100\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\70\\0\\-140\\-140\\-280\\-175
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-30\\0\\6\\0\\-42\\-120\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-140\\-140\\0\\140\\0\\-140\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-70\\-140\\-42\\0\\14\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-100\\-280\\-120\\-140\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-175\\-105\\-210\\-35\\0\\35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[141540, 68460, -2100; 68460, 33110, -1015; -2100, -1015, 31],[420, 35, 1],[-23, -5, -12, -4, -1, 2, 3, 0; 48, 11, 27, 9, 2, -5, -8, -1; 0, 20, 70, 24, 0, -28, -60, -35],71,56]
%
% END OF LATTICE L72.56
%
\hbox{}\par\smallskip%
%
% LATTICE L72.57
%
\leavevmode%
\llap{\spacer}%
${L_{72.57}}=\pdualwitharg{3.7}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{s}{2}{420}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19320\\-8400\\3360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8400\\1911\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3360\\-756\\299
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\160\\392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-220\\-546
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-210\\-520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-204\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\170\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\180\\444
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\179\\441
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-168\\-280\\-504\\-840\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-30\\-84\\-210\\-60\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\84\\0\\-168\\-840\\-336\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-30\\0\\20\\0\\-420\\-240\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-84\\-168\\0\\168\\0\\-168\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-210\\-840\\-420\\0\\420\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-60\\-336\\-240\\-168\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-210\\-210\\-252\\-210\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19320, -8400, 3360; -8400, 1911, -756; 3360, -756, 299],[840, 21, 1],[1, 0, -1, -1, -1, 1, 1, 1; 160, -7, -220, -210, -204, 170, 180, 179; 392, -18, -546, -520, -504, 420, 444, 441],71,57]
%
% END OF LATTICE L72.57
%
\hbox{}\par\smallskip%
%
% LATTICE L72.58
%
\leavevmode%
\llap{\spacer}%
${L_{72.58}}=\pdualwitharg{3.7}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-615720\\-41160\\-12600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41160\\-2751\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12600\\-840\\-247
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\204\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\206\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\334\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\255\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\400\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-15\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-112\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-81\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-84\\-140\\-504\\-420\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-30\\-168\\-210\\-120\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\21\\0\\-84\\-210\\-168\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-30\\0\\5\\0\\-105\\-120\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-168\\-84\\0\\168\\0\\-168\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-210\\-210\\-105\\0\\105\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-120\\-168\\-120\\-168\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-105\\-105\\-252\\-105\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-615720, -41160, -12600; -41160, -2751, -840; -12600, -840, -247],[840, 21, 1],[-13, -13, -21, -16, -25, 1, 7, 5; 204, 206, 334, 255, 400, -15, -112, -81; -28, -36, -63, -50, -84, 0, 24, 21],71,58]
%
% END OF LATTICE L72.58
%
\hbox{}\par\smallskip%
%
% LATTICE L72.59
%
\leavevmode%
\llap{\spacer}%
${L_{72.59}}=\pdualwitharg{2.3}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{280}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21560280\\0\\-77280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77280\\0\\277
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-1\\2510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\6972
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\558
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-10\\10320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-7\\8925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-5\\8088
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-24\\-280\\-36\\-120\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-840\\-168\\-840\\-420\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24\\-240\\-168\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-840\\0\\280\\0\\-840\\-840\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-168\\-24\\0\\12\\0\\-42\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-840\\-240\\-840\\0\\120\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-420\\-168\\-840\\-42\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-120\\-840\\-72\\-120\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[21560280, 0, -77280; 0, 24, 0; -77280, 0, 277],[840, 24, 1],[9, 25, 0, 1, 2, 37, 32, 29; -1, 0, 1, 0, -1, -10, -7, -5; 2510, 6972, 0, 280, 558, 10320, 8925, 8088],71,59]
%
% END OF LATTICE L72.59
%
\hbox{}\par\smallskip%
%
% LATTICE L72.60
%
\leavevmode%
\llap{\spacer}%
${L_{72.60}}=\pdualwitharg{2.3}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}{280}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19320\\4200\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4200\\1392\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-14\\-672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\30\\1500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\35\\1722
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\9\\432
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-12\\-140\\-18\\-60\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-840\\-168\\-840\\-840\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0\\-24\\-240\\-336\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-840\\0\\280\\0\\-840\\-1680\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-168\\-24\\0\\12\\0\\-84\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-840\\-240\\-840\\0\\120\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-840\\-336\\-1680\\-84\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-120\\-840\\-72\\-120\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19320, 4200, 0; 4200, 1392, -48; 0, -48, 1],[840, 24, 1],[0, -3, -1, 1, 1, 7, 8, 2; 0, -14, -5, 0, 4, 30, 35, 9; -1, -672, -228, 140, 210, 1500, 1722, 432],71,60]
%
% END OF LATTICE L72.60
%
\hbox{}\par\smallskip%
%
% LATTICE L72.61
%
\leavevmode%
\llap{\spacer}%
${L_{72.61}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60617695530\\198538620\\21887871180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
198538620\\-650265\\-71688435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21887871180\\-71688435\\-7903284686
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132829\\-268416\\370300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23524\\-47527\\65580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9077\\18363\\-25305
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-589\\1189\\-1642
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32843\\-66418\\91560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59901\\-121079\\166992
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
145093\\-293252\\404490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
224742\\-454196\\626535
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\0\\-210\\-70\\-630\\-210\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-15\\-210\\-105\\-150\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\105\\0\\-210\\-210\\-420\\-525
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-15\\0\\1\\0\\-21\\-60\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-210\\-210\\0\\210\\0\\-210\\-630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-105\\-210\\-21\\0\\21\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-150\\-420\\-60\\-210\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-525\\-105\\-630\\-105\\0\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-60617695530, 198538620, 21887871180; 198538620, -650265, -71688435; 21887871180, -71688435, -7903284686],[210, 105, 1],[132829, 23524, -9077, -589, 32843, 59901, 145093, 224742; -268416, -47527, 18363, 1189, -66418, -121079, -293252, -454196; 370300, 65580, -25305, -1642, 91560, 166992, 404490, 626535],71,61]
%
% END OF LATTICE L72.61
%
\hbox{}\par\smallskip%
%
% LATTICE L72.62
%
\leavevmode%
\llap{\spacer}%
${L_{72.62}}=\pdualwitharg{2.3.5}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
494576040\\10834740\\122456040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10834740\\237360\\2682660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
122456040\\2682660\\30319871
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\2\\-190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
935\\42\\-3780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1173\\28\\-4740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2481\\42\\-10024
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
776\\10\\-3135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1111\\10\\-4488
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4523\\28\\-18270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
260\\-1\\-1050
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-120\\-280\\-90\\-120\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\840\\0\\-840\\-420\\-840\\-4200\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\120\\0\\-60\\-240\\-1680\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-840\\0\\56\\0\\-168\\-1680\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-420\\-60\\0\\15\\0\\-210\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-840\\-240\\-168\\0\\24\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-4200\\-1680\\-1680\\-210\\0\\420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-300\\-420\\-90\\-60\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[494576040, 10834740, 122456040; 10834740, 237360, 2682660; 122456040, 2682660, 30319871],[420, 60, 1],[47, 935, 1173, 2481, 776, 1111, 4523, 260; 2, 42, 28, 42, 10, 10, 28, -1; -190, -3780, -4740, -10024, -3135, -4488, -18270, -1050],71,62]
%
% END OF LATTICE L72.62
%
\hbox{}\par\smallskip%
%
% LATTICE L72.63
%
\leavevmode%
\llap{\spacer}%
${L_{72.63}}=\pdualwitharg{2.3.5}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
139436220\\-6403320\\34738200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6403320\\294060\\-1595280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34738200\\-1595280\\8654441
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\-10\\-335
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
519\\-91\\-2100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89\\-15\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
195\\-7\\-784
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
157\\1\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
221\\-5\\-888
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
757\\-35\\-3045
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
283\\-22\\-1140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-60\\-140\\-90\\-60\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\840\\0\\-840\\-840\\-840\\-2100\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\0\\-120\\-240\\-840\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-840\\0\\56\\0\\-168\\-840\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-840\\-120\\0\\60\\0\\-210\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-840\\-240\\-168\\0\\24\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-2100\\-840\\-840\\-210\\0\\105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-300\\-420\\-180\\-60\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[139436220, -6403320, 34738200; -6403320, 294060, -1595280; 34738200, -1595280, 8654441],[420, 60, 1],[83, 519, 89, 195, 157, 221, 757, 283; -10, -91, -15, -7, 1, -5, -35, -22; -335, -2100, -360, -784, -630, -888, -3045, -1140],71,63]
%
% END OF LATTICE L72.63
%
\hbox{}\par\smallskip%
%
% LATTICE L72.64
%
\leavevmode%
\llap{\spacer}%
${L_{72.64}}=\pdualwitharg{5.7}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${840}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51240\\9240\\3360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9240\\70\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3360\\-35\\-37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-2364\\5880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-406\\1010
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-408\\1015
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-117\\291
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\56\\-140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\59\\-147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\64\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-169\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\0\\-420\\-420\\-2520\\-420\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-30\\-280\\-70\\-200\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\35\\0\\-140\\-70\\-280\\-175
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-30\\0\\3\\0\\-21\\-120\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-280\\-140\\0\\280\\0\\-280\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-70\\-70\\-21\\0\\7\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-200\\-280\\-120\\-280\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-175\\-105\\-420\\-35\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[51240, 9240, 3360; 9240, 70, -35; 3360, -35, -37],[840, 35, 1],[41, 7, 7, 2, -1, -1, -1, 3; -2364, -406, -408, -117, 56, 59, 64, -169; 5880, 1010, 1015, 291, -140, -147, -160, 420],71,64]
%
% END OF LATTICE L72.64
%
\hbox{}\par\smallskip%
%
% LATTICE L72.65
%
\leavevmode%
\llap{\spacer}%
${L_{72.65}}=\pdualwitharg{5.7}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${840}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
283080\\-73080\\-4200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73080\\18865\\1085
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4200\\1085\\62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\96\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\11\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\54\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\18\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-16\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\0\\-840\\-840\\-2520\\-840\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-30\\-140\\-70\\-100\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\0\\140\\0\\-280\\-280\\-560\\-350
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-30\\0\\12\\0\\-84\\-240\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-140\\-280\\0\\280\\0\\-280\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-70\\-280\\-84\\0\\28\\0\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-100\\-560\\-240\\-280\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-350\\-210\\-420\\-70\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[283080, -73080, -4200; -73080, 18865, 1085; -4200, 1085, 62],[840, 35, 1],[25, 3, 15, 5, 1, -3, -5, -1; 96, 11, 54, 18, 4, -10, -16, -2; 0, 10, 70, 24, 0, -28, -60, -35],71,65]
%
% END OF LATTICE L72.65
%
\hbox{}\par\smallskip%
%
% LATTICE L72.66
%
\leavevmode%
\llap{\spacer}%
${L_{72.66}}=\pdualwitharg{2.5}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{280}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1953840\\-170520\\5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-170520\\-14840\\440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\440\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-7\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-21\\3528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\670
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-120\\-840\\-180\\-120\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\280\\0\\-840\\-280\\-280\\-700\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\40\\0\\-40\\-80\\-280\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-840\\0\\168\\0\\-168\\-840\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-280\\-40\\0\\20\\0\\-70\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-280\\-80\\-168\\0\\8\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-700\\-280\\-840\\-70\\0\\35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-200\\-840\\-120\\-40\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1953840, -170520, 5040; -170520, -14840, 440; 5040, 440, -13],[840, 40, 1],[-1, 1, 2, 11, 2, 1, 1, -1; 0, -7, -5, -21, -3, -1, 0, 2; -390, 140, 600, 3528, 670, 352, 385, -320],71,66]
%
% END OF LATTICE L72.66
%
\hbox{}\par\smallskip%
%
% LATTICE L72.67
%
\leavevmode%
\llap{\spacer}%
${L_{72.67}}=\pdualwitharg{2.5}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{280}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36669360\\-442680\\59640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-442680\\-5320\\720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59640\\720\\-97
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-615
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-7\\560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-5\\2420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-21\\15204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-3\\3050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\1836
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\5530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-60\\-420\\-90\\-60\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\280\\0\\-840\\-280\\-280\\-1400\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\0\\-40\\-80\\-560\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-840\\0\\168\\0\\-168\\-1680\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-280\\-40\\0\\20\\0\\-140\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-280\\-80\\-168\\0\\8\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-1400\\-560\\-1680\\-140\\0\\140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-200\\-840\\-120\\-40\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36669360, -442680, 59640; -442680, -5320, 720; 59640, 720, -97],[840, 40, 1],[-1, 1, 4, 25, 5, 3, 9, -1; 0, -7, -5, -21, -3, -1, 0, 2; -615, 560, 2420, 15204, 3050, 1836, 5530, -600],71,67]
%
% END OF LATTICE L72.67
%
\hbox{}\par\smallskip%
%
% LATTICE L72.68
%
\leavevmode%
\llap{\spacer}%
${L_{72.68}}=\pdualwitharg{2.3.7}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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12762960\\-205380\\3158400
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3158400\\-50820\\781597
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3429\\-161\\13846
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\HardButStrongLineBreak\kern3pt%
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-1899\\-89\\7668
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-1477\\-69\\5964
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\HardButStrongLineBreak\kern3pt%
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-317\\-15\\1280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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26\\1\\-105
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\HardButStrongLineBreak\kern3pt%
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-7073\\-335\\28560
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2269\\-107\\9162
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4379\\-206\\17682
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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28\\0\\-168\\-280\\-126\\-840\\-84\\0
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0\\24\\0\\-120\\-84\\-840\\-120\\-84
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\kern3pt%
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-84\\-120\\-336\\-240\\-42\\0\\12\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-420\\-420\\-126\\-420\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12762960, -205380, 3158400; -205380, 3696, -50820; 3158400, -50820, 781597],[420, 84, 1],[-3429, -1899, -1477, -317, 26, -7073, -2269, -4379; -161, -89, -69, -15, 1, -335, -107, -206; 13846, 7668, 5964, 1280, -105, 28560, 9162, 17682],71,68]
%
% END OF LATTICE L72.68
%
\hbox{}\par\smallskip%
%
% LATTICE L72.69
%
\leavevmode%
\llap{\spacer}%
${L_{72.69}}=\pdualwitharg{2.3.7}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9934260\\-196140\\2459100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-196140\\3864\\-48552
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2459100\\-48552\\608719
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
220\\-3\\-889
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
392\\-5\\-1584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
686\\-9\\-2772
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
292\\-5\\-1180
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-1\\210
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\HardButStrongLineBreak\kern3pt%
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-312\\-5\\1260
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\HardButStrongLineBreak\kern3pt%
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23\\-1\\-93
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\HardButStrongLineBreak\kern3pt%
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291\\-5\\-1176
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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7\\0\\-84\\-140\\-126\\-420\\-21\\0
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\kern3pt%
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0\\-84\\-420\\-420\\-252\\-420\\0\\84
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9934260, -196140, 2459100; -196140, 3864, -48552; 2459100, -48552, 608719],[420, 84, 1],[220, 392, 686, 292, -52, -312, 23, 291; -3, -5, -9, -5, -1, -5, -1, -5; -889, -1584, -2772, -1180, 210, 1260, -93, -1176],71,69]
%
% END OF LATTICE L72.69
%
\hbox{}\par\smallskip%
%
% LATTICE L72.70
%
\leavevmode%
\llap{\spacer}%
${L_{72.70}}=\pdualwitharg{3.5.7}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{420}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
171780\\639240\\17640
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
639240\\2249310\\62055
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17640\\62055\\1712
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\322\\-11620
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\133\\-4800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\29\\-1050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-57\\2056
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-780\\28140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-305\\11004
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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5\\-306\\11040
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\64\\-2310
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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140\\0\\-420\\-140\\-1260\\-420\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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-420\\0\\210\\0\\-420\\-420\\-840\\-1050
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\kern3pt%
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-140\\-30\\0\\2\\0\\-42\\-120\\-210
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-210\\-420\\-42\\0\\42\\0\\-210
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-300\\-840\\-120\\-420\\0\\60\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-1050\\-210\\-1260\\-210\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[171780, 639240, 17640; 639240, 2249310, 62055; 17640, 62055, 1712],[420, 105, 1],[-5, -2, 0, 1, 13, 5, 5, -1; 322, 133, 29, -57, -780, -305, -306, 64; -11620, -4800, -1050, 2056, 28140, 11004, 11040, -2310],71,70]
%
% END OF LATTICE L72.70
%
\hbox{}\par\smallskip%
%
% LATTICE L72.71
%
\leavevmode%
\llap{\spacer}%
${L_{72.71}}=\pdualwitharg{2.3.5.7}(\pfillwitharg{2}({L_{72.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{420}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9414938820\\3663236010\\1769890500
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\kern3pt%
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3663236010\\1425330690\\688647750
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1769890500\\688647750\\332719787
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1932\\8306058\\-17201765
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
707\\3039459\\-6294690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-212\\-911593\\1887900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-81673\\169144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
423\\1818775\\-3766665
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1667\\7167047\\-14842884
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2049\\8809286\\-18243945
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6431\\27648542\\-57259860
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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35\\0\\-210\\-70\\-315\\-210\\-105\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-1050\\-210\\-630\\-210\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9414938820, 3663236010, 1769890500; 3663236010, 1425330690, 688647750; 1769890500, 688647750, 332719787],[210, 210, 1],[1932, 707, -212, -19, 423, 1667, 2049, 6431; 8306058, 3039459, -911593, -81673, 1818775, 7167047, 8809286, 27648542; -17201765, -6294690, 1887900, 169144, -3766665, -14842884, -18243945, -57259860],71,71]
%
% END OF LATTICE L72.71
%
\hbox{}\par\smallskip%
%
% LATTICE L72.72
%
\leavevmode%
\llap{\spacer}%
${L_{72.72}}=\pdualwitharg{3.5.7}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
370860\\-1941240\\-53760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1941240\\10161165\\281400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53760\\281400\\7793
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\44\\-1540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\7\\-225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-7\\315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\43
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\19\\-693
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\48\\-1740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\38\\-1365
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-420\\-140\\-1260\\-420\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-30\\-420\\-210\\-300\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\210\\0\\-420\\-420\\-840\\-525
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-30\\0\\2\\0\\-42\\-120\\-105
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-420\\-420\\0\\420\\0\\-420\\-630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-210\\-420\\-42\\0\\42\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-300\\-840\\-120\\-420\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-525\\-105\\-630\\-105\\0\\105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[370860, -1941240, -53760; -1941240, 10161165, 281400; -53760, 281400, 7793],[420, 105, 1],[7, 4, 9, 1, 1, -1, -1, 1; 44, 7, -7, -1, 6, 19, 48, 38; -1540, -225, 315, 43, -210, -693, -1740, -1365],71,72]
%
% END OF LATTICE L72.72
%
\hbox{}\par\smallskip%
%
% LATTICE L72.73
%
\leavevmode%
\llap{\spacer}%
${L_{72.73}}=\pdualwitharg{2.7}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{l}{2}{280}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40616520\\0\\64680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64680\\0\\-103
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-628
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\0\\8160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-1\\3766
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-10\\18200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\1255
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-5\\3136
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-168\\-840\\-252\\-840\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-120\\-56\\-280\\-20\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\56\\0\\-56\\-560\\-56\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-120\\0\\120\\0\\-840\\-120\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-56\\-56\\0\\28\\0\\-14\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-280\\-560\\-840\\0\\280\\0\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-20\\-56\\-120\\-14\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-280\\-840\\-168\\-280\\0\\112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-40616520, 0, 64680; 0, 56, 0; 64680, 0, -103],[840, 56, 1],[-1, -1, 0, 13, 6, 29, 2, 5; -3, 0, 1, 0, -1, -10, -1, -5; -630, -628, 0, 8160, 3766, 18200, 1255, 3136],71,73]
%
% END OF LATTICE L72.73
%
\hbox{}\par\smallskip%
%
% LATTICE L72.74
%
\leavevmode%
\llap{\spacer}%
${L_{72.74}}=\pdualwitharg{2.7}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4454520\\-256200\\-106680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-256200\\-14672\\-6104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-106680\\-6104\\-2539
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-96\\189
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\104\\-208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\99\\-196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-1350\\2700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-722\\1442
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\-4000\\7980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-613\\1222
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-1013\\2016
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-84\\-420\\-126\\-420\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-120\\-56\\-280\\-40\\-56
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\56\\0\\-56\\-560\\-112\\-280
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-120\\0\\120\\0\\-840\\-240\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-56\\-56\\0\\28\\0\\-28\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-280\\-560\\-840\\0\\280\\0\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-40\\-112\\-240\\-28\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-280\\-840\\-168\\-280\\0\\112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4454520, -256200, -106680; -256200, -14672, -6104; -106680, -6104, -2539],[840, 56, 1],[1, -1, -1, 13, 7, 39, 6, 10; -96, 104, 99, -1350, -722, -4000, -613, -1013; 189, -208, -196, 2700, 1442, 7980, 1222, 2016],71,74]
%
% END OF LATTICE L72.74
%
\hbox{}\par\smallskip%
%
% LATTICE L72.75
%
\leavevmode%
\llap{\spacer}%
${L_{72.75}}=\pdualwitharg{2.5.7}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82511520\\1189020\\-19544280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1189020\\17080\\-281680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19544280\\-281680\\4629371
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-260\\783\\-1050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
594\\-1771\\2400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2980\\-8891\\12040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1690\\-5043\\6828
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
771\\-2301\\3115
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
298\\-889\\1204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-250\\747\\-1010
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-849\\2536\\-3430
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-840\\-840\\-630\\-840\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\40\\0\\-120\\-140\\-280\\-200\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\0\\280\\0\\-140\\-560\\-560\\-700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-120\\0\\24\\0\\-168\\-240\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-140\\-140\\0\\35\\0\\-70\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-280\\-560\\-168\\0\\56\\0\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-200\\-560\\-240\\-70\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-700\\-420\\-210\\-140\\0\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[82511520, 1189020, -19544280; 1189020, 17080, -281680; -19544280, -281680, 4629371],[420, 140, 1],[-260, 594, 2980, 1690, 771, 298, -250, -849; 783, -1771, -8891, -5043, -2301, -889, 747, 2536; -1050, 2400, 12040, 6828, 3115, 1204, -1010, -3430],71,75]
%
% END OF LATTICE L72.75
%
\hbox{}\par\smallskip%
%
% LATTICE L72.76
%
\leavevmode%
\llap{\spacer}%
${L_{72.76}}=\pdualwitharg{2.5.7}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{140}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3294060\\162120\\-697200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
162120\\7420\\-34720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-697200\\-34720\\147269
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1793\\-5274\\7245
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
787\\-2315\\3180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
693\\-2039\\2800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\279\\-384
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-537\\1579\\-2170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-97\\285\\-392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
172\\-506\\695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1178\\-3465\\4760
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\0\\-420\\-420\\-630\\-420\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\40\\0\\-120\\-280\\-280\\-100\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\280\\0\\-280\\-560\\-280\\-700
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\kern3pt%
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-420\\-120\\0\\24\\0\\-168\\-120\\-420
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-280\\-280\\0\\140\\0\\-70\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-280\\-560\\-168\\0\\56\\0\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-100\\-280\\-120\\-70\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-700\\-420\\-420\\-140\\0\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3294060, 162120, -697200; 162120, 7420, -34720; -697200, -34720, 147269],[420, 140, 1],[1793, 787, 693, -95, -537, -97, 172, 1178; -5274, -2315, -2039, 279, 1579, 285, -506, -3465; 7245, 3180, 2800, -384, -2170, -392, 695, 4760],71,76]
%
% END OF LATTICE L72.76
%
\hbox{}\par\smallskip%
%
% LATTICE L72.77
%
\leavevmode%
\llap{\spacer}%
${L_{72.77}}=\pdualwitharg{3.5.7}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${280}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1517880\\-1074360\\54600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1074360\\737205\\-37485
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54600\\-37485\\1906
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-1224\\-24500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-742\\-14850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-829\\-16590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-83\\-1661
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\84\\1680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\169\\3381
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\432\\8640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-68\\-1365
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\0\\-420\\-140\\-2520\\-420\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-30\\-840\\-210\\-600\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\105\\0\\-420\\-210\\-840\\-525
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-30\\0\\1\\0\\-21\\-120\\-105
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-840\\-420\\0\\840\\0\\-840\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-210\\-210\\-21\\0\\21\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-600\\-840\\-120\\-840\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-525\\-105\\-1260\\-105\\0\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1517880, -1074360, 54600; -1074360, 737205, -37485; 54600, -37485, 1906],[840, 105, 1],[15, 9, 10, 1, -1, -2, -5, 1; -1224, -742, -829, -83, 84, 169, 432, -68; -24500, -14850, -16590, -1661, 1680, 3381, 8640, -1365],71,77]
%
% END OF LATTICE L72.77
%
\hbox{}\par\smallskip%
%
% LATTICE L72.78
%
\leavevmode%
\llap{\spacer}%
${L_{72.78}}=\pdualwitharg{3.5.7}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${280}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{420}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{840}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
741720\\-1735440\\-47880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1735440\\4060770\\112035
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47880\\112035\\3091
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\88\\-3080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\17\\-585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\56\\-1890
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\-202
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\-378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\36\\-1320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\41\\-1470
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\0\\-840\\-280\\-2520\\-840\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-30\\-420\\-210\\-300\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\0\\420\\0\\-840\\-840\\-1680\\-1050
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\kern3pt%
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-280\\-30\\0\\4\\0\\-84\\-240\\-210
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-420\\-840\\0\\840\\0\\-840\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-210\\-840\\-84\\0\\84\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-300\\-1680\\-240\\-840\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-1050\\-210\\-1260\\-210\\0\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[741720, -1735440, -47880; -1735440, 4060770, 112035; -47880, 112035, 3091],[840, 105, 1],[7, 2, 9, 1, 1, -1, -1, 1; 88, 17, 56, 6, 12, 10, 36, 41; -3080, -585, -1890, -202, -420, -378, -1320, -1470],71,78]
%
% END OF LATTICE L72.78
%
\hbox{}\par\smallskip%
%
% LATTICE L72.79
%
\leavevmode%
\llap{\spacer}%
${L_{72.79}}=\pdualwitharg{2.3.5}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{840}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35449680\\-1848840\\52920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848840\\-96360\\2760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52920\\2760\\-79
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-670
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-7\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-5\\3840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-7\\8456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-3\\5250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-1\\3312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\5355
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-120\\-280\\-180\\-120\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\840\\0\\-840\\-840\\-840\\-2100\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\120\\0\\-120\\-240\\-840\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-840\\0\\56\\0\\-168\\-840\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-840\\-120\\0\\60\\0\\-210\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-840\\-240\\-168\\0\\24\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-2100\\-840\\-840\\-210\\0\\105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-600\\-840\\-360\\-120\\0\\240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35449680, -1848840, 52920; -1848840, -96360, 2760; 52920, 2760, -79],[840, 120, 1],[-1, 1, 6, 13, 8, 5, 8, -1; 0, -7, -5, -7, -3, -1, 0, 2; -670, 420, 3840, 8456, 5250, 3312, 5355, -600],71,79]
%
% END OF LATTICE L72.79
%
\hbox{}\par\smallskip%
%
% LATTICE L72.80
%
\leavevmode%
\llap{\spacer}%
${L_{72.80}}=\pdualwitharg{2.3.5}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{840}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37143120\\-5265960\\110880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5265960\\-746520\\15720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110880\\15720\\-331
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-335
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-7\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-5\\3780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-7\\8708
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-3\\5550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-1\\3636
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\0\\12390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-60\\-140\\-90\\-60\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\840\\0\\-840\\-840\\-840\\-4200\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\0\\-120\\-240\\-1680\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-840\\0\\56\\0\\-168\\-1680\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-840\\-120\\0\\60\\0\\-420\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-840\\-240\\-168\\0\\24\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-4200\\-1680\\-1680\\-420\\0\\420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-600\\-840\\-360\\-120\\0\\240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-37143120, -5265960, 110880; -5265960, -746520, 15720; 110880, 15720, -331],[840, 120, 1],[-1, 1, 12, 27, 17, 11, 37, -1; 0, -7, -5, -7, -3, -1, 0, 2; -335, 0, 3780, 8708, 5550, 3636, 12390, -240],71,80]
%
% END OF LATTICE L72.80
%
\hbox{}\par\smallskip%
%
% LATTICE L72.81
%
\leavevmode%
\llap{\spacer}%
${L_{72.81}}=\pdualwitharg{2.3.7}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{l}{2}{840}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{336}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3503640\\0\\10080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10080\\0\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-348
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\1040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\1386
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-10\\5880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\345
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\336
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-168\\-280\\-252\\-840\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-120\\-168\\-840\\-60\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\168\\0\\-168\\-1680\\-168\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-120\\0\\40\\0\\-840\\-120\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-168\\-168\\0\\84\\0\\-42\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-840\\-1680\\-840\\0\\840\\0\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-60\\-168\\-120\\-42\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-840\\-840\\-504\\-840\\0\\336
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3503640, 0, 10080; 0, 168, 0; 10080, 0, -29],[840, 168, 1],[-1, -1, 0, 3, 4, 17, 1, 1; -1, 0, 1, 0, -1, -10, -1, -5; -350, -348, 0, 1040, 1386, 5880, 345, 336],71,81]
%
% END OF LATTICE L72.81
%
\hbox{}\par\smallskip%
%
% LATTICE L72.82
%
\leavevmode%
\llap{\spacer}%
${L_{72.82}}=\pdualwitharg{2.3.7}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{s}{2}{840}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17086440\\-257880\\-106680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-257880\\-3696\\-1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-106680\\-1512\\-617
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\384\\-768
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\379\\-756
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1150\\2300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-1912\\3822
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-10300\\20580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1523\\3042
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-2273\\4536
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-84\\-140\\-126\\-420\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-120\\-168\\-840\\-120\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\168\\0\\-168\\-1680\\-336\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-120\\0\\40\\0\\-840\\-240\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-168\\-168\\0\\84\\0\\-84\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-840\\-1680\\-840\\0\\840\\0\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-120\\-336\\-240\\-84\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-840\\-840\\-504\\-840\\0\\336
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17086440, -257880, -106680; -257880, -3696, -1512; -106680, -1512, -617],[840, 168, 1],[0, -1, -1, 3, 5, 27, 4, 6; 3, 384, 379, -1150, -1912, -10300, -1523, -2273; -7, -768, -756, 2300, 3822, 20580, 3042, 4536],71,82]
%
% END OF LATTICE L72.82
%
\hbox{}\par\smallskip%
%
% LATTICE L72.83
%
\leavevmode%
\llap{\spacer}%
${L_{72.83}}=\pdualwitharg{2.3.5.7}({L_{72.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{s}{2}{840}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17082240\\615300420\\152357520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
615300420\\22163053080\\5487901020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
152357520\\5487901020\\1358885777
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\17\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
150\\-1267\\5100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
688\\-5947\\23940
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181\\-1487\\5985
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94\\-585\\2352
\end{tabular}\endgroup%
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-26\\439\\-1770
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-145\\1512\\-6090
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17082240, 615300420, 152357520; 615300420, 22163053080, 5487901020; 152357520, 5487901020, 1358885777],[420, 420, 1],[12, 150, 688, 130, 181, 94, -26, -145; 17, -1267, -5947, -1105, -1487, -585, 439, 1512; -70, 5100, 23940, 4448, 5985, 2352, -1770, -6090],71,83]
%
% END OF LATTICE L72.83
%
\hbox{}\par\smallskip%
%
% LATTICE L72.84
%
\leavevmode%
\llap{\spacer}%
${L_{72.84}}=\pdualwitharg{2.3.5.7}(\main({L_{72.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{420}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25620\\981120\\242760
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\kern3pt%
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981120\\37459380\\9268980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
242760\\9268980\\2293523
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
349\\727\\-2975
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
459\\953\\-3900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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397\\821\\-3360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-21\\-43\\176
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\HardButStrongLineBreak\kern3pt%
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-325\\-667\\2730
\end{tabular}\endgroup%
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-65\\-123\\504
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\HardButStrongLineBreak\kern3pt%
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98\\209\\-855
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\HardButStrongLineBreak\kern3pt%
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686\\1437\\-5880
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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35\\0\\-420\\-140\\-630\\-420\\-105\\0
\end{tabular}\endgroup%
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0\\120\\0\\-120\\-840\\-840\\-300\\-420
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-840\\-1680\\-168\\0\\168\\0\\-420
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-105\\-300\\-840\\-120\\-210\\0\\15\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-2100\\-420\\-1260\\-420\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25620, 981120, 242760; 981120, 37459380, 9268980; 242760, 9268980, 2293523],[420, 420, 1],[349, 459, 397, -21, -325, -65, 98, 686; 727, 953, 821, -43, -667, -123, 209, 1437; -2975, -3900, -3360, 176, 2730, 504, -855, -5880],71,84]
%
% END OF LATTICE L72.84
%
\hbox{}\par\smallskip%
%
% LATTICE L72.85
%
\leavevmode%
\llap{\spacer}%
${L_{72.85}}=\pdualwitharg{2.5.7}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{l}{2}{280}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{560}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-182280\\-151200\\-21000
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-151200\\-97160\\-13440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21000\\-13440\\-1859
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\621\\-4410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\183\\-1300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\118\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-159\\1128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-424\\3010
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-347\\2464
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\HardButStrongLineBreak\kern3pt%
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1\\-88\\625
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-1\\79\\-560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-840\\-840\\-1260\\-840\\-210\\0
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\kern3pt%
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\kern3pt%
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-840\\-280\\-560\\-168\\0\\56\\0\\-280
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\kern3pt%
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-210\\-100\\-280\\-120\\-70\\0\\5\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-1400\\-840\\-840\\-280\\0\\560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-182280, -151200, -21000; -151200, -97160, -13440; -21000, -13440, -1859],[840, 280, 1],[-7, -2, -1, 2, 5, 4, 1, -1; 621, 183, 118, -159, -424, -347, -88, 79; -4410, -1300, -840, 1128, 3010, 2464, 625, -560],71,85]
%
% END OF LATTICE L72.85
%
\hbox{}\par\smallskip%
%
% LATTICE L72.86
%
\leavevmode%
\llap{\spacer}%
${L_{72.86}}=\pdualwitharg{2.5.7}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{560}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1157520\\-578760\\-190680
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\kern3pt%
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-578760\\-289240\\-95200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190680\\-95200\\-31271
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-176\\525\\-525
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-147\\439\\-440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-326\\975\\-980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-91\\273\\-276
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-73\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
85\\-253\\252
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57\\-170\\170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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105\\0\\-420\\-420\\-630\\-420\\-210\\0
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\kern3pt%
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0\\40\\0\\-120\\-280\\-280\\-200\\-280
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-420\\0\\280\\0\\-280\\-560\\-560\\-1400
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\kern3pt%
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-420\\-120\\0\\24\\0\\-168\\-240\\-840
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-280\\-280\\0\\140\\0\\-140\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-280\\-560\\-168\\0\\56\\0\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-200\\-560\\-240\\-140\\0\\20\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-1400\\-840\\-840\\-280\\0\\560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1157520, -578760, -190680; -578760, -289240, -95200; -190680, -95200, -31271],[840, 280, 1],[-176, -147, -326, -91, 25, 85, 57, -1; 525, 439, 975, 273, -73, -253, -170, 2; -525, -440, -980, -276, 70, 252, 170, 0],71,86]
%
% END OF LATTICE L72.86
%
\hbox{}\par\smallskip%
%
% LATTICE L72.87
%
\leavevmode%
\llap{\spacer}%
${L_{72.87}}=\pdualwitharg{2.3.5.7}({L_{72.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{l}{2}{840}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{420}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{1680}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38640\\-19320\\1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19320\\-9240\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\840\\-73
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-3\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-1\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-840\\-280\\-1260\\-840\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\120\\0\\-120\\-840\\-840\\-300\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\0\\840\\0\\-840\\-1680\\-840\\-4200
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-120\\0\\8\\0\\-168\\-120\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-840\\-840\\0\\420\\0\\-210\\-2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-840\\-1680\\-168\\0\\168\\0\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-300\\-840\\-120\\-210\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-4200\\-840\\-2520\\-840\\0\\1680
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38640, -19320, 1680; -19320, -9240, 840; 1680, 840, -73],[840, 840, 1],[-3, -2, 3, 2, 11, 8, 2, -1; 0, -1, -5, -1, -3, -1, 0, 2; -70, -60, 0, 32, 210, 168, 45, 0],71,87]
%
% END OF LATTICE L72.87
%
\hbox{}\par\smallskip%
%
% LATTICE L72.88
%
\leavevmode%
\llap{\spacer}%
${L_{72.88}}=\pdualwitharg{2.3.5.7}({L_{72.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{420}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{1680}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2659440\\-3799320\\5880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3799320\\-5427240\\8400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5880\\8400\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-455
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-1560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5\\-4620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-652
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\-1050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\840
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-420\\-140\\-630\\-420\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\120\\0\\-120\\-840\\-840\\-600\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\840\\0\\-840\\-1680\\-1680\\-4200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-120\\0\\8\\0\\-168\\-240\\-840
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-840\\-840\\0\\420\\0\\-420\\-2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-840\\-1680\\-168\\0\\168\\0\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-600\\-1680\\-240\\-420\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-4200\\-840\\-2520\\-840\\0\\1680
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2659440, -3799320, 5880; -3799320, -5427240, 8400; 5880, 8400, -13],[840, 840, 1],[-1, -2, -3, 0, 2, 2, 1, -1; 0, -1, -5, -1, -3, -1, 0, 2; -455, -1560, -4620, -652, -1050, 252, 450, 840],71,88]
%
% END OF LATTICE L72.88
%

\medskip%
%------------------------------WEYL-GROUP-W73--------------------------------
%
\leavevmode\llap{}%
$W_{73\phantom{0}}$%
\qquad\llap{32} lattices, $\chi=72$%
\hfill%
$\infty222222\infty222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L73.1
%
\leavevmode%
\llap{\spacer}%
${L_{73.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{25\rightarrow N_{73},5,2*}%
\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{20,19}{\infty a}{440}\above{1pt}{1pt}{b}{2}{50}\above{1pt}{1pt}{l}{2}{88}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{550}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2584117800\\-1286540200\\1826000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1286540200\\-640522510\\909115
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1826000\\909115\\-1282
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2951517359\\-5913532680\\10443879600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1469189856\\-2943605329\\5198696160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2235998\\4479949\\-7912031
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1411\\2827\\-5005
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15175\\-30404\\53680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
282\\-565\\1000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33161\\66440\\-117304
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5621\\11262\\-19885
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-79828\\159940\\-282425
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5060\\10138\\-17903
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\-220\\-275\\-1760\\-165\\-1375\\-55\\-715\\-4620\\-1925\\-7920\\-440\\-1650\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\440\\0\\-1320\\-220\\-3300\\-220\\-4620\\-39160\\-17600\\-76120\\-4620\\-20900\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-275\\0\\50\\0\\-25\\-825\\-75\\-1925\\-17600\\-8050\\-35200\\-2175\\-10175\\-125
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1760\\-1320\\0\\88\\0\\-2200\\-264\\-7920\\-76120\\-35200\\-154968\\-9680\\-46200\\-616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-220\\-25\\0\\10\\0\\-10\\-440\\-4620\\-2175\\-9680\\-615\\-3025\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1375\\-3300\\-825\\-2200\\0\\550\\0\\-1650\\-20900\\-10175\\-46200\\-3025\\-15675\\-275
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-220\\-75\\-264\\-10\\0\\2\\0\\-220\\-125\\-616\\-45\\-275\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\-220\\-275\\-1760\\-165\\-1375\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\440\\0\\-1320\\-220\\-3300\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-275\\0\\50\\0\\-25\\-825\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1760\\-1320\\0\\88\\0\\-2200\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-220\\-25\\0\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1375\\-3300\\-825\\-2200\\0\\550\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-220\\-75\\-264\\-10\\0\\2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-715\\-4620\\-1925\\-7920\\-440\\-1650\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-39160\\-17600\\-76120\\-4620\\-20900\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1925\\-17600\\-8050\\-35200\\-2175\\-10175\\-125
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7920\\-76120\\-35200\\-154968\\-9680\\-46200\\-616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\-4620\\-2175\\-9680\\-615\\-3025\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1650\\-20900\\-10175\\-46200\\-3025\\-15675\\-275
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-220\\-125\\-616\\-45\\-275\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2584117800, -1286540200, 1826000; -1286540200, -640522510, 909115; 1826000, 909115, -1282],[2200, 5, 1],[-1411, 15175, 282, -33161, -5621, -79828, -5060, -100647, -831639, -371402, -1599767, -96437, -430462, -4218; 2827, -30404, -565, 66440, 11262, 159940, 10138, 201652, 1666236, 744125, 3205224, 193217, 862455, 8451; -5005, 53680, 1000, -117304, -19885, -282425, -17903, -356125, -2942720, -1314200, -5660776, -341245, -1523225, -14927],72,1]
%
% END OF LATTICE L73.1
%
\hbox{}\par\smallskip%
%
% LATTICE L73.2
%
\leavevmode%
\llap{\spacer}%
${L_{73.2}}=\pfillwitharg{2.5}({L_{73.1}})$%
{${}={}$}Nikulin $73$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{2,1}{\infty b}{110}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{22}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[110, 0, 0; 0, 0, -1; 0, -1, 0],[110, 1, 1],[-1, 1, 0, -3, -1, -3, -1, -21, -89, -16, -173, -21, -19, -1; 0, 0, -1, -22, -5, -11, -3, -55, -220, -39, -418, -50, -44, -2; -55, 0, 1, -22, -10, -44, -18, -440, -1980, -361, -3938, -485, -451, -27],72,2]
%
% END OF LATTICE L73.2
%
\hbox{}\par\smallskip%
%
% LATTICE L73.3
%
\leavevmode%
\llap{\spacer}%
${L_{73.3}}=\pfillwitharg{5}({L_{73.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-150449640\\-74774480\\-459360
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\kern3pt%
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-74774480\\-37163418\\-228305
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-150449640, -74774480, -459360; -74774480, -37163418, -228305; -459360, -228305, -1402],[440, 1, 1],[-42314, -360707, -32466, -702651, -42704, -38733, -2063, -2214, 4483, 82, -9797, -1824, -5795, -1985; 85140, 725780, 65325, 1413808, 85925, 77935, 4151, 4455, -9020, -165, 19712, 3670, 11660, 3994; -385, -3520, -322, -7040, -435, -407, -25, -55, 0, 2, 0, -5, -33, -15],72,3]
%
% END OF LATTICE L73.3
%
\hbox{}\par\smallskip%
%
% LATTICE L73.4
%
\leavevmode%
\llap{\spacer}%
${L_{73.4}}=\pfillwitharg{2}({L_{73.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{10,9}{\infty b}{110}\above{1pt}{1pt}{r}{2}{50}\above{1pt}{1pt}{l}{2}{22}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{550}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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719\\22275\\28325
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-18072450, 1613150, -809600; 1613150, -143990, 72265; -809600, 72265, -36268],[550, 5, 1],[133, 615, 564, 1235, 153, 719, 9, 21, 1, -4, -3, 1, 51, 5; 4334, 19536, 17815, 38874, 4789, 22275, 267, 539, -44, -95, 110, 84, 2090, 176; 5665, 25190, 22900, 49874, 6125, 28325, 331, 605, -110, -100, 286, 145, 3025, 239],72,4]
%
% END OF LATTICE L73.4
%
\hbox{}\par\smallskip%
%
% LATTICE L73.5
%
\leavevmode%
\llap{\spacer}%
${L_{73.5}}=\pdualwitharg{2}(\pfillwitharg{2.5}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${220}\above{1pt}{1pt}{4,1}{\infty z}{55}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{44}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9147380\\6050\\-4490530
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6050\\-4\\2970
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4490530\\2970\\-2204441
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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2212451\\16070175\\-4484700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1480\\-10751\\3000
\end{tabular}\endgroup%
\kern3pt%
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1086172\\7889425\\-2201701
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1519\\10340\\-3080
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\HardButStrongLineBreak\kern3pt%
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3283\\23650\\-6655
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1186\\8655\\-2404
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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6431\\47311\\-13035
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1569\\11695\\-3180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1433\\10945\\-2904
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\HardButStrongLineBreak\kern3pt%
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79\\673\\-160
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9147380, 6050, -4490530; 6050, -4, 2970; -4490530, 2970, -2204441],[110, 2, 1],[1519, 3283, 1186, 6431, 1569, 1433, 79, 109, -27, -2, 81, 59, 195, 69; 10340, 23650, 8655, 47311, 11695, 10945, 673, 1485, 0, -55, -11, 130, 880, 402; -3080, -6655, -2404, -13035, -3180, -2904, -160, -220, 55, 4, -165, -120, -396, -140],72,5]
%
% END OF LATTICE L73.5
%
\hbox{}\par\smallskip%
%
% LATTICE L73.6
%
\leavevmode%
\llap{\spacer}%
${L_{73.6}}=\pdualwitharg{5}(\pfillwitharg{2.5}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{2,1}{\infty b}{22}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{110}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3658490\\6050\\1496660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6050\\-10\\-2475
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1496660\\-2475\\-612272
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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1700\\-10751\\4200
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417622\\-2640845\\1031771
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3658490, 6050, 1496660; 6050, -10, -2475; 1496660, -2475, -612272],[110, 5, 1],[-343, -1505, -1364, -14825, -363, -1669, -95, -31, 9, 4, -135, -11, -201, -75; 2068, 9460, 8655, 94622, 2339, 10945, 673, 297, 0, -55, -22, 26, 880, 402; -847, -3718, -3370, -36630, -897, -4125, -235, -77, 22, 10, -330, -27, -495, -185],72,6]
%
% END OF LATTICE L73.6
%
\hbox{}\par\smallskip%
%
% LATTICE L73.7
%
\leavevmode%
\llap{\spacer}%
${L_{73.7}}=\pdualwitharg{2.5}(\pfillwitharg{2.5}({L_{73.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6820, 6050, 3080; 6050, 15120, 2750; 3080, 2750, 1391],[110, 10, 1],[24366, 54103, 98351, 535460, 26301, 121631, 7115, 2683, -5, 9, 5520, 748, 13614, 5180; 99, 220, 400, 2178, 107, 495, 29, 11, 0, 0, 22, 3, 55, 21; -54142, -120219, -218540, -1189815, -58442, -270270, -15810, -5962, 11, -20, -12265, -1662, -30250, -11510],72,7]
%
% END OF LATTICE L73.7
%
\hbox{}\par\smallskip%
%
% LATTICE L73.8
%
\leavevmode%
\llap{\spacer}%
${L_{73.8}}=\pdualwitharg{11}(\pfillwitharg{2.5}({L_{73.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-990, -6050, -110; -6050, -16632, -605; -110, -605, -12],[110, 11, 1],[-919, -4211, -8479, -8429, -11464, -976, -662, -134, -1, 55, 5, -119, -77, -391; -35, -160, -322, -320, -435, -37, -25, -5, 0, 2, 0, -5, -3, -15; 10130, 46410, 93445, 92892, 126335, 10755, 7293, 1475, 10, -605, -52, 1320, 850, 4312],72,8]
%
% END OF LATTICE L73.8
%
\hbox{}\par\smallskip%
%
% LATTICE L73.9
%
\leavevmode%
\llap{\spacer}%
${L_{73.9}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{4,3}{\infty a}{88}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{440}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-957000, -391160, -18480; -391160, -158650, -7865; -18480, -7865, -278],[440, 5, 1],[-80, 875, 16, -9561, -324, -4599, -1457, -5794, -47867, -21376, -460359, -5550, -24771, -1213; 165, -1804, -33, 19712, 668, 9482, 3004, 11946, 98692, 44073, 949168, 11443, 51073, 2501; 649, -7128, -130, 77880, 2639, 37455, 11865, 47179, 389752, 174050, 3748360, 45189, 201685, 9875],72,9]
%
% END OF LATTICE L73.9
%
\hbox{}\par\smallskip%
%
% LATTICE L73.10
%
\leavevmode%
\llap{\spacer}%
${L_{73.10}}=\pdualwitharg{2.11}(\pfillwitharg{2.5}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{4,1}{\infty z}{5}\above{1pt}{1pt}{r}{2}{44}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{220}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{44}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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140627520\\78650\\69232020
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78650\\44\\38720
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[140627520, 78650, 69232020; 78650, 44, 38720; 69232020, 38720, 34083461],[110, 22, 1],[-4269, -9348, -37264, -18406, -49559, -4141, -2589, -389, 32, 0, -226, -1679, -517, -2069; 1220, 2710, 10839, 5365, 14495, 1219, 785, 135, 0, 1, 55, 410, 136, 570; 8670, 18985, 75680, 37381, 100650, 8410, 5258, 790, -65, 0, 459, 3410, 1050, 4202],72,10]
%
% END OF LATTICE L73.10
%
\hbox{}\par\smallskip%
%
% LATTICE L73.11
%
\leavevmode%
\llap{\spacer}%
${L_{73.11}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{10,1}{\infty b}{110}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{550}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{s}{2}{50}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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41667\\468776\\-1760
\end{tabular}\endgroup%
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7612\\85639\\-322
\end{tabular}\endgroup%
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415683\\4676650\\-17600
\end{tabular}\endgroup%
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10259\\115419\\-435
\end{tabular}\endgroup%
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9573\\107701\\-407
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2907\\32705\\-125
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11490050, 1020800, -133100; 1020800, -90690, 11825; -133100, 11825, -1512],[550, 5, 1],[9175, 41667, 7612, 415683, 10259, 9573, 2907, 1231, -43, -44, 557, 147, 833, 1823; 103224, 468776, 85639, 4676650, 115419, 107701, 32705, 13849, -484, -495, 6270, 1654, 9372, 20510; -385, -1760, -322, -17600, -435, -407, -125, -55, 0, 2, 0, -5, -33, -75],72,11]
%
% END OF LATTICE L73.11
%
\hbox{}\par\smallskip%
%
% LATTICE L73.12
%
\leavevmode%
\llap{\spacer}%
${L_{73.12}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${880}\above{1pt}{1pt}{8,5}{\infty z}{220}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}{176}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35111120\\3159640\\626560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3159640\\-284272\\-56384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
626560\\-56384\\-11181
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-18319159\\-569360\\-1023709280
\end{tabular}\endgroup%
\kern3pt%
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1656369\\51479\\92561040
\end{tabular}\endgroup%
\kern3pt%
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326898\\10160\\18267679
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1591\\55\\88880
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\HardButStrongLineBreak\kern3pt%
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-53\\0\\-2970
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\HardButStrongLineBreak\kern3pt%
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-55\\-2\\-3072
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
74\\0\\4147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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236\\5\\13200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1198\\33\\66968
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
508\\15\\28392
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\kern3pt%
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-5720\\-9240\\-3080\\-7920\\-3520\\-2640\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35111120, 3159640, 626560; 3159640, -284272, -56384; 626560, -56384, -11181],[440, 8, 1],[1591, -53, -55, 74, 236, 1198, 508, 12566, 28367, 10351, 28240, 13921, 12959, 779; 55, 0, -2, 0, 5, 33, 15, 385, 880, 322, 880, 435, 407, 25; 88880, -2970, -3072, 4147, 13200, 66968, 28392, 702240, 1585210, 578432, 1578093, 777920, 724152, 43528],72,12]
%
% END OF LATTICE L73.12
%
\hbox{}\par\smallskip%
%
% LATTICE L73.13
%
\leavevmode%
\llap{\spacer}%
${L_{73.13}}=\pdualwitharg{11}(\pfillwitharg{5}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{4,3}{\infty a}{40}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7255160\\-1648240\\29920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1648240\\-374374\\6787
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29920\\6787\\-122
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1679689\\8535000\\62840360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-376749\\1914374\\14094905
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6273\\-31875\\-234686
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\10\\65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\-300\\-2220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-129\\656\\4856
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-237\\1205\\8910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\305\\2252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-206\\1047\\7722
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-20\\-55\\-160\\-165\\-25\\-55\\-65\\-420\\-385\\-720\\-440\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\40\\0\\-120\\-220\\-60\\-220\\-420\\-3560\\-3520\\-6920\\-4620\\-380\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-120\\0\\8\\0\\-40\\-264\\-720\\-6920\\-7040\\-14088\\-9680\\-840\\-616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-60\\-33\\-40\\0\\2\\0\\-30\\-380\\-407\\-840\\-605\\-57\\-55
\end{tabular}\endgroup%
\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-120\\0\\8\\0\\-40\\-264
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-165\\-220\\-55\\0\\110\\0\\-110
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-60\\-33\\-40\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-220\\-165\\-264\\-110\\0\\22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-420\\-385\\-720\\-440\\-30\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-3560\\-3520\\-6920\\-4620\\-380\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-3520\\-3542\\-7040\\-4785\\-407\\-275
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-6920\\-7040\\-14088\\-9680\\-840\\-616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\-4620\\-4785\\-9680\\-6765\\-605\\-495
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-380\\-407\\-840\\-605\\-57\\-55
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-220\\-275\\-616\\-495\\-55\\-77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7255160, -1648240, 29920; -1648240, -374374, 6787; 29920, 6787, -122],[440, 11, 1],[-2, 59, 1, -129, -237, -60, -206, -367, -3011, -2953, -5775, -3822, -309, -163; 10, -300, -5, 656, 1205, 305, 1047, 1865, 15300, 15005, 29344, 19420, 1570, 828; 65, -2220, -33, 4856, 8910, 2252, 7722, 13740, 112660, 110473, 216024, 142945, 11553, 6083],72,13]
%
% END OF LATTICE L73.13
%
\hbox{}\par\smallskip%
%
% LATTICE L73.14
%
\leavevmode%
\llap{\spacer}%
${L_{73.14}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${220}\above{1pt}{1pt}{20,9}{\infty z}{55}\above{1pt}{1pt}{r}{2}{100}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{1100}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42549100\\-601150\\21193700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-601150\\9880\\-299410
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21193700\\-299410\\10556579
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2252354171\\-74048040\\4519792200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33471751\\1100411\\-67167660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1121870336\\-36882432\\2251253759
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-745120\\-24497\\1495230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1660911\\-54604\\3332945
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3021981\\-99350\\6064200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3292402\\-108240\\6606853
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-809325\\-26607\\1624070
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3749171\\-123255\\7523450
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44199\\-1453\\88694
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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\kern3pt%
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-330\\-110\\-50\\0\\20\\0\\-20
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-110\\-110\\-150\\-132\\-20\\0\\4
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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0\\-110\\-250\\-308\\-90\\-550\\-14
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[42549100, -601150, 21193700; -601150, 9880, -299410; 21193700, -299410, 10556579],[550, 10, 1],[-745120, -1660911, -3021981, -3292402, -809325, -3749171, -44199, -86007, -1343, -299, -32106, -21802, -406464, -31358; -24497, -54604, -99350, -108240, -26607, -123255, -1453, -2827, -44, -10, -1056, -717, -13365, -1031; 1495230, 3332945, 6064200, 6606853, 1624070, 7523450, 88694, 172590, 2695, 600, 64427, 43750, 815650, 62926],72,14]
%
% END OF LATTICE L73.14
%
\hbox{}\par\smallskip%
%
% LATTICE L73.15
%
\leavevmode%
\llap{\spacer}%
${L_{73.15}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${220}\above{1pt}{1pt}{20,1}{\infty z}{55}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{275}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{44}\above{1pt}{1pt}{s}{2}{100}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
622538400\\62277050\\311244450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62277050\\6230080\\31136050
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
311244450\\31136050\\155609851
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6025260779\\209757240\\-12093450600
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\kern3pt%
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603490029\\21009281\\-1211279830
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3012403578\\104870724\\-6046270061
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-558406\\-19437\\1120790
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-453623\\-15792\\910478
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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-166407\\-5795\\334000
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[622538400, 62277050, 311244450; 62277050, 6230080, 31136050; 311244450, 31136050, 155609851],[550, 10, 1],[-558406, -1246233, -453623, -12357514, -607705, -563327, -166407, -65327, -1233, -1, -117146, -16028, -60406, -117108; -19437, -43384, -15792, -430210, -21157, -19613, -5795, -2277, -44, 0, -4070, -557, -2101, -4075; 1120790, 2501345, 910478, 24803075, 1219740, 1130668, 334000, 131120, 2475, 2, 235125, 32170, 121242, 235050],72,15]
%
% END OF LATTICE L73.15
%
\hbox{}\par\smallskip%
%
% LATTICE L73.16
%
\leavevmode%
\llap{\spacer}%
${L_{73.16}}=\pdualwitharg{5.11}(\pfillwitharg{2.5}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{2,1}{\infty b}{2}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84823530\\-96580\\34538020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96580\\110\\-39325
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34538020\\-39325\\14063018
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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32178959\\-73479780\\-79235640
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3837\\8758\\9448
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-38240\\87319\\94160
\end{tabular}\endgroup%
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-37777\\86284\\93020
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-10172\\23243\\25047
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[84823530, -96580, 34538020; -96580, 110, -39325; 34538020, -39325, 14063018],[110, 55, 1],[-876, -3837, -38240, -37777, -10172, -4250, -2658, -80, 13, 0, -463, -344, -530, -2122; 1996, 8758, 87319, 86284, 23243, 9719, 6101, 187, -26, 1, 1036, 770, 1196, 4814; 2157, 9448, 94160, 93020, 25047, 10465, 6545, 197, -32, 0, 1140, 847, 1305, 5225],72,16]
%
% END OF LATTICE L73.16
%
\hbox{}\par\smallskip%
%
% LATTICE L73.17
%
\leavevmode%
\llap{\spacer}%
${L_{73.17}}=\pdualwitharg{5}({L_{73.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{20,11}{\infty a}{440}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{2200}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{s}{2}{50}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-39509800\\14410000\\235400
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\kern3pt%
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14410000\\-5255610\\-85855
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
235400\\-85855\\-1402
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
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23393\\64196\\-3520
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2100\\5763\\-322
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226869\\622600\\-35200
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2750\\7547\\-435
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2481\\6809\\-407
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643\\1765\\-125
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1100\\-125\\-15400\\-225\\-275\\-175
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-39509800, 14410000, 235400; 14410000, -5255610, -85855; 235400, -85855, -1402],[2200, 5, 1],[2774, 23393, 2100, 226869, 2750, 2481, 643, 108, -337, -4, 3691, 132, 401, 667; 7612, 64196, 5763, 622600, 7547, 6809, 1765, 297, -924, -11, 10120, 362, 1100, 1830; -385, -3520, -322, -35200, -435, -407, -125, -55, 0, 2, 0, -5, -33, -75],72,17]
%
% END OF LATTICE L73.17
%
\hbox{}\par\smallskip%
%
% LATTICE L73.18
%
\leavevmode%
\llap{\spacer}%
${L_{73.18}}=\pdualwitharg{2.5.11}(\pfillwitharg{2.5}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{220}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{44}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}{220}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3693140\\6061770\\3000470
\end{tabular}\endgroup%
\kern3pt%
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6061770\\-9492120\\-4698430
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\kern3pt%
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3000470\\-4698430\\-2325639
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3693140, 6061770, 3000470; 6061770, -9492120, -4698430; 3000470, -4698430, -2325639],[110, 110, 1],[-7, -16, -322, -160, -87, -37, -25, -1, 0, 2, 0, -1, -3, -15; -290258, -664934, -13388377, -6654619, -3620207, -1540987, -1045037, -42283, -148, 86737, 3799, -37744, -121718, -617668; 586392, 1343329, 27047790, 13443955, 7313702, 3113170, 2111230, 85422, 299, -175230, -7675, 76252, 245900, 1247840],72,18]
%
% END OF LATTICE L73.18
%
\hbox{}\par\smallskip%
%
% LATTICE L73.19
%
\leavevmode%
\llap{\spacer}%
${L_{73.19}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{8,5}{\infty z}{44}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{880}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-57111120\\-7520040\\22487080
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\kern3pt%
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-7520040\\-990160\\2960960
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22487080\\2960960\\-8854121
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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82930598\\-127856\\210578831
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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667\\0\\1694
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709\\-2\\1800
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-4656\\0\\-11825
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-1594\\1\\-4048
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-29976\\33\\-76120
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-57111120, -7520040, 22487080; -7520040, -990160, 2960960; 22487080, 2960960, -8854121],[440, 40, 1],[-4887, 667, 709, -4656, -1594, -29976, -11326, -52192, -114825, -208269, -566134, -55485, -255419, -14619; 11, 0, -2, 0, 1, 33, 15, 77, 176, 322, 880, 87, 407, 25; -12408, 1694, 1800, -11825, -4048, -76120, -28760, -132528, -291566, -528840, -1437535, -140888, -648560, -37120],72,19]
%
% END OF LATTICE L73.19
%
\hbox{}\par\smallskip%
%
% LATTICE L73.20
%
\leavevmode%
\llap{\spacer}%
${L_{73.20}}=\pdualwitharg{5.11}(\pfillwitharg{5}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{4,3}{\infty a}{8}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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\GramMatrix{%
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0\\0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13814680, -70400, 55000; -70400, -330, 275; 55000, 275, -218],[440, 55, 1],[13, 113, 562, 1109, 149, 62, 38, 1, -1, -2, 11, 5, 8, 32; 786, 6836, 34003, 67104, 9017, 3753, 2303, 61, -60, -123, 656, 300, 482, 1932; 4269, 37112, 184580, 364240, 48939, 20365, 12485, 329, -328, -660, 3600, 1639, 2625, 10505],72,20]
%
% END OF LATTICE L73.20
%
\hbox{}\par\smallskip%
%
% LATTICE L73.21
%
\leavevmode%
\llap{\spacer}%
${L_{73.21}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{10,9}{\infty b}{10}\above{1pt}{1pt}{r}{2}{550}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{s}{2}{50}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1632950\\-5744750\\-290950
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5744750\\-20201060\\-1023165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-290950\\-1023165\\-51822
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-9294732\\-18660689\\420622235
\end{tabular}\endgroup%
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-469800\\-943200\\21260249
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-51\\-103\\2320
\end{tabular}\endgroup%
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49\\98\\-2210
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54\\110\\-2475
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-127\\-254\\5728
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-491\\-983\\22165
\end{tabular}\endgroup%
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-691\\-1385\\31225
\end{tabular}\endgroup%
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-515\\-1033\\23287
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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10\\-10\\-275\\-80\\-165\\-125\\-55
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-275\\0\\550\\0\\-275\\-825\\-825
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-110\\-825\\-132\\-110\\0\\22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-210\\-1925\\-360\\-440\\-150\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
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-1925\\-8800\\-88550\\-17600\\-23925\\-10175\\-1375
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1632950, -5744750, -290950; -5744750, -20201060, -1023165; -290950, -1023165, -51822],[550, 55, 1],[-51, 49, 54, -127, -491, -691, -515, -993, -4225, -41814, -8225, -10993, -4529, -529; -103, 98, 110, -254, -983, -1385, -1033, -1993, -8482, -83950, -16514, -22073, -9095, -1063; 2320, -2210, -2475, 5728, 22165, 31225, 23287, 44925, 191190, 1892275, 372232, 497530, 205000, 23958],72,21]
%
% END OF LATTICE L73.21
%
\hbox{}\par\smallskip%
%
% LATTICE L73.22
%
\leavevmode%
\llap{\spacer}%
${L_{73.22}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{10,1}{\infty b}{10}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{50}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{550}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1573550\\-469150\\44000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-469150\\-139040\\13035
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44000\\13035\\-1222
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91759\\-5091440\\-51013600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26492\\-1469949\\-14728120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2479\\137551\\1378189
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-169\\-1695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\54\\540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\57\\572
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-700\\-7000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-654\\-6545
\end{tabular}\endgroup%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1573550, -469150, 44000; -469150, -139040, 13035; 44000, 13035, -1222],[550, 55, 1],[3, -1, -1, 13, 12, 4, 82, 34, 149, 297, 1467, 395, 33, 103; -169, 54, 57, -700, -654, -220, -4530, -1884, -8266, -16481, -81420, -21929, -1833, -5735; -1695, 540, 572, -7000, -6545, -2203, -45375, -18875, -82820, -165132, -815800, -219725, -18367, -57475],72,22]
%
% END OF LATTICE L73.22
%
\hbox{}\par\smallskip%
%
% LATTICE L73.23
%
\leavevmode%
\llap{\spacer}%
${L_{73.23}}=\pdualwitharg{2.11}(\pfillwitharg{5}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{8,5}{\infty z}{20}\above{1pt}{1pt}{*}{2}{176}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{880}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{176}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2837332080, -972455880, -13234232000; -972455880, -333295600, -4535846304; -13234232000, -4535846304, -61728676015],[440, 88, 1],[-208257, 50741, 53249, -70842, -1170382, -357972, -1405010, -2825852, -6118959, -24326121, -5997376, -32204817, -2676137, -1629099; 1057275, -257600, -270334, 359648, 5941755, 1817341, 7132913, 14346205, 31064560, 123498174, 30447312, 163496525, 13586139, 8270567; -33040, 8050, 8448, -11239, -185680, -56792, -222904, -448320, -970770, -3859328, -951481, -5109280, -424568, -258456],72,23]
%
% END OF LATTICE L73.23
%
\hbox{}\par\smallskip%
%
% LATTICE L73.24
%
\leavevmode%
\llap{\spacer}%
${L_{73.24}}=\pdualwitharg{2.11}(\pfillwitharg{2}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{20,9}{\infty z}{5}\above{1pt}{1pt}{r}{2}{1100}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{220}\above{1pt}{1pt}{s}{2}{100}\above{1pt}{1pt}{s}{2}{44}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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80878430600\\686881250\\40298405400
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-250\\-150\\-1650\\-100\\0\\100\\0\\-300\\-950\\-20350\\-2100\\-6050\\-2850\\-550
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\-110\\-1650\\-132\\-220\\0\\44\\0\\-110\\-2750\\-308\\-990\\-550\\-154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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20\\-10\\-550\\-80\\-330\\-250\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\5\\0\\-15\\-110\\-150\\-110
\end{tabular}\endgroup%
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-550\\0\\1100\\0\\-550\\-1650\\-1650
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-15\\0\\1\\0\\-100\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-110\\-550\\0\\220\\0\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-250\\-150\\-1650\\-100\\0\\100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\-110\\-1650\\-132\\-220\\0\\44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-210\\-3850\\-360\\-880\\-300\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-445\\-8800\\-865\\-2310\\-950\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3850\\-8800\\-177100\\-17600\\-47850\\-20350\\-2750
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-865\\-17600\\-1761\\-4840\\-2100\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-880\\-2310\\-47850\\-4840\\-13530\\-6050\\-990
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\-950\\-20350\\-2100\\-6050\\-2850\\-550
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-110\\-2750\\-308\\-990\\-550\\-154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[80878430600, 686881250, 40298405400; 686881250, 5833520, 342244760; 40298405400, 342244760, 20079042901],[550, 110, 1],[9719, 187, 274, 3531, 26439, 45129, 38441, 83257, 185765, 3718766, 368373, 996297, 419751, 54535; -934, -18, 5, -333, -2509, -4305, -3675, -7969, -17788, -356125, -35279, -95424, -40210, -5228; -19490, -375, -550, -7081, -53020, -90500, -77088, -166960, -372525, -7457450, -738719, -1997930, -841750, -109362],72,24]
%
% END OF LATTICE L73.24
%
\hbox{}\par\smallskip%
%
% LATTICE L73.25
%
\leavevmode%
\llap{\spacer}%
${L_{73.25}}=\pdualwitharg{2.5.11}(\pfillwitharg{2}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{20,1}{\infty z}{5}\above{1pt}{1pt}{r}{2}{44}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{r}{2}{220}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{1100}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6624534400\\-131853150\\3300266200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-131853150\\2624380\\-65687710
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3300266200\\-65687710\\1644154341
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6633639741\\73905520\\13318476600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
131940151\\-1469949\\-264898590
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3304801941\\36818868\\6635109689
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6749\\-73\\-13550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\-1\\-265
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-2\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11991\\-145\\-24075
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18080\\-213\\-36300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6210\\-71\\-12468
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132588\\-1495\\-266200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
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20\\-10\\-110\\-400\\-330\\-50\\-550\\-130\\-210\\-770\\-1800\\-880\\-60\\0
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\kern3pt%
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{$\left[\!\llap{\phantom{%
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-400\\-75\\0\\25\\0\\-100\\-3300
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-50\\-30\\-66\\-100\\0\\4\\0
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-550\\-550\\-1650\\-3300\\-1100\\0\\1100
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\kern3pt%
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-130\\-210\\-770\\-1800\\-880\\-60\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-445\\-1760\\-4325\\-2310\\-190\\-550
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-770\\-1760\\-7084\\-17600\\-9570\\-814\\-2750
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-4325\\-17600\\-44025\\-24200\\-2100\\-7700
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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0\\-550\\-2750\\-7700\\-4950\\-550\\-3850
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6624534400, -131853150, 3300266200; -131853150, 2624380, -65687710; 3300266200, -65687710, 1644154341],[550, 110, 1],[6749, 132, -11, 11991, 18080, 6210, 132588, 57518, 128402, 514147, 1273349, 688857, 58057, 188747; -73, -1, -2, -145, -213, -71, -1495, -643, -1431, -5726, -14175, -7663, -645, -2085; -13550, -265, 22, -24075, -36300, -12468, -266200, -115480, -257795, -1032262, -2556525, -1383030, -116562, -378950],72,25]
%
% END OF LATTICE L73.25
%
\hbox{}\par\smallskip%
%
% LATTICE L73.26
%
\leavevmode%
\llap{\spacer}%
${L_{73.26}}=\pdualwitharg{2}({L_{73.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${880}\above{1pt}{1pt}{40,29}{\infty z}{220}\above{1pt}{1pt}{*}{2}{400}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}{4400}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-583796400\\-245337400\\-20550200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-245337400\\-103101680\\-8635960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20550200\\-8635960\\-723109
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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2822250419\\-7043941080\\3918340800
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\kern3pt%
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1186357687\\-2960982339\\1647108880
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99923983\\-249396242\\138731919
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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45955\\-114697\\63800
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-6417\\16016\\-8910
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-6627\\16540\\-9200
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8960\\-22363\\12441
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15240\\-38037\\21160
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285218\\-711865\\396000
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21504\\-53671\\29856
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-583796400, -245337400, -20550200; -245337400, -103101680, -8635960; -20550200, -8635960, -723109],[2200, 40, 1],[45955, -6417, -6627, 8960, 15240, 285218, 21504, 494706, 1087739, 1972667, 1072362, 525421, 2418087, 27647; -114697, 16016, 16540, -22363, -38037, -711865, -53671, -1234717, -2714844, -4923500, -2676465, -1311377, -6035205, -69003; 63800, -8910, -9200, 12441, 21160, 396000, 29856, 686840, 1510190, 2738800, 1488839, 729480, 3357200, 38384],72,26]
%
% END OF LATTICE L73.26
%
\hbox{}\par\smallskip%
%
% LATTICE L73.27
%
\leavevmode%
\llap{\spacer}%
${L_{73.27}}=\pdualwitharg{2.5}({L_{73.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${880}\above{1pt}{1pt}{40,21}{\infty z}{220}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{275}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}{176}\above{1pt}{1pt}{s}{2}{400}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-285555600, -130686600, -20114600; -130686600, -59808080, -9203680; -20114600, -9203680, -1414581],[2200, 40, 1],[-963124, -2174245, -793377, -10822644, -1067017, -993291, -298559, -121971, 4055, 4217, -28306, -18078, -91804, -194666; 2414093, 5449796, 1988618, 27127210, 2674503, 2489707, 748345, 305723, -10164, -10570, 70950, 45313, 230109, 487935; -2011680, -4541350, -1657128, -22605275, -2228680, -2074688, -623600, -254760, 8470, 8808, -59125, -37760, -191752, -406600],72,27]
%
% END OF LATTICE L73.27
%
\hbox{}\par\smallskip%
%
% LATTICE L73.28
%
\leavevmode%
\llap{\spacer}%
${L_{73.28}}=\pdualwitharg{11}({L_{73.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{20,19}{\infty a}{40}\above{1pt}{1pt}{b}{2}{550}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{b}{2}{50}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-58231800, -258944400, -12826000; -258944400, -1150030310, -56969165; -12826000, -56969165, -2822062],[2200, 55, 1],[-2806, 2799, 5628, -6145, -13986, -22397, -17925, -36420, -316607, -1574668, -621471, -417500, -173733, -21301; -6853, 6836, 13745, -15008, -34158, -54700, -43778, -88948, -773244, -3845785, -1517808, -1019653, -424305, -52023; 151095, -150720, -303050, 330896, 753115, 1206025, 965217, 1961125, 17048480, 84791850, 33464624, 22481305, 9355075, 1147003],72,28]
%
% END OF LATTICE L73.28
%
\hbox{}\par\smallskip%
%
% LATTICE L73.29
%
\leavevmode%
\llap{\spacer}%
${L_{73.29}}=\pdualwitharg{5.11}({L_{73.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{20,11}{\infty a}{40}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{200}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{550}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73191800\\-3115200\\94600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3115200\\-132110\\4015
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
94600\\4015\\-122
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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82319\\7120680\\298067000
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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0\\0\\0
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6\\517\\21660
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49\\4236\\177340
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48\\4153\\173833
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\HardButStrongLineBreak\kern3pt%
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469\\40600\\1699200
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62\\5372\\224785
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5\\434\\18153
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left[\!\llap{\phantom{%
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-55\\0\\22\\0\\-55\\-33\\-825
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\kern3pt%
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-800\\-600\\0\\200\\0\\-200\\-6600
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-25\\-60\\-33\\-200\\0\\2\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-275\\-1100\\-825\\-6600\\-550\\0\\550
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-3560\\-3520\\-34600\\-4620\\-380\\-1100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3600\\-34600\\-35200\\-352200\\-48400\\-4200\\-15400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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0\\-1100\\-1375\\-15400\\-2475\\-275\\-1925
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-73191800, -3115200, 94600; -3115200, -132110, 4015; 94600, 4015, -122],[2200, 55, 1],[6, 49, 48, 469, 62, 5, 13, 0, -1, 0, 11, 4, 1, 17; 517, 4236, 4153, 40600, 5372, 434, 1140, 2, -84, -1, 920, 337, 85, 1455; 21660, 177340, 173833, 1699200, 224785, 18153, 47575, 65, -3540, -33, 38800, 14190, 3572, 61050],72,29]
%
% END OF LATTICE L73.29
%
\hbox{}\par\smallskip%
%
% LATTICE L73.30
%
\leavevmode%
\llap{\spacer}%
${L_{73.30}}=\pdualwitharg{2.5.11}(\pfillwitharg{5}({L_{73.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{8,5}{\infty z}{4}\above{1pt}{1pt}{*}{2}{880}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{176}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{s}{2}{880}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-15720468720\\-10613273880\\-7812000680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10613273880\\-7165281200\\-5274073200
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7812000680\\-5274073200\\-3882031491
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-36912662551\\1624400\\74278938800
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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173259\\-7\\-348648
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367793\\-16\\-740106
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1788514\\-80\\-3599005
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1911322\\-87\\-3846128
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786668\\-37\\-1583000
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15720468720, -10613273880, -7812000680; -10613273880, -7165281200, -5274073200; -7812000680, -5274073200, -3882031491],[440, 440, 1],[173259, 367793, 7276233, 1788514, 1911322, 786668, 457214, 8532, -4211, -4593, 29396, 88379, 122287, 451741; -7, -16, -322, -80, -87, -37, -25, -1, 0, 2, 0, -1, -3, -15; -348648, -740106, -14641880, -3599005, -3846128, -1583000, -920040, -17168, 8474, 9240, -59155, -177848, -246080, -909040],72,30]
%
% END OF LATTICE L73.30
%
\hbox{}\par\smallskip%
%
% LATTICE L73.31
%
\leavevmode%
\llap{\spacer}%
${L_{73.31}}=\pdualwitharg{2.11}({L_{73.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{40,29}{\infty z}{20}\above{1pt}{1pt}{*}{2}{4400}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{880}\above{1pt}{1pt}{*}{2}{400}\above{1pt}{1pt}{s}{2}{176}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9655540400, 3722415400, -1587141600; 3722415400, -1435069680, 611876320; -1587141600, 611876320, -260887919],[2200, 440, 1],[-100449, -213235, -4218547, -207386, -1108134, -456092, -53018, -4948, 2441, 2667, -3408, -51233, -70893, -52379; -401803, -852956, -16874510, -829560, -4432623, -1824405, -212077, -19793, 9764, 10670, -13632, -204933, -283575, -209519; -331280, -703250, -13912800, -683961, -3654640, -1504200, -174856, -16320, 8050, 8800, -11239, -168960, -233800, -172744],72,31]
%
% END OF LATTICE L73.31
%
\hbox{}\par\smallskip%
%
% LATTICE L73.32
%
\leavevmode%
\llap{\spacer}%
${L_{73.32}}=\pdualwitharg{2.5.11}({L_{73.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-78602343600, -10658016600, -1795694073800; -10658016600, -1445163280, -243485376640; -1795694073800, -243485376640, -41023135194311],[2200, 440, 1],[86374, -21159, -22017, 147706, 487581, 148993, 2922411, 1175097, 2544101, 10113785, 12467004, 13388600, 1112478, 3384944; 28503807, -6982576, -7265702, 48743720, 160904127, 49168409, 964409585, 387787577, 839565344, 3337596774, 4114170120, 4418301097, 367122975, 1117047335; -172960, 42370, 44088, -295775, -976360, -298352, -5852000, -2353080, -5094450, -20252408, -24964625, -26810080, -2227688, -6778200],72,32]
%
% END OF LATTICE L73.32
%

\medskip%
%------------------------------WEYL-GROUP-W74--------------------------------
%
\leavevmode\llap{}%
$W_{74\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=38$%
\hfill%
$22642264\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L74.1
%
\leavevmode%
\llap{\spacer}%
${L_{74.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}37\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{74}}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{148}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3066708\\21312\\20868
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21312\\-146\\-145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20868\\-145\\-142
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
124319\\-124320\\18381600
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\kern3pt%
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{$\left.\llap{\phantom{%
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\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-40\\5474
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
327\\-370\\48396
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-149
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-40\\-40\\-444\\-12\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\148\\0\\-370\\-444\\-5624\\-222\\-148
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\-3\\-12\\-222\\-15\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\148\\0\\-370
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-370\\-3\\2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-444\\-12\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-444\\-5624\\-222\\-148
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-222\\-15\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-148\\-18\\-51
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3066708, 21312, 20868; 21312, -146, -145; 20868, -145, -142],[444, 1, 1],[37, 327, 1, -1, 5, 191, 20, 50; -40, -370, -3, 2, -2, -148, -18, -51; 5474, 48396, 150, -149, 736, 28194, 2955, 7394],73,1]
%
% END OF LATTICE L74.1
%
\hbox{}\par\smallskip%
%
% LATTICE L74.2
%
\leavevmode%
\llap{\spacer}%
${L_{74.2}}=\pfillwitharg{2}({L_{74.1}})$%
{${}={}$}Nikulin $74$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}37\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{37}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3774\\-1221\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1221\\-395\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8879\\-27195\\-1665
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2896\\-8870\\-543
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-147\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\37\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\-248\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-20\\-10\\-111\\-6\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\37\\0\\-185\\-111\\-1406\\-111\\-74
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\37\\0\\-185
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-185\\-3\\2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-111\\-6\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111\\-1406\\-111\\-74
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-111\\-15\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-74\\-18\\-51
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3774, -1221, 0; -1221, -395, 0; 0, 0, 1],[111, 1, 1],[0, -12, 1, 81, 48, 604, 47, 31; 0, 37, -3, -248, -147, -1850, -144, -95; 1, 0, -3, -20, -10, -111, -6, -1],73,2]
%
% END OF LATTICE L74.2
%
\hbox{}\par\smallskip%
%
% LATTICE L74.3
%
\leavevmode%
\llap{\spacer}%
${L_{74.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{74.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}37\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{111}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-492951\\7437\\-162060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7437\\-105\\2445
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-162060\\2445\\-53278
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1487549\\-23828\\4523916
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24909\\398\\-75753
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-489003\\-7833\\1487150
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-1\\438
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5803\\-74\\17649
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-412\\-6\\1253
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3133\\-51\\9528
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-60\\-30\\-333\\-6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\111\\0\\-555\\-333\\-4218\\-111\\-222
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-555\\-3\\6\\-3\\-222\\-18\\-153
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\111\\0\\-555
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-555\\-3\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-333\\-6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-333\\-4218\\-111\\-222
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-111\\-5\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-222\\-18\\-153
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-492951, 7437, -162060; 7437, -105, 2445; -162060, 2445, -53278],[111, 3, 1],[-144, -5803, -412, -3133, -1167, -10366, -25, 74; -1, -74, -6, -51, -20, -185, -1, 2; 438, 17649, 1253, 9528, 3549, 31524, 76, -225],73,3]
%
% END OF LATTICE L74.3
%
\hbox{}\par\smallskip%
%
% LATTICE L74.4
%
\leavevmode%
\llap{\spacer}%
${L_{74.4}}=\pdualwitharg{3}({L_{74.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}37\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{444}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19330428\\-78588\\52392
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78588\\-318\\213
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52392\\213\\-142
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112479\\-157472\\41235168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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31\\-40\\11370
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
317\\-370\\116328
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\736
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-120\\-120\\-1332\\-12\\-6
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\444\\0\\-1110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-1110\\-3\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-1332\\-12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1332\\-16872\\-222\\-444
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-222\\-5\\-18
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19330428, -78588, 52392; -78588, -318, 213; 52392, 213, -142],[444, 3, 1],[31, 317, 2, 0, -1, 53, 3, 35; -40, -370, -1, 2, -2, -148, -6, -51; 11370, 116328, 736, 3, -372, 19314, 1097, 12828],73,4]
%
% END OF LATTICE L74.4
%
\hbox{}\par\smallskip%
%
% LATTICE L74.5
%
\leavevmode%
\llap{\spacer}%
${L_{74.5}}=\pdualwitharg{2}({L_{74.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}37\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{148}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
679161048\\64380\\-171362244
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64380\\8\\-16244
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171362244\\-16244\\43237195
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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-487235944\\14642787\\-1931059452
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-30513\\916\\-120932
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122936877\\-3694593\\487235027
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{$\left[\!\llap{\phantom{%
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-10952\\327\\-43406
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-94645\\2812\\-375106
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-40\\-444\\-24\\-4
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-444\\-5624\\-444\\-296
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-24\\-444\\-60\\-72
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[679161048, 64380, -171362244; 64380, 8, -16244; -171362244, -16244, 43237195],[444, 4, 1],[-10952, -94645, -109, 0, -2125, -66638, -12968, -30513; 327, 2812, 0, 1, 66, 2035, 393, 916; -43406, -375106, -432, 0, -8422, -264106, -51396, -120932],73,5]
%
% END OF LATTICE L74.5
%
\hbox{}\par\smallskip%
%
% LATTICE L74.6
%
\leavevmode%
\llap{\spacer}%
${L_{74.6}}=\pdualwitharg{37}(\pfillwitharg{2}({L_{74.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}37\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1753023\\-44844\\-569874
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44844\\-1147\\-14578
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-569874\\-14578\\-185255
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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253367\\-405756\\-747252
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6486\\-10388\\-19129
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82386\\-131937\\-242980
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1753023, -44844, -569874; -44844, -1147, -14578; -569874, -14578, -185255],[111, 37, 1],[-62, -56, -413, -979, -352, -82, -1, 13; 72, 79, 624, 1585, 591, 142, 39, -38; 185, 166, 1221, 2886, 1036, 241, 0, -37],73,6]
%
% END OF LATTICE L74.6
%
\hbox{}\par\smallskip%
%
% LATTICE L74.7
%
\leavevmode%
\llap{\spacer}%
${L_{74.7}}=\pdualwitharg{2.3}({L_{74.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}37\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{444}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1049776728\\-1332\\-264834456
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1332\\24\\336
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264834456\\336\\66811625
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1049776728, -1332, -264834456; -1332, 24, 336; -264834456, 336, 66811625],[444, 12, 1],[-50246, -500521, -5429, 0, -277, -122596, -11412, -117887; -283, -2812, -30, 1, -2, -703, -65, -666; -199170, -1984014, -21520, 0, -1098, -485958, -45236, -467292],73,7]
%
% END OF LATTICE L74.7
%
\hbox{}\par\smallskip%
%
% LATTICE L74.8
%
\leavevmode%
\llap{\spacer}%
${L_{74.8}}=\pdualwitharg{3.37}(\pfillwitharg{2}({L_{74.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}37\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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-283494\\1084470\\356643
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[100011, -283494, -93240; -283494, 1084470, 356643; -93240, 356643, 117287],[111, 111, 1],[22, 19, 46, 329, 119, 28, 1, 0; -20765, -17880, -43233, -308774, -111598, -26241, -888, -73; 63159, 54384, 131498, 939171, 339438, 79815, 2701, 222],73,8]
%
% END OF LATTICE L74.8
%
\hbox{}\par\smallskip%
%
% LATTICE L74.9
%
\leavevmode%
\llap{\spacer}%
${L_{74.9}}=\pdualwitharg{37}({L_{74.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}37\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45732\\61716\\11544
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61716\\-83102\\-15577
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11544\\-15577\\-2914
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{$\left.\llap{\phantom{%
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0\\0\\0
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81647\\-2352\\335664
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-45732, 61716, 11544; 61716, -83102, -15577; 11544, -15577, -2914],[444, 37, 1],[147, 169, 677, 1745, 1311, 317, 52, -44; -2, -4, -18, -51, -40, -10, -3, 2; 592, 690, 2775, 7178, 5402, 1308, 222, -185],73,9]
%
% END OF LATTICE L74.9
%
\hbox{}\par\smallskip%
%
% LATTICE L74.10
%
\leavevmode%
\llap{\spacer}%
${L_{74.10}}=\pdualwitharg{3.37}({L_{74.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}37\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${444}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{74}\above{1pt}{1pt}{}{6}{222}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48396\\-24420\\23976
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24420\\-12210\\12099
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23976\\12099\\-11878
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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53815\\1736\\110112
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24738\\797\\50616
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-56\\2\\-111
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-4440\\-1110\\-111\\222
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-4440\\-1332\\-444\\-222
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-1332\\-456\\-222\\-444
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-48396, -24420, 23976; -24420, -12210, 12099; 23976, 12099, -11878],[444, 111, 1],[-1413, -415, -128, -56, 111, -19, -89, -1463; -40, -10, -1, 2, -2, -4, -6, -51; -2886, -846, -259, -111, 222, -42, -185, -2997],73,10]
%
% END OF LATTICE L74.10
%
\hbox{}\par\smallskip%
%
% LATTICE L74.11
%
\leavevmode%
\llap{\spacer}%
${L_{74.11}}=\pdualwitharg{2.37}({L_{74.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}37\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${148}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{888}\above{1pt}{1pt}{}{6}{296}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32856\\-3108\\12432
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3108\\296\\-1184
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12432\\-1184\\4735
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{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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539\\-29640\\-8880
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-4\\205\\62
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\1473\\444
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\3457\\1036
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
148\\0\\-444\\-2960\\-1480\\-444\\-888\\-148
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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148\\0\\-444\\-2960
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-740
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-444\\0\\888\\-444
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2960\\-740\\-444\\296
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1480\\-444\\-888\\-148
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-444\\-152\\-444\\-296
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-888\\-444\\-2220\\-2664
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-148\\-296\\-2664\\-7548
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[32856, -3108, 12432; -3108, 296, -1184; 12432, -1184, 4735],[444, 148, 1],[-5, -4, -28, -63, -22, -5, 1, 0; 243, 205, 1473, 3457, 1239, 289, 9, 1; 74, 62, 444, 1036, 370, 86, 0, 0],73,11]
%
% END OF LATTICE L74.11
%
\hbox{}\par\smallskip%
%
% LATTICE L74.12
%
\leavevmode%
\llap{\spacer}%
${L_{74.12}}=\pdualwitharg{2.3.37}({L_{74.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}37\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${444}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{296}\above{1pt}{1pt}{}{6}{888}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
888\\-50172\\12432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50172\\5423016\\-1343544
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12432\\-1343544\\332861
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
797\\-46963\\-189588
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48336\\-2844617\\-11483616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11970\\704445\\2843819
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
343\\-20182\\-81474
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93\\-5465\\-22062
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\-2273\\-9176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
444\\0\\-444\\-8880\\-4440\\-1332\\-888\\-444
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0\\12\\0\\-2220\\-1332\\-456\\-444\\-888
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
444\\0\\-444\\-8880
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-2220
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-444\\0\\296\\-444
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8880\\-2220\\-444\\888
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4440\\-1332\\-888\\-444
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1332\\-456\\-444\\-888
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-888\\-444\\-740\\-2664
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-444\\-888\\-2664\\-22644
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[888, -50172, 12432; -50172, 5423016, -1343544; 12432, -1343544, 332861],[444, 444, 1],[343, 93, 39, 1, -1, 21, 75, 797; -20182, -5465, -2273, 0, 55, -1244, -4436, -46963; -81474, -22062, -9176, 0, 222, -5022, -17908, -189588],73,12]
%
% END OF LATTICE L74.12
%

\medskip%
%------------------------------WEYL-GROUP-W75--------------------------------
%
\leavevmode\llap{}%
$W_{75\phantom{0}}$%
\qquad\llap{16} lattices, $\chi=48$%
\hfill%
$222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L75.1
%
\leavevmode%
\llap{\spacer}%
${L_{75.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{75}}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{104}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-305240\\-121680\\2080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121680\\-48506\\829
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2080\\829\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18199\\-46800\\-70200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7245\\-18631\\-27945
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\288\\431
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-44\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
278\\-715\\-1079
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
607\\-1560\\-2280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
95\\-244\\-347
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1661\\-4264\\-5928
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-105\\-130
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-40\\-12\\-312\\-20\\-24\\-156\\-280\\-36\\-520\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\26\\0\\-13\\-624\\-65\\-156\\-1547\\-3120\\-455\\-7488\\-130
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\kern3pt%
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-40\\0\\40\\0\\-520\\-80\\-280\\-3120\\-6440\\-960\\-16120\\-320
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-12\\-13\\0\\2\\0\\-5\\-36\\-455\\-960\\-146\\-2496\\-55
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8\\0\\-40\\-12\\-312\\-20
\end{tabular}\endgroup%
\kern3pt%
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0\\26\\0\\-13\\-624\\-65
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\0\\-520\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-13\\0\\2\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-624\\-520\\0\\104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-65\\-80\\-5\\0\\10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-156\\-280\\-36\\-520\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-1547\\-3120\\-455\\-7488\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-3120\\-6440\\-960\\-16120\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-455\\-960\\-146\\-2496\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-7488\\-16120\\-2496\\-43368\\-1040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-130\\-320\\-55\\-1040\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-305240, -121680, 2080; -121680, -48506, 829; 2080, 829, -14],[520, 1, 1],[17, 278, 607, 95, 1661, 41, 11, -5, -47, -11, -205, -6; -44, -715, -1560, -244, -4264, -105, -28, 13, 120, 28, 520, 15; -84, -1079, -2280, -347, -5928, -130, -24, 26, 120, 23, 312, -5],74,1]
%
% END OF LATTICE L75.1
%
\hbox{}\par\smallskip%
%
% LATTICE L75.2
%
\leavevmode%
\llap{\spacer}%
${L_{75.2}}=\pfillwitharg{2}({L_{75.1}})$%
{${}={}$}Nikulin $75$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{26}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7410\\3640\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3640\\1788\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-65\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144949\\-294450\\20800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72252\\-146773\\10368
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12711\\-25821\\1823
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[7410, 3640, -130; 3640, 1788, -65; -130, -65, -16],[130, 1, 1],[-1, 96, -59, -93, -1945, -379, -445, -8793, -8861, -2583, -21247, -736; 2, -195, 120, 189, 3952, 770, 904, 17862, 18000, 5247, 43160, 1495; 0, 13, -10, -14, -286, -55, -64, -1261, -1270, -370, -3042, -105],74,2]
%
% END OF LATTICE L75.2
%
\hbox{}\par\smallskip%
%
% LATTICE L75.3
%
\leavevmode%
\llap{\spacer}%
${L_{75.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{75.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9144980, -151970, 4577040; -151970, 2524, -76060; 4577040, -76060, 2290797],[130, 2, 1],[2599, 7541, -476, -586, 16140, 11493, 12189, 569191, 296236, 178042, 752836, 62447; -2294, -6656, 420, 517, -14248, -10145, -10759, -502411, -261480, -157153, -664508, -55120; -5269, -15288, 965, 1188, -32721, -23300, -24711, -1153932, -600565, -360948, -1526239, -126600],74,3]
%
% END OF LATTICE L75.3
%
\hbox{}\par\smallskip%
%
% LATTICE L75.4
%
\leavevmode%
\llap{\spacer}%
${L_{75.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{75.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24667370, -394550, 4938180; -394550, 6310, -78985; 4938180, -78985, 988578],[130, 5, 1],[2429, 3524, -89, -274, 15081, 1074, 11391, 265966, 55369, 83194, 703559, 5836; -4588, -6656, 168, 517, -28496, -2029, -21518, -502411, -104592, -157153, -1329016, -11024; -12500, -18135, 458, 1410, -77610, -5527, -58620, -1368705, -284938, -428130, -3620630, -30033],74,4]
%
% END OF LATTICE L75.4
%
\hbox{}\par\smallskip%
%
% LATTICE L75.5
%
\leavevmode%
\llap{\spacer}%
${L_{75.5}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{75.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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-9304750\\185580\\-4600840
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[466359140, -9304750, 230596600; -9304750, 185580, -4600840; 230596600, -4600840, 114021121],[130, 10, 1],[4468, -29314, -2116, 5489, 154512, 16305, 59967, 2542279, 259856, 767731, 3196108, 48130; 27, -117, -10, 24, 754, 81, 307, 13143, 1346, 3984, 16614, 253; -9035, 59280, 4279, -11100, -312455, -32972, -121265, -5140980, -525479, -1552500, -6463145, -97328],74,5]
%
% END OF LATTICE L75.5
%
\hbox{}\par\smallskip%
%
% LATTICE L75.6
%
\leavevmode%
\llap{\spacer}%
${L_{75.6}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{75.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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90015770\\-72280\\-27603680
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-27603680\\22165\\8464774
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[90015770, -72280, -27603680; -72280, 52, 22165; -27603680, 22165, 8464774],[130, 13, 1],[-21121, -2357, 3867, 2380, -10091, -46703, -99059, -177913, -2407467, -723460, -470629, -253747; -654, -73, 120, 74, -312, -1445, -3066, -5507, -74520, -22394, -14568, -7855; -68874, -7686, 12610, 7761, -32906, -152295, -323024, -580161, -7850570, -2359149, -1534686, -827450],74,6]
%
% END OF LATTICE L75.6
%
\hbox{}\par\smallskip%
%
% LATTICE L75.7
%
\leavevmode%
\llap{\spacer}%
${L_{75.7}}=\pdualwitharg{5}({L_{75.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2096120, -949520, 41080; -949520, -430110, 18615; 41080, 18615, -802],[520, 5, 1],[-2497, -28609, -11871, -8890, -149921, -611, -455, -173, 63, 34, -3583, -127; 5072, 58110, 24112, 18057, 304512, 1241, 924, 351, -128, -69, 7280, 258; -10180, -116675, -48416, -36260, -611520, -2493, -1860, -715, 256, 140, -14560, -517],74,7]
%
% END OF LATTICE L75.7
%
\hbox{}\par\smallskip%
%
% LATTICE L75.8
%
\leavevmode%
\llap{\spacer}%
${L_{75.8}}=\pdualwitharg{2.13}(\pfillwitharg{2}({L_{75.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[406374540, -2146690, 195103090; -2146690, 11284, -1030640; 195103090, -1030640, 93670277],[130, 26, 1],[-4575, 1665, 8894, -4344, -10246, -71277, -53695, -176475, -1174294, -694896, -222834, -220073; -13, -2, 5, 1, -9, -75, -72, -253, -1705, -1021, -331, -350; 9529, -3468, -18525, 9048, 21341, 148460, 111839, 367572, 2445885, 1447368, 464131, 458380],74,8]
%
% END OF LATTICE L75.8
%
\hbox{}\par\smallskip%
%
% LATTICE L75.9
%
\leavevmode%
\llap{\spacer}%
${L_{75.9}}=\pdualwitharg{2}({L_{75.1}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1411280, -16120, 5200; -16120, -112, 56; 5200, 56, -19],[520, 8, 1],[-1, -1, 2, 5, 16, 8, 2, 40, 8, 7, 10, -3; -15, -13, 30, 74, 234, 115, 27, 481, 90, 70, 78, -55; -318, -312, 635, 1584, 5057, 2520, 622, 12168, 2405, 2064, 2847, -1000],74,9]
%
% END OF LATTICE L75.9
%
\hbox{}\par\smallskip%
%
% LATTICE L75.10
%
\leavevmode%
\llap{\spacer}%
${L_{75.10}}=\pdualwitharg{13}({L_{75.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[520, 0, 0; 0, -26, 13; 0, 13, -6],[520, 13, 1],[-1, 0, 1, 0, -1, -2, -7, -6, -161, -24, -31, -8; 4, 1, 0, -5, -16, -20, -40, -28, -720, -103, -128, -25; 0, 2, 0, -13, -48, -65, -156, -119, -3120, -455, -576, -130],74,10]
%
% END OF LATTICE L75.10
%
\hbox{}\par\smallskip%
%
% LATTICE L75.11
%
\leavevmode%
\llap{\spacer}%
${L_{75.11}}=\pdualwitharg{5.13}(\pfillwitharg{2}({L_{75.1}}))$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[297478610, -2904330, -65031980; -2904330, 28210, 634855; -65031980, 634855, 14216654],[130, 65, 1],[-5631, 1026, 2191, -2676, -12619, -8778, -66121, -108654, -289199, -427836, -274389, -27098; 11236, -2054, -4380, 5353, 25220, 17541, 132098, 217055, 577716, 854651, 548116, 54126; -26260, 4785, 10218, -12480, -58850, -40937, -308360, -506715, -1348698, -1995240, -1279630, -126373],74,11]
%
% END OF LATTICE L75.11
%
\hbox{}\par\smallskip%
%
% LATTICE L75.12
%
\leavevmode%
\llap{\spacer}%
${L_{75.12}}=\pdualwitharg{2.5.13}(\pfillwitharg{2}({L_{75.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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0\\20\\0\\-130\\-120\\-130\\-390\\-1190\\-1560\\-4550\\-1440\\-260
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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65\\0\\-65\\-390\\-195\\-130
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-65\\0\\13\\0\\-65\\-104
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\kern3pt%
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-390\\-130\\0\\260\\0\\-130
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-195\\-120\\-65\\0\\5\\0
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-195\\-390\\-455\\-1170\\-325\\0
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1170\\-4550\\-6240\\-18980\\-6240\\-1430
\end{tabular}\endgroup%
\kern3pt%
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-325\\-1440\\-2015\\-6240\\-2085\\-520
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-260\\-416\\-1430\\-520\\-182
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7951949460, 10593424010, 5154038760; 10593424010, 14112342140, 6866104700; 5154038760, 6866104700, 3340578997],[130, 130, 1],[-327, -73, 12, 74, -156, -289, -1533, -5507, -7452, -22394, -7284, -1571; -642524, -143404, 23526, 144785, -306994, -568317, -3013525, -10824743, -14647722, -44017373, -14317202, -3087732; 1321125, 294860, -48373, -297700, 631225, 1168544, 6196255, 22257280, 30117893, 90506260, 29438295, 6348836],74,12]
%
% END OF LATTICE L75.12
%
\hbox{}\par\smallskip%
%
% LATTICE L75.13
%
\leavevmode%
\llap{\spacer}%
${L_{75.13}}=\pdualwitharg{2.5}({L_{75.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{1040}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{l}{2}{65}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1040\\-520\\-2080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-240\\-1040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2080\\-1040\\-4159
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12715\\936\\6240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5379\\395\\2640
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-25102\\1848\\12319
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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21\\-3\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-13\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\2\\0
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\HardButStrongLineBreak\kern3pt%
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128\\0\\-65
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-1\\-16
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-10\\-120\\-390\\-40\\-60\\-1560\\-70\\-360\\-650\\0
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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20\\0\\-10\\-120\\-390\\-40
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0\\1040\\0\\-520\\-3120\\-520
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-120\\-520\\0\\80\\0\\-40
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-390\\-3120\\-65\\0\\65\\0
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-60\\-1560\\-70\\-360\\-650\\0
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0\\-1040\\-64\\-440\\-1040\\-56
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1040, -520, -2080; -520, -240, -1040; -2080, -1040, -4159],[520, 40, 1],[21, 5, -2, -1, 128, 32, 142, 6352, 328, 1957, 4110, 131; -3, -13, 0, 2, 0, -1, -9, -455, -24, -146, -312, -11; -10, 0, 1, 0, -65, -16, -70, -3120, -161, -960, -2015, -64],74,13]
%
% END OF LATTICE L75.13
%
\hbox{}\par\smallskip%
%
% LATTICE L75.14
%
\leavevmode%
\llap{\spacer}%
${L_{75.14}}=\pdualwitharg{5.13}({L_{75.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${520}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{104}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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293070440\\-91787280\\-704600
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\kern3pt%
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-91787280\\28747030\\220675
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-704600\\220675\\1694
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
821439\\3370320\\-97364800
\end{tabular}\endgroup%
\kern3pt%
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-257176\\-1055179\\30482920
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1972\\-8091\\233739
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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47\\188\\-4940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\11\\-185
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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0\\-1\\130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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27\\112\\-3360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\46\\-1417
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\right.$}%
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520\\0\\-520\\-780\\-1560\\-260
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0\\10\\0\\-65\\-240\\-65
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-780\\-65\\0\\130\\0\\-65
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-1560\\-780\\-3640\\-2340\\-2600\\0
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\kern3pt%
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-780\\-595\\-3120\\-2275\\-2880\\-130
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-3640\\-3120\\-16744\\-12480\\-16120\\-832
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2340\\-2275\\-12480\\-9490\\-12480\\-715
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-130\\-832\\-715\\-1040\\-91
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[293070440, -91787280, -704600; -91787280, 28747030, 220675; -704600, 220675, 1694],[520, 65, 1],[47, 3, 1, 0, 27, 11, 225, 201, 1087, 816, 1061, 57; 188, 11, 0, -1, 112, 46, 928, 826, 4464, 3349, 4352, 233; -4940, -185, 416, 130, -3360, -1417, -27300, -23995, -129376, -96850, -125600, -6643],74,14]
%
% END OF LATTICE L75.14
%
\hbox{}\par\smallskip%
%
% LATTICE L75.15
%
\leavevmode%
\llap{\spacer}%
${L_{75.15}}=\pdualwitharg{2.13}({L_{75.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[63440, -56680, 1040; -56680, 49712, -936; 1040, -936, 17],[520, 104, 1],[3, 3, -1, -18, -7, -69, -35, -99, -319, -366, -57, -91; 1, 1, 0, -5, -2, -20, -10, -28, -90, -103, -16, -25; -130, -128, 65, 832, 319, 3120, 1586, 4496, 14495, 16640, 2593, 4160],74,15]
%
% END OF LATTICE L75.15
%
\hbox{}\par\smallskip%
%
% LATTICE L75.16
%
\leavevmode%
\llap{\spacer}%
${L_{75.16}}=\pdualwitharg{2.5.13}({L_{75.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-34297640\\-49055760\\122720
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-23979280, -34297640, 85800; -34297640, -49055760, 122720; 85800, 122720, -307],[520, 520, 1],[15, -1, -2, -1, 8, 26, 112, 382, 256, 1525, 246, 101; -3, -1, 0, 2, 0, -1, -9, -35, -24, -146, -24, -11; 2990, -680, -559, 520, 2235, 6864, 27690, 92720, 61919, 367640, 59125, 23816],74,16]
%
% END OF LATTICE L75.16
%

\medskip%
%------------------------------WEYL-GROUP-W76--------------------------------
%
\leavevmode\llap{}%
$W_{76\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=48$%
\hfill%
$222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L76.1
%
\leavevmode%
\llap{\spacer}%
${L_{76.1}}$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}47\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{76}}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{188}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{282}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{94}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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1692\\-2\\-7
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1188348, 1692, 3384; 1692, -2, -7; 3384, -7, 2],[564, 1, 1],[-29, -195, -2, -35, -1, -1, 1, 7, 0, -59, -7, -93; -14832, -99734, -1023, -17907, -512, -517, 510, 3572, 0, -30174, -3580, -47564; -2826, -18988, -194, -3384, -96, -94, 96, 658, -1, -5781, -684, -9071],75,1]
%
% END OF LATTICE L76.1
%
\hbox{}\par\smallskip%
%
% LATTICE L76.2
%
\leavevmode%
\llap{\spacer}%
${L_{76.2}}=\pfillwitharg{2}({L_{76.1}})$%
{${}={}$}Nikulin $76$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}47\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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7050\\-2397\\0
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{$\left.\llap{\phantom{%
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10339\\30456\\4512
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-3575\\-10531\\-1560
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440\\1296\\191
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\3\\0
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16\\47\\0
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-1\\-3\\-1
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\HardButStrongLineBreak\kern3pt%
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-1579\\-4653\\-705
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-93\\-274\\-41
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[7050, -2397, 0; -2397, 815, 0; 0, 0, -1],[141, 1, 1],[1, 16, -1, -1579, -93, -2473, -386, -2601, -54, -1006, -17, -112; 3, 47, -3, -4653, -274, -7285, -1137, -7661, -159, -2961, -50, -329; 0, 0, -1, -705, -41, -1081, -168, -1128, -23, -423, -7, -47],75,2]
%
% END OF LATTICE L76.2
%
\hbox{}\par\smallskip%
%
% LATTICE L76.3
%
\leavevmode%
\llap{\spacer}%
${L_{76.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{76.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}47\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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1833\\3\\-618
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
133245\\-618\\-44666
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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13034321\\12361188\\38705205
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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95\\94\\282
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101\\93\\300
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-397479, 1833, 133245; 1833, 3, -618; 133245, -618, -44666],[141, 3, 1],[-67, 95, 101, -10415, -1962, -54701, -2915, -60161, -1379, -9607, -594, -5365; -62, 94, 93, -9917, -1865, -51935, -2766, -57058, -1305, -9071, -559, -5029; -199, 282, 300, -30926, -5826, -162432, -8656, -178647, -4095, -28529, -1764, -15933],75,3]
%
% END OF LATTICE L76.3
%
\hbox{}\par\smallskip%
%
% LATTICE L76.4
%
\leavevmode%
\llap{\spacer}%
${L_{76.4}}=\pdualwitharg{3}({L_{76.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}47\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{564}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{94}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{282}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1868532, 6768, 3384; 6768, -6, -21; 3384, -21, -2],[564, 3, 1],[-43, -853, -8, -43, -3, -1, 1, 7, -1, -98, -33, -422; -5776, -114586, -1075, -5781, -404, -141, 134, 940, -134, -13160, -4432, -56682; -12226, -242520, -2274, -12220, -852, -282, 284, 1974, -285, -27871, -9384, -119991],75,4]
%
% END OF LATTICE L76.4
%
\hbox{}\par\smallskip%
%
% LATTICE L76.5
%
\leavevmode%
\llap{\spacer}%
${L_{76.5}}=\pdualwitharg{2}({L_{76.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}47\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[62943528, -2484420, 15619416; -2484420, 98072, -616508; 15619416, -616508, 3875953],[564, 4, 1],[24412, 158409, 2655, 39059, 173, 1073, 67, 5950, 842, 125300, 3324, 163286; 189, 1222, 20, 282, 1, 0, 0, 47, 7, 987, 26, 1269; -98346, -638166, -10696, -157356, -697, -4324, -270, -23970, -3392, -504780, -13391, -657812],75,5]
%
% END OF LATTICE L76.5
%
\hbox{}\par\smallskip%
%
% LATTICE L76.6
%
\leavevmode%
\llap{\spacer}%
${L_{76.6}}=\pdualwitharg{47}(\pfillwitharg{2}({L_{76.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3087759, -34122, 1050873; -34122, -47, 11609; 1050873, 11609, -357649],[141, 47, 1],[-3070, 33, 1535, -10447, -30780, -18237, -136951, -20036, -21538, -9556, -9226, -1766; -105, 1, 53, -339, -1002, -595, -4473, -655, -707, -315, -306, -59; -9024, 97, 4512, -30708, -90475, -53606, -402555, -58894, -63309, -28089, -27119, -5191],75,6]
%
% END OF LATTICE L76.6
%
\hbox{}\par\smallskip%
%
% LATTICE L76.7
%
\leavevmode%
\llap{\spacer}%
${L_{76.7}}=\pdualwitharg{2.3}({L_{76.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}47\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1580464488, -16106148, 392273844; -16106148, 164136, -3997572; 392273844, -3997572, 97363003],[564, 12, 1],[14512, 289013, 5549, 32291, 726, 9799, 69, 4690, 700, 65610, 5523, 283904; 63, 1222, 20, 94, 1, 0, 0, 47, 7, 329, 26, 1269; -58466, -1164378, -22356, -130096, -2925, -39480, -278, -18894, -2820, -264328, -22251, -1143792],75,7]
%
% END OF LATTICE L76.7
%
\hbox{}\par\smallskip%
%
% LATTICE L76.8
%
\leavevmode%
\llap{\spacer}%
${L_{76.8}}=\pdualwitharg{3.47}(\pfillwitharg{2}({L_{76.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}47\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[143538, 249288, 2679; 249288, 433011, 4653; 2679, 4653, 50],[141, 141, 1],[-1, 0, 1, -5, -43, -25, -62, -27, -28, -4, -11, -2; -5, 1, 4, -58, -491, -282, -695, -301, -304, -42, -109, -18; 517, -93, -423, 5659, 47940, 27549, 67915, 29421, 29751, 4117, 10716, 1779],75,8]
%
% END OF LATTICE L76.8
%
\hbox{}\par\smallskip%
%
% LATTICE L76.9
%
\leavevmode%
\llap{\spacer}%
${L_{76.9}}=\pdualwitharg{47}({L_{76.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}47\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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127\\870\\-268
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2820, 564, 564; 564, -94, -47; 564, -47, 118],[564, 47, 1],[937, 127, 44, 11, -1, -1, 1, 7, 23, 56, 269, 68; 6420, 870, 301, 75, -8, -7, 6, 48, 158, 384, 1844, 466; -1974, -268, -94, -24, 0, 2, 0, -14, -47, -117, -564, -143],75,9]
%
% END OF LATTICE L76.9
%
\hbox{}\par\smallskip%
%
% LATTICE L76.10
%
\leavevmode%
\llap{\spacer}%
${L_{76.10}}=\pdualwitharg{3.47}({L_{76.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}47\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-919884\\27072\\4512
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4512\\-141\\-22
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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113\\334\\20976
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38\\113\\7050
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3\\9\\556
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-1\\-3\\-186
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-919884, 27072, 4512; 27072, -282, -141; 4512, -141, -22],[564, 141, 1],[279, 113, 38, 3, -3, -1, 1, 7, 22, 17, 243, 61; 824, 334, 113, 9, -8, -3, 2, 20, 64, 50, 716, 180; 51794, 20976, 7050, 556, -564, -186, 188, 1302, 4089, 3157, 45120, 11325],75,10]
%
% END OF LATTICE L76.10
%
\hbox{}\par\smallskip%
%
% LATTICE L76.11
%
\leavevmode%
\llap{\spacer}%
${L_{76.11}}=\pdualwitharg{2.47}({L_{76.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}47\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[39480, 93060, 59784; 93060, 216952, 139120; 59784, 139120, 89183],[564, 188, 1],[-19832, -2753, -2243, -713, -152, 1, 211, -50, -560, -2090, -2651, -2804; 59685, 8285, 6749, 2145, 457, -3, -633, 151, 1687, 6291, 7979, 8439; -79806, -11078, -9024, -2868, -611, 4, 846, -202, -2256, -8412, -10669, -11284],75,11]
%
% END OF LATTICE L76.11
%
\hbox{}\par\smallskip%
%
% LATTICE L76.12
%
\leavevmode%
\llap{\spacer}%
${L_{76.12}}=\pdualwitharg{2.3.47}({L_{76.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}47\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120696, -7392348, -1834692; -7392348, 453346584, 112515180; -1834692, 112515180, 27924917],[564, 564, 1],[63, 26, 20, 2, 1, 0, 0, 1, 7, 7, 26, 27; 78552, 31765, 21137, 1785, 105, 137, 2123, 2810, 13438, 9740, 34470, 34438; -316498, -127986, -85164, -7192, -423, -552, -8554, -11322, -54144, -39244, -138885, -138756],75,12]
%
% END OF LATTICE L76.12
%

\medskip%
%------------------------------WEYL-GROUP-W77--------------------------------
%
\leavevmode\llap{}%
$W_{77\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=64$%
\hfill%
$22222232222223\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L77.1
%
\leavevmode%
\llap{\spacer}%
${L_{77.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}31\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{77}}%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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156860\\-31620\\-620
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\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[156860, -31620, -620; -31620, 6374, 125; -620, 125, 2],[620, 1, 1],[-18, -4546, -481, -8208, -7744, -99349, -2670, -249, -3731, -53, -69, 1, 25, 0; -89, -22475, -2378, -40579, -38285, -491164, -13200, -1231, -18445, -262, -341, 5, 124, 0; -17, -4185, -442, -7533, -7105, -91140, -2449, -228, -3410, -48, -62, 0, 0, -1],76,1]
%
% END OF LATTICE L77.1
%
\hbox{}\par\smallskip%
%
% LATTICE L77.2
%
\leavevmode%
\llap{\spacer}%
${L_{77.2}}=\pfillwitharg{2}({L_{77.1}})$%
{${}={}$}Nikulin $77$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-8835\\-1705\\155
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\kern3pt%
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-1705\\-329\\30
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155\\30\\-2
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-8835, -1705, 155; -1705, -329, 30; 155, 30, -2],[155, 1, 1],[142, 2148, 16, 48, 2, -6, -1, 9, 2533, 135, 4633, 4377, 28092, 1511; -738, -11160, -83, -248, -10, 31, 5, -47, -13175, -702, -24087, -22755, -146041, -7855; -179, -2635, -17, -31, 5, 0, -3, -16, -3410, -178, -6014, -5660, -36270, -1947],76,2]
%
% END OF LATTICE L77.2
%
\hbox{}\par\smallskip%
%
% LATTICE L77.3
%
\leavevmode%
\llap{\spacer}%
${L_{77.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{77.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[109585, 14725, 21700; 14725, 1955, 2915; 21700, 2915, 4297],[155, 5, 1],[-1782, -5324, -186, -448, 6, 180, 1, -133, -6549, -1729, -58917, -11113, -356370, -19151; -363, -1085, -38, -93, 1, 31, 0, -27, -1333, -352, -11997, -2263, -72571, -3900; 9245, 27621, 965, 2325, -31, -930, -5, 690, 33976, 8970, 305660, 57654, 1848840, 99355],76,3]
%
% END OF LATTICE L77.3
%
\hbox{}\par\smallskip%
%
% LATTICE L77.4
%
\leavevmode%
\llap{\spacer}%
${L_{77.4}}=\pdualwitharg{2}({L_{77.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6173960, 53940, -1542560; 53940, 456, -13480; -1542560, -13480, 385407],[620, 4, 1],[313, 43207, 2149, 69847, 65093, 207723, 22186, 1921, 26047, 85, -593, -307, 39, 154; -251, -34255, -1701, -55211, -51435, -164114, -17525, -1514, -20460, -64, 496, 250, -31, -125; 1244, 171740, 8542, 277636, 258740, 825685, 88188, 7636, 103540, 338, -2356, -1220, 155, 612],76,4]
%
% END OF LATTICE L77.4
%
\hbox{}\par\smallskip%
%
% LATTICE L77.5
%
\leavevmode%
\llap{\spacer}%
${L_{77.5}}=\pdualwitharg{5}({L_{77.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2429780, -49600, 19840; -49600, -1010, 405; 19840, 405, -162],[620, 5, 1],[16, 56, 7, 26, 2, 71, 0, -1, 37, 23, 439, 85, 5509, 150; -49, -155, -14, -31, -1, 0, 2, -1, -155, -86, -1519, -289, -18600, -502; 1830, 6448, 820, 3100, 242, 8680, 5, -125, 4123, 2590, 49755, 9647, 625580, 17045],76,5]
%
% END OF LATTICE L77.5
%
\hbox{}\par\smallskip%
%
% LATTICE L77.6
%
\leavevmode%
\llap{\spacer}%
${L_{77.6}}=\pdualwitharg{31}(\pfillwitharg{2}({L_{77.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[310, 155, 0; 155, 62, 0; 0, 0, 1],[155, 31, 1],[101, 49, 11, 1, 1, 0, 0, 7, 59, 97, 107, 3131, 648, 1080; -228, -110, -24, -2, 0, 0, -1, -17, -135, -221, -243, -7105, -1470, -2449; -868, -420, -93, -8, 0, 1, 0, -62, -510, -837, -922, -26970, -5581, -9300],76,6]
%
% END OF LATTICE L77.6
%
\hbox{}\par\smallskip%
%
% LATTICE L77.7
%
\leavevmode%
\llap{\spacer}%
${L_{77.7}}=\pdualwitharg{2.5}({L_{77.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[176448280, -3100, -44406260; -3100, 40, 780; -44406260, 780, 11175603],[620, 20, 1],[-33809, -109879, -5509, -32299, -2113, -18451, 0, -151, -100673, -28451, -1020461, -194855, -3139829, -339600; -592, -1922, -96, -558, -36, -310, 1, -3, -1767, -499, -17887, -3415, -55025, -5951; -134340, -436604, -21890, -128340, -8396, -73315, 0, -600, -400024, -113050, -4054800, -774256, -12476105, -1349400],76,7]
%
% END OF LATTICE L77.7
%
\hbox{}\par\smallskip%
%
% LATTICE L77.8
%
\leavevmode%
\llap{\spacer}%
${L_{77.8}}=\pdualwitharg{31}({L_{77.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1860, -620, 0; -620, 186, 31; 0, 31, -46],[620, 31, 1],[-53, -21, 1, 1, 1, -7, -14, -14, -46, -135, -68, -1944, -797, -656; -163, -65, 2, 3, 5, -20, -41, -42, -140, -412, -208, -5950, -2440, -2009; -124, -50, 0, 2, 0, -16, -31, -31, -105, -310, -157, -4495, -1844, -1519],76,8]
%
% END OF LATTICE L77.8
%
\hbox{}\par\smallskip%
%
% LATTICE L77.9
%
\leavevmode%
\llap{\spacer}%
${L_{77.9}}=\pdualwitharg{5.31}(\pfillwitharg{2}({L_{77.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-310, 775, 155; 775, 3255, 620; 155, 620, 118],[155, 155, 1],[49, 5, 7, 1, 1, 0, -2, 1, 5, 43, 49, 289, 300, 502; 1648, 156, 153, 8, -12, -1, 30, 152, 204, 1647, 1792, 10452, 10801, 17970; -8370, -792, -775, -40, 62, 5, -155, -775, -1037, -8370, -9105, -53103, -54875, -91295],76,9]
%
% END OF LATTICE L77.9
%
\hbox{}\par\smallskip%
%
% LATTICE L77.10
%
\leavevmode%
\llap{\spacer}%
${L_{77.10}}=\pdualwitharg{2.31}({L_{77.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${248}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}{124}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{1240}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{248}\above{1pt}{1pt}{+}{3}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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109\\-4\\434
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[184760, 56420, -45880; 56420, 17112, -14012; -45880, -14012, 11393],[620, 124, 1],[1275, 597, 109, 1, -157, -10, -1, 124, 802, 1290, 1398, 40728, 4207, 13991; -42, -20, -4, 0, 10, 1, 3, -3, -25, -41, -45, -1315, -136, -453; 5084, 2380, 434, 4, -620, -39, 0, 496, 3200, 5146, 5576, 162440, 16779, 55800],76,10]
%
% END OF LATTICE L77.10
%
\hbox{}\par\smallskip%
%
% LATTICE L77.11
%
\leavevmode%
\llap{\spacer}%
${L_{77.11}}=\pdualwitharg{5.31}({L_{77.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${310}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{620}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{62}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{310}\above{1pt}{1pt}{-}{3}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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3\\-14\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\-1\\210
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3\\-1\\744
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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3\\0\\800
\end{tabular}\endgroup%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1085\\-155\\-930\\-155\\-465\\-620\\-155
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-155\\-33\\-310\\-105\\-527\\-1020\\-775
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-930\\-310\\-3720\\-1550\\-8370\\-16740\\-13330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-155\\-105\\-1550\\-785\\-4495\\-9220\\-7595
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-465\\-527\\-8370\\-4495\\-26133\\-53940\\-44795
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-620\\-1020\\-16740\\-9220\\-53940\\-111620\\-93000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-155\\-775\\-13330\\-7595\\-44795\\-93000\\-77810
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4142220, -869860, 15500; -869860, -182590, 3255; 15500, 3255, -58],[620, 155, 1],[-15, -1, 3, 1, 3, 3, -1, -5, -3, -43, -21, -119, -243, -199; -49, -5, -14, -1, -1, 0, 2, -1, -5, -86, -49, -289, -600, -502; -6820, -554, 0, 210, 744, 800, -155, -1395, -1089, -16430, -8425, -48391, -99380, -81995],76,11]
%
% END OF LATTICE L77.11
%
\hbox{}\par\smallskip%
%
% LATTICE L77.12
%
\leavevmode%
\llap{\spacer}%
${L_{77.12}}=\pdualwitharg{2.5.31}({L_{77.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1240}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{620}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{248}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{1240}\above{1pt}{1pt}{-}{3}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1240\\93620\\-23560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93620\\318552280\\-80159800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23560\\-80159800\\20171237
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13751\\-2466600\\-9802200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2237125\\401315819\\1594817940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-562925\\-100982604\\-401302069
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1318\\-236363\\-939300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-23667\\-94052
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-172\\-30813\\-122450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-4303\\-17100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-6147\\-24428
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-1652\\-6565
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1240\\0\\-1240\\-2480\\-16120\\-8680\\-30380\\-4340\\-620\\-1860\\-620\\-1860\\-620\\-620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-200\\-1488\\-840\\-3100\\-620\\-132\\-620\\-420\\-2108\\-1020\\-3100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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0\\8\\0\\-200\\-1488\\-840\\-3100
\end{tabular}\endgroup%
\kern3pt%
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-1240\\0\\620\\0\\-1240\\-930\\-4340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2480\\-200\\0\\40\\0\\-80\\-620
\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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-8680\\-840\\-930\\-80\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30380\\-3100\\-4340\\-620\\-620\\0\\1240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4340\\-620\\-1860\\-620\\-1860\\-620\\-620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-620\\-132\\-620\\-420\\-2108\\-1020\\-3100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1860\\-620\\-3720\\-3100\\-16740\\-8370\\-26660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-620\\-420\\-3100\\-3140\\-17980\\-9220\\-30380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-620\\-1020\\-8370\\-9220\\-53940\\-27905\\-93000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-620\\-3100\\-26660\\-30380\\-179180\\-93000\\-311240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1240, 93620, -23560; 93620, 318552280, -80159800; -23560, -80159800, 20171237],[620, 620, 1],[-1318, -132, -172, -24, -34, -9, 1, -57, -143, -1203, -1351, -7941, -4116, -13751; -236363, -23667, -30813, -4303, -6147, -1652, 0, -10297, -25665, -215847, -242357, -1424481, -738328, -2466600; -939300, -94052, -122450, -17100, -24428, -6565, 0, -40920, -101992, -857770, -963120, -5660848, -2934095, -9802200],76,12]
%
% END OF LATTICE L77.12
%

\medskip%
%------------------------------WEYL-GROUP-W78--------------------------------
%
\leavevmode\llap{}%
$W_{78\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=10$%
\hfill%
$26222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L78.1
%
\leavevmode%
\llap{\spacer}%
${L_{78.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{78}}%
\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{b}{2}{330}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34021020\\23100\\36960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23100\\-14\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36960\\-27\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1221\\-1078
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-609\\-540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\610\\539
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\15840\\14025
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\608\\540
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\0\\-11\\-165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\-3\\-165\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-3\\2\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-165\\0\\330\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-4\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-34021020, 23100, 36960; 23100, -14, -27; 36960, -27, -38],[660, 1, 1],[-2, -1, 1, 26, 1; -1221, -609, 610, 15840, 608; -1078, -540, 539, 14025, 540],77,1]
%
% END OF LATTICE L78.1
%
\hbox{}\par\smallskip%
%
% LATTICE L78.2
%
\leavevmode%
\llap{\spacer}%
${L_{78.2}}=\pfillwitharg{2}({L_{78.1}})$%
{${}={}$}Nikulin $78$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{s}{2}{330}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
330\\165\\825
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
165\\79\\330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
825\\330\\118
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
104\\-264\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-66\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\160\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1429\\3630\\-165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\61\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\0\\-11\\-165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\-3\\-165\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-3\\2\\0\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-165\\0\\330\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-2\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[330, 165, 825; 165, 79, 330; 825, 330, 118],[165, 1, 1],[104, 26, -63, -1429, -24; -264, -66, 160, 3630, 61; 11, 3, -7, -165, -3],77,2]
%
% END OF LATTICE L78.2
%
\hbox{}\par\smallskip%
%
% LATTICE L78.3
%
\leavevmode%
\llap{\spacer}%
${L_{78.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{78.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7347615\\18810\\-2439360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18810\\-42\\6243
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2439360\\6243\\-809851
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6318\\-5621\\-19074
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
106\\-93\\-320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-317\\280\\957
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14301\\-12760\\-43175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2683\\-2391\\-8100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\0\\-33\\-165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-3\\-55\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-3\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-55\\0\\110\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7347615, 18810, -2439360; 18810, -42, 6243; -2439360, 6243, -809851],[165, 3, 1],[6318, 106, -317, 14301, 2683; -5621, -93, 280, -12760, -2391; -19074, -320, 957, -43175, -8100],77,3]
%
% END OF LATTICE L78.3
%
\hbox{}\par\smallskip%
%
% LATTICE L78.4
%
\leavevmode%
\llap{\spacer}%
${L_{78.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{78.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10172085\\28380\\-4059825
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28380\\-70\\11325
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4059825\\11325\\-1620334
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10916\\-5621\\-27390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
550\\-279\\-1380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-548\\280\\1375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14822\\-7656\\-37191
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4635\\-2391\\-11630
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\0\\-55\\-165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\-15\\-165\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-15\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-165\\0\\66\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10172085, 28380, -4059825; 28380, -70, 11325; -4059825, 11325, -1620334],[165, 5, 1],[10916, 550, -548, 14822, 4635; -5621, -279, 280, -7656, -2391; -27390, -1380, 1375, -37191, -11630],77,4]
%
% END OF LATTICE L78.4
%
\hbox{}\par\smallskip%
%
% LATTICE L78.5
%
\leavevmode%
\llap{\spacer}%
${L_{78.5}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{78.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{}{6}{22}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18946455\\57090\\-5158890
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57090\\-154\\15543
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5158890\\15543\\-1404703
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1294\\-511\\-4758
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
718\\-279\\-2640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-715\\280\\2629
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8783\\-3480\\-32295
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6043\\-2391\\-22220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-11\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\66\\-33\\-165\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-33\\22\\0\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-165\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-66\\-22\\0\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18946455, 57090, -5158890; 57090, -154, 15543; -5158890, 15543, -1404703],[165, 11, 1],[1294, 718, -715, 8783, 6043; -511, -279, 280, -3480, -2391; -4758, -2640, 2629, -32295, -22220],77,5]
%
% END OF LATTICE L78.5
%
\hbox{}\par\smallskip%
%
% LATTICE L78.6
%
\leavevmode%
\llap{\spacer}%
${L_{78.6}}=\pdualwitharg{3}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41137140\\67320\\94380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67320\\-102\\-159
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
94380\\-159\\-214
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\3476\\6237
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\522\\935
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-521\\-936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-1045\\-1870
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\868\\1560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\0\\-33\\-165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-3\\-55\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-3\\6\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-55\\0\\110\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-41137140, 67320, 94380; 67320, -102, -159; 94380, -159, -214],[660, 3, 1],[20, 3, -3, -6, 5; 3476, 522, -521, -1045, 868; 6237, 935, -936, -1870, 1560],77,6]
%
% END OF LATTICE L78.6
%
\hbox{}\par\smallskip%
%
% LATTICE L78.7
%
\leavevmode%
\llap{\spacer}%
${L_{78.7}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{78.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${330}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4951794705\\-41653260\\1636929855
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41653260\\350385\\-13769445
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1636929855\\-13769445\\541124887
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-438051\\-440\\1325115
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1519\\2\\-4595
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7552\\7\\-22845
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-249558\\-253\\754919
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-213752\\-216\\646605
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
330\\0\\-165\\-165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-15\\-55\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-15\\30\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-55\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-30\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4951794705, -41653260, 1636929855; -41653260, 350385, -13769445; 1636929855, -13769445, 541124887],[165, 15, 1],[-438051, 1519, 7552, -249558, -213752; -440, 2, 7, -253, -216; 1325115, -4595, -22845, 754919, 646605],77,7]
%
% END OF LATTICE L78.7
%
\hbox{}\par\smallskip%
%
% LATTICE L78.8
%
\leavevmode%
\llap{\spacer}%
${L_{78.8}}=\pdualwitharg{2}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${88}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{*}{2}{1320}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9317350680\\-64311060\\2315115000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64311060\\443896\\-15979596
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2315115000\\-15979596\\575244793
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1465\\0\\-5896
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1309\\6\\-5268
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2124\\7\\-8548
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68878\\165\\-277200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1101\\2\\-4431
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\0\\-44\\-660\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\-12\\-660\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-12\\8\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-660\\0\\1320\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-4\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9317350680, -64311060, 2315115000; -64311060, 443896, -15979596; 2315115000, -15979596, 575244793],[660, 4, 1],[1465, 1309, 2124, 68878, 1101; 0, 6, 7, 165, 2; -5896, -5268, -8548, -277200, -4431],77,8]
%
% END OF LATTICE L78.8
%
\hbox{}\par\smallskip%
%
% LATTICE L78.9
%
\leavevmode%
\llap{\spacer}%
${L_{78.9}}=\pdualwitharg{5}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-743820\\7260\\91080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7260\\-70\\-915
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91080\\-915\\-10366
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\-3333\\-110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-1809\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\1666\\55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
124\\8976\\297
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\1808\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\0\\-55\\-165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\-15\\-165\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-15\\10\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-165\\0\\66\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-20\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-743820, 7260, 91080; 7260, -70, -915; 91080, -915, -10366],[660, 5, 1],[-46, -25, 23, 124, 25; -3333, -1809, 1666, 8976, 1808; -110, -60, 55, 297, 60],77,9]
%
% END OF LATTICE L78.9
%
\hbox{}\par\smallskip%
%
% LATTICE L78.10
%
\leavevmode%
\llap{\spacer}%
${L_{78.10}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{78.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{}{6}{66}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5630720865\\-65881035\\1858114335
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65881035\\770847\\-21740466
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1858114335\\-21740466\\613169959
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67677\\-40\\205083
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2639\\2\\-7997
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12763\\7\\-38676
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192907\\-115\\584570
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-363419\\-216\\1101276
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-33\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\-33\\-55\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-33\\66\\0\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-55\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-66\\-66\\0\\33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5630720865, -65881035, 1858114335; -65881035, 770847, -21740466; 1858114335, -21740466, 613169959],[165, 33, 1],[-67677, 2639, 12763, -192907, -363419; -40, 2, 7, -115, -216; 205083, -7997, -38676, 584570, 1101276],77,10]
%
% END OF LATTICE L78.10
%
\hbox{}\par\smallskip%
%
% LATTICE L78.11
%
\leavevmode%
\llap{\spacer}%
${L_{78.11}}=\pdualwitharg{11}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{}{6}{22}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11220\\1320\\660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1320\\-154\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
660\\-77\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\15\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-11\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\66\\-33\\-165\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-33\\22\\0\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-165\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-132\\-44\\0\\44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11220, 1320, 660; 1320, -154, -77; 660, -77, -38],[660, 11, 1],[0, -1, 0, 1, 1; -1, -9, 5, 15, 8; 2, 0, -11, -15, 0],77,11]
%
% END OF LATTICE L78.11
%
\hbox{}\par\smallskip%
%
% LATTICE L78.12
%
\leavevmode%
\llap{\spacer}%
${L_{78.12}}=\pdualwitharg{2.3}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${264}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{2}{440}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18008782440\\-184743900\\4474555140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-184743900\\1895208\\-45902424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4474555140\\-45902424\\1111771091
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22270\\55\\-89628
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6180\\17\\-24872
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
489\\2\\-1968
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4209\\0\\-16940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1177\\2\\-4737
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
264\\0\\-132\\-660\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\-12\\-220\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-12\\24\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-220\\0\\440\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-12\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18008782440, -184743900, 4474555140; -184743900, 1895208, -45902424; 4474555140, -45902424, 1111771091],[660, 12, 1],[22270, 6180, 489, 4209, 1177; 55, 17, 2, 0, 2; -89628, -24872, -1968, -16940, -4737],77,12]
%
% END OF LATTICE L78.12
%
\hbox{}\par\smallskip%
%
% LATTICE L78.13
%
\leavevmode%
\llap{\spacer}%
${L_{78.13}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{78.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{s}{2}{330}\above{1pt}{1pt}{}{6}{110}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6230763330\\89469270\\-2369629515
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89469270\\1284745\\-34026135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2369629515\\-34026135\\901196854
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105942\\-317866\\266565
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12394\\37188\\-31185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19979\\59944\\-50270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
181187\\-543630\\455892
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
568899\\-1706913\\1431430
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-55\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\330\\-165\\-165\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-165\\110\\0\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-165\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-330\\-110\\0\\55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6230763330, 89469270, -2369629515; 89469270, 1284745, -34026135; -2369629515, -34026135, 901196854],[165, 55, 1],[105942, -12394, -19979, 181187, 568899; -317866, 37188, 59944, -543630, -1706913; 266565, -31185, -50270, 455892, 1431430],77,13]
%
% END OF LATTICE L78.13
%
\hbox{}\par\smallskip%
%
% LATTICE L78.14
%
\leavevmode%
\llap{\spacer}%
${L_{78.14}}=\pdualwitharg{3.5}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${330}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-259525860\\-22168740\\178200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22168740\\-1893630\\15225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
178200\\15225\\-122
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
189\\-2068\\17985
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-22\\175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\547\\-4770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\11\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\-908\\7920
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
330\\0\\-165\\-165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-15\\-55\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-15\\30\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-55\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-60\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-259525860, -22168740, 178200; -22168740, -1893630, 15225; 178200, 15225, -122],[660, 15, 1],[189, 2, -50, -1, 83; -2068, -22, 547, 11, -908; 17985, 175, -4770, -88, 7920],77,14]
%
% END OF LATTICE L78.14
%
\hbox{}\par\smallskip%
%
% LATTICE L78.15
%
\leavevmode%
\llap{\spacer}%
${L_{78.15}}=\pdualwitharg{2.5}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${440}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{}{6}{40}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9486840\\12969660\\2393160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12969660\\17731160\\3271740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2393160\\3271740\\603701
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111\\0\\440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
461\\6\\-1860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
500\\7\\-2020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2086\\33\\-8448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107\\2\\-435
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
440\\0\\-220\\-660\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\120\\-60\\-660\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-60\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-660\\0\\264\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-20\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9486840, 12969660, 2393160; 12969660, 17731160, 3271740; 2393160, 3271740, 603701],[660, 20, 1],[-111, 461, 500, 2086, 107; 0, 6, 7, 33, 2; 440, -1860, -2020, -8448, -435],77,15]
%
% END OF LATTICE L78.15
%
\hbox{}\par\smallskip%
%
% LATTICE L78.16
%
\leavevmode%
\llap{\spacer}%
${L_{78.16}}=\pdualwitharg{3.11}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{}{6}{66}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{132}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135365340\\32117580\\-194040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32117580\\-7620294\\46035
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-194040\\46035\\-278
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\91\\2505
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\25\\649
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-253\\-6996
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\420\\11616
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-33\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\-33\\-55\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-33\\66\\0\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-55\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-132\\-132\\0\\132
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-135365340, 32117580, -194040; 32117580, -7620294, 46035; -194040, 46035, -278],[660, 33, 1],[18, 5, -50, -1, 83; 91, 25, -253, -5, 420; 2505, 649, -6996, -130, 11616],77,16]
%
% END OF LATTICE L78.16
%
\hbox{}\par\smallskip%
%
% LATTICE L78.17
%
\leavevmode%
\llap{\spacer}%
${L_{78.17}}=\pdualwitharg{3.5.11}(\pfillwitharg{2}({L_{78.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{}{6}{330}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-272655428595\\43971031665\\16042815855
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43971031665\\-7091190645\\-2587218480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16042815855\\-2587218480\\-943945777
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-511\\-102384\\271935
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93\\-19018\\50545
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\56699\\-150645
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-232\\-46287\\122923
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2391\\-477886\\1269180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\0\\-165\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\110\\-165\\-55\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-165\\330\\0\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-55\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-330\\-330\\0\\165
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-272655428595, 43971031665, 16042815855; 43971031665, -7091190645, -2587218480; 16042815855, -2587218480, -943945777],[165, 165, 1],[-511, -93, 280, -232, -2391; -102384, -19018, 56699, -46287, -477886; 271935, 50545, -150645, 122923, 1269180],77,17]
%
% END OF LATTICE L78.17
%
\hbox{}\par\smallskip%
%
% LATTICE L78.18
%
\leavevmode%
\llap{\spacer}%
${L_{78.18}}=\pdualwitharg{2.11}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{}{6}{88}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
236280\\9900\\58740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9900\\616\\2464
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58740\\2464\\14603
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
131\\6\\-528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
131\\7\\-528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
253\\15\\-1020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\2\\-121
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-44\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\264\\-132\\-660\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-132\\88\\0\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-660\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-132\\-44\\0\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[236280, 9900, 58740; 9900, 616, 2464; 58740, 2464, 14603],[660, 44, 1],[-1, 131, 131, 253, 30; 0, 6, 7, 15, 2; 4, -528, -528, -1020, -121],77,18]
%
% END OF LATTICE L78.18
%
\hbox{}\par\smallskip%
%
% LATTICE L78.19
%
\leavevmode%
\llap{\spacer}%
${L_{78.19}}=\pdualwitharg{5.11}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{b}{2}{330}\above{1pt}{1pt}{}{6}{110}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{220}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78479940\\248160\\-295680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
248160\\-770\\935
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-295680\\935\\-1114
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-9\\1320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\5\\-2915
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\3\\-1059
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\8\\-1320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-55\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\330\\-165\\-165\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-165\\110\\0\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-165\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-660\\-220\\0\\220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-78479940, 248160, -295680; 248160, -770, 935; -295680, 935, -1114],[660, 55, 1],[-2, -5, 11, 4, 5; -1, -9, 5, 3, 8; 530, 1320, -2915, -1059, -1320],77,19]
%
% END OF LATTICE L78.19
%
\hbox{}\par\smallskip%
%
% LATTICE L78.20
%
\leavevmode%
\llap{\spacer}%
${L_{78.20}}=\pdualwitharg{2.3.5}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1320}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{}{6}{120}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71475316440\\777715620\\17755793880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
777715620\\8462280\\193198980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17755793880\\193198980\\4410868423
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
125458\\-2893\\-504900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38276\\-883\\-154040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5397\\-124\\-21720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2941\\-66\\-11836
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5356\\-123\\-21555
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1320\\0\\-660\\-660\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\40\\-60\\-220\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-60\\120\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-220\\0\\88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-60\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[71475316440, 777715620, 17755793880; 777715620, 8462280, 193198980; 17755793880, 193198980, 4410868423],[660, 60, 1],[125458, 38276, 5397, 2941, 5356; -2893, -883, -124, -66, -123; -504900, -154040, -21720, -11836, -21555],77,20]
%
% END OF LATTICE L78.20
%
\hbox{}\par\smallskip%
%
% LATTICE L78.21
%
\leavevmode%
\llap{\spacer}%
${L_{78.21}}=\pdualwitharg{2.3.11}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{}{6}{264}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71984493240\\-536019660\\17883845760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-536019660\\3991416\\-133168860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17883845760\\-133168860\\4443067177
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13359\\-383\\53760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45189\\-1295\\181852
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5773\\-166\\23232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-661\\-20\\2660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5732\\-165\\23067
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-132\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\88\\-132\\-220\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-132\\264\\0\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-220\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-132\\-132\\0\\33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[71984493240, -536019660, 17883845760; -536019660, 3991416, -133168860; 17883845760, -133168860, 4443067177],[660, 132, 1],[-13359, -45189, -5773, -661, -5732; -383, -1295, -166, -20, -165; 53760, 181852, 23232, 2660, 23067],77,21]
%
% END OF LATTICE L78.21
%
\hbox{}\par\smallskip%
%
% LATTICE L78.22
%
\leavevmode%
\llap{\spacer}%
${L_{78.22}}=\pdualwitharg{3.5.11}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{}{6}{330}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{660}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1747020\\99000\\49500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99000\\-5610\\-2805
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49500\\-2805\\-1402
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\25\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\27\\-55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-53\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\88\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\0\\-165\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\110\\-165\\-55\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-165\\330\\0\\-660
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-55\\0\\2\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-660\\-660\\0\\660
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1747020, 99000, 49500; 99000, -5610, -2805; 49500, -2805, -1402],[660, 165, 1],[1, 0, -3, 0, 5; 25, 27, -53, -1, 88; -15, -55, 0, 2, 0],77,22]
%
% END OF LATTICE L78.22
%
\hbox{}\par\smallskip%
%
% LATTICE L78.23
%
\leavevmode%
\llap{\spacer}%
${L_{78.23}}=\pdualwitharg{2.5.11}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{*}{2}{1320}\above{1pt}{1pt}{}{6}{440}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3549832440\\-2828100\\-884268000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2828100\\3080\\705100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-884268000\\705100\\220272799
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
333\\-999\\1340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33131\\99399\\-133320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33131\\99400\\-133320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12653\\37962\\-50916
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7367\\22103\\-29645
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-220\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1320\\-660\\-660\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-660\\440\\0\\-220
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-660\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-660\\-220\\0\\55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3549832440, -2828100, -884268000; -2828100, 3080, 705100; -884268000, 705100, 220272799],[660, 220, 1],[333, -33131, -33131, -12653, -7367; -999, 99399, 99400, 37962, 22103; 1340, -133320, -133320, -50916, -29645],77,23]
%
% END OF LATTICE L78.23
%
\hbox{}\par\smallskip%
%
% LATTICE L78.24
%
\leavevmode%
\llap{\spacer}%
${L_{78.24}}=\pdualwitharg{2.3.5.11}({L_{78.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{*}{2}{440}\above{1pt}{1pt}{}{6}{1320}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
927960\\306900\\76560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
306900\\262680\\65340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76560\\65340\\16253
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\3203\\-12900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\10925\\-44000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1311\\-5280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1270\\-5115
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\-660\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\440\\-660\\-220\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-660\\1320\\0\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-220\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-660\\-660\\0\\165
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[927960, 306900, 76560; 306900, 262680, 65340; 76560, 65340, 16253],[660, 660, 1],[5, 17, 2, 0, 2; 3203, 10925, 1311, -1, 1270; -12900, -44000, -5280, 4, -5115],77,24]
%
% END OF LATTICE L78.24
%

\medskip%
%------------------------------WEYL-GROUP-W79--------------------------------
%
\leavevmode\llap{}%
$W_{79\phantom{0}}$%
\qquad\llap{48} lattices, $\chi=60$%
\hfill%
$22222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L79.1
%
\leavevmode%
\llap{\spacer}%
${L_{79.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{79},3,2}%
\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}{990}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24993540\\79200\\-77220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79200\\246\\-243
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77220\\-243\\238
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121441\\-19005360\\-58655520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\-68861\\-212520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
394\\61661\\190301
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\804\\2439
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\3630\\11143
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\4552\\14022
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
989\\154770\\477675
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\9848\\30416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\16379\\50655
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\1235\\3845
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-36\\-1980\\-144\\-297\\-45\\-45\\-99\\-90\\-2475\\-144\\-198\\0
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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18\\0\\-36\\-1980\\-144\\-297\\-45
\end{tabular}\endgroup%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[24993540, 79200, -77220; 79200, 246, -243; -77220, -243, 238],[1980, 3, 1],[5, 23, 29, 989, 63, 105, 8, 1, -1, -1, 1, 1, 5, 2; 804, 3630, 4552, 154770, 9848, 16379, 1235, 135, -187, -170, 165, 168, 836, 330; 2439, 11143, 14022, 477675, 30416, 50655, 3845, 459, -517, -498, 495, 496, 2475, 985],78,1]
%
% END OF LATTICE L79.1
%
\hbox{}\par\smallskip%
%
% LATTICE L79.2
%
\leavevmode%
\llap{\spacer}%
${L_{79.2}}=\pfillwitharg{2.3}({L_{79.1}})$%
{${}={}$}Nikulin $79$%
\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[13035, 3465, 165; 3465, 921, 44; 165, 44, 2],[165, 1, 1],[0, -6, 4, 266, 32, 151, 31, 17, 193, 115, 2539, 240, 784, 54; 0, 22, -15, -990, -119, -561, -115, -63, -715, -426, -9405, -889, -2904, -200; 1, 11, -3, -275, -34, -165, -35, -19, -209, -123, -2695, -254, -825, -55],78,2]
%
% END OF LATTICE L79.2
%
\hbox{}\par\smallskip%
%
% LATTICE L79.3
%
\leavevmode%
\llap{\spacer}%
${L_{79.3}}=\pfillwitharg{3}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-514140, 1320, 0; 1320, -2, -5; 0, -5, 18],[660, 1, 1],[-1, -3, -1, 1, 1, 4, 1, 0, -8, -13, -166, -33, -59, -6; -395, -1177, -390, 385, 388, 1551, 385, -7, -3245, -5238, -66715, -13252, -23661, -2395; -112, -330, -108, 110, 108, 429, 105, -5, -957, -1530, -19415, -3852, -6864, -690],78,3]
%
% END OF LATTICE L79.3
%
\hbox{}\par\smallskip%
%
% LATTICE L79.4
%
\leavevmode%
\llap{\spacer}%
${L_{79.4}}=\pfillwitharg{2}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\GramMatrix{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[990, 0, -495; 0, -15, -3; -495, -3, 247],[495, 3, 1],[-25, -21, -3, -1, 2, 17, 3, -11, -111, -81, -5939, -194, -677, -63; -6, 0, 1, 0, -1, -11, -5, -15, -77, -50, -3465, -111, -374, -30; -54, -44, -6, 0, 4, 33, 5, -27, -253, -183, -13365, -436, -1518, -140],78,4]
%
% END OF LATTICE L79.4
%
\hbox{}\par\smallskip%
%
% LATTICE L79.5
%
\leavevmode%
\llap{\spacer}%
${L_{79.5}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{79.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5775\\285\\-1983
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[36630, -5775, 13860; -5775, 285, -1983; 13860, -1983, 5179],[165, 3, 1],[288, 1070, 74, 54, -166, -533, -491, -153, 415, 241, 22221, 2326, 3008, 1166; -277, -1045, -73, -55, 164, 528, 490, 160, -242, -200, -19250, -2036, -2673, -1075; -879, -3267, -226, -165, 507, 1628, 1500, 468, -1254, -733, -67650, -7083, -9163, -3555],78,5]
%
% END OF LATTICE L79.5
%
\hbox{}\par\smallskip%
%
% LATTICE L79.6
%
\leavevmode%
\llap{\spacer}%
${L_{79.6}}=\pdualwitharg{5}(\pfillwitharg{2.3}({L_{79.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[489390, -16005, 198000; -16005, 475, -6460; 198000, -6460, 80103],[165, 5, 1],[361, 1323, 272, 13, -203, -1951, -119, -177, 701, 1016, 6059, 3147, 12071, 303; -277, -1045, -219, -11, 164, 1584, 98, 160, -242, -600, -3850, -2036, -8019, -215; -915, -3355, -690, -33, 515, 4950, 302, 450, -1760, -2565, -15312, -7955, -30525, -767],78,6]
%
% END OF LATTICE L79.6
%
\hbox{}\par\smallskip%
%
% LATTICE L79.7
%
\leavevmode%
\llap{\spacer}%
${L_{79.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{79.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12482415, 231165, -80190; 231165, -4281, 1485; -80190, 1485, -514],[495, 3, 1],[-43, -193, 0, 19, -1, -34, -98, -54, -3008, -679, -16024, -4655, -5301, -1357; -2373, -10659, -1, 1045, -54, -1870, -5400, -2978, -165924, -37456, -883960, -256794, -292435, -74865; -149, -693, -3, 55, 0, -99, -315, -181, -10197, -2307, -54505, -15840, -18051, -4635],78,7]
%
% END OF LATTICE L79.7
%
\hbox{}\par\smallskip%
%
% LATTICE L79.8
%
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\llap{\spacer}%
${L_{79.8}}=\pdualwitharg{11}(\pfillwitharg{2.3}({L_{79.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2061510, -46695, 564960; -46695, 1045, -12793; 564960, -12793, 154827],[165, 11, 1],[274, 88, 194, 4, -144, -125, -411, -105, 81, 989, 2531, 2848, 970, 1256; -277, -95, -219, -5, 164, 144, 490, 160, -22, -600, -1750, -2036, -729, -1075; -1023, -329, -726, -15, 539, 468, 1540, 396, -298, -3663, -9390, -10571, -3603, -4675],78,8]
%
% END OF LATTICE L79.8
%
\hbox{}\par\smallskip%
%
% LATTICE L79.9
%
\leavevmode%
\llap{\spacer}%
${L_{79.9}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{79.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7169580, 3300, -22440; 3300, -6, -9; -22440, -9, 70],[660, 3, 1],[5, 69, 29, 989, 189, 105, 24, 1, -3, -1, 1, 3, 5, 6; 278, 3784, 1584, 53900, 10292, 5709, 1295, 49, -187, -58, 55, 172, 286, 340; 1635, 22539, 9470, 322905, 61704, 34276, 7830, 324, -990, -328, 330, 984, 1639, 1965],78,9]
%
% END OF LATTICE L79.9
%
\hbox{}\par\smallskip%
%
% LATTICE L79.10
%
\leavevmode%
\llap{\spacer}%
${L_{79.10}}=\pdualwitharg{3.5}(\pfillwitharg{2.3}({L_{79.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1241874150, -16436145, 410509605; -16436145, 217530, -5433075; 410509605, -5433075, 135696629],[165, 15, 1],[48695, 74183, 200, -3011, 719, 12582, 7214, 60358, 1125400, 254257, 3601760, 1744129, 1986723, 101839; 304, 462, 1, -22, 3, 77, 45, 377, 7029, 1588, 22495, 10893, 12408, 636; -147300, -224400, -605, 9108, -2175, -38060, -21822, -182580, -3404280, -769115, -10895148, -5275905, -6009740, -308058],78,10]
%
% END OF LATTICE L79.10
%
\hbox{}\par\smallskip%
%
% LATTICE L79.11
%
\leavevmode%
\llap{\spacer}%
${L_{79.11}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{79.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[109278840, -2475660, 27147780; -2475660, 56088, -615020; 27147780, -615020, 6744233],[660, 4, 1],[1845, 2843, 79, 1421, 122, 2329, 1511, 2351, 43313, 29293, 690919, 33446, 228451, 19469; 30, 44, 0, 0, 1, 33, 25, 39, 715, 483, 11385, 551, 3762, 320; -7424, -11440, -318, -5720, -491, -9372, -6080, -9460, -174284, -117870, -2780140, -134581, -919248, -78340],78,11]
%
% END OF LATTICE L79.11
%
\hbox{}\par\smallskip%
%
% LATTICE L79.12
%
\leavevmode%
\llap{\spacer}%
${L_{79.12}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{79.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-20460, -9900, 660; -9900, -4790, 325; 660, 325, 78],[660, 5, 1],[-97, -395, -233, -267, -167, -16, 17, 20, -452, -845, -2274, -2297, -4219, -94; 200, 814, 480, 550, 344, 33, -35, -41, 935, 1746, 4697, 4744, 8712, 194; -15, -55, -30, -33, -20, 0, 2, 0, -110, -180, -462, -460, -825, -17],78,12]
%
% END OF LATTICE L79.12
%
\hbox{}\par\smallskip%
%
% LATTICE L79.13
%
\leavevmode%
\llap{\spacer}%
${L_{79.13}}=\pdualwitharg{3.11}(\pfillwitharg{2.3}({L_{79.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[424301955, -14250225, 140006625; -14250225, 478566, -4702137; 140006625, -4702137, 46197890],[165, 33, 1],[99590, 13774, 363, -3074, 1187, 2312, 73726, 123472, 209288, 520115, 3349004, 3567805, 369458, 1041584; 304, 42, 1, -10, 3, 7, 225, 377, 639, 1588, 10225, 10893, 1128, 3180; -301785, -41739, -1100, 9315, -3597, -7006, -223410, -374154, -634200, -1576091, -10148400, -10811427, -1119559, -3156285],78,13]
%
% END OF LATTICE L79.13
%
\hbox{}\par\smallskip%
%
% LATTICE L79.14
%
\leavevmode%
\llap{\spacer}%
${L_{79.14}}=\pdualwitharg{3}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-324072540, -1174140, 409860; -1174140, -4254, 1485; 409860, 1485, -514],[1980, 3, 1],[-47, -215, -1, 19, -1, -34, -103, -58, -3250, -1469, -17344, -10079, -5741, -1472; 12921, 59103, 274, -5225, 276, 9350, 28320, 15946, 893508, 403864, 4768280, 2770956, 1578335, 404685; -149, -693, -6, 55, 0, -99, -315, -181, -10197, -4614, -54505, -31680, -18051, -4635],78,14]
%
% END OF LATTICE L79.14
%
\hbox{}\par\smallskip%
%
% LATTICE L79.15
%
\leavevmode%
\llap{\spacer}%
${L_{79.15}}=\pdualwitharg{11}(\pfillwitharg{3}({L_{79.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30128340\\29700\\55440
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29700\\-22\\-55
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{$\left.\llap{\phantom{%
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-1\\-24\\-531
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30128340, 29700, 55440; 29700, -22, -55; 55440, -55, -102],[660, 11, 1],[-21, -3, -1, 1, 1, -1, -14, -25, -43, -643, -691, -1473, -229, -216; -535, -77, -30, 25, 28, -24, -350, -632, -1090, -16308, -17530, -37372, -5811, -5485; -11132, -1590, -528, 530, 528, -531, -7425, -13255, -22797, -340890, -366335, -780912, -121404, -114510],78,15]
%
% END OF LATTICE L79.15
%
\hbox{}\par\smallskip%
%
% LATTICE L79.16
%
\leavevmode%
\llap{\spacer}%
${L_{79.16}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{79.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[232650, -20295, 3960; -20295, 1770, -345; 3960, -345, 67],[495, 15, 1],[1, 17, 2, 5, 5, 1, -1, -1, -17, -2, -5, -5, -1, 1; 8, 198, 26, 66, 66, 11, -15, -19, -561, -103, -429, -594, -616, -18; -20, 0, 15, 44, 45, 0, -18, -40, -1980, -435, -2024, -2925, -3300, -162],78,16]
%
% END OF LATTICE L79.16
%
\hbox{}\par\smallskip%
%
% LATTICE L79.17
%
\leavevmode%
\llap{\spacer}%
${L_{79.17}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{79.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[42075, 3465, 1485; 3465, 285, 120; 1485, 120, 38],[495, 15, 1],[-14, -20, -6, -40, -4, 1, 1, 5, -13, -15, -257, -44, -166, -4; 186, 264, 79, 528, 53, -11, -13, -63, 187, 208, 3531, 603, 2266, 54; -45, -55, -15, -99, -10, 0, 2, 0, -110, -90, -1386, -230, -825, -17],78,17]
%
% END OF LATTICE L79.17
%
\hbox{}\par\smallskip%
%
% LATTICE L79.18
%
\leavevmode%
\llap{\spacer}%
${L_{79.18}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{79.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2893042680, -9409620, 718823160; -9409620, 30600, -2337972; 718823160, -2337972, 178603219],[660, 12, 1],[-18608, -346438, -78236, -5539958, -268244, -611003, -156473, -14887, -22007, 81, 13283, 284, -3072, -11002; 39, 715, 161, 11385, 551, 1254, 320, 30, 44, 0, 0, 1, 11, 25; 74892, 1394316, 314878, 22296780, 1079607, 2459116, 629760, 59916, 88572, -326, -53460, -1143, 12364, 44280],78,18]
%
% END OF LATTICE L79.18
%
\hbox{}\par\smallskip%
%
% LATTICE L79.19
%
\leavevmode%
\llap{\spacer}%
${L_{79.19}}=\pdualwitharg{5.11}(\pfillwitharg{2.3}({L_{79.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[546052485, 20870025, -192122700; 20870025, 797610, -7342940; -192122700, -7342940, 67596253],[165, 55, 1],[-155965, -21571, -1705, 963, -1858, -10862, -23092, -193366, -327760, -2443612, -1048956, -5587438, -1735793, -326239; 468199, 64755, 5118, -2891, 5577, 32607, 69321, 580475, 983919, 7335600, 3148913, 16773207, 5210763, 979353; -392425, -54275, -4290, 2423, -4675, -27330, -58102, -486530, -824680, -6148395, -2639288, -14058605, -4367445, -820853],78,19]
%
% END OF LATTICE L79.19
%
\hbox{}\par\smallskip%
%
% LATTICE L79.20
%
\leavevmode%
\llap{\spacer}%
${L_{79.20}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{79.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-325380, 27060, -5940; 27060, -2250, 495; -5940, 495, -106],[660, 15, 1],[97, 149, 1, -7, 1, 23, 14, 119, 2227, 1007, 7135, 6911, 3937, 202; 1055, 1617, 10, -77, 12, 253, 153, 1297, 24255, 10966, 77693, 75252, 42867, 2199; -525, -825, -10, 33, 0, -110, -72, -630, -11880, -5380, -38148, -36960, -21065, -1083],78,20]
%
% END OF LATTICE L79.20
%
\hbox{}\par\smallskip%
%
% LATTICE L79.21
%
\leavevmode%
\llap{\spacer}%
${L_{79.21}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{79.1}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[238448760, -3831300, 59623080; -3831300, 61560, -958000; 59623080, -958000, 14908493],[660, 20, 1],[1502, 3720, 634, 1028, 118, -329, -1, 945, 23197, 16495, 79723, 19458, 134914, 2448; -515, -1287, -225, -385, -48, 66, 0, -322, -7920, -5634, -27236, -6648, -46101, -837; -6040, -14960, -2550, -4136, -475, 1320, 4, -3800, -93280, -66330, -320584, -78245, -542520, -9844],78,21]
%
% END OF LATTICE L79.21
%
\hbox{}\par\smallskip%
%
% LATTICE L79.22
%
\leavevmode%
\llap{\spacer}%
${L_{79.22}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{79.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2824965, 13365, -6930; 13365, -165, -33; -6930, -33, 17],[495, 33, 1],[-5, -1, -3, -1, 2, 2, 8, 14, 4, 24, 136, 46, 13, 7; -6, 0, 1, 0, -1, -1, -5, -15, -7, -50, -315, -111, -34, -30; -2079, -409, -1221, -405, 814, 813, 3245, 5643, 1597, 9537, 53865, 18194, 5127, 2695],78,22]
%
% END OF LATTICE L79.22
%
\hbox{}\par\smallskip%
%
% LATTICE L79.23
%
\leavevmode%
\llap{\spacer}%
${L_{79.23}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{79.1}}))$%
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{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10891485, -1815165, 16830; -1815165, -302511, 2805; 16830, 2805, -26],[495, 33, 1],[16, 10, 5, 2, -1, -1, 1, 11, 71, 184, 403, 1297, 136, 404; -88, -54, -26, -10, 6, 5, -15, -63, -399, -1031, -2255, -7254, -760, -2250; 847, 639, 429, 215, 0, -108, -990, 308, 2826, 7656, 17110, 55440, 5883, 18315],78,23]
%
% END OF LATTICE L79.23
%
\hbox{}\par\smallskip%
%
% LATTICE L79.24
%
\leavevmode%
\llap{\spacer}%
${L_{79.24}}=\pdualwitharg{3.11}(\pfillwitharg{3}({L_{79.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1337820, 9900, 13200; 9900, -66, -99; 13200, -99, -130],[660, 33, 1],[-19, -3, -1, 1, 3, 0, -9, -20, -36, -181, -586, -1251, -65, -186; -291, -47, -18, 15, 52, 1, -125, -297, -541, -2726, -8835, -18868, -981, -2815; -1716, -270, -88, 90, 264, -1, -825, -1815, -3261, -16390, -53055, -113256, -5884, -16830],78,24]
%
% END OF LATTICE L79.24
%
\hbox{}\par\smallskip%
%
% LATTICE L79.25
%
\leavevmode%
\llap{\spacer}%
${L_{79.25}}=\pdualwitharg{2}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20169820440, -42053220, -5052724380; -42053220, 87672, 10534716; -5052724380, 10534716, 1265753641],[1980, 12, 1],[75394, 466076, 315532, 22335472, 360451, 2462573, 210029, 59729, 29375, -245, -40177, -123, 14682, 15176; 369, 2277, 1541, 109065, 1760, 12023, 1025, 291, 143, -1, -165, 0, 77, 75; 300960, 1860496, 1259550, 89159400, 1438859, 9830172, 838400, 238428, 117260, -978, -160380, -491, 58608, 60580],78,25]
%
% END OF LATTICE L79.25
%
\hbox{}\par\smallskip%
%
% LATTICE L79.26
%
\leavevmode%
\llap{\spacer}%
${L_{79.26}}=\pdualwitharg{2.3}({L_{79.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4201560, -5835060, -1045440; -5835060, 8557176, 1451340; -1045440, 1451340, 260129],[1980, 12, 1],[-180189, -905903, -12591, 11353, 620, -79460, -318886, -202892, -11735770, -2668976, -63219718, -9194564, -20989995, -5427329; -873, -4389, -61, 55, 3, -385, -1545, -983, -56859, -12931, -306295, -44547, -101695, -26295; -719296, -3616272, -50262, 45320, 2475, -317196, -1272960, -809924, -46847988, -10654278, -252366620, -36703755, -83789904, -21665340],78,26]
%
% END OF LATTICE L79.26
%
\hbox{}\par\smallskip%
%
% LATTICE L79.27
%
\leavevmode%
\llap{\spacer}%
${L_{79.27}}=\pdualwitharg{3.5.11}(\pfillwitharg{2.3}({L_{79.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[29927951295, -4824026625, -1760021835; -4824026625, 777575205, 283694400; -1760021835, 283694400, 103504474],[165, 165, 1],[-277, -95, -73, -1, 164, 48, 98, 160, -22, -200, -350, -2036, -243, -215; -12735, -3769, -2600, -27, 5713, 1624, 3028, 2016, -6950, -22411, -32130, -177217, -19489, -13747; 30195, 8715, 5885, 57, -12870, -3635, -6633, -2805, 18675, 58025, 82113, 451110, 49285, 34023],78,27]
%
% END OF LATTICE L79.27
%
\hbox{}\par\smallskip%
%
% LATTICE L79.28
%
\leavevmode%
\llap{\spacer}%
${L_{79.28}}=\pdualwitharg{2.11}(\pfillwitharg{3}({L_{79.1}}))$%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-576\\-4488\\-9840\\-5280\\-3324\\-3300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-440\\-3960\\-9240\\-5060\\-3300\\-3740
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3239526840, -10411500, 804936000; -10411500, 33528, -2586980; 804936000, -2586980, 200005123],[660, 44, 1],[86927, 12697, 4313, 651, 1506, 5698, 59256, 103370, 177600, 1327766, 2853804, 1520870, 945787, 892229; 30, 4, 0, 0, 1, 3, 25, 39, 65, 483, 1035, 551, 342, 320; -349844, -51100, -17358, -2620, -6061, -22932, -238480, -416020, -714764, -5343690, -11485340, -6120851, -3806388, -3590840],78,28]
%
% END OF LATTICE L79.28
%
\hbox{}\par\smallskip%
%
% LATTICE L79.29
%
\leavevmode%
\llap{\spacer}%
${L_{79.29}}=\pdualwitharg{3.5}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{s}{2}{990}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{180}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4773780\\-437580\\-19800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-437580\\40110\\1815
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19800\\1815\\82
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44351\\443520\\914760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4072\\-40721\\-83985
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-1760\\-3631
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-29\\-85
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-33\\-495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\34\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\55\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\120\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\11\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-21\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-60\\-220\\-720\\-495\\-45\\-25\\-495\\-150\\-275\\-720\\-330\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\990\\0\\-990\\-3960\\-3465\\-495\\-495\\-17325\\-6930\\-15345\\-43560\\-23760\\-990
\end{tabular}\endgroup%
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-60\\0\\60\\0\\-180\\-330\\-90\\-150\\-6930\\-3000\\-6930\\-19980\\-11220\\-540
\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-45\\-495\\-90\\-99\\-180\\0\\18\\0\\-990\\-540\\-1386\\-4140\\-2475\\-153
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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10\\0\\-60\\-220\\-720\\-495\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\990\\0\\-990\\-3960\\-3465\\-495
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\60\\0\\-180\\-330\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-990\\0\\22\\0\\-165\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-3960\\-180\\0\\180\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-495\\-3465\\-330\\-165\\0\\330\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-495\\-90\\-99\\-180\\0\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-495\\-150\\-275\\-720\\-330\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-495\\-17325\\-6930\\-15345\\-43560\\-23760\\-990
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-150\\-6930\\-3000\\-6930\\-19980\\-11220\\-540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-275\\-15345\\-6930\\-16357\\-47520\\-27060\\-1386
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-43560\\-19980\\-47520\\-138420\\-79200\\-4140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-23760\\-11220\\-27060\\-79200\\-45705\\-2475
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-990\\-540\\-1386\\-4140\\-2475\\-153
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4773780, -437580, -19800; -437580, 40110, 1815; -19800, 1815, 82],[1980, 15, 1],[-3, -5, 3, 5, 11, 1, -2, -5, -259, -115, -269, -779, -441, -22; -29, -33, 34, 55, 120, 11, -21, -51, -2607, -1154, -2695, -7800, -4411, -219; -85, -495, -30, -11, 0, 0, -18, -80, -4950, -2280, -5434, -15840, -9075, -477],78,29]
%
% END OF LATTICE L79.29
%
\hbox{}\par\smallskip%
%
% LATTICE L79.30
%
\leavevmode%
\llap{\spacer}%
${L_{79.30}}=\pdualwitharg{5}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${90}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{b}{2}{198}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9900\\-1980\\-1980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1980\\-390\\-345
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1980\\-345\\38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11087\\-62568\\11880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1946\\-10982\\2085
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-98\\553\\-106
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\294\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\396\\-55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\232\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\792\\-99
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\164\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\11\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-17\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
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0\\110\\0\\-990\\-440\\-1155\\-55\\-495\\-1925\\-2310\\-15345\\-4840\\-7920\\-110
\end{tabular}\endgroup%
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-180\\0\\60\\0\\-60\\-330\\-30\\-450\\-2310\\-3000\\-20790\\-6660\\-11220\\-180
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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90\\0\\-180\\-1980\\-720\\-1485\\-45
\end{tabular}\endgroup%
\kern3pt%
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0\\110\\0\\-990\\-440\\-1155\\-55
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-180\\0\\60\\0\\-60\\-330\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1980\\-990\\0\\198\\0\\-495\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-440\\-60\\0\\20\\0\\-20
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-55\\-30\\-99\\-20\\0\\2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-225\\-495\\-450\\-2475\\-720\\-990\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-495\\-1925\\-2310\\-15345\\-4840\\-7920\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-450\\-2310\\-3000\\-20790\\-6660\\-11220\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2475\\-15345\\-20790\\-147213\\-47520\\-81180\\-1386
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-4840\\-6660\\-47520\\-15380\\-26400\\-460
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-990\\-7920\\-11220\\-81180\\-26400\\-45705\\-825
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-110\\-180\\-1386\\-460\\-825\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9900, -1980, -1980; -1980, -390, -345; -1980, -345, 38],[1980, 15, 1],[-52, -70, -41, -140, -29, -2, 3, 10, -84, -155, -1246, -419, -768, -17; 294, 396, 232, 792, 164, 11, -17, -57, 473, 874, 7029, 2364, 4334, 96; -45, -55, -30, -99, -20, 0, 2, 0, -110, -180, -1386, -460, -825, -17],78,30]
%
% END OF LATTICE L79.30
%
\hbox{}\par\smallskip%
%
% LATTICE L79.31
%
\leavevmode%
\llap{\spacer}%
${L_{79.31}}=\pdualwitharg{5.11}(\pfillwitharg{3}({L_{79.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{660}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{220}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1492260, 735900, -7920; 735900, -362890, 3905; -7920, 3905, -42],[660, 55, 1],[-8, 0, 13, 2, 17, 1, -2, -15, -23, -335, -71, -753, -116, -21; -25, -1, 30, 5, 44, 3, -5, -41, -65, -954, -203, -2156, -333, -61; -825, -95, 330, 87, 880, 90, -88, -990, -1720, -25740, -5532, -58960, -9165, -1727],78,31]
%
% END OF LATTICE L79.31
%
\hbox{}\par\smallskip%
%
% LATTICE L79.32
%
\leavevmode%
\llap{\spacer}%
${L_{79.32}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{79.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{s}{2}{1320}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{440}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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41\\1\\-165
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5286600, 614460, 1317360; 614460, 72120, 153120; 1317360, 153120, 328271],[660, 60, 1],[-27771, -46265, -609, 525, 41, -4211, -3323, -31455, -605221, -137585, -1954983, -473849, -1081599, -55903; -587, -979, -13, 11, 1, -88, -70, -664, -12782, -2906, -41294, -10009, -22847, -1181; 111720, 186120, 2450, -2112, -165, 16940, 13368, 126540, 2434740, 553490, 7864692, 1906245, 4351160, 224892],78,32]
%
% END OF LATTICE L79.32
%
\hbox{}\par\smallskip%
%
% LATTICE L79.33
%
\leavevmode%
\llap{\spacer}%
${L_{79.33}}=\pdualwitharg{11}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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1\\45\\1080
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1162260, -211860, 9900; -211860, -37290, 1749; 9900, 1749, -82],[1980, 33, 1],[-19, -1, -1, 1, 1, 0, -3, -20, -12, -181, -586, -417, -65, -62; -885, -47, -50, 45, 48, 1, -135, -921, -555, -8378, -27135, -19312, -3011, -2875; -21186, -1124, -1188, 1080, 1144, 21, -3245, -22077, -13297, -200706, -650025, -462616, -72126, -68860],78,33]
%
% END OF LATTICE L79.33
%
\hbox{}\par\smallskip%
%
% LATTICE L79.34
%
\leavevmode%
\llap{\spacer}%
${L_{79.34}}=\pdualwitharg{3.11}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-174055860, -7252740, 67320; -7252740, -302214, 2805; 67320, 2805, -26],[1980, 33, 1],[-12, -8, -9, -2, 1, 1, 4, -7, -49, -257, -283, -1825, -96, -289; 296, 198, 224, 50, -24, -25, -105, 171, 1203, 6314, 6955, 44856, 2360, 7110; 847, 639, 858, 215, 0, -108, -990, 308, 2826, 15312, 17110, 110880, 5883, 18315],78,34]
%
% END OF LATTICE L79.34
%
\hbox{}\par\smallskip%
%
% LATTICE L79.35
%
\leavevmode%
\llap{\spacer}%
${L_{79.35}}=\pdualwitharg{3.5.11}(\pfillwitharg{2}({L_{79.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21646350, -6493905, 59400; -6493905, -1948155, 17820; 59400, 17820, -163],[495, 165, 1],[-14, -8, -3, 0, 5, 1, -1, -11, -67, -172, -75, -1205, -126, -74; 11, 9, 7, 1, 6, -1, -3, 3, 33, 91, 41, 666, 71, 45; -3905, -1935, -330, 109, 2475, 255, -693, -3685, -20835, -52800, -22879, -366795, -38205, -22077],78,35]
%
% END OF LATTICE L79.35
%
\hbox{}\par\smallskip%
%
% LATTICE L79.36
%
\leavevmode%
\llap{\spacer}%
${L_{79.36}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{79.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-495, 44055, -495; 44055, -3818265, 42900; -495, 42900, -482],[495, 165, 1],[59, 5, 11, 5, 4, -1, -1, 25, 23, 185, 247, 444, 141, 29; -27, -1, 2, 3, 5, 1, -1, -33, -19, -142, -183, -325, -101, -19; -2475, -95, 165, 261, 440, 90, -88, -2970, -1720, -12870, -16596, -29480, -9165, -1727],78,36]
%
% END OF LATTICE L79.36
%
\hbox{}\par\smallskip%
%
% LATTICE L79.37
%
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\llap{\spacer}%
${L_{79.37}}=\pdualwitharg{2.3.11}(\pfillwitharg{3}({L_{79.1}}))$%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[44311080, -1684980, 11007480; -1684980, 64680, -418572; 11007480, -418572, 2734409],[660, 132, 1],[27119, 3783, 235, 313, 828, 789, 20823, 33907, 57243, 142159, 915077, 487380, 100921, 284307; 30, 4, 0, 0, 1, 1, 25, 39, 65, 161, 1035, 551, 114, 320; -109164, -15228, -946, -1260, -3333, -3176, -83820, -136488, -230424, -572242, -3683520, -1961883, -406244, -1144440],78,37]
%
% END OF LATTICE L79.37
%
\hbox{}\par\smallskip%
%
% LATTICE L79.38
%
\leavevmode%
\llap{\spacer}%
${L_{79.38}}=\pdualwitharg{3.5.11}(\pfillwitharg{3}({L_{79.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${330}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{220}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{660}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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660\\0\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[660, 0, 0; 0, -235290, 1815; 0, 1815, -14],[660, 165, 1],[-12, -22, -111, -72, -769, -40, -23, -12, -2, -1, 0, 1, 0, -1; -7, -3, -6, -1, 0, 1, 3, 7, 3, 6, 1, 0, -1, -3; -990, -540, -1540, -624, -5280, -145, 231, 825, 375, 770, 129, 0, -130, -396],78,38]
%
% END OF LATTICE L79.38
%
\hbox{}\par\smallskip%
%
% LATTICE L79.39
%
\leavevmode%
\llap{\spacer}%
${L_{79.39}}=\pdualwitharg{2.3.5}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{s}{2}{3960}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{r}{2}{1320}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-104940\\17880\\26400
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\kern3pt%
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91763\\-2277\\366300
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4637\\-115\\18510
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4883\\-121\\19492
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124\\-3\\495
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7052760, -104940, 1766160; -104940, 17880, 26400; 1766160, 26400, -442283],[1980, 60, 1],[12691, 91763, 4637, 4883, 124, -9755, -487, 7811, 584803, 138877, 671683, 491924, 1137365, 61993; -315, -2277, -115, -121, -3, 242, 12, -194, -14520, -3448, -16676, -12213, -28237, -1539; 50660, 366300, 18510, 19492, 495, -38940, -1944, 31180, 2334420, 554370, 2681228, 1963665, 4540140, 247464],78,39]
%
% END OF LATTICE L79.39
%
\hbox{}\par\smallskip%
%
% LATTICE L79.40
%
\leavevmode%
\llap{\spacer}%
${L_{79.40}}=\pdualwitharg{2.5}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[49107960, -2528460, -12272040; -2528460, 130200, 631860; -12272040, 631860, 3066773],[1980, 60, 1],[5276, 4354, 744, 3670, 144, -331, 1, 3337, 27227, 19353, 280559, 22824, 158236, 2870; 255, 209, 35, 165, 6, -22, 0, 162, 1320, 938, 13596, 1106, 7667, 139; 21060, 17380, 2970, 14652, 575, -1320, 4, 13320, 108680, 77250, 1119888, 91105, 631620, 11456],78,40]
%
% END OF LATTICE L79.40
%
\hbox{}\par\smallskip%
%
% LATTICE L79.41
%
\leavevmode%
\llap{\spacer}%
${L_{79.41}}=\pdualwitharg{2.5.11}(\pfillwitharg{3}({L_{79.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[64515000, 1024980, -15259860; 1024980, 17160, -241780; -15259860, -241780, 3609943],[660, 220, 1],[18917, 4185, 8119, 1507, 2638, 671, 503, 13395, 28127, 218065, 95429, 255858, 160889, 31809; -57416, -12702, -24642, -4574, -8007, -2037, -1527, -40657, -85371, -661869, -289645, -776577, -488328, -96546; 76120, 16840, 32670, 6064, 10615, 2700, 2024, 53900, 113180, 877470, 383996, 1029545, 647400, 127996],78,41]
%
% END OF LATTICE L79.41
%
\hbox{}\par\smallskip%
%
% LATTICE L79.42
%
\leavevmode%
\llap{\spacer}%
${L_{79.42}}=\pdualwitharg{2.11}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${792}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{*}{2}{360}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{440}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22995399240\\-64391580\\-5760505080
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22995399240, -64391580, -5760505080; -64391580, 180312, 16130532; -5760505080, 16130532, 1443045995],[1980, 132, 1],[-39385, -1919, -711, -947, -361, -965, -9149, -47123, -26917, -201141, -1296673, -230327, -143215, -135031; -828, -42, -22, -30, -7, -17, -175, -951, -551, -4129, -26655, -4737, -2948, -2790; -157212, -7660, -2838, -3780, -1441, -3852, -36520, -188100, -107444, -802890, -5175900, -919391, -571668, -539000],78,42]
%
% END OF LATTICE L79.42
%
\hbox{}\par\smallskip%
%
% LATTICE L79.43
%
\leavevmode%
\llap{\spacer}%
${L_{79.43}}=\pdualwitharg{2.3.11}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1933307640\\-28814940\\-484268400
\end{tabular}\endgroup%
\kern3pt%
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-28814940\\435336\\7217760
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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0\\-3960\\-11880\\-27720\\-45540\\-9900\\-33660
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1933307640, -28814940, -484268400; -28814940, 435336, 7217760; -484268400, 7217760, 121302931],[1980, 132, 1],[-280318, -144616, -43830, -12876, 620, -1981, -333313, -273385, -1516493, -3832091, -8292669, -13289492, -2766534, -7972232; -1413, -729, -221, -65, 3, -10, -1680, -1378, -7644, -19316, -41800, -66987, -13945, -40185; -1119008, -577296, -174966, -51400, 2475, -7908, -1330560, -1091332, -6053724, -15297414, -33103700, -53050635, -11043792, -31824540],78,43]
%
% END OF LATTICE L79.43
%
\hbox{}\par\smallskip%
%
% LATTICE L79.44
%
\leavevmode%
\llap{\spacer}%
${L_{79.44}}=\pdualwitharg{3.5.11}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{b}{2}{660}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{1980}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{198}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1950300\\164340\\-1980
\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21889\\-112176\\-30246480
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-2200\\-11276\\-3040125
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-4\\-22\\-5665
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\HardButStrongLineBreak\kern3pt%
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-2\\-12\\-2925
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\HardButStrongLineBreak\kern3pt%
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-1\\-8\\-1650
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\1980
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\HardButStrongLineBreak\kern3pt%
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0\\1\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-1188
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1950300, 164340, -1980; 164340, 13530, -165; -1980, -165, 2],[1980, 165, 1],[-4, -2, -1, 0, 1, 0, -1, -4, -22, -111, -24, -769, -40, -23; -22, -12, -8, 0, 12, 1, -3, -19, -111, -566, -123, -3948, -206, -120; -5665, -2925, -1650, -1, 1980, 90, -1188, -5390, -30240, -153120, -33164, -1063260, -55365, -31977],78,44]
%
% END OF LATTICE L79.44
%
\hbox{}\par\smallskip%
%
% LATTICE L79.45
%
\leavevmode%
\llap{\spacer}%
${L_{79.45}}=\pdualwitharg{5.11}({L_{79.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${990}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{660}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}{220}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-141718500, 23530320, -261360; 23530320, -3906870, 43395; -261360, 43395, -482],[1980, 165, 1],[16, 2, 13, 4, 9, 0, -1, -4, 2, 43, 32, 119, 20, 5; 69, 11, 82, 27, 64, 1, -7, -57, -7, -26, 9, 64, 19, 11; -2475, -95, 330, 261, 880, 90, -88, -2970, -1720, -25740, -16596, -58960, -9165, -1727],78,45]
%
% END OF LATTICE L79.45
%
\hbox{}\par\smallskip%
%
% LATTICE L79.46
%
\leavevmode%
\llap{\spacer}%
${L_{79.46}}=\pdualwitharg{2.3.5.11}(\pfillwitharg{3}({L_{79.1}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9240, -1197900, -297660; -1197900, 155294040, 38588220; -297660, 38588220, 9588589],[660, 660, 1],[-908, -1662, -4192, -5438, -14520, -3021, -1737, -907, -153, -43, -7, 0, -4, -78; 813, 1493, 3767, 4887, 13049, 2715, 1561, 813, 133, 27, -9, -41, -5, 65; -3300, -6060, -15290, -19836, -52965, -11020, -6336, -3300, -540, -110, 36, 165, 20, -264],78,46]
%
% END OF LATTICE L79.46
%
\hbox{}\par\smallskip%
%
% LATTICE L79.47
%
\leavevmode%
\llap{\spacer}%
${L_{79.47}}=\pdualwitharg{2.3.5.11}({L_{79.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L79.47
%
\hbox{}\par\smallskip%
%
% LATTICE L79.48
%
\leavevmode%
\llap{\spacer}%
${L_{79.48}}=\pdualwitharg{2.5.11}({L_{79.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[639456840, 10448460, 152343180; 10448460, 244200, 2434080; 152343180, 2434080, 36335383],[1980, 660, 1],[-747461, -56959, -113169, -57959, -30104, -1999, -1499, -475657, -350747, -2740773, -3611395, -3231544, -2036531, -406207; 2239404, 170650, 339056, 173646, 90192, 5989, 4491, 1425075, 1050843, 8211395, 10819791, 9681752, 6101476, 1217002; 2983860, 227380, 451770, 231372, 120175, 7980, 5984, 1898820, 1400180, 10941150, 14416668, 12900305, 8129820, 1621576],78,48]
%
% END OF LATTICE L79.48
%

\medskip%
%------------------------------WEYL-GROUP-W80--------------------------------
%
\leavevmode\llap{}%
$W_{80\phantom{0}}$%
\qquad\llap{88} lattices, $\chi=72$%
\hfill%
$2222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L80.1
%
\leavevmode%
\llap{\spacer}%
${L_{80.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{80}}%
\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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248820\\0\\-660
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0\\4\\-1
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-660\\-1\\2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-2\\-193\\-744
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2\\192\\743
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[248820, 0, -660; 0, 4, -1; -660, -1, 2],[660, 1, 1],[-21, -9, -17, -1, -29, -2, -1, 0, -1, -1, -5, -1, -81, -8, -9, -2; -2024, -870, -1650, -98, -2860, -198, -100, 0, -88, -90, -462, -94, -7700, -762, -860, -192; -7810, -3350, -6336, -374, -10890, -753, -380, -1, -374, -370, -1848, -370, -30030, -2967, -3340, -743],79,1]
%
% END OF LATTICE L80.1
%
\hbox{}\par\smallskip%
%
% LATTICE L80.2
%
\leavevmode%
\llap{\spacer}%
${L_{80.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N'_{36}}%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3198360\\531960\\-2640
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
531960\\88477\\-439
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\kern3pt%
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-2640\\-439\\2
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{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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21303\\-128480\\-80740
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[3198360, 531960, -2640; 531960, 88477, -439; -2640, -439, 2],[1320, 1, 1],[1605, 514, 1543, 193, 21303, 1931, 1867, 161, 2641, 451, 580, 0, -73, -1, 63, 32; -9680, -3100, -9306, -1164, -128480, -11646, -11260, -971, -15928, -2720, -3498, 0, 440, 6, -380, -193; -6072, -1945, -5841, -731, -80740, -7320, -7080, -611, -10032, -1715, -2211, -1, 220, 0, -240, -121],79,2]
%
% END OF LATTICE L80.2
%
\hbox{}\par\smallskip%
%
% LATTICE L80.3
%
\leavevmode%
\llap{\spacer}%
${L_{80.3}}$%
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{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[86152440, 195360, 0; 195360, 443, 0; 0, 0, -1],[1320, 1, 1],[107, 37, 119, 16, 1869, 86, 171, 31, 267, 48, 68, 1, 1, -1, -1, 3; -47212, -16325, -52503, -7059, -824560, -37941, -75440, -13676, -117788, -21175, -29997, -441, -440, 441, 440, -1324; -572, -185, -561, -71, -7920, -360, -700, -122, -1012, -175, -231, -1, 0, 0, -20, -22],79,3]
%
% END OF LATTICE L80.3
%
\hbox{}\par\smallskip%
%
% LATTICE L80.4
%
\leavevmode%
\llap{\spacer}%
${L_{80.4}}=\pfillwitharg{2}({L_{80.1}})$%
{${}={}$}Nikulin $80$%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1155, -495, 495; -495, -211, 198; 495, 198, -37],[165, 1, 1],[-52, -2, 455, 158, 16688, 3433, 2057, 997, 5618, 2532, 5111, 348, 11142, 1627, 473, 15; 132, 5, -1155, -401, -42350, -8712, -5220, -2530, -14256, -6425, -12969, -883, -28270, -4128, -1200, -38; 11, 0, -99, -34, -3575, -735, -440, -213, -1199, -540, -1089, -74, -2365, -345, -100, -3],79,4]
%
% END OF LATTICE L80.4
%
\hbox{}\par\smallskip%
%
% LATTICE L80.5
%
\leavevmode%
\llap{\spacer}%
${L_{80.5}}=\pfillwitharg{2}({L_{80.2}})$%
{${}={}$}Nikulin $36'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[37290, 1650, 990; 1650, 73, 44; 990, 44, 23],[330, 1, 1],[1, -3, -17, -3, -183, -17, -17, -3, -23, -7, -5, 1, 73, 7, 7, 1; -22, 65, 363, 63, 3740, 345, 340, 58, 418, 115, 33, -27, -1760, -165, -160, -22; 0, 5, 33, 7, 550, 54, 60, 13, 132, 55, 99, 5, 110, 6, 0, -1],79,5]
%
% END OF LATTICE L80.5
%
\hbox{}\par\smallskip%
%
% LATTICE L80.6
%
\leavevmode%
\llap{\spacer}%
${L_{80.6}}=\main({L_{80.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[292380, 97020, -660; 97020, 32194, -219; -660, -219, 1],[660, 1, 1],[2641, 451, 580, 0, -73, -1, 63, 64, 1605, 514, 1543, 193, 21303, 1931, 1867, 322; -7964, -1360, -1749, 0, 220, 3, -190, -193, -4840, -1550, -4653, -582, -64240, -5823, -5630, -971; -1012, -175, -231, -1, 0, 0, -20, -22, -572, -185, -561, -71, -7920, -720, -700, -122],79,6]
%
% END OF LATTICE L80.6
%
\hbox{}\par\smallskip%
%
% LATTICE L80.7
%
\leavevmode%
\llap{\spacer}%
${L_{80.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{80.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1248060, 8745, -414645; 8745, -51, 2904; -414645, 2904, -137758],[165, 3, 1],[-4198, -3071, -3451, -1457, -88116, -5454, -8264, -3080, -13622, -5029, -2489, -163, -984, 54, 164, -160; 1705, 1245, 1397, 589, 35585, 2202, 3335, 1242, 5489, 2025, 1001, 65, 385, -22, -65, 66; 12672, 9270, 10417, 4398, 265980, 16463, 24945, 9297, 41118, 15180, 7513, 492, 2970, -163, -495, 483],79,7]
%
% END OF LATTICE L80.7
%
\hbox{}\par\smallskip%
%
% LATTICE L80.8
%
\leavevmode%
\llap{\spacer}%
${L_{80.8}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{80.2}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[441870, -18480, 220440; -18480, 742, -9218; 220440, -9218, 109973],[330, 2, 1],[-257, -169, 1247, 529, 28958, 5984, 3607, 1759, 9937, 8969, 18113, 1231, 19442, 2816, 793, 1; 22, 15, -99, -43, -2365, -489, -295, -144, -814, -735, -1485, -101, -1595, -231, -65, 0; 517, 340, -2508, -1064, -58245, -12036, -7255, -3538, -19987, -18040, -36432, -2476, -39105, -5664, -1595, -2],79,8]
%
% END OF LATTICE L80.8
%
\hbox{}\par\smallskip%
%
% LATTICE L80.9
%
\leavevmode%
\llap{\spacer}%
${L_{80.9}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{80.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1787280, 13365, -713625; 13365, -85, 5335; -713625, 5335, -284936],[165, 5, 1],[-7278, -1065, -17954, -2527, -30568, -28381, -2867, -5343, -23632, -1745, -12956, -283, -342, 281, 57, -277; 1705, 249, 4191, 589, 7117, 6606, 667, 1242, 5489, 405, 3003, 65, 77, -66, -13, 66; 18260, 2672, 45045, 6340, 76692, 71205, 7193, 13405, 59290, 4378, 32505, 710, 858, -705, -143, 695],79,9]
%
% END OF LATTICE L80.9
%
\hbox{}\par\smallskip%
%
% LATTICE L80.10
%
\leavevmode%
\llap{\spacer}%
${L_{80.10}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{80.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[508530, -24090, -169950; -24090, 1113, 8052; -169950, 8052, 56797],[330, 3, 1],[355, 116, -292, -370, -40479, -1394, -5041, -1229, -13885, -6266, -4218, -860, -27171, -656, -1109, -1; 44, 15, -33, -43, -4730, -163, -590, -144, -1628, -735, -495, -101, -3190, -77, -130, 0; 1056, 345, -869, -1101, -120450, -4148, -15000, -3657, -41316, -18645, -12551, -2559, -80850, -1952, -3300, -3],79,10]
%
% END OF LATTICE L80.10
%
\hbox{}\par\smallskip%
%
% LATTICE L80.11
%
\leavevmode%
\llap{\spacer}%
${L_{80.11}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{80.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[663630, -35310, 132330; -35310, 1855, -7040; 132330, -7040, 26387],[330, 5, 1],[-217, -14, 554, 232, 5069, 2618, 631, 769, 8687, 784, 7916, 538, 3401, 1232, 139, 1; 44, 3, -99, -43, -946, -489, -118, -144, -1628, -147, -1485, -101, -638, -231, -26, 0; 1100, 71, -2805, -1175, -25674, -13260, -3196, -3895, -44000, -3971, -40095, -2725, -17226, -6240, -704, -5],79,11]
%
% END OF LATTICE L80.11
%
\hbox{}\par\smallskip%
%
% LATTICE L80.12
%
\leavevmode%
\llap{\spacer}%
${L_{80.12}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{80.1}}))$%
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{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3436620, 27225, -936045; 27225, -187, 7414; -936045, 7414, -254954],[165, 11, 1],[-866, -6971, -2137, -3309, -18196, -37168, -18774, -6998, -2814, -11429, -1543, -371, -204, 368, 374, -362; 155, 1245, 381, 589, 3235, 6606, 3335, 1242, 499, 2025, 273, 65, 35, -66, -65, 66; 3184, 25630, 7857, 12166, 66900, 136653, 69025, 25729, 10346, 42020, 5673, 1364, 750, -1353, -1375, 1331],79,12]
%
% END OF LATTICE L80.12
%
\hbox{}\par\smallskip%
%
% LATTICE L80.13
%
\leavevmode%
\llap{\spacer}%
${L_{80.13}}=\pdualwitharg{3}({L_{80.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[640860, -2640, -2640; -2640, 12, 9; -2640, 9, 14],[660, 3, 1],[-61, -29, -21, -5, -189, -5, -11, -1, -5, -1, -1, -1, -141, -5, -19, -5; -9284, -4410, -3190, -758, -28600, -756, -1660, -150, -748, -150, -154, -154, -21560, -764, -2900, -762; -5610, -2670, -1936, -462, -17490, -463, -1020, -93, -462, -90, -88, -90, -12870, -457, -1740, -459],79,13]
%
% END OF LATTICE L80.13
%
\hbox{}\par\smallskip%
%
% LATTICE L80.14
%
\leavevmode%
\llap{\spacer}%
${L_{80.14}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{80.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[84670410, -192720, 41552280; -192720, 438, -94578; 41552280, -94578, 20391917],[330, 6, 1],[-81, 751, 1501, 857, 29398, 1872, 2967, 1175, 5361, 4049, 2019, 103, -2998, -272, -567, -215; -11, 65, 121, 63, 1870, 115, 170, 58, 209, 115, 11, -27, -880, -55, -80, -22; 165, -1530, -3058, -1746, -59895, -3814, -6045, -2394, -10923, -8250, -4114, -210, 6105, 554, 1155, 438],79,14]
%
% END OF LATTICE L80.14
%
\hbox{}\par\smallskip%
%
% LATTICE L80.15
%
\leavevmode%
\llap{\spacer}%
${L_{80.15}}=\pdualwitharg{3}(\main({L_{80.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[119460, 47520, -1320; 47520, 18903, -525; -1320, -525, 14],[660, 3, 1],[761, 167, 108, 20, 1119, 25, 31, -4, -35, -2, 13, 13, 2511, 85, 299, 70; -1936, -425, -275, -51, -2860, -64, -80, 10, 88, 5, -33, -33, -6380, -216, -760, -178; -924, -210, -143, -30, -1980, -49, -90, -3, 0, 0, -11, -12, -2640, -91, -330, -81],79,15]
%
% END OF LATTICE L80.15
%
\hbox{}\par\smallskip%
%
% LATTICE L80.16
%
\leavevmode%
\llap{\spacer}%
${L_{80.16}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{80.1}}))$%
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{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1416286410, -5400120, 468186345; -5400120, 20595, -1785135; 468186345, -1785135, 154769863],[165, 15, 1],[131781, 18106, 96963, 39208, 461255, 141761, 42444, 77573, 337743, 24578, 59545, 3476, 8269, 519, 240, 7795; 341, 46, 242, 96, 1111, 340, 101, 182, 781, 56, 132, 6, 11, 0, 1, 22; -398640, -54771, -293315, -118605, -1395306, -428830, -128394, -234660, -1021680, -74349, -180125, -10515, -25014, -1570, -726, -23580],79,16]
%
% END OF LATTICE L80.16
%
\hbox{}\par\smallskip%
%
% LATTICE L80.17
%
\leavevmode%
\llap{\spacer}%
${L_{80.17}}=\pdualwitharg{2}(\main({L_{80.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[42278280, -192060, 10569240; -192060, 868, -48012; 10569240, -48012, 2642227],[660, 4, 1],[-10958, -7877, -11152, -163, -41, 325, 77, -497, -4409, -6093, -19582, -2631, -76794, -28265, -14047, -5091; 14608, 10505, 14883, 219, 55, -438, -110, 656, 5852, 8095, 26037, 3501, 102245, 37638, 18710, 6784; 44099, 31700, 44880, 656, 165, -1308, -310, 2000, 17743, 24520, 78804, 10588, 309045, 113748, 56530, 20488],79,17]
%
% END OF LATTICE L80.17
%
\hbox{}\par\smallskip%
%
% LATTICE L80.18
%
\leavevmode%
\llap{\spacer}%
${L_{80.18}}=\pdualwitharg{2}({L_{80.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-160754880, 246180, -39942540; 246180, -376, 61168; -39942540, 61168, -9924467],[660, 4, 1],[8642, 3471, 5887, 241, 3416, 152, -123, -162, 246, 569, 3001, 567, 41024, 7928, 2143, 1778; -143, -65, -132, -9, -275, -39, -5, 1, 11, 5, 0, -3, -385, -81, -25, -25; -34782, -13970, -23694, -970, -13750, -612, 495, 652, -990, -2290, -12078, -2282, -165110, -31908, -8625, -7156],79,18]
%
% END OF LATTICE L80.18
%
\hbox{}\par\smallskip%
%
% LATTICE L80.19
%
\leavevmode%
\llap{\spacer}%
${L_{80.19}}=\pdualwitharg{5}(\main({L_{80.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1848660, 614460, 5940; 614460, -204230, -1975; 5940, -1975, -19],[660, 5, 1],[837, 33, 272, 10, 47, 1, -5, -6, 65, 8, 179, 31, 855, 409, 87, 88; 2332, 92, 759, 28, 132, 3, -14, -17, 176, 22, 495, 86, 2376, 1137, 242, 245; 19140, 749, 6105, 215, 968, 0, -108, -110, 1980, 211, 4455, 745, 20152, 9600, 2028, 2030],79,19]
%
% END OF LATTICE L80.19
%
\hbox{}\par\smallskip%
%
% LATTICE L80.20
%
\leavevmode%
\llap{\spacer}%
${L_{80.20}}=\pdualwitharg{5}({L_{80.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15055260, 19800, -6615840; 19800, -20, 8965; -6615840, 8965, -2895694],[660, 5, 1],[-49581, -3751, -29027, -815, -1629, 82, 163, -82, -11425, -1795, -31979, -4731, -59377, -27812, -5709, -5464; -4363436, -330110, -2554530, -71722, -143352, 7218, 14344, -7220, -1005532, -157978, -2814438, -416366, -5225616, -2447658, -502432, -480868; 99770, 7548, 58410, 1640, 3278, -165, -328, 165, 22990, 3612, 64350, 9520, 119482, 55965, 11488, 10995],79,20]
%
% END OF LATTICE L80.20
%
\hbox{}\par\smallskip%
%
% LATTICE L80.21
%
\leavevmode%
\llap{\spacer}%
${L_{80.21}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{80.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1011724230, -860640, 503663820; -860640, 730, -428450; 503663820, -428450, 250737539],[330, 10, 1],[-575, 919, 27107, 5053, 33640, 31916, 3329, 6427, 28339, 4129, 28421, -5, -5876, -6676, -805, -1379; -11, 13, 363, 63, 374, 345, 34, 58, 209, 23, 33, -27, -176, -165, -16, -22; 1155, -1846, -54450, -10150, -67573, -64110, -6687, -12910, -56925, -8294, -57090, 10, 11803, 13410, 1617, 2770],79,21]
%
% END OF LATTICE L80.21
%
\hbox{}\par\smallskip%
%
% LATTICE L80.22
%
\leavevmode%
\llap{\spacer}%
${L_{80.22}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{80.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1160610, -68970, -422400; -68970, 4081, 25102; -422400, 25102, 153731],[330, 11, 1],[41, 141, -113, -511, -5059, -5746, -6921, -1686, -1731, -8591, -1577, -1179, -3391, -2704, -1529, -4; 4, 15, -9, -43, -430, -489, -590, -144, -148, -735, -135, -101, -290, -231, -130, 0; 112, 385, -309, -1397, -13830, -15708, -18920, -4609, -4732, -23485, -4311, -3223, -9270, -7392, -4180, -11],79,22]
%
% END OF LATTICE L80.22
%
\hbox{}\par\smallskip%
%
% LATTICE L80.23
%
\leavevmode%
\llap{\spacer}%
${L_{80.23}}=\pdualwitharg{3}({L_{80.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[238920, 95040, -1320; 95040, 37806, -525; -1320, -525, 7],[1320, 3, 1],[761, 167, 108, 20, 1119, 25, 31, -2, -35, -2, 13, 13, 2511, 85, 299, 35; -1936, -425, -275, -51, -2860, -64, -80, 5, 88, 5, -33, -33, -6380, -216, -760, -89; -1848, -420, -286, -60, -3960, -98, -180, -3, 0, 0, -22, -24, -5280, -182, -660, -81],79,23]
%
% END OF LATTICE L80.23
%
\hbox{}\par\smallskip%
%
% LATTICE L80.24
%
\leavevmode%
\llap{\spacer}%
${L_{80.24}}=\pdualwitharg{3}({L_{80.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4584189720, 26192760, -233640; 26192760, -149658, 1335; -233640, 1335, -11],[1320, 3, 1],[-783, -224, -196, -61, -5453, -77, -387, -47, -207, -1, 31, 16, 503, 2, -63, -43; -136972, -39185, -34287, -10671, -953920, -13470, -67700, -8222, -36212, -175, 5423, 2799, 88000, 350, -11020, -7522; 7392, 2100, 1826, 564, 50160, 707, 3540, 426, 1848, 0, -286, -144, -3960, -7, 660, 414],79,24]
%
% END OF LATTICE L80.24
%
\hbox{}\par\smallskip%
%
% LATTICE L80.25
%
\leavevmode%
\llap{\spacer}%
${L_{80.25}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{80.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[878842470, -982410, -294620370; -982410, 1095, 329340; -294620370, 329340, 98767601],[330, 15, 1],[885, -707, -6951, -3887, -51751, -8183, -5121, -4943, -43587, -3175, -7283, 5, 9049, 1713, 1239, 1061; -22, 13, 121, 63, 748, 115, 68, 58, 418, 23, 11, -27, -352, -55, -32, -22; 2640, -2109, -20735, -11595, -154374, -24410, -15276, -14745, -130020, -9471, -21725, 15, 26994, 5110, 3696, 3165],79,25]
%
% END OF LATTICE L80.25
%
\hbox{}\par\smallskip%
%
% LATTICE L80.26
%
\leavevmode%
\llap{\spacer}%
${L_{80.26}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{80.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1625030385, -8579505, 536253465; -8579505, 45309, -2831202; 536253465, -2831202, 176961478],[165, 33, 1],[20682, 156053, 15173, 67387, 359888, 243256, 363942, 132890, 52542, 210067, 9235, 5837, 6232, 824, 2178, 13558; 31, 230, 22, 96, 505, 340, 505, 182, 71, 280, 12, 6, 5, 0, 5, 22; -62673, -472890, -45979, -204204, -1090575, -737143, -1102860, -402699, -159219, -636570, -27985, -17688, -18885, -2497, -6600, -41085],79,26]
%
% END OF LATTICE L80.26
%
\hbox{}\par\smallskip%
%
% LATTICE L80.27
%
\leavevmode%
\llap{\spacer}%
${L_{80.27}}=\pdualwitharg{5}({L_{80.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35931720, 5983560, 36960; 5983560, -996415, -6155; 36960, -6155, -38],[1320, 5, 1],[837, 33, 272, 10, 47, 1, -5, -3, 65, 8, 179, 31, 855, 409, 87, 44; 4664, 184, 1518, 56, 264, 6, -28, -17, 352, 44, 990, 172, 4752, 2274, 484, 245; 58520, 2289, 18645, 655, 2948, 0, -328, -165, 6160, 651, 13695, 2285, 61732, 29400, 6208, 3105],79,27]
%
% END OF LATTICE L80.27
%
\hbox{}\par\smallskip%
%
% LATTICE L80.28
%
\leavevmode%
\llap{\spacer}%
${L_{80.28}}=\pdualwitharg{5}({L_{80.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4672378920, 53705520, -497640; 53705520, -617305, 5720; -497640, 5720, -53],[1320, 5, 1],[487, 18, 134, 3, 9, -1, -1, 3, 151, 11, 185, 26, 629, 146, 59, 55; 42548, 1573, 11715, 263, 792, -87, -88, 260, 13156, 959, 16137, 2269, 54912, 12747, 5152, 4804; 19140, 749, 6105, 215, 968, 0, -108, -110, 1980, 211, 4455, 745, 20152, 4800, 2028, 2030],79,28]
%
% END OF LATTICE L80.28
%
\hbox{}\par\smallskip%
%
% LATTICE L80.29
%
\leavevmode%
\llap{\spacer}%
${L_{80.29}}=\pdualwitharg{11}(\main({L_{80.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-94380, -660, -1320; -660, 11, 0; -1320, 0, -13],[660, 11, 1],[9, 17, 2, 0, -1, -1, 1, 2, 5, 18, 5, 7, 71, 71, 69, 12; 544, 1025, 120, -1, -60, -57, 70, 125, 308, 1105, 306, 427, 4320, 4317, 4190, 727; -932, -1760, -207, 0, 100, 99, -110, -209, -520, -1870, -519, -726, -7360, -7359, -7150, -1243],79,29]
%
% END OF LATTICE L80.29
%
\hbox{}\par\smallskip%
%
% LATTICE L80.30
%
\leavevmode%
\llap{\spacer}%
${L_{80.30}}=\pdualwitharg{2.11}(\pfillwitharg{2}({L_{80.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[21991530, -188760, 10747110; -188760, 1606, -92246; 10747110, -92246, 5252039],[330, 22, 1],[-22, 1667, 877, 1753, 5101, 10550, 5404, 2011, 764, 5753, 607, -269, -1391, -3130, -1694, -527; -1, 65, 33, 63, 170, 345, 170, 58, 19, 115, 3, -27, -80, -165, -80, -22; 45, -3410, -1794, -3586, -10435, -21582, -11055, -4114, -1563, -11770, -1242, 550, 2845, 6402, 3465, 1078],79,30]
%
% END OF LATTICE L80.30
%
\hbox{}\par\smallskip%
%
% LATTICE L80.31
%
\leavevmode%
\llap{\spacer}%
${L_{80.31}}=\pdualwitharg{11}({L_{80.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[677820, -5940, -3300; -5940, 44, 33; -3300, 33, 14],[660, 11, 1],[9, 51, 11, 11, 41, 37, 29, 3, 1, -1, -1, -1, 9, 13, 21, 7; 496, 2810, 606, 606, 2260, 2040, 1600, 166, 56, -50, -54, -54, 500, 720, 1160, 386; 950, 5390, 1164, 1166, 4350, 3927, 3080, 319, 106, -110, -108, -110, 930, 1353, 2200, 737],79,31]
%
% END OF LATTICE L80.31
%
\hbox{}\par\smallskip%
%
% LATTICE L80.32
%
\leavevmode%
\llap{\spacer}%
${L_{80.32}}=\pdualwitharg{2.3}({L_{80.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36231360, 1149060, -9000420; 1149060, -36408, 285444; -9000420, 285444, -2235841],[660, 12, 1],[-27762, -12947, -9079, -2005, -67312, -3358, -1554, -152, 1230, 887, 235, -407, -65348, -4682, -4476, -4672; -143, -65, -44, -9, -275, -13, -5, 1, 11, 5, 0, -3, -385, -27, -25, -25; 111738, 52110, 36542, 8070, 270930, 13516, 6255, 612, -4950, -3570, -946, 1638, 263010, 18844, 18015, 18804],79,32]
%
% END OF LATTICE L80.32
%
\hbox{}\par\smallskip%
%
% LATTICE L80.33
%
\leavevmode%
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${L_{80.33}}=\pdualwitharg{2.3}(\main({L_{80.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11298540, -120780, 2823480; -120780, 1272, -30180; 2823480, -30180, 705581],[660, 12, 1],[-7813, -7453, -5389, -1267, -22914, -2374, -2452, -316, -41, 313, 153, -161, -16356, -2386, -4688, -2540; 4301, 4105, 2970, 699, 12650, 1311, 1355, 175, 22, -175, -88, 87, 8965, 1309, 2575, 1397; 31449, 30000, 21692, 5100, 92235, 9556, 9870, 1272, 165, -1260, -616, 648, 65835, 9604, 18870, 10224],79,33]
%
% END OF LATTICE L80.33
%
\hbox{}\par\smallskip%
%
% LATTICE L80.34
%
\leavevmode%
\llap{\spacer}%
${L_{80.34}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{80.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1714231860, 11376090, -667726785; 11376090, 75515, -4431185; -667726785, -4431185, 260092651],[165, 55, 1],[-32377, -48859, -71258, -105491, -112677, -1142411, -113946, -208031, -82251, -65769, -43370, -9137, -1951, -3869, -682, -21225; 97162, 146623, 213840, 316569, 338132, 3428253, 341939, 624275, 246824, 197363, 130146, 27417, 5854, 11607, 2047, 63697; -81465, -122936, -179295, -265430, -283511, -2874465, -286704, -523435, -206955, -165484, -109125, -22990, -4909, -9735, -1716, -53405],79,34]
%
% END OF LATTICE L80.34
%
\hbox{}\par\smallskip%
%
% LATTICE L80.35
%
\leavevmode%
\llap{\spacer}%
${L_{80.35}}=\pdualwitharg{3.5}(\main({L_{80.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8156940, -127380, 25080; -127380, -1965, 390; 25080, 390, -77],[660, 15, 1],[179, 7, 19, 2, 9, 0, -1, -1, 19, 2, 14, 7, 189, 30, 19, 19; 5720, 225, 616, 67, 308, 1, -34, -39, 484, 57, 418, 215, 5896, 939, 598, 603; 87120, 3414, 9295, 990, 4488, 5, -498, -525, 8580, 936, 6655, 3360, 91212, 14495, 9198, 9225],79,35]
%
% END OF LATTICE L80.35
%
\hbox{}\par\smallskip%
%
% LATTICE L80.36
%
\leavevmode%
\llap{\spacer}%
${L_{80.36}}=\pdualwitharg{3.5}({L_{80.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8548980, 423060, 79200; 423060, -20730, -3885; 79200, -3885, -728],[660, 15, 1],[949, 61, 115, -7, 9, 6, 13, 40, 833, 101, 479, 169, 1773, 264, 149, 122; -157278, -10112, -19074, 1156, -1474, -989, -2148, -6617, -137874, -16720, -79310, -27988, -293678, -43731, -24684, -20215; 942480, 60594, 114290, -6930, 8844, 5930, 12876, 39660, 826320, 100206, 475310, 167730, 1759956, 262070, 147924, 121140],79,36]
%
% END OF LATTICE L80.36
%
\hbox{}\par\smallskip%
%
% LATTICE L80.37
%
\leavevmode%
\llap{\spacer}%
${L_{80.37}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{80.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3341101170, 6136680, 1639827420; 6136680, 11130, 3011910; 1639827420, 3011910, 804834643],[330, 30, 1],[46727, 6031, -79471, -99859, -1090950, -375632, -135805, -331013, -1869647, -337471, -1135809, -231581, -731970, -176768, -29915, -427; 22, 3, -33, -43, -473, -163, -59, -144, -814, -147, -495, -101, -319, -77, -13, 0; -95205, -12288, 161920, 203460, 2222781, 765340, 276699, 674430, 3809355, 687588, 2314180, 471840, 1491369, 360160, 60951, 870],79,37]
%
% END OF LATTICE L80.37
%
\hbox{}\par\smallskip%
%
% LATTICE L80.38
%
\leavevmode%
\llap{\spacer}%
${L_{80.38}}=\pdualwitharg{2}({L_{80.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[361680, -3960, 1320; -3960, 8, 24; 1320, 24, -37],[1320, 8, 1],[1, -1, -13, -3, -113, -22, -12, -5, -12, -19, -31, -1, 3, 2, 2, 1; 132, -130, -1716, -398, -15070, -2937, -1605, -671, -1617, -2570, -4224, -142, 220, 237, 255, 131; 121, -120, -1584, -368, -13970, -2724, -1490, -624, -1507, -2400, -3960, -136, 110, 204, 230, 120],79,38]
%
% END OF LATTICE L80.38
%
\hbox{}\par\smallskip%
%
% LATTICE L80.39
%
\leavevmode%
\llap{\spacer}%
${L_{80.39}}=\pdualwitharg{2}({L_{80.3}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-905520, 9240, 2640; 9240, 88, 8; 2640, 8, -1],[1320, 8, 1],[1, 1, -1, -1, -31, -13, -8, -4, -23, -21, -43, -3, -24, -7, -2, 0; -198, -200, 198, 200, 6215, 2607, 1605, 803, 4620, 4220, 8646, 604, 4840, 1413, 405, 1; 1034, 1040, -1056, -1048, -32505, -13632, -8390, -4196, -24134, -22040, -45144, -3152, -25245, -7368, -2110, -4],79,39]
%
% END OF LATTICE L80.39
%
\hbox{}\par\smallskip%
%
% LATTICE L80.40
%
\leavevmode%
\llap{\spacer}%
${L_{80.40}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{80.2}}))$%
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{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1959870, 68970, -592680; 68970, 2409, -20856; -592680, -20856, 179231],[330, 33, 1],[-9, 347, 61, 367, 2147, 741, 2283, 427, 327, 1243, 45, -49, -557, -211, -693, -109; -2, 65, 11, 63, 340, 115, 340, 58, 38, 115, 1, -27, -160, -55, -160, -22; -30, 1155, 203, 1221, 7140, 2464, 7590, 1419, 1086, 4125, 149, -165, -1860, -704, -2310, -363],79,40]
%
% END OF LATTICE L80.40
%
\hbox{}\par\smallskip%
%
% LATTICE L80.41
%
\leavevmode%
\llap{\spacer}%
${L_{80.41}}=\pdualwitharg{2.5}(\main({L_{80.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[202140840, -984060, 50158680; -984060, 4820, -244180; 50158680, -244180, 12446239],[660, 20, 1],[43880, 6471, 48212, 1133, 1741, 1297, 81, 1779, 15729, 4367, 71006, 9705, 57868, 107083, 10757, 19897; 11858, 1749, 13035, 307, 473, 354, 22, 480, 4246, 1179, 19173, 2621, 15631, 28926, 2906, 5376; -176605, -26044, -194040, -4560, -7007, -5220, -326, -7160, -63305, -17576, -285780, -39060, -232903, -430980, -43294, -80080],79,41]
%
% END OF LATTICE L80.41
%
\hbox{}\par\smallskip%
%
% LATTICE L80.42
%
\leavevmode%
\llap{\spacer}%
${L_{80.42}}=\pdualwitharg{2.5}({L_{80.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26788539360, 11881831500, 6683550060; 11881831500, 5270086520, 2964432460; 6683550060, 2964432460, 1667498209],[660, 20, 1],[1255798, 91373, 663281, 12655, 25392, 328, 123, 27442, 481410, 66555, 1073927, 145273, 1711816, 1578952, 78841, 288414; 297, 22, 165, 4, 11, 3, 0, 5, 99, 14, 231, 32, 385, 357, 18, 67; -5033930, -366274, -2658810, -50730, -101794, -1320, -493, -110000, -1929730, -266786, -4304850, -582330, -6861866, -6329280, -316037, -1156120],79,42]
%
% END OF LATTICE L80.42
%
\hbox{}\par\smallskip%
%
% LATTICE L80.43
%
\leavevmode%
\llap{\spacer}%
${L_{80.43}}=\pdualwitharg{11}({L_{80.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-267960, 0, -1320; 0, 22, 11; -1320, 11, -1],[1320, 11, 1],[5, 18, 5, 7, 71, 71, 69, 6, 9, 17, 2, 0, -1, -1, 1, 1; 528, 1895, 525, 733, 7420, 7416, 7200, 625, 936, 1765, 207, -1, -100, -96, 120, 107; -1040, -3740, -1038, -1452, -14720, -14718, -14300, -1243, -1864, -3520, -414, 0, 200, 198, -220, -209],79,43]
%
% END OF LATTICE L80.43
%
\hbox{}\par\smallskip%
%
% LATTICE L80.44
%
\leavevmode%
\llap{\spacer}%
${L_{80.44}}=\pdualwitharg{11}({L_{80.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[649884840, -16663680, -137280; -16663680, 427273, 3520; -137280, 3520, 29],[1320, 11, 1],[39, 142, 40, 57, 589, 296, 581, 103, 79, 153, 19, 2, 1, -1, 9, 15; 1548, 5635, 1587, 2261, 23360, 11739, 23040, 4084, 3132, 6065, 753, 79, 40, -39, 360, 596; -3292, -11825, -3291, -4631, -47400, -23760, -46420, -8162, -6212, -11935, -1461, -121, -120, 0, -1100, -1342],79,44]
%
% END OF LATTICE L80.44
%
\hbox{}\par\smallskip%
%
% LATTICE L80.45
%
\leavevmode%
\llap{\spacer}%
${L_{80.45}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{80.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[84365490, 584430, 17181120; 584430, 4015, 119020; 17181120, 119020, 3498953],[330, 55, 1],[55, -419, -1103, -2207, -2573, -13309, -2729, -2543, -1937, -1463, -779, 325, 691, 3899, 847, 661; -2, 13, 33, 63, 68, 345, 68, 58, 38, 23, 3, -27, -32, -165, -32, -22; -270, 2057, 5415, 10835, 12632, 65340, 13398, 12485, 9510, 7183, 3825, -1595, -3392, -19140, -4158, -3245],79,45]
%
% END OF LATTICE L80.45
%
\hbox{}\par\smallskip%
%
% LATTICE L80.46
%
\leavevmode%
\llap{\spacer}%
${L_{80.46}}=\pdualwitharg{3.5}({L_{80.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16313880, -8660520, 50160; -8660520, -4597545, 26625; 50160, 26625, -154],[1320, 15, 1],[-465, -55, -404, -208, -5707, -909, -579, -292, -5541, -218, -597, -65, -299, -1, 33, 19; 968, 114, 836, 430, 11792, 1878, 1196, 603, 11440, 450, 1232, 134, 616, 2, -68, -39; 15840, 1791, 12925, 6585, 179652, 28580, 18168, 9135, 172920, 6789, 18535, 1995, 9108, 20, -1008, -555],79,46]
%
% END OF LATTICE L80.46
%
\hbox{}\par\smallskip%
%
% LATTICE L80.47
%
\leavevmode%
\llap{\spacer}%
${L_{80.47}}=\pdualwitharg{3.5}({L_{80.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1851960, 85800, 17160; 85800, -3795, -765; 17160, -765, -154],[1320, 15, 1],[371, 22, 102, 35, 717, 53, 59, 47, 355, 11, 19, -2, 9, 2, 7, 19; -40744, -2418, -11220, -3854, -79024, -5843, -6508, -5190, -39248, -1218, -2112, 218, -968, -217, -764, -2082; 243540, 14451, 67045, 23025, 472032, 34900, 38868, 30990, 234300, 7269, 12595, -1305, 5808, 1300, 4572, 12450],79,47]
%
% END OF LATTICE L80.47
%
\hbox{}\par\smallskip%
%
% LATTICE L80.48
%
\leavevmode%
\llap{\spacer}%
${L_{80.48}}=\pdualwitharg{3.11}({L_{80.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4594260, -34320, -12540; -34320, 132, 99; -12540, 99, 34],[660, 33, 1],[5, -1, -1, -1, 69, 30, 131, 40, 49, 271, 19, 55, 201, 60, 139, 14; 72, -10, -14, -14, 980, 426, 1860, 568, 696, 3850, 270, 782, 2860, 854, 1980, 200; 1638, -330, -328, -330, 22590, 9823, 42900, 13101, 16050, 88770, 6224, 18018, 65850, 19657, 45540, 4587],79,48]
%
% END OF LATTICE L80.48
%
\hbox{}\par\smallskip%
%
% LATTICE L80.49
%
\leavevmode%
\llap{\spacer}%
${L_{80.49}}=\pdualwitharg{2.3.11}(\pfillwitharg{2}({L_{80.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28331490, -834900, 13950750; -834900, 24486, -411114; 13950750, -411114, 6869509],[330, 66, 1],[-353, -2437, 643, 8741, 43284, 32776, 59221, 28855, 14813, 147037, 8997, 20179, 29016, 15424, 13079, 65; 2, 15, -3, -43, -215, -163, -295, -144, -74, -735, -45, -101, -145, -77, -65, 0; 717, 4950, -1306, -17754, -87915, -66572, -120285, -58608, -30087, -298650, -18274, -40986, -58935, -31328, -26565, -132],79,49]
%
% END OF LATTICE L80.49
%
\hbox{}\par\smallskip%
%
% LATTICE L80.50
%
\leavevmode%
\llap{\spacer}%
${L_{80.50}}=\pdualwitharg{3.11}(\main({L_{80.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-660, 0, 0; 0, 33, 33; 0, 33, 34],[660, 33, 1],[1, 1, 0, 0, 7, 3, 13, 4, 5, 14, 1, 3, 23, 7, 17, 2; 16, 5, -1, -1, 180, 78, 340, 104, 128, 355, 25, 73, 540, 162, 380, 40; -12, 0, 1, 0, -180, -77, -330, -99, -120, -330, -23, -66, -480, -143, -330, -33],79,50]
%
% END OF LATTICE L80.50
%
\hbox{}\par\smallskip%
%
% LATTICE L80.51
%
\leavevmode%
\llap{\spacer}%
${L_{80.51}}=\pdualwitharg{3.5.11}(\pfillwitharg{2}({L_{80.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-49489624635, 7979956545, 2911482585; 7979956545, -1286728410, -469462125; 2911482585, -469462125, -171282988],[165, 165, 1],[155, 249, 127, 589, 647, 2202, 667, 1242, 499, 405, 91, 65, 7, -22, -13, 66; 78614, 126592, 64691, 300560, 330594, 1125513, 341125, 635831, 255700, 207722, 46747, 33754, 3720, -11133, -6811, 32797; -212835, -342738, -175150, -813780, -895113, -3047440, -923637, -1721610, -692355, -562452, -126580, -91410, -10077, 30140, 18447, -88770],79,51]
%
% END OF LATTICE L80.51
%
\hbox{}\par\smallskip%
%
% LATTICE L80.52
%
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\llap{\spacer}%
${L_{80.52}}=\pdualwitharg{2.11}(\main({L_{80.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[23883420, 905520, 5936040; 905520, 34364, 225060; 5936040, 225060, 1475357],[660, 44, 1],[-1412, -10936, -1363, -164, -128, -361, -629, -1225, -737, -10554, -2935, -4134, -10617, -42619, -20861, -7371; 1, 5, 0, -1, 0, 3, 5, 5, 2, 25, 6, 7, 15, 57, 25, 7; 5681, 44000, 5484, 660, 515, 1452, 2530, 4928, 2965, 42460, 11808, 16632, 42715, 171468, 83930, 29656],79,52]
%
% END OF LATTICE L80.52
%
\hbox{}\par\smallskip%
%
% LATTICE L80.53
%
\leavevmode%
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${L_{80.53}}=\pdualwitharg{2.11}({L_{80.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[75907920, -2663100, 18857520; -2663100, 93368, -661584; 18857520, -661584, 4684703],[660, 44, 1],[-2362, -11179, -1883, -1077, -1484, -854, 642, 1268, 558, 2159, 79, -727, -7536, -17186, -5152, -4876; -13, -65, -12, -9, -25, -39, -5, 1, 1, 5, 0, -3, -35, -81, -25, -25; 9506, 44990, 7578, 4334, 5970, 3432, -2585, -5104, -2246, -8690, -318, 2926, 30330, 69168, 20735, 19624],79,53]
%
% END OF LATTICE L80.53
%
\hbox{}\par\smallskip%
%
% LATTICE L80.54
%
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\llap{\spacer}%
${L_{80.54}}=\pdualwitharg{2.3}({L_{80.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-868560, 43560, -5280; 43560, -2184, 264; -5280, 264, -31],[1320, 24, 1],[2, -1, -5, -3, -83, -5, -7, -2, -2, 1, 5, 3, 83, 5, 7, 2; 44, -20, -110, -68, -1980, -121, -175, -55, -77, -40, 66, 56, 1650, 101, 145, 43; 33, 0, -88, -72, -2970, -196, -330, -144, -363, -600, -352, -48, -330, -4, 30, 24],79,54]
%
% END OF LATTICE L80.54
%
\hbox{}\par\smallskip%
%
% LATTICE L80.55
%
\leavevmode%
\llap{\spacer}%
${L_{80.55}}=\pdualwitharg{2.3}({L_{80.3}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240240, 17160, -5280; 17160, 744, -240; -5280, -240, 77],[1320, 24, 1],[3, 1, -1, -1, -16, -2, -3, -1, -3, -1, 1, 1, 16, 2, 3, 1; 506, 170, -176, -178, -2970, -375, -575, -203, -704, -350, 44, 142, 2475, 315, 485, 167; 1782, 600, -616, -624, -10395, -1312, -2010, -708, -2442, -1200, 176, 504, 8745, 1112, 1710, 588],79,55]
%
% END OF LATTICE L80.55
%
\hbox{}\par\smallskip%
%
% LATTICE L80.56
%
\leavevmode%
\llap{\spacer}%
${L_{80.56}}=\pdualwitharg{5.11}(\main({L_{80.3}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[613140, 199320, -6600; 199320, 64790, -2145; -6600, -2145, 71],[660, 55, 1],[-11, -5, -4, -2, -1, -1, 1, 2, 1, 0, -1, -3, -9, -49, -11, -12; 12, 8, 9, 8, 4, 3, -6, -17, -24, -14, -15, -14, -16, -63, -6, 5; -680, -231, -105, 55, 28, 0, -88, -330, -640, -429, -555, -715, -1348, -6600, -1232, -990],79,56]
%
% END OF LATTICE L80.56
%
\hbox{}\par\smallskip%
%
% LATTICE L80.57
%
\leavevmode%
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${L_{80.57}}=\pdualwitharg{5.11}({L_{80.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[759660, 249480, 5280; 249480, 81620, 1705; 5280, 1705, 34],[660, 55, 1],[-561, -441, -281, -41, -1, 35, -1, -81, -241, -361, -521, -761, -801, -4045, -801, -721; 2216, 1742, 1110, 162, 4, -138, 4, 320, 952, 1426, 2058, 3006, 3164, 15978, 3164, 2848; -24010, -18876, -12030, -1760, -46, 1485, -44, -3465, -10310, -15444, -22290, -32560, -34274, -173085, -34276, -30855],79,57]
%
% END OF LATTICE L80.57
%
\hbox{}\par\smallskip%
%
% LATTICE L80.58
%
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\llap{\spacer}%
${L_{80.58}}=\pdualwitharg{2.5.11}(\pfillwitharg{2}({L_{80.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16822458510, -26259420, 8375008290; -26259420, 40810, -13073170; 8375008290, -13073170, 4169471651],[330, 110, 1],[-10751, -14797, 59189, 267737, 265078, 3010768, 362647, 883439, 453511, 900317, 826331, 617783, 177682, 1416832, 80113, 2081; 2, 3, -9, -43, -43, -489, -59, -144, -74, -147, -135, -101, -29, -231, -13, 0; 21595, 29722, -118890, -537790, -532449, -6047580, -728431, -1774520, -910945, -1808422, -1659810, -1240910, -356901, -2845920, -160919, -4180],79,58]
%
% END OF LATTICE L80.58
%
\hbox{}\par\smallskip%
%
% LATTICE L80.59
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L80.59
%
\hbox{}\par\smallskip%
%
% LATTICE L80.60
%
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${L_{80.60}}=\pdualwitharg{2.3.5}({L_{80.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19197859560, 292381980, 4770259560; 292381980, 4452960, 72650700; 4770259560, 72650700, 1185307987],[660, 60, 1],[789395, 50946, 97093, -5284, 10313, 10563, 5404, 65823, 685441, 83130, 394519, 139360, 1464523, 436357, 61634, 202329; 10736, 691, 1309, -75, 154, 152, 76, 914, 9460, 1145, 5423, 1911, 20042, 5968, 842, 2758; -3177570, -205074, -390830, 21270, -41514, -42520, -21753, -264960, -2759130, -334626, -1588070, -560970, -5895186, -1756480, -248097, -814440],79,60]
%
% END OF LATTICE L80.60
%
\hbox{}\par\smallskip%
%
% LATTICE L80.61
%
\leavevmode%
\llap{\spacer}%
${L_{80.61}}=\pdualwitharg{3.11}({L_{80.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1320, 0, -1320; 0, 66, 297; -1320, 297, 17],[1320, 33, 1],[211, 584, 41, 119, 873, 261, 607, 31, 23, 1, -2, -2, 297, 129, 563, 86; 952, 2635, 185, 537, 3940, 1178, 2740, 140, 104, 5, -9, -9, 1340, 582, 2540, 388; -240, -660, -46, -132, -960, -286, -660, -33, -24, 0, 2, 0, -360, -154, -660, -99],79,61]
%
% END OF LATTICE L80.61
%
\hbox{}\par\smallskip%
%
% LATTICE L80.62
%
\leavevmode%
\llap{\spacer}%
${L_{80.62}}=\pdualwitharg{3.11}({L_{80.3}})$%
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{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25080, -1320, 1320; -1320, 66, -99; 1320, -99, -181],[1320, 33, 1],[115, 316, 22, 63, 457, 68, 313, 31, 11, -1, -1, 0, 173, 37, 317, 95; 1948, 5355, 373, 1069, 7760, 1155, 5320, 528, 188, -15, -17, -1, 2920, 625, 5360, 1608; -240, -660, -46, -132, -960, -143, -660, -66, -24, 0, 2, 0, -360, -77, -660, -198],79,62]
%
% END OF LATTICE L80.62
%
\hbox{}\par\smallskip%
%
% LATTICE L80.63
%
\leavevmode%
\llap{\spacer}%
${L_{80.63}}=\pdualwitharg{2.5}({L_{80.2}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-135960, -151800, 7920; -151800, -148720, 7800; 7920, 7800, -409],[1320, 40, 1],[35, 22, 392, 58, 364, 341, 35, 67, 152, 46, 356, 10, 10, -1, -1, 1; 781, 493, 8811, 1307, 8217, 7701, 791, 1516, 3443, 1043, 8085, 229, 231, -21, -23, 20; 15565, 9824, 175560, 26040, 163702, 153420, 15758, 30200, 68585, 20776, 161040, 4560, 4598, -420, -458, 400],79,63]
%
% END OF LATTICE L80.63
%
\hbox{}\par\smallskip%
%
% LATTICE L80.64
%
\leavevmode%
\llap{\spacer}%
${L_{80.64}}=\pdualwitharg{2.5}({L_{80.3}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-53714760, -1379400, 76560; -1379400, -35120, 1960; 76560, 1960, -109],[1320, 40, 1],[70, 22, 392, 58, 182, 341, 35, 67, 304, 46, 356, 10, 5, -1, -1, 1; 1507, 475, 8481, 1257, 3949, 7401, 760, 1456, 6611, 1001, 7755, 219, 110, -21, -22, 20; 76230, 23984, 427680, 63320, 198781, 372480, 38238, 73220, 332310, 50296, 389400, 10960, 5489, -1080, -1098, 1060],79,64]
%
% END OF LATTICE L80.64
%
\hbox{}\par\smallskip%
%
% LATTICE L80.65
%
\leavevmode%
\llap{\spacer}%
${L_{80.65}}=\pdualwitharg{3.5.11}(\pfillwitharg{2}({L_{80.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14816745570, 30182460, 4946548860; 30182460, 61215, 10076550; 4946548860, 10076550, 1651397989],[330, 165, 1],[-16573, -11405, 15207, 206363, 408627, 773534, 559033, 680926, 699103, 693935, 212303, 476167, 273903, 364016, 123497, 1604; 33150, 22813, -30417, -412769, -817340, -1547231, -1118184, -1361996, -1398354, -1388017, -424651, -952435, -547864, -728109, -247020, -3208; 49440, 34023, -45365, -615615, -1219002, -2307580, -1667688, -2031315, -2085540, -2070123, -633335, -1420485, -817098, -1085920, -368412, -4785],79,65]
%
% END OF LATTICE L80.65
%
\hbox{}\par\smallskip%
%
% LATTICE L80.66
%
\leavevmode%
\llap{\spacer}%
${L_{80.66}}=\pdualwitharg{5.11}({L_{80.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1226280, -199320, 13200; -199320, 32395, -2145; 13200, -2145, 142],[1320, 55, 1],[-1, 0, 1, 3, 9, 49, 11, 6, 11, 5, 4, 2, 1, 1, -1, -1; -48, -28, -30, -28, -32, -126, -12, 5, 24, 16, 18, 16, 8, 6, -12, -17; -640, -429, -555, -715, -1348, -6600, -1232, -495, -680, -231, -105, 55, 28, 0, -88, -165],79,66]
%
% END OF LATTICE L80.66
%
\hbox{}\par\smallskip%
%
% LATTICE L80.67
%
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${L_{80.67}}=\pdualwitharg{5.11}({L_{80.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-115213560, -12801360, 51480; -12801360, -1422355, 5720; 51480, 5720, -23],[1320, 55, 1],[-9, -5, -5, -4, -3, -4, 1, 5, 7, 4, 4, 3, 1, -1, -3, -7; 60, 31, 27, 13, -16, -69, -48, -76, -84, -43, -39, -25, -8, 9, 24, 52; -5260, -3509, -4515, -5775, -10808, -26400, -9812, -7810, -5300, -1771, -765, 495, 248, 0, -748, -2750],79,67]
%
% END OF LATTICE L80.67
%
\hbox{}\par\smallskip%
%
% LATTICE L80.68
%
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${L_{80.68}}=\pdualwitharg{2.3.11}({L_{80.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2787829440, -24160620, 692681220; -24160620, 209352, -6003096; 692681220, -6003096, 172107829],[660, 132, 1],[4942, 13857, 81, -3001, -11308, -6680, -2911, 2394, 4274, 32223, 2991, 12217, 57388, 37400, 25951, 16038; 1, 5, 0, -3, -35, -27, -25, -25, -13, -65, -4, -9, -25, -13, -5, 1; -19890, -55770, -326, 12078, 45510, 26884, 11715, -9636, -17202, -129690, -12038, -49170, -230970, -150524, -104445, -64548],79,68]
%
% END OF LATTICE L80.68
%
\hbox{}\par\smallskip%
%
% LATTICE L80.69
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[106260, 21780, 26400; 21780, 4488, 5412; 26400, 5412, 6559],[660, 132, 1],[11, -1, -1, 33, 332, 536, 1066, 590, 169, 1801, 121, 327, 568, 664, 734, 130; 1, 5, 0, -1, 0, 1, 5, 5, 2, 25, 2, 7, 15, 19, 25, 7; -45, 0, 4, -132, -1335, -2156, -4290, -2376, -681, -7260, -488, -1320, -2295, -2684, -2970, -528],79,69]
%
% END OF LATTICE L80.69
%
\hbox{}\par\smallskip%
%
% LATTICE L80.70
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7260, -1980, 0; -1980, -96690, 825; 0, 825, -7],[660, 165, 1],[-21, -14, -6, -23, -43, -70, -39, -31, -21, -7, -1, 2, 1, 0, -3, -11; -76, -51, -22, -85, -160, -261, -146, -117, -80, -27, -4, 7, 4, 1, -10, -39; -9000, -6039, -2605, -10065, -18948, -30910, -17292, -13860, -9480, -3201, -475, 825, 468, 110, -1188, -4620],79,70]
%
% END OF LATTICE L80.70
%
\hbox{}\par\smallskip%
%
% LATTICE L80.71
%
\leavevmode%
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${L_{80.71}}=\pdualwitharg{3.5.11}({L_{80.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-28380, -97020, 33000; -97020, -331320, 112695; 33000, 112695, -38332],[660, 165, 1],[53, 69, 29, 109, 101, 164, 91, 72, 49, 33, 5, -7, 1, 6, 11, 30; -678, -962, -440, -1826, -1838, -3057, -1776, -1535, -1146, -862, -168, 2, 14, 17, -48, -289; -1950, -2772, -1270, -5280, -5322, -8855, -5148, -4455, -3330, -2508, -490, 0, 42, 55, -132, -825],79,71]
%
% END OF LATTICE L80.71
%
\hbox{}\par\smallskip%
%
% LATTICE L80.72
%
\leavevmode%
\llap{\spacer}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1786544640870, -4450804708020, -2217328207380; -4450804708020, 11088254999910, 5524012441530; -2217328207380, 5524012441530, 2751985182017],[330, 330, 1],[-1, 13, 11, 63, 34, 115, 34, 58, 19, 23, 1, -27, -16, -55, -16, -22; 59894, -912225, -800401, -4804111, -2799979, -9655026, -2969420, -5533233, -2106848, -3181683, -564235, 710203, 753025, 2831846, 922466, 1439325; -120225, 1831104, 1606640, 9643260, 5620377, 19380460, 5960493, 11106810, 4229055, 6386556, 1132580, -1425600, -1511547, -5684360, -1851663, -2889150],79,72]
%
% END OF LATTICE L80.72
%
\hbox{}\par\smallskip%
%
% LATTICE L80.73
%
\leavevmode%
\llap{\spacer}%
${L_{80.73}}=\pdualwitharg{2.11}({L_{80.2}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12514920, -2502720, 844800; -2502720, -500456, 168872; 844800, 168872, -56887],[1320, 88, 1],[-50, -1, 151, 205, -473, -1513, -1337, -956, -320, -7399, -1631, -1685, -3227, -5887, -2363, -524; 301, 5, -909, -1233, 2855, 9126, 8060, 5761, 1928, 44575, 9825, 10149, 19435, 35454, 14230, 3155; 151, 0, -456, -616, 1450, 4620, 4070, 2904, 971, 22440, 4944, 5104, 9770, 17820, 7150, 1584],79,73]
%
% END OF LATTICE L80.73
%
\hbox{}\par\smallskip%
%
% LATTICE L80.74
%
\leavevmode%
\llap{\spacer}%
${L_{80.74}}=\pdualwitharg{2.11}({L_{80.3}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2077680, 658680, 319440; 658680, 207592, 100672; 319440, 100672, 48821],[1320, 88, 1],[1, 1, -1, -1, 4, 22, 17, 11, 7, 79, 17, 17, 16, 58, 23, 5; -460, -430, 492, 558, -1625, -9159, -7255, -4793, -3086, -35030, -7584, -7650, -7240, -26301, -10475, -2299; 942, 880, -1008, -1144, 3325, 18744, 14850, 9812, 6318, 71720, 15528, 15664, 14825, 53856, 21450, 4708],79,74]
%
% END OF LATTICE L80.74
%
\hbox{}\par\smallskip%
%
% LATTICE L80.75
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13320780, 374880, -3032040; 374880, 9020, -86460; -3032040, -86460, 689309],[660, 220, 1],[8050, 12723, 8168, 1257, -39, -3607, -235, 3771, 3181, 9739, 14294, 21205, 11270, 228227, 22697, 41153; -23952, -37856, -24303, -3740, 116, 10731, 699, -11221, -9465, -28978, -42531, -63094, -33533, -679071, -67533, -122447; 32405, 51216, 32880, 5060, -157, -14520, -946, 15180, 12805, 39204, 57540, 85360, 45367, 918720, 91366, 165660],79,75]
%
% END OF LATTICE L80.75
%
\hbox{}\par\smallskip%
%
% LATTICE L80.76
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[469059360, -17903820, -129818040; -17903820, 683320, 4955060; -129818040, 4955060, 35928731],[660, 220, 1],[-11796, -10163, -7537, -2925, -586, 164, 380, 1586, -1160, -2793, -5419, -10031, -12370, -129724, -6858, -27498; 35157, 30287, 22458, 8711, 1745, -489, -1131, -4709, 3471, 8341, 16170, 29917, 36883, 386769, 20445, 81965; -47470, -40898, -30330, -11770, -2358, 660, 1529, 6380, -4670, -11242, -21810, -40370, -49782, -522060, -27599, -110660],79,76]
%
% END OF LATTICE L80.76
%
\hbox{}\par\smallskip%
%
% LATTICE L80.77
%
\leavevmode%
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${L_{80.77}}=\pdualwitharg{2.3.5}({L_{80.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1320, 26400, -1320; 26400, -523560, 26160; -1320, 26160, -1307],[1320, 120, 1],[464, 221, 1029, 355, 1827, 541, 151, 242, 460, 115, 203, -19, 21, 19, 17, 94; 132, 61, 275, 91, 451, 132, 36, 55, 99, 23, 33, -7, 11, 8, 6, 29; 2145, 984, 4400, 1440, 7062, 2060, 558, 840, 1485, 336, 440, -120, 198, 140, 102, 480],79,77]
%
% END OF LATTICE L80.77
%
\hbox{}\par\smallskip%
%
% LATTICE L80.78
%
\leavevmode%
\llap{\spacer}%
${L_{80.78}}=\pdualwitharg{2.3.5}({L_{80.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15537720, -2851200, 71280; -2851200, -462360, 11520; 71280, 11520, -287],[1320, 120, 1],[433, 103, 479, 165, 424, 251, 70, 112, 425, 53, 93, -9, 5, 9, 8, 44; -94501, -22481, -104555, -36019, -92565, -54798, -15283, -24455, -92807, -11575, -20317, 1963, -1089, -1962, -1745, -9601; -3685770, -876816, -4077920, -1404840, -3610299, -2137280, -596082, -953820, -3619770, -451464, -792440, 76560, -42471, -76520, -68058, -374460],79,78]
%
% END OF LATTICE L80.78
%
\hbox{}\par\smallskip%
%
% LATTICE L80.79
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14520, 5280, 0; 5280, -46695, 825; 0, 825, -14],[1320, 165, 1],[59, 20, 3, -5, -3, -1, 7, 14, 55, 37, 16, 62, 117, 191, 107, 43; -160, -54, -8, 14, 8, 2, -20, -39, -152, -102, -44, -170, -320, -522, -292, -117; -9480, -3201, -475, 825, 468, 110, -1188, -2310, -9000, -6039, -2605, -10065, -18948, -30910, -17292, -6930],79,79]
%
% END OF LATTICE L80.79
%
\hbox{}\par\smallskip%
%
% LATTICE L80.80
%
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\llap{\spacer}%
${L_{80.80}}=\pdualwitharg{3.5.11}({L_{80.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4267560, -943800, 17160; -943800, -48345, 825; 17160, 825, -14],[1320, 165, 1],[-5, -3, -1, -2, 1, 2, 5, 9, 11, 6, 2, 5, 5, 3, 1, -3; 1232, 742, 248, 498, -248, -497, -1244, -2242, -2744, -1498, -500, -1254, -1264, -763, -268, 730; 66420, 40029, 13385, 26895, -13392, -26840, -67188, -121110, -148260, -80949, -27025, -67815, -68448, -41360, -14652, 39270],79,80]
%
% END OF LATTICE L80.80
%
\hbox{}\par\smallskip%
%
% LATTICE L80.81
%
\leavevmode%
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${L_{80.81}}=\pdualwitharg{2.3.11}({L_{80.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2640, -1320, 0; -1320, -778008, 4752; 0, 4752, -29],[1320, 264, 1],[1, -1, -1, -3, 17, 16, 38, 25, 8, 181, 13, 39, 73, 44, 52, 11; 2, 0, -2, -8, 20, 21, 55, 39, 13, 300, 22, 68, 130, 79, 95, 21; 327, 0, -328, -1320, 3210, 3388, 8910, 6336, 2115, 48840, 3584, 11088, 21210, 12892, 15510, 3432],79,81]
%
% END OF LATTICE L80.81
%
\hbox{}\par\smallskip%
%
% LATTICE L80.82
%
\leavevmode%
\llap{\spacer}%
${L_{80.82}}=\pdualwitharg{2.3.11}({L_{80.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2640, -43560, -21120; -43560, 587928, 285120; -21120, 285120, 138271],[1320, 264, 1],[3, 1, -1, -1, 19, 33, 72, 44, 27, 299, 21, 61, 56, 67, 78, 16; -154, 0, 62, 128, -835, -1491, -3355, -2109, -1316, -14700, -1042, -3068, -2840, -3409, -3995, -831; 318, 0, -128, -264, 1725, 3080, 6930, 4356, 2718, 30360, 2152, 6336, 5865, 7040, 8250, 1716],79,82]
%
% END OF LATTICE L80.82
%
\hbox{}\par\smallskip%
%
% LATTICE L80.83
%
\leavevmode%
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${L_{80.83}}=\pdualwitharg{2.3.5.11}(\main({L_{80.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[50820, -5934060, -1474440; -5934060, 692882520, 172161000; -1474440, 172161000, 42776963],[660, 660, 1],[-53, -141, -61, -239, -116, -764, -218, -366, -67, -99, -19, -1, -4, -36, -22, -114; 48, -34, -85, -658, -446, -3177, -1035, -2135, -459, -788, -189, -164, 35, 437, 213, 491; -195, 132, 340, 2640, 1791, 12760, 4158, 8580, 1845, 3168, 760, 660, -141, -1760, -858, -1980],79,83]
%
% END OF LATTICE L80.83
%
\hbox{}\par\smallskip%
%
% LATTICE L80.84
%
\leavevmode%
\llap{\spacer}%
${L_{80.84}}=\pdualwitharg{2.3.5.11}({L_{80.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1137840, -3233340, -803220; -3233340, 9224160, 2291520; -803220, 2291520, 569273],[660, 660, 1],[-795, -1076, -471, -1862, -1805, -5941, -846, -2821, -1017, -736, -133, 50, -7, -99, -60, -803; -35329, -47811, -20926, -82715, -80173, -263873, -37573, -125271, -45155, -32673, -5902, 2231, -311, -4407, -2669, -35697; 141090, 190938, 83570, 330330, 320178, 1053800, 150051, 500280, 180330, 130482, 23570, -8910, 1242, 17600, 10659, 142560],79,84]
%
% END OF LATTICE L80.84
%
\hbox{}\par\smallskip%
%
% LATTICE L80.85
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1320, 0, 0; 0, -1531640, 11000; 0, 11000, -79],[1320, 440, 1],[-7, -11, -7, -1, 0, 1, 0, -2, -3, -9, -13, -19, -5, -101, -10, -18; 27, 49, 39, 19, 1, 0, -3, -11, -3, -1, 9, 29, 11, 240, 27, 59; 3730, 6776, 5400, 2640, 139, 0, -418, -1540, -430, -176, 1200, 3960, 1511, 33000, 3718, 8140],79,85]
%
% END OF LATTICE L80.85
%
\hbox{}\par\smallskip%
%
% LATTICE L80.86
%
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\llap{\spacer}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[77973720, 19060800, -118800; 19060800, 4659160, -29040; -118800, -29040, 181],[1320, 440, 1],[9, 31, 23, 9, 1, 1, -1, -2, 1, 9, 17, 31, 19, 199, 21, 42; 26, 91, 69, 29, 3, 0, -4, -11, 1, 17, 39, 79, 51, 540, 58, 119; 10075, 34936, 26160, 10560, 1138, 660, -1298, -3080, 815, 8624, 17400, 33000, 20642, 217140, 23078, 46640],79,86]
%
% END OF LATTICE L80.86
%
\hbox{}\par\smallskip%
%
% LATTICE L80.87
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4594920, 14520000, 33000; 14520000, 39688440, 89760; 33000, 89760, 203],[1320, 1320, 1],[73, 189, 79, 295, 68, 441, 122, 192, 65, 87, 13, -19, 1, 19, 16, 84; -10151, -26281, -10985, -41019, -9455, -61318, -16963, -26695, -9037, -12095, -1807, 2643, -139, -2642, -2225, -11681; 4476570, 11589864, 4844360, 18089280, 4169631, 27041080, 7480638, 11772420, 3985290, 5333856, 796880, -1165560, 61299, 1165120, 981222, 5151300],79,87]
%
% END OF LATTICE L80.87
%
\hbox{}\par\smallskip%
%
% LATTICE L80.88
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[33407880, 30228000, -27720; 30228000, 27346440, -25080; -27720, -25080, 23],[1320, 1320, 1],[14, 71, 29, 105, 47, 151, 41, 62, 10, 25, 3, -9, 1, 9, 7, 34; 12, 61, 25, 91, 41, 132, 36, 55, 9, 23, 3, -7, 1, 8, 6, 29; 29955, 152064, 62200, 225720, 101322, 325820, 88638, 134640, 21855, 55176, 6880, -18480, 2298, 19580, 14982, 72600],79,88]
%
% END OF LATTICE L80.88
%

\medskip%
%------------------------------WEYL-GROUP-W81--------------------------------
%
\leavevmode\llap{}%
$W_{81\phantom{0}}$%
\qquad\llap{48} lattices, $\chi=48$%
\hfill%
$222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L81.1
%
\leavevmode%
\llap{\spacer}%
${L_{81.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{23\rightarrow N_{81},3,2}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-7446780, 17820, 17820; 17820, -42, -45; 17820, -45, -34],[1980, 3, 1],[31, 967, 8, 23, 2, -1, -1, 23, 1, 13, 13, 67; 10178, 317460, 2626, 7548, 655, -330, -328, 7590, 329, 4272, 4270, 22000; 2760, 86130, 713, 2052, 180, -88, -90, 1980, 88, 1152, 1155, 5962],80,1]
%
% END OF LATTICE L81.1
%
\hbox{}\par\smallskip%
%
% LATTICE L81.2
%
\leavevmode%
\llap{\spacer}%
${L_{81.2}}=\pfillwitharg{2.3}({L_{81.1}})$%
{${}={}$}Nikulin $81$%
\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[9570, 3135, 0; 3135, 1027, 0; 0, 0, -1],[165, 1, 1],[-1, -18, 1, 21, 182, 595, 296, 3263, 176, 97, 113, 54; 3, 55, -3, -64, -555, -1815, -903, -9955, -537, -296, -345, -165; 0, 0, -1, -9, -75, -242, -120, -1320, -71, -39, -45, -22],80,2]
%
% END OF LATTICE L81.2
%
\hbox{}\par\smallskip%
%
% LATTICE L81.3
%
\leavevmode%
\llap{\spacer}%
${L_{81.3}}=\pfillwitharg{3}({L_{81.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-149820, 2640, 660; 2640, -34, -15; 660, -15, -2],[660, 1, 1],[41, 417, 10, 9, 1, -3, -1, 23, 2, 7, 19, 91; 1200, 12210, 293, 264, 30, -88, -30, 660, 58, 204, 555, 2662; 4458, 45320, 1086, 976, 105, -330, -108, 2530, 219, 764, 2070, 9900],80,3]
%
% END OF LATTICE L81.3
%
\hbox{}\par\smallskip%
%
% LATTICE L81.4
%
\leavevmode%
\llap{\spacer}%
${L_{81.4}}=\pfillwitharg{2}({L_{81.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-12271545, -24255, -69795; -24255, -42, -132; -69795, -132, -391],[495, 3, 1],[-46, -1417, -23, -32, -4, 2, 1, -68, -4, -22, -41, -101; -12431, -382965, -6217, -8652, -1085, 539, 271, -18315, -1079, -5940, -11075, -27291; 12405, 382140, 6203, 8631, 1080, -539, -270, 18315, 1078, 5931, 11055, 27236],80,4]
%
% END OF LATTICE L81.4
%
\hbox{}\par\smallskip%
%
% LATTICE L81.5
%
\leavevmode%
\llap{\spacer}%
${L_{81.5}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{81.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[165, 2310, 165; 2310, 333, 771; 165, 771, 91],[165, 3, 1],[-33, 109, 107, 260, 533, 4111, 633, 19691, 973, 460, 67, -151; -5, 0, 14, 36, 75, 583, 90, 2805, 139, 66, 10, -22; 101, -330, -327, -795, -1630, -12573, -1936, -60225, -2976, -1407, -205, 462],80,5]
%
% END OF LATTICE L81.5
%
\hbox{}\par\smallskip%
%
% LATTICE L81.6
%
\leavevmode%
\llap{\spacer}%
${L_{81.6}}=\pdualwitharg{5}(\pfillwitharg{2.3}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19635, 3960, 7920; 3960, 555, 1585; 7920, 1585, 3194],[165, 5, 1],[-112, 35, 128, 305, 373, 4782, 2207, 4574, 1129, 533, 46, -173; -15, 0, 14, 36, 45, 583, 270, 561, 139, 66, 6, -22; 285, -88, -325, -775, -948, -12155, -5610, -11627, -2870, -1355, -117, 440],80,6]
%
% END OF LATTICE L81.6
%
\hbox{}\par\smallskip%
%
% LATTICE L81.7
%
\leavevmode%
\llap{\spacer}%
${L_{81.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-907830, 8415, 17820; 8415, -78, -165; 17820, -165, -329],[495, 3, 1],[26, 333, 64, 15, 33, 104, 9, 52, -1, -1, 2, 127; 2846, 36465, 7011, 1644, 3620, 11418, 989, 5720, -108, -110, 215, 13893; -21, -275, -54, -13, -30, -99, -9, -55, 0, 1, 0, -99],80,7]
%
% END OF LATTICE L81.7
%
\hbox{}\par\smallskip%
%
% LATTICE L81.8
%
\leavevmode%
\llap{\spacer}%
${L_{81.8}}=\pdualwitharg{11}(\pfillwitharg{2.3}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[62205, 8910, 16995; 8910, 1221, 2431; 16995, 2431, 4643],[165, 11, 1],[-61, 19, 85, 190, 1139, 263, 1331, 1251, 677, 318, 131, -9; -15, 0, 14, 36, 225, 53, 270, 255, 139, 66, 30, -2; 231, -70, -319, -715, -4290, -991, -5016, -4715, -2552, -1199, -495, 34],80,8]
%
% END OF LATTICE L81.8
%
\hbox{}\par\smallskip%
%
% LATTICE L81.9
%
\leavevmode%
\llap{\spacer}%
${L_{81.9}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{81.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1246740, -623700, 5940; -623700, -312006, 2967; 5940, 2967, -26],[660, 3, 1],[-159, -2429, 27, 53, -54, -3261, -641, -23971, -747, -1013, -346, -2019; 324, 4950, -55, -108, 110, 6644, 1306, 48840, 1522, 2064, 705, 4114; 646, 9900, -108, -216, 215, 13134, 2584, 96690, 3015, 4092, 1400, 8184],80,9]
%
% END OF LATTICE L81.9
%
\hbox{}\par\smallskip%
%
% LATTICE L81.10
%
\leavevmode%
\llap{\spacer}%
${L_{81.10}}=\pdualwitharg{3.5}(\pfillwitharg{2.3}({L_{81.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[249987870, 526845, 82634145; 526845, 1110, 174150; 82634145, 174150, 27314933],[165, 15, 1],[319, 7058, 3828, 4904, 1475, 46635, 6656, 38977, 8727, 3466, -80, -600; -8, -176, -95, -119, -35, -1078, -152, -880, -193, -73, 3, 22; -965, -21351, -11580, -14835, -4462, -141075, -20135, -117909, -26400, -10485, 242, 1815],80,10]
%
% END OF LATTICE L81.10
%
\hbox{}\par\smallskip%
%
% LATTICE L81.11
%
\leavevmode%
\llap{\spacer}%
${L_{81.11}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[287257080, -486420, 71362500; -486420, 824, -120840; 71362500, -120840, 17728393],[660, 4, 1],[3946, 40203, 1935, 444, 328, 82, 79, 4072, 480, 361, 3697, 8773; 285, 2970, 148, 36, 45, 11, 0, 165, 23, 21, 240, 616; -15882, -161810, -7788, -1787, -1320, -330, -318, -16390, -1932, -1453, -14880, -35310],80,11]
%
% END OF LATTICE L81.11
%
\hbox{}\par\smallskip%
%
% LATTICE L81.12
%
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\llap{\spacer}%
${L_{81.12}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[407220, -10560, 2640; -10560, 270, -65; 2640, -65, 14],[660, 5, 1],[-1, 5, 2, 5, 2, 31, 11, 17, 1, -1, -1, -9; -54, 264, 106, 264, 105, 1606, 564, 858, 47, -60, -54, -484; -60, 286, 115, 280, 108, 1540, 510, 704, 20, -100, -63, -550],80,12]
%
% END OF LATTICE L81.12
%
\hbox{}\par\smallskip%
%
% LATTICE L81.13
%
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\llap{\spacer}%
${L_{81.13}}=\pdualwitharg{3.11}(\pfillwitharg{2.3}({L_{81.1}}))$%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[55288035, 367950, 18242565; 367950, 2442, 121407; 18242565, 121407, 6019226],[165, 33, 1],[461, 4643, 5532, 7013, 10436, 5925, 9249, 24487, 11946, 4639, -726, -99; -8, -80, -95, -119, -175, -98, -152, -400, -193, -73, 15, 2; -1397, -14070, -16764, -21252, -31625, -17955, -28028, -74205, -36201, -14058, 2200, 300],80,13]
%
% END OF LATTICE L81.13
%
\hbox{}\par\smallskip%
%
% LATTICE L81.14
%
\leavevmode%
\llap{\spacer}%
${L_{81.14}}=\pdualwitharg{3}({L_{81.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-396651420\\180180\\-178144560
\end{tabular}\endgroup%
\kern3pt%
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180180\\-78\\81345
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-178144560\\81345\\-79962134
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-99106921\\-21803694660\\198615780
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-396651420, 180180, -178144560; 180180, -78, 81345; -178144560, 81345, -79962134],[1980, 3, 1],[-12823, -164227, -15781, -7397, -8136, -51277, -4437, -25633, 247, 493, -494, -62639; -2821088, -36130270, -3471849, -1627352, -1789930, -11280984, -976142, -5639260, 54342, 108460, -108685, -13780734; 25698, 329120, 31626, 14824, 16305, 102762, 8892, 51370, -495, -988, 990, 125532],80,14]
%
% END OF LATTICE L81.14
%
\hbox{}\par\smallskip%
%
% LATTICE L81.15
%
\leavevmode%
\llap{\spacer}%
${L_{81.15}}=\pdualwitharg{11}(\pfillwitharg{3}({L_{81.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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10\\499\\3278
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31\\1544\\10164
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33\\1640\\10822
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{$\left.\llap{\phantom{%
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0\\0\\0
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-924\\-1100\\-396\\-572\\-660\\-132
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7667220, 19140, 20460; 19140, -22, -55; 20460, -55, -54],[660, 11, 1],[-1, 13, 10, 31, 79, 33, 161, 147, 38, 33, 1, -1; -48, 650, 499, 1544, 3930, 1640, 7998, 7300, 1886, 1636, 45, -50; -330, 4260, 3278, 10164, 25905, 10822, 52800, 48210, 12463, 10824, 330, -328],80,15]
%
% END OF LATTICE L81.15
%
\hbox{}\par\smallskip%
%
% LATTICE L81.16
%
\leavevmode%
\llap{\spacer}%
${L_{81.16}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{81.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40590, 7920, 2475; 7920, -345, -150; 2475, -150, -61],[495, 15, 1],[7, 8, -1, -1, 0, 85, 18, 47, 46, 11, 5, 80; -538, -616, 75, 77, 1, -6468, -1372, -3586, -3513, -841, -383, -6138; 1605, 1837, -225, -230, -3, 19305, 4095, 10703, 10485, 2510, 1143, 18315],80,16]
%
% END OF LATTICE L81.16
%
\hbox{}\par\smallskip%
%
% LATTICE L81.17
%
\leavevmode%
\llap{\spacer}%
${L_{81.17}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[133155, 0, -2475; 0, 15, 0; -2475, 0, 46],[495, 15, 1],[0, 13, 3, 16, 6, 76, 35, 218, 18, 26, 1, 1; 1, 0, -1, -9, -4, -55, -26, -165, -14, -21, -1, 0; 0, 693, 160, 855, 321, 4070, 1875, 11682, 965, 1395, 54, 55],80,17]
%
% END OF LATTICE L81.17
%
\hbox{}\par\smallskip%
%
% LATTICE L81.18
%
\leavevmode%
\llap{\spacer}%
${L_{81.18}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2462152440, -11331540, 613661400; -11331540, 52152, -2824248; 613661400, -2824248, 152947603],[660, 12, 1],[1204, 21563, -3, 41, 1456, 27828, 5241, 191122, 11604, 3826, 4989, 14709; -827, -14850, -2, -27, -985, -18931, -3568, -130185, -7909, -2610, -3410, -10076; -4846, -86790, 12, -165, -5860, -112002, -21094, -769230, -46704, -15399, -20080, -59202],80,18]
%
% END OF LATTICE L81.18
%
\hbox{}\par\smallskip%
%
% LATTICE L81.19
%
\leavevmode%
\llap{\spacer}%
${L_{81.19}}=\pdualwitharg{5.11}(\pfillwitharg{2.3}({L_{81.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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52813695\\-464310\\-21541245
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-464310\\4070\\189365
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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154955\\-463234\\389895
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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-8656\\25873\\-21780
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-10973\\32800\\-27610
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-9797\\29286\\-24651
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[52813695, -464310, -21541245; -464310, 4070, 189365; -21541245, 189365, 8786061],[165, 55, 1],[-2164, -1453, -8656, -10973, -9797, -9270, -43411, -7662, -18689, -7257, 682, 155; 6468, 4343, 25873, 32800, 29286, 27712, 129777, 22906, 55874, 21698, -2037, -463; -5445, -3656, -21780, -27610, -24651, -23325, -109230, -19279, -47025, -18260, 1716, 390],80,19]
%
% END OF LATTICE L81.19
%
\hbox{}\par\smallskip%
%
% LATTICE L81.20
%
\leavevmode%
\llap{\spacer}%
${L_{81.20}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{660}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-40260\\3300\\-1980
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1980\\165\\-82
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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3168\\34703\\-7200
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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191\\2094\\-430
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1373\\15048\\-3102
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205\\2246\\-465
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263\\2880\\-600
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16\\175\\-37
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421\\4598\\-990
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-40260, 3300, -1980; 3300, -270, 165; -1980, 165, -82],[660, 15, 1],[191, 1373, 205, 263, 16, 421, 29, 79, -7, 1, 6, 1031; 2094, 15048, 2246, 2880, 175, 4598, 316, 858, -77, 12, 66, 11308; -430, -3102, -465, -600, -37, -990, -70, -198, 15, 0, -13, -2310],80,20]
%
% END OF LATTICE L81.20
%
\hbox{}\par\smallskip%
%
% LATTICE L81.21
%
\leavevmode%
\llap{\spacer}%
${L_{81.21}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{81.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{220}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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44883960\\-1314060\\11148060
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[44883960, -1314060, 11148060; -1314060, 38440, -326380; 11148060, -326380, 2768901],[660, 20, 1],[-589, -432, 328, 789, 2329, 34033, 16394, 35203, 9155, 2372, 832, 738; -12, -11, 5, 14, 42, 616, 297, 638, 166, 43, 15, 11; 2370, 1738, -1320, -3175, -9372, -136950, -65970, -141658, -36840, -9545, -3348, -2970],80,21]
%
% END OF LATTICE L81.21
%
\hbox{}\par\smallskip%
%
% LATTICE L81.22
%
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${L_{81.22}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{81.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[52470, 25245, 1980; 25245, 9042, 693; 1980, 693, 53],[495, 33, 1],[4, 2, -1, -1, -3, 2, 6, 8, 19, 5, 13, 4; -130, -65, 33, 34, 110, -54, -175, -240, -582, -156, -415, -129; 1551, 775, -396, -407, -1320, 639, 2079, 2855, 6930, 1859, 4950, 1539],80,22]
%
% END OF LATTICE L81.22
%
\hbox{}\par\smallskip%
%
% LATTICE L81.23
%
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-33\\0\\22\\0\\-165\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-297\\-495\\0\\99\\0\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-1485\\-165\\0\\330\\0
\end{tabular}\endgroup%
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-165\\-450\\-77\\-99\\0\\1
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-858\\-2475\\-462\\-693\\-165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\-1485\\-319\\-594\\-330\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693\\-2475\\-594\\-1287\\-990\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-990\\-330\\-990\\-1155\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-22\\-99\\-165\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-495, -495, -990; -495, -462, -990; -990, -990, -1979],[495, 33, 1],[-1, 88, 44, 202, 341, 71, 346, 947, 163, 212, 4, -2; 1, 0, -1, -9, -20, -5, -26, -75, -14, -21, -5, 0; 0, -45, -22, -99, -165, -34, -165, -450, -77, -99, 0, 1],80,23]
%
% END OF LATTICE L81.23
%
\hbox{}\par\smallskip%
%
% LATTICE L81.24
%
\leavevmode%
\llap{\spacer}%
${L_{81.24}}=\pdualwitharg{3.11}(\pfillwitharg{3}({L_{81.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{l}{2}{132}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33660\\-8580\\1980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8580\\-2178\\495
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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7199\\-36480\\-36960
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1755\\-8893\\-9009
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-330\\1672\\1693
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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15\\-78\\-88
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\HardButStrongLineBreak\kern3pt%
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13\\-70\\-90
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\HardButStrongLineBreak\kern3pt%
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1\\-4\\0
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23\\-115\\-110
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67\\-338\\-336
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\60\\0\\-660\\-990\\-1800
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-33660, -8580, 1980; -8580, -2178, 495; 1980, 495, -106],[660, 33, 1],[15, 13, -7, 1, 23, 67, 135, 437, 142, 179, 52, 23; -78, -70, 35, -4, -115, -338, -682, -2210, -719, -908, -265, -118; -88, -90, 33, 0, -110, -336, -682, -2220, -726, -924, -275, -126],80,24]
%
% END OF LATTICE L81.24
%
\hbox{}\par\smallskip%
%
% LATTICE L81.25
%
\leavevmode%
\llap{\spacer}%
${L_{81.25}}=\pdualwitharg{2.3}({L_{81.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{220}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{396}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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2260030437720\\501890655420\\-565388600040
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\kern3pt%
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501890655420\\111456122808\\-125557271400
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-565388600040\\-125557271400\\141442461889
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-46148875359\\8733628\\-184463357832
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-487862\\89\\-1950054
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-1093301\\204\\-4370076
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-482176\\95\\-1927320
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-1441834\\297\\-5763186
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2260030437720, 501890655420, -565388600040; 501890655420, 111456122808, -125557271400; -565388600040, -125557271400, 141442461889],[1980, 12, 1],[-487862, -5964751, -1093301, -121162, -482176, -1441834, -119803, -719564, -496, -371, -125489, -2568755; 89, 1100, 204, 23, 95, 297, 26, 165, 3, 0, 20, 462; -1950054, -23841950, -4370076, -484301, -1927320, -5763186, -478866, -2876170, -1980, -1483, -501600, -10267686],80,25]
%
% END OF LATTICE L81.25
%
\hbox{}\par\smallskip%
%
% LATTICE L81.26
%
\leavevmode%
\llap{\spacer}%
${L_{81.26}}=\pdualwitharg{2}({L_{81.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{1980}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720969480\\-16768620\\-180623520
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\kern3pt%
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-16768620\\390072\\4201020
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180623520\\4201020\\45251369
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[720969480, -16768620, -180623520; -16768620, 390072, 4201020; -180623520, 4201020, 45251369],[1980, 12, 1],[-36667, -1126186, -18204, -12664, -3427, -237, -248, -92009, -3945, -9485, -33488, -80976; 124, 3795, 61, 42, 10, 0, 1, 330, 14, 33, 115, 275; -146370, -4495590, -72668, -50553, -13680, -946, -990, -367290, -15748, -37863, -133680, -323246],80,26]
%
% END OF LATTICE L81.26
%
\hbox{}\par\smallskip%
%
% LATTICE L81.27
%
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${L_{81.27}}=\pdualwitharg{3.5.11}(\pfillwitharg{2.3}({L_{81.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[890244630, -144258180, -52632195; -144258180, 23376045, 8528685; -52632195, 8528685, 3111667],[165, 165, 1],[-5, 0, 14, 36, 15, 53, 90, 51, 139, 66, 2, -2; 2337, -382, -9365, -21209, -8512, -29543, -49872, -28139, -76198, -35833, -995, 1022; -6490, 1047, 25905, 58740, 23584, 81870, 138215, 77988, 211200, 99330, 2761, -2835],80,27]
%
% END OF LATTICE L81.27
%
\hbox{}\par\smallskip%
%
% LATTICE L81.28
%
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\kern3pt%
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-660\\-600\\-308\\-66\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3432\\-3300\\-1848\\-462\\-660\\0
\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-1980\\-1276\\-396\\-1320\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\-550\\-396\\-143\\-660\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-1320\\-1320\\-660\\-4620\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-88\\-66\\-660\\-136
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2893042680, -29725740, 718823160; -29725740, 305448, -7385840; 718823160, -7385840, 178603219],[660, 44, 1],[410, 9007, 13141, 10826, 116432, 25862, 129305, 120708, 64688, 15117, 13283, 81; 3, 10, 12, 8, 75, 15, 72, 65, 33, 7, 0, 0; -1650, -36250, -52888, -43571, -468600, -104086, -520410, -485810, -260348, -60841, -53460, -326],80,28]
%
% END OF LATTICE L81.28
%
\hbox{}\par\smallskip%
%
% LATTICE L81.29
%
\leavevmode%
\llap{\spacer}%
${L_{81.29}}=\pdualwitharg{5}({L_{81.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{396}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{180}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{220}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1213740\\1980\\1980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1980\\510\\135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1980\\135\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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2375\\-532224\\1978020
\end{tabular}\endgroup%
\kern3pt%
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-34\\7615\\-28305
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-12\\2688\\-9991
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
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-1\\226\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-1122\\4158
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-225\\835
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\HardButStrongLineBreak\kern3pt%
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11\\-2472\\9180
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\HardButStrongLineBreak\kern3pt%
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2\\-449\\1668
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\-10538\\39160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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60\\0\\-30\\-540\\-120\\-3300\\-1560\\-9900\\-420\\-1260\\-30\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\right.$}%
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60\\0\\-30\\-540\\-120\\-3300
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\kern3pt%
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0\\396\\0\\-1980\\-594\\-19800
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-540\\-1980\\0\\180\\0\\-1980
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\kern3pt%
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-120\\-594\\-15\\0\\6\\0
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-3300\\-19800\\-770\\-1980\\0\\220
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\kern3pt%
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-1560\\-9900\\-420\\-1260\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9900\\-65736\\-2970\\-9900\\-396\\-1980
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\kern3pt%
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-420\\-2970\\-145\\-540\\-30\\-220
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-396\\-30\\-180\\-21\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1980\\-220\\-1980\\-330\\-7480
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1213740, 1980, 1980; 1980, 510, 135; 1980, 135, 34],[1980, 15, 1],[-1, 5, 1, 11, 2, 47, 21, 127, 5, 13, 0, -3; 226, -1122, -225, -2472, -449, -10538, -4706, -28446, -1119, -2904, 1, 682; -840, 4158, 835, 9180, 1668, 39160, 17490, 105732, 4160, 10800, -3, -2530],80,29]
%
% END OF LATTICE L81.29
%
\hbox{}\par\smallskip%
%
% LATTICE L81.30
%
\leavevmode%
\llap{\spacer}%
${L_{81.30}}=\pdualwitharg{3.5}({L_{81.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{1980}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-19635660\\-9733680\\7931880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9733680\\-4825110\\3931605
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7931880\\3931605\\-3196606
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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62916479\\-131825760\\-6019200
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\kern3pt%
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31148424\\-65263739\\-2979960
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24535044\\51407053\\2347259
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-85603\\179360\\8190
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\HardButStrongLineBreak\kern3pt%
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-223743\\468798\\21406
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109595\\229629\\10485
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52473\\109944\\5020
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\HardButStrongLineBreak\kern3pt%
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-11948\\25034\\1143
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-382913\\802296\\36630
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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0\\44\\0\\-220\\-198\\-19800\\-3300\\-7304\\-2970\\-1100\\-132\\-1980
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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60\\0\\-90\\-180\\-120\\-9900
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-90\\0\\90\\0\\-45\\-6930
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\kern3pt%
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-1560\\-3300\\-1260\\-420\\-30\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-2970\\-1305\\-540\\-90\\-1980
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\kern3pt%
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-420\\-1100\\-540\\-260\\-60\\-1980
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\kern3pt%
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-30\\-132\\-90\\-60\\-21\\-990
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1980\\-1980\\-1980\\-990\\-67320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19635660, -9733680, 7931880; -9733680, -4825110, 3931605; 7931880, 3931605, -3196606],[1980, 15, 1],[-85603, -223743, -109595, -52473, -11948, -382913, -33557, -38409, 2351, 4809, 32, -403543; 179360, 468798, 229629, 109944, 25034, 802296, 70310, 80476, -4926, -10076, -67, 845526; 8190, 21406, 10485, 5020, 1143, 36630, 3210, 3674, -225, -460, -3, 38610],80,30]
%
% END OF LATTICE L81.30
%
\hbox{}\par\smallskip%
%
% LATTICE L81.31
%
\leavevmode%
\llap{\spacer}%
${L_{81.31}}=\pdualwitharg{5.11}(\pfillwitharg{3}({L_{81.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${660}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{l}{2}{220}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-652740\\-18480\\1980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18480\\-330\\55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1980\\55\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1279\\960\\422400
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\41\\18480
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-3\\-1321
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-652740, -18480, 1980; -18480, -330, 55; 1980, 55, -6],[660, 55, 1],[41, 7, 8, 5, -1, -1, -1, 1, 4, 11, 5, 9; 30, 4, 2, -4, -3, -2, 0, 2, 7, 16, 6, 8; 13530, 2294, 2585, 1540, -363, -350, -330, 346, 1375, 3740, 1683, 2990],80,31]
%
% END OF LATTICE L81.31
%
\hbox{}\par\smallskip%
%
% LATTICE L81.32
%
\leavevmode%
\llap{\spacer}%
${L_{81.32}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{81.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1983960, 333300, 495000; 333300, 56280, 83160; 495000, 83160, 123503],[660, 60, 1],[-10233, -74398, -22500, -7352, -1869, -26013, -1952, -6023, 567, 41, -568, -54408; -201, -1463, -443, -145, -37, -517, -39, -121, 11, 1, -11, -1067; 41150, 299178, 90480, 29565, 7516, 104610, 7850, 24222, -2280, -165, 2284, 218790],80,32]
%
% END OF LATTICE L81.32
%
\hbox{}\par\smallskip%
%
% LATTICE L81.33
%
\leavevmode%
\llap{\spacer}%
${L_{81.33}}=\pdualwitharg{11}({L_{81.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-511970580, 502920, 502920; 502920, -462, -495; 502920, -495, -494],[1980, 33, 1],[181, 497, 43, 113, 2, -1, -1, 43, 11, 103, 88, 37; 5378, 14760, 1276, 3348, 55, -30, -28, 1290, 329, 3072, 2620, 1100; 178860, 491130, 42493, 111672, 1980, -988, -990, 42480, 10868, 101772, 86955, 36562],80,33]
%
% END OF LATTICE L81.33
%
\hbox{}\par\smallskip%
%
% LATTICE L81.34
%
\leavevmode%
\llap{\spacer}%
${L_{81.34}}=\pdualwitharg{3.11}({L_{81.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-88884180, -5983560, -156420; -5983560, -402798, -10527; -156420, -10527, -274],[1980, 33, 1],[-803, -869, -850, -359, -317, -143, -97, -31, 40, -1, -133, -397; 12706, 13750, 13449, 5680, 5015, 2262, 1534, 490, -633, 16, 2105, 6282; -29766, -32200, -31482, -13288, -11715, -5274, -3564, -1130, 1485, -44, -4950, -14724],80,34]
%
% END OF LATTICE L81.34
%
\hbox{}\par\smallskip%
%
% LATTICE L81.35
%
\leavevmode%
\llap{\spacer}%
${L_{81.35}}=\pdualwitharg{3.5.11}(\pfillwitharg{2}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[990, -186615, 3465; -186615, 35332770, -656040; 3465, -656040, 12181],[495, 165, 1],[-19, -3, -17, 0, 3, 7, 9, 1, -1, -4, -5, -13; 199, 44, 441, 97, 38, 51, 46, 5, 0, 1, 11, 96; 10725, 2371, 23760, 5225, 2046, 2745, 2475, 269, 0, 55, 594, 5175],80,35]
%
% END OF LATTICE L81.35
%
\hbox{}\par\smallskip%
%
% LATTICE L81.36
%
\leavevmode%
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${L_{81.36}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15250455, 17759610, -77220; 17759610, -20681430, 89925; -77220, 89925, -391],[495, 165, 1],[-332, -182, -157, -205, -1, 2, 2, -16, -41, -191, -65, -68; -26, -15, -14, -21, -1, 0, 1, 0, -1, -9, -4, -5; 59565, 32481, 27775, 35640, -33, -395, -165, 3159, 7865, 35640, 11913, 12275],80,36]
%
% END OF LATTICE L81.36
%
\hbox{}\par\smallskip%
%
% LATTICE L81.37
%
\leavevmode%
\llap{\spacer}%
${L_{81.37}}=\pdualwitharg{2.3.11}(\pfillwitharg{3}({L_{81.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5270760, -609180, 1313400; -609180, 70488, -151800; 1313400, -151800, 327281],[660, 132, 1],[1021, 1571, -131, -41, 1037, 2059, 4329, 14479, 9761, 3251, 4313, 1151; -36, -55, 5, 1, -40, -76, -159, -530, -356, -118, -155, -41; -4114, -6330, 528, 165, -4180, -8298, -17446, -58350, -39336, -13101, -17380, -4638],80,37]
%
% END OF LATTICE L81.37
%
\hbox{}\par\smallskip%
%
% LATTICE L81.38
%
\leavevmode%
\llap{\spacer}%
${L_{81.38}}=\pdualwitharg{3.5.11}(\pfillwitharg{3}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[660, 0, 0; 0, -13530, 165; 0, 165, -2],[660, 165, 1],[-3, -1, 0, 1, 0, -3, -7, -5, -9, -13, -1, -3; -2, 0, 2, 0, -1, -14, -28, -18, -29, -36, -2, -4; -220, -18, 165, 0, -88, -1260, -2530, -1632, -2640, -3300, -187, -390],80,38]
%
% END OF LATTICE L81.38
%
\hbox{}\par\smallskip%
%
% LATTICE L81.39
%
\leavevmode%
\llap{\spacer}%
${L_{81.39}}=\pdualwitharg{2.3.5}({L_{81.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18567519723720, -287839393380, -4642098949620; -287839393380, 4462174680, 71963243640; -4642098949620, 71963243640, 1160579494661],[1980, 60, 1],[-44902, -108891, -98909, -10842, -8448, -125050, -10323, -12604, -496, -203, -2597, -239435; 10813, 26103, 23589, 2565, 1961, 28281, 2257, 2651, -63, 49, 653, 57981; -180270, -437162, -397080, -43525, -33912, -501930, -41430, -50578, -1980, -815, -10428, -961290],80,39]
%
% END OF LATTICE L81.39
%
\hbox{}\par\smallskip%
%
% LATTICE L81.40
%
\leavevmode%
\llap{\spacer}%
${L_{81.40}}=\pdualwitharg{2.5}({L_{81.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20361064680, 166975380, -5100598800; 166975380, 1369320, -41828580; -5100598800, -41828580, 1277738101],[1980, 60, 1],[-263, -7291, -1493, -4295, -3415, -45937, -21607, -137051, -11629, -8827, -959, -2177; -4, -33, -3, 6, 10, 176, 89, 594, 54, 45, 7, 11; -1050, -29106, -5960, -17145, -13632, -183370, -86250, -547074, -46420, -35235, -3828, -8690],80,40]
%
% END OF LATTICE L81.40
%
\hbox{}\par\smallskip%
%
% LATTICE L81.41
%
\leavevmode%
\llap{\spacer}%
${L_{81.41}}=\pdualwitharg{2.5.11}(\pfillwitharg{3}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[373759320, 685740, -92355780; 685740, 1320, -169400; -92355780, -169400, 22821111],[660, 220, 1],[-64693, -12388, -34492, -8623, -2591, -405, -82, -567, -4373, -4332, -10364, -12550; 193866, 37123, 103361, 25840, 7764, 1214, 249, 1700, 13108, 12983, 31059, 37609; -260370, -49858, -138820, -34705, -10428, -1630, -330, -2282, -17600, -17435, -41712, -50510],80,41]
%
% END OF LATTICE L81.41
%
\hbox{}\par\smallskip%
%
% LATTICE L81.42
%
\leavevmode%
\llap{\spacer}%
${L_{81.42}}=\pdualwitharg{2.3.11}({L_{81.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7406083080, 236140740, -1855713420; 236140740, 7530072, -59168868; -1855713420, -59168868, 464978891],[1980, 132, 1],[363381, 394357, 774245, 82547, 300737, 73691, 58709, 27733, -14483, 1871, 121353, 179563; 2939, 3190, 6264, 668, 2435, 597, 476, 225, -117, 15, 980, 1452; 1450614, 1574270, 3090780, 329527, 1200540, 294174, 234366, 110710, -57816, 7469, 484440, 716814],80,42]
%
% END OF LATTICE L81.42
%
\hbox{}\par\smallskip%
%
% LATTICE L81.43
%
\leavevmode%
\llap{\spacer}%
${L_{81.43}}=\pdualwitharg{2.11}({L_{81.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[243984244680, -494899020, -61119786420; -494899020, 1003992, 123975720; -61119786420, 123975720, 15310940659],[1980, 132, 1],[-1210167, -3390886, -606194, -425249, -124497, -247, -248, -240359, -120055, -301000, -1085918, -241836; 124, 345, 61, 42, 10, 0, 1, 30, 14, 33, 115, 25; -4830870, -13536090, -2419868, -1697553, -496980, -986, -990, -959490, -479248, -1201563, -4334880, -965386],80,43]
%
% END OF LATTICE L81.43
%
\hbox{}\par\smallskip%
%
% LATTICE L81.44
%
\leavevmode%
\llap{\spacer}%
${L_{81.44}}=\pdualwitharg{5.11}({L_{81.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-181666980, -8488260, 130680; -8488260, -396330, 6105; 130680, 6105, -94],[1980, 165, 1],[141, 77, 33, 85, 0, -1, -1, 7, 9, 83, 14, 29; 794, 432, 184, 468, -1, -6, -4, 42, 53, 480, 80, 164; 247500, 135054, 57805, 148500, -66, -1780, -1650, 12456, 15950, 146520, 24651, 50950],80,44]
%
% END OF LATTICE L81.44
%
\hbox{}\par\smallskip%
%
% LATTICE L81.45
%
\leavevmode%
\llap{\spacer}%
${L_{81.45}}=\pdualwitharg{3.5.11}({L_{81.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[89086140, 29858400, -138600; 29858400, 10002630, -46365; -138600, -46365, 214],[1980, 165, 1],[-2537, -545, -2642, -1099, -186, -385, -213, -5, 167, -1, -89, -1265; 10120, 2174, 10539, 4384, 742, 1536, 850, 20, -666, 4, 355, 5046; 549450, 118036, 572220, 238040, 40293, 83430, 46200, 1094, -36135, 220, 19272, 273960],80,45]
%
% END OF LATTICE L81.45
%
\hbox{}\par\smallskip%
%
% LATTICE L81.46
%
\leavevmode%
\llap{\spacer}%
${L_{81.46}}=\pdualwitharg{2.3.5.11}(\pfillwitharg{3}({L_{81.1}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1320, -108900, -27060; -108900, 8932440, 2219580; -27060, 2219580, 551533],[660, 660, 1],[-28, -9, 1, 0, -6, -56, -117, -78, -262, -87, -23, -31; 191, 64, 0, -41, -11, 141, 354, 263, 981, 368, 120, 186; -770, -258, 0, 165, 44, -570, -1430, -1062, -3960, -1485, -484, -750],80,46]
%
% END OF LATTICE L81.46
%
\hbox{}\par\smallskip%
%
% LATTICE L81.47
%
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\llap{\spacer}%
${L_{81.47}}=\pdualwitharg{2.3.5.11}({L_{81.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1381701027960, 8186850540, 339958101780; 8186850540, 48508680, 2014318680; 339958101780, 2014318680, 83644369231],[1980, 660, 1],[-211382, -52917, -597689, -75462, -74864, -116990, -120943, -13548, -496, 8997, 8399, -82405; 635981, 159207, 1798185, 227028, 225223, 351951, 363839, 40757, 1491, -27064, -25259, 247941; 843810, 211238, 2385900, 301235, 298848, 467010, 482790, 54082, 1980, -35915, -33528, 328950],80,47]
%
% END OF LATTICE L81.47
%
\hbox{}\par\smallskip%
%
% LATTICE L81.48
%
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\llap{\spacer}%
${L_{81.48}}=\pdualwitharg{2.5.11}({L_{81.1}})$%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[151782392520, 169090020, 37895354640; 169090020, 188760, 42216240; 37895354640, 42216240, 9461294639],[1980, 660, 1],[1426587, 802934, 722566, 513239, 34125, 1731, 248, 53167, 133535, 342862, 251242, 283636; -4281407, -2409729, -2168533, -1540311, -102415, -5195, -743, -159561, -400757, -1028979, -754015, -851235; -5694810, -3205242, -2884420, -2048805, -136224, -6910, -990, -212238, -533060, -1368675, -1002936, -1132250],80,48]
%
% END OF LATTICE L81.48
%

\medskip%
%------------------------------WEYL-GROUP-W82--------------------------------
%
\leavevmode\llap{}%
$W_{82\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=18$%
\hfill%
$2222222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L82.1
%
\leavevmode%
\llap{\spacer}%
${L_{82.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{82}}%
\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{132}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13682460\\-6811200\\14520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6811200\\-3390650\\7227
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14520\\7227\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2285\\4598\\3674
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2773\\5580\\4470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-326\\656\\527
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4723\\9504\\7656
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82\\-165\\-130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
163\\-328\\-264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82\\165\\129
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\-22\\-1188\\-110\\-44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-660\\-90\\-60\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\2\\0\\-5\\-8\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1188\\-660\\0\\132\\0\\-132\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\-90\\-5\\0\\10\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-60\\-8\\-132\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-9\\-264\\-15\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-13682460, -6811200, 14520; -6811200, -3390650, 7227; 14520, 7227, -14],[660, 1, 1],[-2285, -2773, -326, -4723, 82, 163, -82; 4598, 5580, 656, 9504, -165, -328, 165; 3674, 4470, 527, 7656, -130, -264, 129],81,1]
%
% END OF LATTICE L82.1
%
\hbox{}\par\smallskip%
%
% LATTICE L82.2
%
\leavevmode%
\llap{\spacer}%
${L_{82.2}}=\pfillwitharg{2}({L_{82.1}})$%
{${}={}$}Nikulin $82$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4290\\1155\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1155\\311\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\11\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\15\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-11\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
125\\-462\\-99
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-100\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-26\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-15\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-11\\-297\\-55\\-11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-165\\-45\\-15\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\2\\0\\-5\\-4\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-297\\-165\\0\\33\\0\\-33\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-45\\-5\\0\\10\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-15\\-4\\-33\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-9\\-132\\-15\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[4290, 1155, 0; 1155, 311, 0; 0, 0, -1],[165, 1, 1],[-3, -4, 3, 125, 27, 7, 4; 11, 15, -11, -462, -100, -26, -15; 0, 0, -3, -99, -20, -5, -3],81,2]
%
% END OF LATTICE L82.2
%
\hbox{}\par\smallskip%
%
% LATTICE L82.3
%
\leavevmode%
\llap{\spacer}%
${L_{82.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{82.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1431870\\36630\\-474705
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36630\\-789\\12144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-474705\\12144\\-157378
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
228572\\649\\-689403
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40421\\115\\-121915
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1487\\-4\\4485
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3844\\-11\\11594
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7251\\20\\-21870
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28640\\81\\-86382
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35274\\100\\-106391
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\0\\-33\\-297\\-165\\-33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-55\\-45\\-15\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\6\\0\\-15\\-12\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-297\\-55\\0\\11\\0\\-33\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-45\\-15\\0\\30\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-15\\-12\\-33\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-9\\-44\\-15\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1431870, 36630, -474705; 36630, -789, 12144; -474705, 12144, -157378],[165, 3, 1],[228572, 40421, -1487, -3844, 7251, 28640, 35274; 649, 115, -4, -11, 20, 81, 100; -689403, -121915, 4485, 11594, -21870, -86382, -106391],81,3]
%
% END OF LATTICE L82.3
%
\hbox{}\par\smallskip%
%
% LATTICE L82.4
%
\leavevmode%
\llap{\spacer}%
${L_{82.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{82.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${55}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1851630\\53790\\-738375
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53790\\-1315\\21450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-738375\\21450\\-294442
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
404297\\649\\-1013815
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42898\\69\\-107571
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2630\\-4\\6595
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20398\\-33\\51150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2565\\4\\-6432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50658\\81\\-127030
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
187177\\300\\-469365
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\0\\-55\\-1485\\-55\\-55\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-165\\-9\\-15\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\0\\10\\0\\-5\\-20\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1485\\-165\\0\\165\\0\\-165\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-9\\-5\\0\\2\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-15\\-20\\-165\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-45\\-660\\-15\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1851630, 53790, -738375; 53790, -1315, 21450; -738375, 21450, -294442],[165, 5, 1],[404297, 42898, -2630, -20398, 2565, 50658, 187177; 649, 69, -4, -33, 4, 81, 300; -1013815, -107571, 6595, 51150, -6432, -127030, -469365],81,4]
%
% END OF LATTICE L82.4
%
\hbox{}\par\smallskip%
%
% LATTICE L82.5
%
\leavevmode%
\llap{\spacer}%
${L_{82.5}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{82.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{165}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3221790\\105270\\-876645
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105270\\-2893\\28644
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-876645\\28644\\-238534
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49154\\59\\-180641
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
286853\\345\\-1054185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3517\\-4\\12925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2480\\-3\\9114
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17151\\20\\-63030
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67748\\81\\-248974
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
250324\\300\\-919941
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-11\\-27\\-55\\-11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\165\\0\\-165\\-495\\-165\\-165
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\22\\0\\-55\\-44\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-165\\0\\3\\0\\-33\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-495\\-55\\0\\110\\0\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-165\\-44\\-33\\0\\11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-99\\-132\\-165\\0\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3221790, 105270, -876645; 105270, -2893, 28644; -876645, 28644, -238534],[165, 11, 1],[49154, 286853, -3517, -2480, 17151, 67748, 250324; 59, 345, -4, -3, 20, 81, 300; -180641, -1054185, 12925, 9114, -63030, -248974, -919941],81,5]
%
% END OF LATTICE L82.5
%
\hbox{}\par\smallskip%
%
% LATTICE L82.6
%
\leavevmode%
\llap{\spacer}%
${L_{82.6}}=\pdualwitharg{3}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-708180\\-350460\\660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-350460\\-173382\\315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
660\\315\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-475\\968\\4290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-211\\430\\1910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-79\\161\\717
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-367\\748\\3344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-55\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\-108\\-480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-66\\-1188\\-330\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-220\\-90\\-60\\-10
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\0\\6\\0\\-15\\-24\\-9
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1188\\-220\\0\\44\\0\\-132\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-90\\-15\\0\\30\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-60\\-24\\-132\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-9\\-88\\-15\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-708180, -350460, 660; -350460, -173382, 315; 660, 315, 2],[660, 3, 1],[-475, -211, -79, -367, 27, 53, 0; 968, 430, 161, 748, -55, -108, 0; 4290, 1910, 717, 3344, -240, -480, -1],81,6]
%
% END OF LATTICE L82.6
%
\hbox{}\par\smallskip%
%
% LATTICE L82.7
%
\leavevmode%
\llap{\spacer}%
${L_{82.7}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{82.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${165}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24328329630\\597960\\8042312025
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
597960\\15\\197670
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8042312025\\197670\\2658578854
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1529215\\-9911\\4625940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54428\\-351\\164647
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8266\\62\\-25005
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26036\\165\\-78760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8865\\-62\\26817
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-189280\\-1239\\572580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234843\\-1528\\710410
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
165\\0\\-165\\-1485\\-165\\-165\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-55\\-9\\-15\\-5
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\0\\30\\0\\-15\\-60\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1485\\-55\\0\\55\\0\\-165\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-9\\-15\\0\\6\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-15\\-60\\-165\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-45\\-220\\-15\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24328329630, 597960, 8042312025; 597960, 15, 197670; 8042312025, 197670, 2658578854],[165, 15, 1],[-1529215, -54428, 8266, 26036, -8865, -189280, -234843; -9911, -351, 62, 165, -62, -1239, -1528; 4625940, 164647, -25005, -78760, 26817, 572580, 710410],81,7]
%
% END OF LATTICE L82.7
%
\hbox{}\par\smallskip%
%
% LATTICE L82.8
%
\leavevmode%
\llap{\spacer}%
${L_{82.8}}=\pdualwitharg{2}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24778489560\\-53802540\\6183816540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53802540\\116824\\-13427172
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6183816540\\-13427172\\1543257385
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16099\\-32351\\-64790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16094\\-32355\\-64770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3266\\-6571\\-13144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13669\\-27522\\-55011
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
323\\-660\\-1300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-82\\-165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3271\\-6567\\-13164
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\-44\\-594\\-220\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-330\\-180\\-30\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\0\\8\\0\\-20\\-8\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-594\\-330\\0\\33\\0\\-33\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-180\\-20\\0\\40\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-30\\-8\\-33\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-36\\-264\\-60\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24778489560, -53802540, 6183816540; -53802540, 116824, -13427172; 6183816540, -13427172, 1543257385],[660, 4, 1],[16099, 16094, 3266, 13669, 323, 41, 3271; -32351, -32355, -6571, -27522, -660, -82, -6567; -64790, -64770, -13144, -55011, -1300, -165, -13164],81,8]
%
% END OF LATTICE L82.8
%
\hbox{}\par\smallskip%
%
% LATTICE L82.9
%
\leavevmode%
\llap{\spacer}%
${L_{82.9}}=\pdualwitharg{5}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${220}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{660}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72720780\\26452140\\132660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26452140\\-9621950\\-48255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132660\\-48255\\-242
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-30\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
533\\1452\\2640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\41\\47
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\140\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\72\\-105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
220\\0\\-110\\-5940\\-110\\-220\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-660\\-18\\-60\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\0\\10\\0\\-5\\-40\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5940\\-660\\0\\660\\0\\-660\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\-18\\-5\\0\\2\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-60\\-40\\-660\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-45\\-1320\\-15\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-72720780, 26452140, 132660; 26452140, -9621950, -48255; 132660, -48255, -242],[660, 5, 1],[-1, -11, -1, 533, 15, 51, 26; 0, -30, -3, 1452, 41, 140, 72; -550, -48, 50, 2640, 47, 40, -105],81,9]
%
% END OF LATTICE L82.9
%
\hbox{}\par\smallskip%
%
% LATTICE L82.10
%
\leavevmode%
\llap{\spacer}%
${L_{82.10}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{82.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{55}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29050263330\\969210\\9586487955
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
969210\\33\\319836
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9586487955\\319836\\3163508374
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224594\\-901\\680595
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-439661\\-1755\\1332320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13351\\62\\-40458
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3824\\15\\-11588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71601\\-310\\216975
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-305788\\-1239\\926640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-379400\\-1528\\1149709
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-33\\-27\\-165\\-33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\55\\0\\-55\\-495\\-165\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\66\\0\\-165\\-132\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-55\\0\\1\\0\\-33\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-495\\-165\\0\\330\\0\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-165\\-132\\-33\\0\\33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-55\\-99\\-44\\-165\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[29050263330, 969210, 9586487955; 969210, 33, 319836; 9586487955, 319836, 3163508374],[165, 33, 1],[-224594, -439661, 13351, 3824, -71601, -305788, -379400; -901, -1755, 62, 15, -310, -1239, -1528; 680595, 1332320, -40458, -11588, 216975, 926640, 1149709],81,10]
%
% END OF LATTICE L82.10
%
\hbox{}\par\smallskip%
%
% LATTICE L82.11
%
\leavevmode%
\llap{\spacer}%
${L_{82.11}}=\pdualwitharg{11}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{660}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28136460\\1772760\\-27060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1772760\\111694\\-1705
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27060\\-1705\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-16\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-450\\660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-77\\154
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-108\\204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\15\\-55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\76\\-220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\60\\-231
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-22\\-108\\-110\\-44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\660\\0\\-660\\-990\\-660\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\0\\22\\0\\-55\\-88\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-660\\0\\12\\0\\-132\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\-990\\-55\\0\\110\\0\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-660\\-88\\-132\\0\\44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-330\\-99\\-264\\-165\\0\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28136460, 1772760, -27060; 1772760, 111694, -1705; -27060, -1705, 26],[660, 11, 1],[1, 29, 5, 7, -1, -5, -4; -16, -450, -77, -108, 15, 76, 60; -10, 660, 154, 204, -55, -220, -231],81,11]
%
% END OF LATTICE L82.11
%
\hbox{}\par\smallskip%
%
% LATTICE L82.12
%
\leavevmode%
\llap{\spacer}%
${L_{82.12}}=\pdualwitharg{2.3}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4178923320\\-11862180\\1040339520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11862180\\33672\\-2953080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1040339520\\-2953080\\258991667
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3526\\-2321\\-14190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1389\\-925\\-5590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1139\\-769\\-4584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2151\\-1463\\-8657
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
969\\-670\\-3900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-27\\-165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
166\\-105\\-668
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-132\\-594\\-660\\-66\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-110\\-180\\-30\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\24\\0\\-60\\-24\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-594\\-110\\0\\11\\0\\-33\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-180\\-60\\0\\120\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-30\\-24\\-33\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-36\\-88\\-60\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4178923320, -11862180, 1040339520; -11862180, 33672, -2953080; 1040339520, -2953080, 258991667],[660, 12, 1],[3526, 1389, 1139, 2151, 969, 41, 166; -2321, -925, -769, -1463, -670, -27, -105; -14190, -5590, -4584, -8657, -3900, -165, -668],81,12]
%
% END OF LATTICE L82.12
%
\hbox{}\par\smallskip%
%
% LATTICE L82.13
%
\leavevmode%
\llap{\spacer}%
${L_{82.13}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{82.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29432433525\\-1259445\\-11698908375
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1259445\\55\\500610
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11698908375\\500610\\4650123718
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
351535\\-1055506\\884515
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
412895\\-1239738\\1038906
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20897\\62753\\-52580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17956\\53913\\-45180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22414\\-67304\\56397
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
478620\\-1437099\\1204280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1781513\\-5349123\\4482555
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-55\\-135\\-55\\-55\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\33\\0\\-165\\-99\\-165\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\0\\110\\0\\-55\\-220\\-495
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-165\\0\\15\\0\\-165\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-99\\-55\\0\\22\\0\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-165\\-220\\-165\\0\\55\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-495\\-660\\-165\\0\\330
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[29432433525, -1259445, -11698908375; -1259445, 55, 500610; -11698908375, 500610, 4650123718],[165, 55, 1],[351535, 412895, -20897, -17956, 22414, 478620, 1781513; -1055506, -1239738, 62753, 53913, -67304, -1437099, -5349123; 884515, 1038906, -52580, -45180, 56397, 1204280, 4482555],81,13]
%
% END OF LATTICE L82.13
%
\hbox{}\par\smallskip%
%
% LATTICE L82.14
%
\leavevmode%
\llap{\spacer}%
${L_{82.14}}=\pdualwitharg{3.5}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${660}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{220}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72698340\\-6623100\\231000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6623100\\-603390\\21045
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
231000\\21045\\-734
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-220\\-330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\23\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-484\\1540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-51\\111
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-220\\300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-54\\25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
660\\0\\-330\\-5940\\-330\\-660\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-220\\-18\\-60\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\0\\30\\0\\-15\\-120\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5940\\-220\\0\\220\\0\\-660\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-18\\-15\\0\\6\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-60\\-120\\-660\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-45\\-440\\-15\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-72698340, -6623100, 231000; -6623100, -603390, 21045; 231000, 21045, -734],[660, 15, 1],[19, -1, -2, 49, 5, 21, 5; -220, 10, 23, -484, -51, -220, -54; -330, -28, 30, 1540, 111, 300, 25],81,14]
%
% END OF LATTICE L82.14
%
\hbox{}\par\smallskip%
%
% LATTICE L82.15
%
\leavevmode%
\llap{\spacer}%
${L_{82.15}}=\pdualwitharg{2.5}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${220}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31647000\\15180\\7864560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15180\\7480\\4320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7864560\\4320\\1954453
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29874\\8811\\-120230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
407\\120\\-1638
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
651\\192\\-2620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89171\\26301\\-358875
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9605\\2833\\-38656
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9483\\2797\\-38165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28490\\8403\\-114660
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
220\\0\\-220\\-2970\\-220\\-110\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-330\\-36\\-30\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\0\\40\\0\\-20\\-40\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2970\\-330\\0\\165\\0\\-165\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-36\\-20\\0\\8\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\-30\\-40\\-165\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-180\\-1320\\-60\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31647000, 15180, 7864560; 15180, 7480, 4320; 7864560, 4320, 1954453],[660, 20, 1],[29874, 407, 651, 89171, 9605, 9483, 28490; 8811, 120, 192, 26301, 2833, 2797, 8403; -120230, -1638, -2620, -358875, -38656, -38165, -114660],81,15]
%
% END OF LATTICE L82.15
%
\hbox{}\par\smallskip%
%
% LATTICE L82.16
%
\leavevmode%
\llap{\spacer}%
${L_{82.16}}=\pdualwitharg{3.11}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{220}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{l}{2}{132}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
763620\\302280\\7920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
302280\\119658\\3135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7920\\3135\\82
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-22\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-20\\-110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\16\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-48\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-25\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-66\\-108\\-330\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\220\\0\\-220\\-990\\-660\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\0\\66\\0\\-165\\-264\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-220\\0\\4\\0\\-132\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-990\\-165\\0\\330\\0\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-660\\-264\\-132\\0\\132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-110\\-99\\-88\\-165\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[763620, 302280, 7920; 302280, 119658, 3135; 7920, 3135, 82],[660, 33, 1],[9, 9, -6, -3, 2, 19, 10; -22, -20, 16, 8, -5, -48, -25; -30, -110, -33, -16, 0, 0, -11],81,16]
%
% END OF LATTICE L82.16
%
\hbox{}\par\smallskip%
%
% LATTICE L82.17
%
\leavevmode%
\llap{\spacer}%
${L_{82.17}}=\pdualwitharg{3.5.11}(\pfillwitharg{2}({L_{82.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46415705325\\7478178345\\2728414590
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7478178345\\-1204832310\\-439583265
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2728414590\\-439583265\\-160382026
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\-4939477\\13539375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-1921723\\5267548
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\353411\\-968715
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\83072\\-227705
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-344694\\944823
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\-6807968\\18661005
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\-8384997\\22983730
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-165\\-135\\-165\\-165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\0\\-55\\-99\\-165\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\0\\330\\0\\-165\\-660\\-495
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-55\\0\\5\\0\\-165\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-99\\-165\\0\\66\\0\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-165\\-660\\-165\\0\\165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-55\\-495\\-220\\-165\\0\\110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-46415705325, 7478178345, 2728414590; 7478178345, -1204832310, -439583265; 2728414590, -439583265, -160382026],[165, 165, 1],[59, 23, -4, -1, 4, 81, 100; -4939477, -1921723, 353411, 83072, -344694, -6807968, -8384997; 13539375, 5267548, -968715, -227705, 944823, 18661005, 22983730],81,17]
%
% END OF LATTICE L82.17
%
\hbox{}\par\smallskip%
%
% LATTICE L82.18
%
\leavevmode%
\llap{\spacer}%
${L_{82.18}}=\pdualwitharg{2.11}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{660}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{440}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{264}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50591640\\1596540\\12561780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1596540\\46376\\396440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12561780\\396440\\3119059
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5310\\123\\21370
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44033\\1020\\177210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3105\\72\\12496
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1429\\-33\\-5751
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14049\\-325\\-56540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\1\\165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17482\\405\\70356
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-44\\-54\\-220\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\660\\0\\-330\\-1980\\-330\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\0\\88\\0\\-220\\-88\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-330\\0\\3\\0\\-33\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-1980\\-220\\0\\440\\0\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-330\\-88\\-33\\0\\11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-660\\-396\\-264\\-660\\0\\264
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[50591640, 1596540, 12561780; 1596540, 46376, 396440; 12561780, 396440, 3119059],[660, 44, 1],[-5310, -44033, -3105, 1429, 14049, -41, -17482; 123, 1020, 72, -33, -325, 1, 405; 21370, 177210, 12496, -5751, -56540, 165, 70356],81,18]
%
% END OF LATTICE L82.18
%
\hbox{}\par\smallskip%
%
% LATTICE L82.19
%
\leavevmode%
\llap{\spacer}%
${L_{82.19}}=\pdualwitharg{5.11}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{220}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-214500\\-19140\\660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19140\\-1210\\55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
660\\55\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\12\\1920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\407
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-6\\-825
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-110\\-540\\-110\\-220\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\132\\0\\-660\\-198\\-660\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\0\\110\\0\\-55\\-440\\-495
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\-660\\0\\60\\0\\-660\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\-198\\-55\\0\\22\\0\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-660\\-440\\-660\\0\\220\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-330\\-495\\-1320\\-165\\0\\330
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-214500, -19140, 660; -19140, -1210, 55; 660, 55, -2],[660, 55, 1],[-1, -1, 1, 5, 1, 1, -2; -4, -6, 1, 12, 3, 4, -6; -450, -528, 330, 1920, 407, 440, -825],81,19]
%
% END OF LATTICE L82.19
%
\hbox{}\par\smallskip%
%
% LATTICE L82.20
%
\leavevmode%
\llap{\spacer}%
${L_{82.20}}=\pdualwitharg{2.3.5}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${660}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20918040\\306900\\5195520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
306900\\38040\\76020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5195520\\76020\\1290439
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51082\\-1265\\-205590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\-2\\-326
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1133\\-28\\-4560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55086\\-1364\\-221705
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17809\\-441\\-71676
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17364\\-430\\-69885
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17000\\-421\\-68420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
660\\0\\-660\\-2970\\-660\\-330\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-110\\-36\\-30\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\0\\120\\0\\-60\\-120\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2970\\-110\\0\\55\\0\\-165\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-36\\-60\\0\\24\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-30\\-120\\-165\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-180\\-440\\-60\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20918040, 306900, 5195520; 306900, 38040, 76020; 5195520, 76020, 1290439],[660, 60, 1],[51082, 81, 1133, 55086, 17809, 17364, 17000; -1265, -2, -28, -1364, -441, -430, -421; -205590, -326, -4560, -221705, -71676, -69885, -68420],81,20]
%
% END OF LATTICE L82.20
%
\hbox{}\par\smallskip%
%
% LATTICE L82.21
%
\leavevmode%
\llap{\spacer}%
${L_{82.21}}=\pdualwitharg{2.3.11}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{220}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{1320}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-658680\\-60060\\-163020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60060\\21912\\-15180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-163020\\-15180\\-40343
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1961\\-91\\-7890
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5714\\-265\\-22990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1706\\-79\\-6864
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\2\\173
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4101\\190\\16500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-2\\-165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2045\\-95\\-8228
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-132\\-54\\-660\\-66\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\220\\0\\-110\\-1980\\-330\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\264\\0\\-660\\-264\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-110\\0\\1\\0\\-33\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-1980\\-660\\0\\1320\\0\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-330\\-264\\-33\\0\\33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-220\\-396\\-88\\-660\\0\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-658680, -60060, -163020; -60060, 21912, -15180; -163020, -15180, -40343],[660, 132, 1],[1961, 5714, 1706, -43, -4101, 41, 2045; -91, -265, -79, 2, 190, -2, -95; -7890, -22990, -6864, 173, 16500, -165, -8228],81,21]
%
% END OF LATTICE L82.21
%
\hbox{}\par\smallskip%
%
% LATTICE L82.22
%
\leavevmode%
\llap{\spacer}%
${L_{82.22}}=\pdualwitharg{3.5.11}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}{330}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{660}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16500\\21120\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21120\\26070\\-825
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-825\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\4\\260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\4\\231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\12\\660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-330\\-540\\-330\\-660\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\44\\0\\-220\\-198\\-660\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\0\\330\\0\\-165\\-1320\\-495
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\-220\\0\\20\\0\\-660\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-198\\-165\\0\\66\\0\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-660\\-1320\\-660\\0\\660\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-110\\-495\\-440\\-165\\0\\110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16500, 21120, -660; 21120, 26070, -825; -660, -825, 26],[660, 165, 1],[-1, -1, 1, 5, 4, 11, 1; -2, -2, -1, 4, 4, 12, 1; -90, -88, 0, 260, 231, 660, 55],81,22]
%
% END OF LATTICE L82.22
%
\hbox{}\par\smallskip%
%
% LATTICE L82.23
%
\leavevmode%
\llap{\spacer}%
${L_{82.23}}=\pdualwitharg{2.5.11}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{*}{2}{440}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{1320}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
319296120\\833580\\-78714240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
833580\\2200\\-205480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78714240\\-205480\\19404983
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-728\\2177\\-2930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2345\\7011\\-9438
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3881\\11603\\-15620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2143\\6408\\-8625
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-809\\2420\\-3256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\124\\-165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-164\\495\\-660
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-220\\-270\\-220\\-110\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\132\\0\\-330\\-396\\-330\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\0\\440\\0\\-220\\-440\\-1980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\-330\\0\\15\\0\\-165\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-396\\-220\\0\\88\\0\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\-330\\-440\\-165\\0\\55\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-660\\-1980\\-1320\\-660\\0\\1320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[319296120, 833580, -78714240; 833580, 2200, -205480; -78714240, -205480, 19404983],[660, 220, 1],[-728, -2345, -3881, -2143, -809, -41, -164; 2177, 7011, 11603, 6408, 2420, 124, 495; -2930, -9438, -15620, -8625, -3256, -165, -660],81,23]
%
% END OF LATTICE L82.23
%
\hbox{}\par\smallskip%
%
% LATTICE L82.24
%
\leavevmode%
\llap{\spacer}%
${L_{82.24}}=\pdualwitharg{2.3.5.11}({L_{82.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{1320}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{264}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{440}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6600\\-308220\\-76560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308220\\14201880\\3527700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76560\\3527700\\876269
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-67\\270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\71\\-286
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\983\\-3960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\247\\-995
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\426\\-1716
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\41\\-165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-273\\1100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-660\\-270\\-660\\-330\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\44\\0\\-110\\-396\\-330\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\0\\1320\\0\\-660\\-1320\\-1980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\-110\\0\\5\\0\\-165\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-396\\-660\\0\\264\\0\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-330\\-1320\\-165\\0\\165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-220\\-1980\\-440\\-660\\0\\440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6600, -308220, -76560; -308220, 14201880, 3527700; -76560, 3527700, 876269],[660, 660, 1],[3, -2, -30, -7, -11, 1, 11; -67, 71, 983, 247, 426, 41, -273; 270, -286, -3960, -995, -1716, -165, 1100],81,24]
%
% END OF LATTICE L82.24
%

\medskip%
%------------------------------WEYL-GROUP-W83--------------------------------
%
\leavevmode\llap{}%
$W_{83\phantom{0}}$%
\qquad\llap{16} lattices, $\chi=64$%
\hfill%
$23222222322222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L83.1
%
\leavevmode%
\llap{\spacer}%
${L_{83.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{83}}%
\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{-}{3}{2}\above{1pt}{1pt}{l}{2}{136}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{34}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156619640\\493680\\-116960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
493680\\-1538\\349
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116960\\349\\-66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1845791\\749136960\\690421680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5510\\-2236301\\-2061025
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1044\\423720\\390509
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-713\\-289380\\-266700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-181\\-73461\\-67703
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-20293\\-18702
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3175\\-1288600\\-1187552
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-309\\-125410\\-115575
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-97\\-39368\\-36280
\end{tabular}\endgroup%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-156619640, 493680, -116960; 493680, -1538, 349; -116960, 349, -66],[680, 1, 1],[-713, -181, -50, -3175, -309, -97, -87, 17, 7, -8, -4713, -619, -367, -899; -289380, -73461, -20293, -1288600, -125410, -39368, -35309, 6900, 2841, -3247, -1912840, -251230, -148952, -364871; -266700, -67703, -18702, -1187552, -115575, -36280, -32538, 6360, 2618, -2993, -1762968, -231545, -137280, -336277],82,1]
%
% END OF LATTICE L83.1
%
\hbox{}\par\smallskip%
%
% LATTICE L83.2
%
\leavevmode%
\llap{\spacer}%
${L_{83.2}}=\pfillwitharg{2}({L_{83.1}})$%
{${}={}$}Nikulin $83$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-10710, 2210, 170; 2210, -456, -35; 170, -35, -2],[170, 1, 1],[-237, -83, 1, 7, -2, -7, -109, -87, -79, -55, -3679, -862, -209, -809; -1140, -399, 5, 34, -10, -34, -527, -420, -381, -265, -17714, -4150, -1006, -3893; -400, -146, -3, 0, 5, -4, -119, -110, -109, -82, -5780, -1365, -336, -1326],82,2]
%
% END OF LATTICE L83.2
%
\hbox{}\par\smallskip%
%
% LATTICE L83.3
%
\leavevmode%
\llap{\spacer}%
${L_{83.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{83.1}}))$%
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{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-522580, 5270, -259760; 5270, -48, 2618; -259760, 2618, -129119],[170, 2, 1],[2068, 2114, 595, 9620, 3771, 302, 1143, -37, -83, 82, 13398, 7061, 1052, 10366; -1185, -1205, -334, -5304, -2065, -162, -578, 30, 50, -51, -7786, -4095, -608, -5967; -4185, -4278, -1204, -19465, -7630, -611, -2312, 75, 168, -166, -27115, -14290, -2129, -20978],82,3]
%
% END OF LATTICE L83.3
%
\hbox{}\par\smallskip%
%
% LATTICE L83.4
%
\leavevmode%
\llap{\spacer}%
${L_{83.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{83.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-630190, 8840, 127160; 8840, -120, -1785; 127160, -1785, -25658],[170, 5, 1],[-347, -889, -252, -8215, -323, -261, -506, 5, 34, -33, -11165, -589, -879, -4339; -474, -1205, -334, -10608, -413, -324, -578, 12, 50, -51, -15572, -819, -1216, -5967; -1686, -4320, -1225, -39950, -1571, -1270, -2465, 24, 165, -160, -54230, -2861, -4270, -21080],82,4]
%
% END OF LATTICE L83.4
%
\hbox{}\par\smallskip%
%
% LATTICE L83.5
%
\leavevmode%
\llap{\spacer}%
${L_{83.5}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{83.1}}))$%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14808020, -466990, 7255770; -466990, 14740, -228820; 7255770, -228820, 3555249],[170, 10, 1],[-13705, -74497, -24584, -465628, -38511, -17818, -92484, -1016, 892, 49, -368562, -40001, -31252, -324611; -109, -592, -195, -3689, -305, -141, -731, -8, 7, 0, -2941, -319, -249, -2584; 27963, 152000, 50160, 950045, 78576, 36355, 188700, 2073, -1820, -100, 751995, 81616, 63765, 662320],82,5]
%
% END OF LATTICE L83.5
%
\hbox{}\par\smallskip%
%
% LATTICE L83.6
%
\leavevmode%
\llap{\spacer}%
${L_{83.6}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{83.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1448230, -17340, -427380; -17340, -136, -5117; -427380, -5117, -126122],[170, 17, 1],[14349, 7300, 2027, 3795, 12569, 1977, 209, -351, -301, 306, 5537, 24759, 7355, 2124; -20, -14, -7, -20, -75, -16, -3, -10, -1, 2, 0, -5, -4, -2; -48620, -24735, -6868, -12858, -42585, -6698, -708, 1190, 1020, -1037, -18762, -83895, -24922, -7197],82,6]
%
% END OF LATTICE L83.6
%
\hbox{}\par\smallskip%
%
% LATTICE L83.7
%
\leavevmode%
\llap{\spacer}%
${L_{83.7}}=\pdualwitharg{5}({L_{83.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{+}{3}{10}\above{1pt}{1pt}{l}{2}{680}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{r}{2}{170}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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-36040\\555\\-38
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30033560, 484840, -36040; 484840, -7690, 555; -36040, 555, -38],[680, 5, 1],[-97, -127, -38, -2563, -51, -85, -87, 1, 7, -2, -2877, -77, -235, -593; -9688, -12684, -3795, -255952, -5093, -8488, -8687, 100, 699, -200, -287368, -7691, -23472, -59228; -49508, -64815, -19390, -1307640, -26019, -43360, -44370, 512, 3570, -1025, -1468800, -39309, -119960, -302685],82,7]
%
% END OF LATTICE L83.7
%
\hbox{}\par\smallskip%
%
% LATTICE L83.8
%
\leavevmode%
\llap{\spacer}%
${L_{83.8}}=\pdualwitharg{2}({L_{83.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{+}{3}{16}\above{1pt}{1pt}{l}{2}{17}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{272}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-1377680\\-132600\\-342040
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\kern3pt%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1377680, -132600, -342040; -132600, -12048, -32912; -342040, -32912, -84919],[680, 8, 1],[-294, -338, 589, 36571, 38197, 2804, 54401, 10671, 10715, 2870, 22232, 17147, 655, 4402; -15, -17, 30, 1853, 1935, 142, 2754, 540, 542, 145, 1122, 865, 33, 221; 1190, 1368, -2384, -148019, -154600, -11349, -220184, -43190, -43368, -11616, -89981, -69400, -2651, -17816],82,8]
%
% END OF LATTICE L83.8
%
\hbox{}\par\smallskip%
%
% LATTICE L83.9
%
\leavevmode%
\llap{\spacer}%
${L_{83.9}}=\pdualwitharg{2.17}(\pfillwitharg{2}({L_{83.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1311305540, 280330, 650801990; 280330, 68, 139128; 650801990, 139128, 322993549],[170, 34, 1],[430082, 467180, 153893, 171268, 1203639, 111294, 33945, 31597, -5501, 0, 136518, 1258649, 196492, 119948; -600, -653, -216, -241, -1695, -157, -48, -45, 8, 1, -189, -1745, -273, -167; -866575, -941324, -310080, -345089, -2425220, -224247, -68396, -63665, 11084, 0, -275071, -2536060, -395913, -241684],82,9]
%
% END OF LATTICE L83.9
%
\hbox{}\par\smallskip%
%
% LATTICE L83.10
%
\leavevmode%
\llap{\spacer}%
${L_{83.10}}=\pdualwitharg{17}({L_{83.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-143480, 70040, -148240; 70040, -26146, 49453; -148240, 49453, -87906],[680, 17, 1],[1457, 595, -686, -23673, -52849, -31327, -4513, -60833, -15439, -4262, -15911, -26319, -8257, -435; 11620, 4745, -5471, -188792, -421470, -249832, -35991, -485140, -123125, -33989, -126888, -209890, -65848, -3469; 4080, 1666, -1921, -66288, -147985, -87720, -12637, -170340, -43231, -11934, -44552, -73695, -23120, -1218],82,10]
%
% END OF LATTICE L83.10
%
\hbox{}\par\smallskip%
%
% LATTICE L83.11
%
\leavevmode%
\llap{\spacer}%
${L_{83.11}}=\pdualwitharg{5.17}(\pfillwitharg{2}({L_{83.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1269344610, -436050, 257061760; -436050, 170, -88315; 257061760, -88315, 52058954],[170, 85, 1],[109205, 296562, 97690, 217439, 152812, 141297, 21548, 8023, -3492, 0, 173321, 159796, 249463, 76142; -218650, -593777, -195596, -435360, -305963, -282908, -43144, -16064, 6992, 1, -347020, -319941, -499472, -152451; -539614, -1465400, -482715, -1074430, -755089, -698190, -106475, -39644, 17255, 0, -856430, -789599, -1232670, -376240],82,11]
%
% END OF LATTICE L83.11
%
\hbox{}\par\smallskip%
%
% LATTICE L83.12
%
\leavevmode%
\llap{\spacer}%
${L_{83.12}}=\pdualwitharg{2.5}({L_{83.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-970074510160, -227319932920, -3788362280; -227319932920, -53268435920, -887736200; -3788362280, -887736200, -14794419],[680, 40, 1],[-209, -239, 2090, 89861, 18453, 6584, 123798, 4688, 22634, 5375, 37044, 5435, 881, 3107; 0, 2, -1, -136, -29, -11, -221, -9, -47, -14, -119, -19, -4, -34; 53518, 61080, -535120, -23002275, -4723464, -1685285, -31687320, -1199902, -5793000, -1375520, -9478605, -1390584, -225355, -793560],82,12]
%
% END OF LATTICE L83.12
%
\hbox{}\par\smallskip%
%
% LATTICE L83.13
%
\leavevmode%
\llap{\spacer}%
${L_{83.13}}=\pdualwitharg{2.5.17}(\pfillwitharg{2}({L_{83.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L83.13
%
\hbox{}\par\smallskip%
%
% LATTICE L83.14
%
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\llap{\spacer}%
${L_{83.14}}=\pdualwitharg{5.17}({L_{83.1}})$%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-641240, 72080, 14960; 72080, -6630, -1615; 14960, -1615, -346],[680, 85, 1],[-449, -586, -174, -687, -232, -385, -23, 5, 31, -11, -793, -360, -1095, -162; 1104, 1442, 429, 1696, 573, 952, 57, -12, -77, 26, 1944, 883, 2688, 398; -24548, -32045, -9520, -37600, -12699, -21080, -1260, 272, 1700, -595, -43320, -19669, -59840, -8855],82,14]
%
% END OF LATTICE L83.14
%
\hbox{}\par\smallskip%
%
% LATTICE L83.15
%
\leavevmode%
\llap{\spacer}%
${L_{83.15}}=\pdualwitharg{2.17}({L_{83.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-4489360\\3529880\\1783640
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3529880\\-2775216\\-1402432
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4489360, 3529880, 1783640; 3529880, -2775216, -1402432; 1783640, -1402432, -708647],[680, 136, 1],[-17879, -17999, -4857, -2229, -29308, -1129, -457, 461, 581, -949, -3577, -63588, -4677, -5349; 540, 542, 145, 66, 865, 33, 13, -15, -17, 30, 109, 1935, 142, 162; -46070, -46376, -12512, -5741, -75480, -2907, -1176, 1190, 1496, -2448, -9219, -163880, -12053, -13784],82,15]
%
% END OF LATTICE L83.15
%
\hbox{}\par\smallskip%
%
% LATTICE L83.16
%
\leavevmode%
\llap{\spacer}%
${L_{83.16}}=\pdualwitharg{2.5.17}({L_{83.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-521708240\\-183723080\\1529320
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-521708240, -183723080, 1529320; -183723080, -64699280, 538560; 1529320, 538560, -4483],[680, 680, 1],[-9, -47, -14, -7, -19, -4, -2, 0, 2, -1, -8, -29, -11, -13; 336, 1622, 385, 156, 389, 63, 13, -15, -17, 150, 379, 1323, 472, 522; 37298, 178840, 41480, 16355, 40256, 6205, 880, -1802, -1360, 17680, 42805, 149056, 52955, 58280],82,16]
%
% END OF LATTICE L83.16
%

\medskip%
%------------------------------WEYL-GROUP-W84--------------------------------
%
\leavevmode\llap{}%
$W_{84\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=12$%
\hfill%
$222222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L84.1
%
\leavevmode%
\llap{\spacer}%
${L_{84.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{84}}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{156}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11003460\\-3664440\\15600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3664440\\-1220354\\5195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15600\\5195\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\-118\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\-130\\-208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\115\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-361\\1092\\1872
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
451\\-1365\\-2535
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-10\\-156\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\52\\0\\-468\\-26\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\0\\-5\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-468\\0\\156\\0\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-26\\-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-390\\-195\\-1560\\0\\390
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-11003460, -3664440, 15600; -3664440, -1220354, 5195; 15600, 5195, -22],[780, 1, 1],[39, 43, -38, -361, -1, 451; -118, -130, 115, 1092, 3, -1365; -210, -208, 210, 1872, -1, -2535],83,1]
%
% END OF LATTICE L84.1
%
\hbox{}\par\smallskip%
%
% LATTICE L84.2
%
\leavevmode%
\llap{\spacer}%
${L_{84.2}}=\pfillwitharg{2}({L_{84.1}})$%
{${}={}$}Nikulin $84$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
390\\-195\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\97\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\13\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-15\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145\\-312\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-28\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-181\\-390\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-5\\-39\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\13\\0\\-117\\-13\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\10\\0\\-5\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-117\\0\\39\\0\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-13\\-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-195\\-195\\-780\\0\\390
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[390, -195, 0; -195, 97, 0; 0, 0, 1],[195, 1, 1],[0, 6, -7, -145, -13, -181; 0, 13, -15, -312, -28, -390; 1, 0, -5, -39, -2, 0],83,2]
%
% END OF LATTICE L84.2
%
\hbox{}\par\smallskip%
%
% LATTICE L84.3
%
\leavevmode%
\llap{\spacer}%
${L_{84.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{84.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{130}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4849845\\25155\\1617915
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25155\\-123\\-8394
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1617915\\-8394\\-539738
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5853\\-5260\\-17463
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19568\\-17589\\-58383
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1644\\-1480\\-4905
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
854\\767\\2548
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
365\\329\\1089
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36099\\-32435\\-107705
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-15\\-39\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\39\\0\\-117\\-39\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\0\\-15\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-117\\0\\13\\0\\-260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-39\\-15\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-195\\-195\\-260\\0\\130
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4849845, 25155, 1617915; 25155, -123, -8394; 1617915, -8394, -539738],[195, 3, 1],[-5853, -19568, -1644, 854, 365, -36099; -5260, -17589, -1480, 767, 329, -32435; -17463, -58383, -4905, 2548, 1089, -107705],83,3]
%
% END OF LATTICE L84.3
%
\hbox{}\par\smallskip%
%
% LATTICE L84.4
%
\leavevmode%
\llap{\spacer}%
${L_{84.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{84.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{65}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{195}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7741695\\40755\\1549665
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40755\\-205\\-8160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1549665\\-8160\\-310198
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5021\\-5260\\-24945
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16786\\-17589\\-83395
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-282\\-296\\-1401
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2198\\2301\\10920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
313\\329\\1555
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18581\\-19461\\-92313
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-5\\-195\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\65\\0\\-585\\-65\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\0\\-5\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\-585\\0\\195\\0\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-65\\-5\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-195\\-39\\-780\\0\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7741695, 40755, 1549665; 40755, -205, -8160; 1549665, -8160, -310198],[195, 5, 1],[-5021, -16786, -282, 2198, 313, -18581; -5260, -17589, -296, 2301, 329, -19461; -24945, -83395, -1401, 10920, 1555, -92313],83,4]
%
% END OF LATTICE L84.4
%
\hbox{}\par\smallskip%
%
% LATTICE L84.5
%
\leavevmode%
\llap{\spacer}%
${L_{84.5}}=\pdualwitharg{3}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{52}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{130}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693420\\-780\\3120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\6\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3120\\3\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-26\\-234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\15\\225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\52\\676
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-65\\-910
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-30\\-156\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\156\\0\\-468\\-78\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\0\\-15\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-468\\0\\52\\0\\-520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-78\\-15\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-390\\-195\\-520\\0\\130
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-693420, -780, 3120; -780, 6, 3; 3120, 3, -14],[780, 3, 1],[-1, -1, 1, 3, 0, -4; -18, -26, 15, 52, 1, -65; -228, -234, 225, 676, 0, -910],83,5]
%
% END OF LATTICE L84.5
%
\hbox{}\par\smallskip%
%
% LATTICE L84.6
%
\leavevmode%
\llap{\spacer}%
${L_{84.6}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{84.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${13}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19293495\\103155\\8906040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
103155\\-533\\-47619
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8906040\\-47619\\-4111103
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25358\\-5260\\-54873
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6521\\-1353\\-14111
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7119\\-1480\\-15405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
854\\177\\1848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1580\\329\\3419
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36094\\-7485\\-78105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\0\\-65\\-39\\-26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-9\\-13\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\0\\130\\0\\-65\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-9\\0\\3\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-13\\-65\\0\\26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-195\\-60\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19293495, 103155, 8906040; 103155, -533, -47619; 8906040, -47619, -4111103],[195, 13, 1],[-25358, -6521, -7119, 854, 1580, -36094; -5260, -1353, -1480, 177, 329, -7485; -54873, -14111, -15405, 1848, 3419, -78105],83,6]
%
% END OF LATTICE L84.6
%
\hbox{}\par\smallskip%
%
% LATTICE L84.7
%
\leavevmode%
\llap{\spacer}%
${L_{84.7}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{84.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{195}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{65}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
150810270\\-6193395\\-52222365
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6193395\\254355\\2144640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52222365\\2144640\\18083486
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39501\\-271\\114105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
129880\\-897\\375180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1914\\-14\\5529
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5828\\39\\-16835
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1807\\14\\-5220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49585\\-338\\143234
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\-195\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\195\\0\\-585\\-195\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\6\\0\\-15\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\-585\\0\\65\\0\\-260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-195\\-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-195\\-39\\-260\\0\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[150810270, -6193395, -52222365; -6193395, 254355, 2144640; -52222365, 2144640, 18083486],[195, 15, 1],[39501, 129880, 1914, -5828, -1807, 49585; -271, -897, -14, 39, 14, -338; 114105, 375180, 5529, -16835, -5220, 143234],83,7]
%
% END OF LATTICE L84.7
%
\hbox{}\par\smallskip%
%
% LATTICE L84.8
%
\leavevmode%
\llap{\spacer}%
${L_{84.8}}=\pdualwitharg{2}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{1560}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2415143640\\-3986580\\-605564700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3986580\\6584\\999580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-605564700\\999580\\151837183
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-877\\-1169\\-3490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2045\\-2730\\-8138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-779\\-1040\\-3100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7262\\-9672\\-28899
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1365\\-1817\\-5432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36457\\-48555\\-145080
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-78\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\52\\0\\-234\\-52\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\40\\0\\-20\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-234\\0\\39\\0\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-52\\-20\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-780\\-780\\-1560\\0\\1560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2415143640, -3986580, -605564700; -3986580, 6584, 999580; -605564700, 999580, 151837183],[780, 4, 1],[-877, -2045, -779, -7262, -1365, -36457; -1169, -2730, -1040, -9672, -1817, -48555; -3490, -8138, -3100, -28899, -5432, -145080],83,8]
%
% END OF LATTICE L84.8
%
\hbox{}\par\smallskip%
%
% LATTICE L84.9
%
\leavevmode%
\llap{\spacer}%
${L_{84.9}}=\pdualwitharg{5}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{260}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{780}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
244140\\-1560\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\10\\5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\5\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-138\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-416\\-130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\156\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-273\\-78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-10\\-780\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\260\\0\\-2340\\-130\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\2\\0\\-5\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\-2340\\0\\780\\0\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-130\\-5\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-390\\-39\\-1560\\0\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[244140, -1560, -780; -1560, 10, 5; -780, 5, 2],[780, 5, 1],[-1, -3, 0, 1, 0, -2; -138, -416, 0, 156, 1, -273; -40, -130, -1, 0, 0, -78],83,9]
%
% END OF LATTICE L84.9
%
\hbox{}\par\smallskip%
%
% LATTICE L84.10
%
\leavevmode%
\llap{\spacer}%
${L_{84.10}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{84.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${39}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
266853990\\13284180\\-85409415
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13284180\\661323\\-4251741
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85409415\\-4251741\\27336178
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88449\\-271\\-276393
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22388\\-69\\-69960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21591\\-70\\-67470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1001\\3\\3128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4106\\14\\12831
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42672\\-130\\-133345
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\0\\-195\\-39\\-78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-9\\-39\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\0\\390\\0\\-195\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-9\\0\\1\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-39\\-195\\0\\78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-195\\-20\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[266853990, 13284180, -85409415; 13284180, 661323, -4251741; -85409415, -4251741, 27336178],[195, 39, 1],[-88449, -22388, -21591, 1001, 4106, -42672; -271, -69, -70, 3, 14, -130; -276393, -69960, -67470, 3128, 12831, -133345],83,10]
%
% END OF LATTICE L84.10
%
\hbox{}\par\smallskip%
%
% LATTICE L84.11
%
\leavevmode%
\llap{\spacer}%
${L_{84.11}}=\pdualwitharg{2.3}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{156}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{520}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
299125320\\-347100\\-75171720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-347100\\408\\87228
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75171720\\87228\\18891037
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-481\\-2\\-1914
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2019\\-13\\-8034
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-769\\-5\\-3060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-722\\0\\-2873
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-193\\2\\-768
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3659\\0\\-14560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-60\\-78\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\156\\0\\-234\\-156\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\0\\-60\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-234\\0\\13\\0\\-520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-156\\-60\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-780\\-780\\-520\\0\\520
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[299125320, -347100, -75171720; -347100, 408, 87228; -75171720, 87228, 18891037],[780, 12, 1],[-481, -2019, -769, -722, -193, -3659; -2, -13, -5, 0, 2, 0; -1914, -8034, -3060, -2873, -768, -14560],83,11]
%
% END OF LATTICE L84.11
%
\hbox{}\par\smallskip%
%
% LATTICE L84.12
%
\leavevmode%
\llap{\spacer}%
${L_{84.12}}=\pdualwitharg{13}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{130}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22207380\\7417800\\56160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7417800\\-2477722\\-18759
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56160\\-18759\\-142
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\-130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-14\\-128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\120\\1152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\84\\767
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\90\\765
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\0\\-130\\-156\\-52\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-36\\-26\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\0\\130\\0\\-65\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-36\\0\\12\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-26\\-65\\0\\26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-195\\-120\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-22207380, 7417800, 56160; 7417800, -2477722, -18759; 56160, -18759, -142],[780, 13, 1],[-3, -5, 2, 43, 30, 32; -8, -14, 5, 120, 84, 90; -130, -128, 130, 1152, 767, 765],83,12]
%
% END OF LATTICE L84.12
%
\hbox{}\par\smallskip%
%
% LATTICE L84.13
%
\leavevmode%
\llap{\spacer}%
${L_{84.13}}=\pdualwitharg{3.5}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{780}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{260}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3785340\\-5460\\13260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5460\\30\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13260\\15\\-46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-34\\-300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\546\\5070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\65\\597
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\884\\8060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-65\\-598
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-30\\-780\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\780\\0\\-2340\\-390\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\6\\0\\-15\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\-2340\\0\\260\\0\\-520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-390\\-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-390\\-39\\-520\\0\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3785340, -5460, 13260; -5460, 30, 15; 13260, 15, -46],[780, 15, 1],[-1, 17, 2, 27, 0, -2; -34, 546, 65, 884, 1, -65; -300, 5070, 597, 8060, 0, -598],83,13]
%
% END OF LATTICE L84.13
%
\hbox{}\par\smallskip%
%
% LATTICE L84.14
%
\leavevmode%
\llap{\spacer}%
${L_{84.14}}=\pdualwitharg{5.13}(\pfillwitharg{2}({L_{84.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${65}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1581898695\\-41755155\\334752990
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41755155\\1102205\\-8836035
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
334752990\\-8836035\\70838666
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
184181\\-552814\\-939315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46620\\-139929\\-237760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8993\\-26993\\-45864
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6253\\18768\\31890
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8552\\25670\\43615
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53314\\-160020\\-271899
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\0\\-65\\-195\\-130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-45\\-65\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\0\\26\\0\\-65\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\-45\\0\\15\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-65\\-65\\0\\130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-39\\-60\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1581898695, -41755155, 334752990; -41755155, 1102205, -8836035; 334752990, -8836035, 70838666],[195, 65, 1],[184181, 46620, 8993, -6253, -8552, 53314; -552814, -139929, -26993, 18768, 25670, -160020; -939315, -237760, -45864, 31890, 43615, -271899],83,14]
%
% END OF LATTICE L84.14
%
\hbox{}\par\smallskip%
%
% LATTICE L84.15
%
\leavevmode%
\llap{\spacer}%
${L_{84.15}}=\pdualwitharg{2.5}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{260}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{195}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{s}{2}{312}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-301080\\-780\\75660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\3080\\200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75660\\200\\-19013
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
299\\-2\\1190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2254\\-13\\8970
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
194\\-1\\772
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\0\\195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-387\\2\\-1540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-353\\0\\-1404
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-390\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\260\\0\\-1170\\-260\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\8\\0\\-20\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390\\-1170\\0\\195\\0\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-260\\-20\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-780\\-156\\-1560\\0\\312
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-301080, -780, 75660; -780, 3080, 200; 75660, 200, -19013],[780, 20, 1],[299, 2254, 194, 49, -387, -353; -2, -13, -1, 0, 2, 0; 1190, 8970, 772, 195, -1540, -1404],83,15]
%
% END OF LATTICE L84.15
%
\hbox{}\par\smallskip%
%
% LATTICE L84.16
%
\leavevmode%
\llap{\spacer}%
${L_{84.16}}=\pdualwitharg{3.13}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{390}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
501540\\-7589400\\584220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7589400\\111247266\\-8562489
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
584220\\-8562489\\659038
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
129\\2062\\26676
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127\\2030\\26262
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-129\\-2065\\-26715
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-381\\-6092\\-78812
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-761\\-12167\\-157404
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-253\\-4045\\-52330
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\-390\\-156\\-156\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-36\\-78\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390\\0\\390\\0\\-195\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-36\\0\\4\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-78\\-195\\0\\78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-195\\-40\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[501540, -7589400, 584220; -7589400, 111247266, -8562489; 584220, -8562489, 659038],[780, 39, 1],[129, 127, -129, -381, -761, -253; 2062, 2030, -2065, -6092, -12167, -4045; 26676, 26262, -26715, -78812, -157404, -52330],83,16]
%
% END OF LATTICE L84.16
%
\hbox{}\par\smallskip%
%
% LATTICE L84.17
%
\leavevmode%
\llap{\spacer}%
${L_{84.17}}=\pdualwitharg{3.5.13}(\pfillwitharg{2}({L_{84.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${195}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28987000770\\78222825135\\-26326845870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78222825135\\-211088079345\\71044267710
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26326845870\\71044267710\\-23910814813
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5260\\7149252\\21247785
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1353\\1838461\\5463960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-296\\401373\\1192893
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\-80260\\-238535
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
329\\-445372\\-1323660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-499\\678414\\2016266
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
195\\0\\-195\\-195\\-390\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-45\\-195\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\0\\78\\0\\-195\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\-45\\0\\5\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390\\-195\\-195\\0\\390\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-39\\-20\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-28987000770, 78222825135, -26326845870; 78222825135, -211088079345, 71044267710; -26326845870, 71044267710, -23910814813],[195, 195, 1],[-5260, -1353, -296, 59, 329, -499; 7149252, 1838461, 401373, -80260, -445372, 678414; 21247785, 5463960, 1192893, -238535, -1323660, 2016266],83,17]
%
% END OF LATTICE L84.17
%
\hbox{}\par\smallskip%
%
% LATTICE L84.18
%
\leavevmode%
\llap{\spacer}%
${L_{84.18}}=\pdualwitharg{2.13}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{520}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
460906680\\-3882060\\-115924380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3882060\\33176\\976404
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-115924380\\976404\\29156579
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2620\\271\\-10426
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-97\\10\\-386
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6207\\640\\-24700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5287\\546\\-21039
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12806\\1323\\-50960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17852\\1845\\-71040
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\0\\-260\\-78\\-104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-18\\-52\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-260\\0\\520\\0\\-260\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-18\\0\\3\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-52\\-260\\0\\104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-780\\-120\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[460906680, -3882060, -115924380; -3882060, 33176, 976404; -115924380, 976404, 29156579],[780, 52, 1],[-2620, -97, -6207, -5287, -12806, -17852; 271, 10, 640, 546, 1323, 1845; -10426, -386, -24700, -21039, -50960, -71040],83,18]
%
% END OF LATTICE L84.18
%
\hbox{}\par\smallskip%
%
% LATTICE L84.19
%
\leavevmode%
\llap{\spacer}%
${L_{84.19}}=\pdualwitharg{2.3.5}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{780}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{65}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{s}{2}{104}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1814278440\\-15410460\\-455927160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15410460\\130920\\3872640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-455927160\\3872640\\114574241
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-867\\-2\\-3450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12839\\-13\\-51090
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2319\\-1\\-9228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9621\\0\\-38285
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1357\\2\\-5400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-967\\0\\-3848
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-390\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\780\\0\\-1170\\-780\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\24\\0\\-60\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390\\-1170\\0\\65\\0\\-520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-780\\-60\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-780\\-156\\-520\\0\\104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1814278440, -15410460, -455927160; -15410460, 130920, 3872640; -455927160, 3872640, 114574241],[780, 60, 1],[-867, -12839, -2319, -9621, -1357, -967; -2, -13, -1, 0, 2, 0; -3450, -51090, -9228, -38285, -5400, -3848],83,19]
%
% END OF LATTICE L84.19
%
\hbox{}\par\smallskip%
%
% LATTICE L84.20
%
\leavevmode%
\llap{\spacer}%
${L_{84.20}}=\pdualwitharg{5.13}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${260}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60567780\\-34320\\117000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34320\\130\\65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
117000\\65\\-226
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\40\\4670
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\54\\6227
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\168\\19200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-1038
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
260\\0\\-130\\-780\\-260\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-180\\-130\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\0\\26\\0\\-65\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\-180\\0\\60\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-260\\-130\\-65\\0\\130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-39\\-120\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-60567780, -34320, 117000; -34320, 130, 65; 117000, 65, -226],[780, 65, 1],[-1, 9, 12, 37, 0, -2; -6, 40, 54, 168, 1, -9; -520, 4670, 6227, 19200, 0, -1038],83,20]
%
% END OF LATTICE L84.20
%
\hbox{}\par\smallskip%
%
% LATTICE L84.21
%
\leavevmode%
\llap{\spacer}%
${L_{84.21}}=\pdualwitharg{2.3.13}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{1560}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{312}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8674618616040\\-136751229420\\-2179995318540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136751229420\\2155817976\\34366587528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2179995318540\\34366587528\\547848821857
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5953\\-1624\\23790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-485\\129\\-1938
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31037\\8255\\-124020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2461\\641\\-9833
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6873\\1706\\-27456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2001\\-600\\8000
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\-780\\-78\\-312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-18\\-156\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\0\\1560\\0\\-780\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-18\\0\\1\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-156\\-780\\0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-780\\-40\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8674618616040, -136751229420, -2179995318540; -136751229420, 2155817976, 34366587528; -2179995318540, 34366587528, 547848821857],[780, 156, 1],[5953, -485, -31037, -2461, -6873, 2001; -1624, 129, 8255, 641, 1706, -600; 23790, -1938, -124020, -9833, -27456, 8000],83,21]
%
% END OF LATTICE L84.21
%
\hbox{}\par\smallskip%
%
% LATTICE L84.22
%
\leavevmode%
\llap{\spacer}%
${L_{84.22}}=\pdualwitharg{3.5.13}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${780}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\0\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\390\\195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\195\\-682
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\14\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\19\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\20\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
780\\0\\-390\\-780\\-780\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-180\\-390\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390\\0\\78\\0\\-195\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\-180\\0\\20\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\-390\\-195\\0\\390\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-39\\-40\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-780, 0, -780; 0, 390, 195; -780, 195, -682],[780, 195, 1],[-1, 29, 38, 39, 0, -2; -2, 14, 19, 20, 1, -1; 0, -30, -39, -40, 0, 2],83,22]
%
% END OF LATTICE L84.22
%
\hbox{}\par\smallskip%
%
% LATTICE L84.23
%
\leavevmode%
\llap{\spacer}%
${L_{84.23}}=\pdualwitharg{2.5.13}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${260}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{520}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27207674520\\78610740\\6777808440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78610740\\227240\\19582940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6777808440\\19582940\\1688445991
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8265\\-24797\\-32890
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11572\\-34717\\-46050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26840\\-80521\\-106808
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23291\\-69873\\-92685
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8559\\-25675\\-34060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
389\\-1167\\-1548
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
260\\0\\-260\\-390\\-520\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-90\\-260\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-260\\0\\104\\0\\-260\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390\\-90\\0\\15\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-260\\-260\\0\\520\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-156\\-120\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[27207674520, 78610740, 6777808440; 78610740, 227240, 19582940; 6777808440, 19582940, 1688445991],[780, 260, 1],[8265, 11572, 26840, 23291, 8559, 389; -24797, -34717, -80521, -69873, -25675, -1167; -32890, -46050, -106808, -92685, -34060, -1548],83,23]
%
% END OF LATTICE L84.23
%
\hbox{}\par\smallskip%
%
% LATTICE L84.24
%
\leavevmode%
\llap{\spacer}%
${L_{84.24}}=\pdualwitharg{2.3.5.13}({L_{84.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${780}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{1560}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76440\\760500\\-190320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
760500\\7601880\\-1902420
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190320\\-1902420\\476093
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\293\\1170
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\383\\1530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\859\\3432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\239\\955
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\195\\780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
780\\0\\-780\\-390\\-1560\\0
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0\\60\\0\\-90\\-780\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\0\\312\\0\\-780\\-156
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-390\\-90\\0\\5\\0\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-780\\-780\\0\\1560\\0
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0\\-60\\-156\\-40\\0\\8
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[76440, 760500, -190320; 760500, 7601880, -1902420; -190320, -1902420, 476093],[780, 780, 1],[-2, -1, -1, 0, 2, 0; 293, 383, 859, 239, 195, 1; 1170, 1530, 3432, 955, 780, 4],83,24]
%
% END OF LATTICE L84.24
%

\medskip%
%------------------------------WEYL-GROUP-W85--------------------------------
%
\leavevmode\llap{}%
$W_{85\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=42$%
\hfill%
$2224222242\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L85.1
%
\leavevmode%
\llap{\spacer}%
${L_{85.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{85}}%
\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{4}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36481380\\30420\\42900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30420\\-22\\-37
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42900\\-37\\-50
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107639\\26479440\\72728760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-18451\\-50675
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-32472\\-89189
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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11\\2710\\7430
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\7631\\20943
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\11565\\31755
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\10578\\29054
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\3443\\9460
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-60\\-100\\-50\\-120\\-130\\-90\\-40\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\right.$}%
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20\\0\\-60\\-100\\-50
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0\\26\\0\\-52\\-39
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-60\\0\\30\\0\\-15
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\kern3pt%
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-100\\-52\\0\\4\\-2
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\kern3pt%
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-50\\-39\\-15\\-2\\2
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\kern3pt%
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-120\\-130\\-90\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-195\\-195\\-130\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-195\\-255\\-210\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-130\\-210\\-200\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-26\\-60\\-68\\-27
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-36481380, 30420, 42900; 30420, -22, -37; 42900, -37, -50],[780, 1, 1],[11, 31, 47, 43, 14, 19, 8, -2, -1, 1; 2710, 7631, 11565, 10578, 3443, 4670, 1963, -495, -246, 247; 7430, 20943, 31755, 29054, 9460, 12840, 5408, -1350, -676, 675],84,1]
%
% END OF LATTICE L85.1
%
\hbox{}\par\smallskip%
%
% LATTICE L85.2
%
\leavevmode%
\llap{\spacer}%
${L_{85.2}}=\pfillwitharg{2}({L_{85.1}})$%
{${}={}$}Nikulin $85$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2730\\-585\\-390
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-585\\125\\78
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390\\78\\-31
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-18136\\-88335\\6435
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-899\\-4379\\318
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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1\\5\\0
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-40\\-195\\13
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34\\165\\-15
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75\\365\\-28
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99\\482\\-36
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-30\\0\\30\\0\\-15
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-25\\-26\\0\\1\\-1
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-10\\-65\\-105\\-50\\-34
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[2730, -585, -390; -585, 125, 78; -390, 78, -31],[195, 1, 1],[1, -40, 34, 75, 99, 154, 443, 431, 142, 56; 5, -195, 165, 365, 482, 750, 2158, 2100, 692, 273; 0, 13, -15, -28, -36, -55, -156, -150, -49, -19],84,2]
%
% END OF LATTICE L85.2
%
\hbox{}\par\smallskip%
%
% LATTICE L85.3
%
\leavevmode%
\llap{\spacer}%
${L_{85.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{85.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{4}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16016910\\-1635075\\5396235
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\kern3pt%
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-1635075\\-166677\\550875
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16016910, -1635075, 5396235; -1635075, -166677, 550875; 5396235, 550875, -1818038],[195, 3, 1],[15896, 90743, 46203, 63762, 41789, 28744, 25321, -1563, -1266, 2851; -815, -4654, -2370, -3271, -2144, -1475, -1300, 80, 65, -146; 46935, 267930, 136420, 188265, 123387, 84870, 74763, -4615, -3738, 8418],84,3]
%
% END OF LATTICE L85.3
%
\hbox{}\par\smallskip%
%
% LATTICE L85.4
%
\leavevmode%
\llap{\spacer}%
${L_{85.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{85.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13881270, -1964625, 2817555; -1964625, -277795, 398775; 2817555, 398775, -571894],[195, 5, 1],[1874, 53485, 16339, 37580, 24629, 3388, 14921, -553, -746, 1681; -163, -4654, -1422, -3271, -2144, -295, -1300, 48, 65, -146; 9119, 260260, 79506, 182865, 119845, 16486, 72605, -2691, -3630, 8180],84,4]
%
% END OF LATTICE L85.4
%
\hbox{}\par\smallskip%
%
% LATTICE L85.5
%
\leavevmode%
\llap{\spacer}%
${L_{85.5}}=\pdualwitharg{3}({L_{85.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-88140, -14820, -1560; -14820, -2454, -285; -1560, -285, -14],[780, 3, 1],[-89, -311, -171, -493, -169, -251, -131, 1, 17, -1; 450, 1573, 865, 2494, 855, 1270, 663, -5, -86, 5; 750, 2613, 1435, 4134, 1416, 2100, 1092, -10, -144, 9],84,5]
%
% END OF LATTICE L85.5
%
\hbox{}\par\smallskip%
%
% LATTICE L85.6
%
\leavevmode%
\llap{\spacer}%
${L_{85.6}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{85.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14917110, -3282825, 6911385; -3282825, -722267, 1521000; 6911385, 1521000, -3202178],[195, 13, 1],[18821, 8261, 164009, 75437, 49434, 33994, 2302, -5564, -1496, 3381; -815, -358, -7110, -3271, -2144, -1475, -100, 240, 65, -146; 40235, 17660, 350610, 161265, 105677, 72670, 4921, -11895, -3198, 7228],84,6]
%
% END OF LATTICE L85.6
%
\hbox{}\par\smallskip%
%
% LATTICE L85.7
%
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${L_{85.7}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{85.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6889923690, 24716250, -2388778665; 24716250, 88665, -8569275; -2388778665, -8569275, 828204167],[195, 15, 1],[2431, 68165, 6957, 48345, 31991, 4517, 22159, -9, 291, 2749; 512, 14261, 1451, 10064, 6646, 935, 4550, -4, 65, 589; 7017, 196755, 20081, 139545, 92340, 13038, 63960, -26, 840, 7935],84,7]
%
% END OF LATTICE L85.7
%
\hbox{}\par\smallskip%
%
% LATTICE L85.8
%
\leavevmode%
\llap{\spacer}%
${L_{85.8}}=\pdualwitharg{2}({L_{85.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[23660602680, -44527860, -5945665440; -44527860, 83800, 11189392; -5945665440, 11189392, 1494084407],[780, 4, 1],[-13753, -83289, -132047, -62511, -42323, -31107, -33347, -2533, -293, -2537; 45, 247, 375, 173, 114, 80, 78, 0, 2, 11; -54730, -331448, -525480, -248762, -168424, -123790, -132704, -10080, -1166, -10096],84,8]
%
% END OF LATTICE L85.8
%
\hbox{}\par\smallskip%
%
% LATTICE L85.9
%
\leavevmode%
\llap{\spacer}%
${L_{85.9}}=\pdualwitharg{5}({L_{85.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3930420, 1300260, 10140; 1300260, -430150, -3355; 10140, -3355, -26],[780, 5, 1],[63, 163, -1, -21, 2, 23, 396, 130, 623, 213; 186, 481, -3, -62, 6, 68, 1170, 384, 1840, 629; 566, 1495, -3, -190, 5, 194, 3445, 1143, 5510, 1895],84,9]
%
% END OF LATTICE L85.9
%
\hbox{}\par\smallskip%
%
% LATTICE L85.10
%
\leavevmode%
\llap{\spacer}%
${L_{85.10}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{85.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2280616455, -22929075, -731209245; -22929075, 230529, 7351500; -731209245, 7351500, 234439666],[195, 39, 1],[-18793, -8083, -53549, -74361, -49162, -34652, -2606, 104, -462, -4283; 2560, 1097, 7255, 10064, 6646, 4675, 350, -20, 65, 589; -58695, -25245, -167245, -232245, -153543, -108225, -8139, 325, -1443, -13377],84,10]
%
% END OF LATTICE L85.10
%
\hbox{}\par\smallskip%
%
% LATTICE L85.11
%
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${L_{85.11}}=\pdualwitharg{2.3}({L_{85.1}})$%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[471277560, -2949180, -118229280; -2949180, 18456, 739860; -118229280, 739860, 29660149],[780, 12, 1],[731, 6153, 3673, 5597, 4061, 3339, 4429, 397, 101, 9; 155, 1391, 845, 1299, 950, 790, 1066, 100, 24, -5; 2910, 24492, 14620, 22278, 16164, 13290, 17628, 1580, 402, 36],84,11]
%
% END OF LATTICE L85.11
%
\hbox{}\par\smallskip%
%
% LATTICE L85.12
%
\leavevmode%
\llap{\spacer}%
${L_{85.12}}=\pdualwitharg{13}({L_{85.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${260}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{*}{4}{26}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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-2420340\\26520\\307320
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\kern3pt%
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26520\\-286\\-3445
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2987\\161865\\9555
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2739\\148422\\8762
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2420340, 26520, 307320; 26520, -286, -3445; 307320, -3445, -37706],[780, 13, 1],[691, 151, 2987, 2739, 894, 1219, 40, -122, -65, 61; 37450, 8183, 161865, 148422, 48443, 66050, 2167, -6615, -3522, 3307; 2210, 483, 9555, 8762, 2860, 3900, 128, -390, -208, 195],84,12]
%
% END OF LATTICE L85.12
%
\hbox{}\par\smallskip%
%
% LATTICE L85.13
%
\leavevmode%
\llap{\spacer}%
${L_{85.13}}=\pdualwitharg{3.5}({L_{85.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{390}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{4}{30}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-471900, 12480, 6240; 12480, -330, -165; 6240, -165, -82],[780, 15, 1],[1, 18, 2, 29, 10, 3, 8, 0, -1, 0; 46, 767, 83, 1186, 403, 118, 299, -1, -38, 7; -18, -195, -17, -210, -60, -12, 0, 2, 0, -15],84,13]
%
% END OF LATTICE L85.13
%
\hbox{}\par\smallskip%
%
% LATTICE L85.14
%
\leavevmode%
\llap{\spacer}%
${L_{85.14}}=\pdualwitharg{5.13}(\pfillwitharg{2}({L_{85.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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-78\\0\\78\\0\\-195
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-65\\-75\\-195\\-325\\-130
\end{tabular}\endgroup%
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-117\\-195\\-663\\-1365\\-780
\end{tabular}\endgroup%
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-130\\-325\\-1365\\-3250\\-2210
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15538444350, 77266020, 3002702625; 77266020, 384215, 14931150; 3002702625, 14931150, 580252622],[195, 65, 1],[7773, 16715, 66439, 153765, 101656, 14330, 5388, -130, 956, 8859; -22807, -49048, -194964, -451231, -298322, -42055, -15814, 378, -2803, -25988; -39637, -85235, -338793, -784095, -518375, -73073, -27475, 663, -4875, -45175],84,14]
%
% END OF LATTICE L85.14
%
\hbox{}\par\smallskip%
%
% LATTICE L85.15
%
\leavevmode%
\llap{\spacer}%
${L_{85.15}}=\pdualwitharg{2.5}({L_{85.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{520}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{4}{40}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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72079800\\-1070940\\-18184920
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\kern3pt%
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-1070940\\16120\\270200
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[72079800, -1070940, -18184920; -1070940, 16120, 270200; -18184920, 270200, 4587851],[780, 20, 1],[-3589, -18425, -193, -485, -2136, -1844, -52204, -16106, -37546, -25029; 955, 4901, 51, 129, 569, 491, 13897, 4287, 9993, 6661; -14282, -73320, -768, -1930, -8500, -7338, -207740, -64092, -149410, -99600],84,15]
%
% END OF LATTICE L85.15
%
\hbox{}\par\smallskip%
%
% LATTICE L85.16
%
\leavevmode%
\llap{\spacer}%
${L_{85.16}}=\pdualwitharg{3.13}({L_{85.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${780}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{130}\above{1pt}{1pt}{b}{2}{156}\above{1pt}{1pt}{*}{4}{78}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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-2044380\\16709160\\-1287000
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\kern3pt%
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16709160\\-135158946\\10409919
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1287000\\10409919\\-801770
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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984559\\25632040\\331217640
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2044380, 16709160, -1287000; 16709160, -135158946, 10409919; -1287000, 10409919, -801770],[780, 39, 1],[-2113, -562, -4002, -11513, -3938, -5827, -232, 32, 397, -32; -55020, -14632, -104190, -299728, -102519, -151690, -6039, 835, 10334, -836; -710970, -189075, -1346345, -3873090, -1324752, -1960140, -78036, 10790, 133536, -10803],84,16]
%
% END OF LATTICE L85.16
%
\hbox{}\par\smallskip%
%
% LATTICE L85.17
%
\leavevmode%
\llap{\spacer}%
${L_{85.17}}=\pdualwitharg{3.5.13}(\pfillwitharg{2}({L_{85.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${39}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{l}{2}{195}\above{1pt}{1pt}{}{4}{390}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21706695075, 59912507070, -20164275690; 59912507070, -165364096170, 55655277795; -20164275690, 55655277795, -18731453914],[195, 195, 1],[-163, -358, -474, -3271, -2144, -295, -100, 16, 65, -146; -1133981, -2488693, -3293955, -22726221, -14892592, -2048236, -693524, 111738, 450702, -1018493; -3369132, -7394070, -9786556, -67521090, -44246865, -6085443, -2060505, 331981, 1339065, -3026010],84,17]
%
% END OF LATTICE L85.17
%
\hbox{}\par\smallskip%
%
% LATTICE L85.18
%
\leavevmode%
\llap{\spacer}%
${L_{85.18}}=\pdualwitharg{2.13}({L_{85.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10437960, 47072220, -2556060; 47072220, 212282824, -11527100; -2556060, -11527100, 625931],[780, 52, 1],[-6729, -2797, -54791, -25159, -16495, -11471, -843, 191, -321, -1705; 45, 19, 375, 173, 114, 80, 6, 0, 2, 11; -26650, -11072, -216840, -99554, -65260, -45370, -3332, 780, -1274, -6760],84,18]
%
% END OF LATTICE L85.18
%
\hbox{}\par\smallskip%
%
% LATTICE L85.19
%
\leavevmode%
\llap{\spacer}%
${L_{85.19}}=\pdualwitharg{2.3.5}({L_{85.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[338520, 36660, -85020; 36660, 18600, -9180; -85020, -9180, 21353],[780, 60, 1],[-1211, -33133, -3349, -23157, -15247, -2137, -10391, 1, -279, -1493; 9, 247, 25, 173, 114, 16, 78, 0, 2, 11; -4818, -131820, -13324, -92130, -60660, -8502, -41340, 4, -1110, -5940],84,19]
%
% END OF LATTICE L85.19
%
\hbox{}\par\smallskip%
%
% LATTICE L85.20
%
\leavevmode%
\llap{\spacer}%
${L_{85.20}}=\pdualwitharg{5.13}({L_{85.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-401700, -376740, 9360; -376740, -353210, 8775; 9360, 8775, -218],[780, 65, 1],[-5, -5, -19, -85, -27, -7, -1, 1, 1, -3; 24, 28, 114, 532, 177, 50, 9, -3, -14, 8; 754, 915, 3783, 17810, 5980, 1716, 320, -78, -520, 195],84,20]
%
% END OF LATTICE L85.20
%
\hbox{}\par\smallskip%
%
% LATTICE L85.21
%
\leavevmode%
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${L_{85.21}}=\pdualwitharg{2.3.13}({L_{85.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14058491823480, -220208555580, -3532863574500; -220208555580, 3449289480, 55337855196; -3532863574500, 55337855196, 887799715129],[780, 156, 1],[75094, 31350, 205192, 283120, 185959, 129771, 9623, -327, 3691, 18906; -17405, -7307, -47965, -66309, -43646, -30580, -2290, -20, -870, -4339; 299910, 125208, 819520, 1130766, 742716, 518310, 38436, -1300, 14742, 75504],84,21]
%
% END OF LATTICE L85.21
%
\hbox{}\par\smallskip%
%
% LATTICE L85.22
%
\leavevmode%
\llap{\spacer}%
${L_{85.22}}=\pdualwitharg{3.5.13}({L_{85.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-818220, 705900, 24180; 705900, -591630, -20475; 24180, -20475, -706],[780, 195, 1],[-291, -50, 8, 85, -57, -151, -171, -229, -3179, -1048; 622, 107, -17, -182, 121, 322, 365, 489, 6790, 2239; -28002, -4815, 767, 8190, -5460, -14508, -16440, -22022, -305760, -100815],84,22]
%
% END OF LATTICE L85.22
%
\hbox{}\par\smallskip%
%
% LATTICE L85.23
%
\leavevmode%
\llap{\spacer}%
${L_{85.23}}=\pdualwitharg{2.5.13}({L_{85.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[157845480, 1426620, 38585820; 1426620, 13000, 348660; 38585820, 348660, 9432487],[780, 260, 1],[-2752, -5506, -21140, -47882, -30919, -4177, -1423, 353, -621, -3726; 8433, 16871, 64773, 146707, 94731, 12797, 4359, -1083, 1903, 11419; 10946, 21900, 84084, 190450, 122980, 16614, 5660, -1404, 2470, 14820],84,23]
%
% END OF LATTICE L85.23
%
\hbox{}\par\smallskip%
%
% LATTICE L85.24
%
\leavevmode%
\llap{\spacer}%
${L_{85.24}}=\pdualwitharg{2.3.5.13}({L_{85.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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7009080\\302487900\\-76026600
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\kern3pt%
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302487900\\13054356120\\-3281051280
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76026600\\-3281051280\\824651741
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138295\\-6850608\\-27243840
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5968703\\-295664695\\-1175814120
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{$\left.\llap{\phantom{%
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0\\0\\0
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7009080, 302487900, -76026600; 302487900, 13054356120, -3281051280; -76026600, -3281051280, 824651741],[780, 780, 1],[-179, -63, 3, -31, -154, -114, -230, -296, -2020, -1311; 9277, 3485, 13, 1275, 6473, 5237, 11029, 14501, 100323, 66097; 36894, 13860, 52, 5070, 25740, 20826, 43860, 57668, 398970, 262860],84,24]
%
% END OF LATTICE L85.24
%

\medskip%
%------------------------------WEYL-GROUP-W86--------------------------------
%
\leavevmode\llap{}%
$W_{86\phantom{0}}$%
\qquad\llap{48} lattices, $\chi=28$%
\hfill%
$22622222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L86.1
%
\leavevmode%
\llap{\spacer}%
${L_{86.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N_{86},3,2}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{2340}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{18}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{234}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-143865540\\159120\\10754640
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\kern3pt%
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159120\\-174\\-12363
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10754640\\-12363\\-693998
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
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3027\\2125402\\9046
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112363\\78895440\\335790
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260\\182558\\777
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-259\\-181857\\-774
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\HardButStrongLineBreak\kern3pt%
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-261\\-183260\\-780
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\HardButStrongLineBreak\kern3pt%
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1292\\907179\\3861
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\HardButStrongLineBreak\kern3pt%
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691\\485185\\2065
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6225\\4370873\\18603
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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0\\2340\\0\\-5850\\-16380\\-19890\\-1170\\-1170
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\kern3pt%
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\kern3pt%
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-20\\-1170\\-15\\-45\\-60\\0\\10\\0
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\kern3pt%
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0\\-1170\\-39\\-351\\-780\\-702\\0\\78
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-143865540, 159120, 10754640; 159120, -174, -12363; 10754640, -12363, -693998],[2340, 3, 1],[3027, 112363, 260, -259, -261, 1292, 691, 6225; 2125402, 78895440, 182558, -181857, -183260, 907179, 485185, 4370873; 9046, 335790, 777, -774, -780, 3861, 2065, 18603],85,1]
%
% END OF LATTICE L86.1
%
\hbox{}\par\smallskip%
%
% LATTICE L86.2
%
\leavevmode%
\llap{\spacer}%
${L_{86.2}}=\pfillwitharg{2.3}({L_{86.1}})$%
{${}={}$}Nikulin $86$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{65}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12090\\4095\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4095\\-1387\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\65\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138\\-409\\-27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-577\\-1710\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-465\\-1378\\-78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81\\-240\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-79\\-234\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-27\\-105\\-78\\-10\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\65\\0\\-325\\-1365\\-1105\\-195\\-195
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\-3\\-30\\-39\\-15\\-39
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-325\\-3\\2\\0\\-13\\-15\\-117
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-1365\\-30\\0\\15\\0\\-30\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-1105\\-39\\-13\\0\\26\\0\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-195\\-15\\-15\\-30\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-195\\-39\\-117\\-390\\-234\\0\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-12090, 4095, 0; 4095, -1387, 0; 0, 0, 1],[195, 1, 1],[0, 22, -1, -138, -577, -465, -81, -79; 0, 65, -3, -409, -1710, -1378, -240, -234; 1, 0, -3, -27, -105, -78, -10, 0],85,2]
%
% END OF LATTICE L86.2
%
\hbox{}\par\smallskip%
%
% LATTICE L86.3
%
\leavevmode%
\llap{\spacer}%
${L_{86.3}}=\pfillwitharg{3}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{260}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8395140\\2811120\\25740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2811120\\-941306\\-8619
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25740\\-8619\\-58
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6579\\19648\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80051\\239070\\-650
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
443\\1323\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-222\\-663\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-221\\-660\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1332\\3978\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1701\\5080\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13973\\41730\\-117
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-54\\-420\\-156\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\260\\0\\-650\\-5460\\-2210\\-390\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\-3\\-60\\-39\\-15\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-650\\-3\\2\\0\\-13\\-15\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-5460\\-60\\0\\60\\0\\-60\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-2210\\-39\\-13\\0\\26\\0\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-390\\-15\\-15\\-60\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-390\\-39\\-117\\-780\\-234\\0\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-8395140, 2811120, 25740; 2811120, -941306, -8619; 25740, -8619, -58],[780, 1, 1],[6579, 80051, 443, -222, -221, 1332, 1701, 13973; 19648, 239070, 1323, -663, -660, 3978, 5080, 41730; -54, -650, -3, 2, 0, -13, -15, -117],85,3]
%
% END OF LATTICE L86.3
%
\hbox{}\par\smallskip%
%
% LATTICE L86.4
%
\leavevmode%
\llap{\spacer}%
${L_{86.4}}=\pfillwitharg{2}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{585}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{18}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{234}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7842510\\41535\\1001520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41535\\-219\\-5343
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1001520\\-5343\\-126353
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
607\\71376\\1793
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22973\\2701335\\67860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
131\\15403\\387
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\-7527\\-189
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-7760\\-195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
317\\37284\\936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
237\\27870\\700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2403\\282568\\7098
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-81\\-105\\-234\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\585\\0\\-2925\\-4095\\-9945\\-585\\-585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\-9\\-30\\-117\\-15\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81\\-2925\\-9\\18\\0\\-117\\-45\\-351
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-4095\\-30\\0\\15\\0\\-30\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-9945\\-117\\-117\\0\\234\\0\\-702
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-585\\-15\\-45\\-30\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-585\\-39\\-351\\-390\\-702\\0\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-7842510, 41535, 1001520; 41535, -219, -5343; 1001520, -5343, -126353],[585, 3, 1],[607, 22973, 131, -64, -66, 317, 237, 2403; 71376, 2701335, 15403, -7527, -7760, 37284, 27870, 282568; 1793, 67860, 387, -189, -195, 936, 700, 7098],85,4]
%
% END OF LATTICE L86.4
%
\hbox{}\par\smallskip%
%
% LATTICE L86.5
%
\leavevmode%
\llap{\spacer}%
${L_{86.5}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{195}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10155405\\-2428335\\3399435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2428335\\-577338\\812916
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3399435\\812916\\-1137931
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
457155\\-21016\\1350684
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5111309\\-234975\\15101580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5897\\271\\-17423
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5942\\-273\\17556
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115082\\-5290\\340015
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
643091\\-29562\\1900041
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
401133\\-18440\\1185165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
755045\\-34710\\2230813
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-81\\-105\\-234\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\195\\0\\-975\\-1365\\-3315\\-585\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\-3\\-10\\-39\\-15\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81\\-975\\-3\\6\\0\\-39\\-45\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-1365\\-10\\0\\5\\0\\-30\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-3315\\-39\\-39\\0\\78\\0\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-585\\-15\\-45\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-195\\-13\\-117\\-130\\-234\\0\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10155405, -2428335, 3399435; -2428335, -577338, 812916; 3399435, 812916, -1137931],[195, 3, 1],[457155, 5111309, -5897, 5942, 115082, 643091, 401133, 755045; -21016, -234975, 271, -273, -5290, -29562, -18440, -34710; 1350684, 15101580, -17423, 17556, 340015, 1900041, 1185165, 2230813],85,5]
%
% END OF LATTICE L86.5
%
\hbox{}\par\smallskip%
%
% LATTICE L86.6
%
\leavevmode%
\llap{\spacer}%
${L_{86.6}}=\pdualwitharg{5}(\pfillwitharg{2.3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13403715\\-3595995\\2693535
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3595995\\-962230\\722670
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2693535\\722670\\-541277
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
277053\\-21016\\1350630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
619527\\-46995\\3020186
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10723\\813\\-52275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3602\\-273\\17560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41848\\-3174\\204009
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
389747\\-29562\\1900015
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48621\\-3688\\237027
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1372763\\-104130\\6692205
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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5\\0\\-15\\-135\\-105\\-390\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\13\\0\\-325\\-273\\-1105\\-39\\-195
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\-15\\-30\\-195\\-15\\-195
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-325\\-15\\10\\0\\-65\\-15\\-585
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-273\\-30\\0\\3\\0\\-6\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390\\-1105\\-195\\-65\\0\\130\\0\\-1170
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-39\\-15\\-15\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-195\\-195\\-585\\-390\\-1170\\0\\390
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13403715, -3595995, 2693535; -3595995, -962230, 722670; 2693535, 722670, -541277],[195, 5, 1],[277053, 619527, -10723, 3602, 41848, 389747, 48621, 1372763; -21016, -46995, 813, -273, -3174, -29562, -3688, -104130; 1350630, 3020186, -52275, 17560, 204009, 1900015, 237027, 6692205],85,6]
%
% END OF LATTICE L86.6
%
\hbox{}\par\smallskip%
%
% LATTICE L86.7
%
\leavevmode%
\llap{\spacer}%
${L_{86.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{65}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5256810\\33930\\18135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33930\\-219\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18135\\-117\\-58
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\8559\\-81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
226\\35165\\-325
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\466\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-156\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-155\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\936\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\4050\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\11830\\-117
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-9\\-81\\-315\\-234\\-90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\65\\0\\-325\\-1365\\-1105\\-585\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\6\\-3\\-30\\-39\\-45\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81\\-325\\-3\\2\\0\\-13\\-45\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-1365\\-30\\0\\15\\0\\-90\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-1105\\-39\\-13\\0\\26\\0\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-585\\-45\\-45\\-90\\0\\90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-195\\-39\\-117\\-390\\-234\\0\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5256810, 33930, 18135; 33930, -219, -117; 18135, -117, -58],[585, 3, 1],[55, 226, 3, -1, -1, 6, 26, 76; 8559, 35165, 466, -156, -155, 936, 4050, 11830; -81, -325, -3, 2, 0, -13, -45, -117],85,7]
%
% END OF LATTICE L86.7
%
\hbox{}\par\smallskip%
%
% LATTICE L86.8
%
\leavevmode%
\llap{\spacer}%
${L_{86.8}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{780}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31731180\\28860\\42900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28860\\-18\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42900\\-39\\-58
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\-54\\25914
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
433\\-650\\320580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-741
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-13\\3705
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-15\\5925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-39\\17771
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-162\\-420\\-468\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\780\\0\\-1950\\-5460\\-6630\\-1170\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\-3\\-20\\-39\\-15\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-162\\-1950\\-3\\6\\0\\-39\\-45\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-5460\\-20\\0\\20\\0\\-60\\-260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-6630\\-39\\-39\\0\\78\\0\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-1170\\-15\\-45\\-60\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-390\\-13\\-117\\-260\\-234\\0\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-31731180, 28860, 42900; 28860, -18, -39; 42900, -39, -58],[780, 3, 1],[35, 433, 1, -1, -1, 5, 8, 24; -54, -650, -1, 2, 0, -13, -15, -39; 25914, 320580, 740, -741, -740, 3705, 5925, 17771],85,8]
%
% END OF LATTICE L86.8
%
\hbox{}\par\smallskip%
%
% LATTICE L86.9
%
\leavevmode%
\llap{\spacer}%
${L_{86.9}}=\pdualwitharg{13}(\pfillwitharg{2.3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${13}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{}{6}{26}\above{1pt}{1pt}{l}{2}{195}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{130}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27301755\\-8266635\\12612210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8266635\\-2501798\\3818841
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12612210\\3818841\\-5826286
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
630195\\-21016\\1350414
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
541993\\-18075\\1161410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24406\\813\\-52299
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8202\\-273\\17576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
476021\\-15870\\1020045
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68202\\-2274\\146147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
553008\\-18440\\1185015
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240200\\-8010\\514713
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\0\\-39\\-351\\-1365\\-78\\-130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-325\\-1365\\-85\\-195\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\78\\-39\\-390\\-39\\-195\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-351\\-325\\-39\\26\\0\\-13\\-195\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1365\\-1365\\-390\\0\\195\\0\\-390\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-85\\-39\\-13\\0\\2\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-195\\-195\\-195\\-390\\0\\130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-39\\-117\\-390\\-18\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-27301755, -8266635, 12612210; -8266635, -2501798, 3818841; 12612210, 3818841, -5826286],[195, 13, 1],[630195, 541993, -24406, 8202, 476021, 68202, 553008, 240200; -21016, -18075, 813, -273, -15870, -2274, -18440, -8010; 1350414, 1161410, -52299, 17576, 1020045, 146147, 1185015, 514713],85,9]
%
% END OF LATTICE L86.9
%
\hbox{}\par\smallskip%
%
% LATTICE L86.10
%
\leavevmode%
\llap{\spacer}%
${L_{86.10}}=\pdualwitharg{3.5}(\pfillwitharg{2.3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{130}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64855240905\\201668220\\-22483132620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
201668220\\627090\\-69911595
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22483132620\\-69911595\\7794146554
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
433335\\161009\\1251450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
980297\\364221\\2831049
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2294\\-857\\-6625
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\0\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18519\\6886\\53482
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
548818\\204009\\1584960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71808\\26687\\207378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
702048\\260871\\2027480
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\-405\\-105\\-1170\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\39\\0\\-975\\-273\\-3315\\-117\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\10\\-15\\-10\\-195\\-15\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-405\\-975\\-15\\30\\0\\-195\\-45\\-585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-273\\-10\\0\\1\\0\\-6\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1170\\-3315\\-195\\-195\\0\\390\\0\\-1170
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-117\\-15\\-45\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-195\\-65\\-585\\-130\\-1170\\0\\130
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[64855240905, 201668220, -22483132620; 201668220, 627090, -69911595; -22483132620, -69911595, 7794146554],[195, 15, 1],[433335, 980297, -2294, -26, 18519, 548818, 71808, 702048; 161009, 364221, -857, 0, 6886, 204009, 26687, 260871; 1251450, 2831049, -6625, -75, 53482, 1584960, 207378, 2027480],85,10]
%
% END OF LATTICE L86.10
%
\hbox{}\par\smallskip%
%
% LATTICE L86.11
%
\leavevmode%
\llap{\spacer}%
${L_{86.11}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{260}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{312}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61707642360\\235749540\\-15427084140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
235749540\\900664\\-58938048
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15427084140\\-58938048\\3856814423
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16126\\21809\\-64170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-201079\\272025\\-800150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3079\\4179\\-12252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-833\\1110\\-3315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6416\\8619\\-25532
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7358\\9925\\-29280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65704\\88803\\-261456
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-108\\-210\\-312\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\260\\0\\-1300\\-2730\\-4420\\-780\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\-12\\-60\\-156\\-60\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-1300\\-12\\8\\0\\-52\\-60\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-2730\\-60\\0\\15\\0\\-60\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-4420\\-156\\-52\\0\\104\\0\\-936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-780\\-60\\-60\\-60\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-780\\-156\\-468\\-780\\-936\\0\\312
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[61707642360, 235749540, -15427084140; 235749540, 900664, -58938048; -15427084140, -58938048, 3856814423],[780, 4, 1],[-16126, -201079, -3079, 1, -833, -6416, -7358, -65704; 21809, 272025, 4179, 0, 1110, 8619, 9925, 88803; -64170, -800150, -12252, 4, -3315, -25532, -29280, -261456],85,11]
%
% END OF LATTICE L86.11
%
\hbox{}\par\smallskip%
%
% LATTICE L86.12
%
\leavevmode%
\llap{\spacer}%
${L_{86.12}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4741620\\37440\\1605240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37440\\-290\\-12675
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1605240\\-12675\\-543442
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3771\\-54\\11140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9479\\-130\\28002
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
391\\-3\\1155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\2\\-195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\0\\-192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
396\\-13\\1170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
153\\-3\\452
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7525\\-117\\22230
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-30\\-270\\-420\\-780\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\52\\0\\-650\\-1092\\-2210\\-78\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\-15\\-60\\-195\\-15\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\-650\\-15\\10\\0\\-65\\-15\\-585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-1092\\-60\\0\\12\\0\\-12\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\-2210\\-195\\-65\\0\\130\\0\\-1170
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-78\\-15\\-15\\-12\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-390\\-195\\-585\\-780\\-1170\\0\\390
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4741620, 37440, 1605240; 37440, -290, -12675; 1605240, -12675, -543442],[780, 5, 1],[3771, 9479, 391, -66, -65, 396, 153, 7525; -54, -130, -3, 2, 0, -13, -3, -117; 11140, 28002, 1155, -195, -192, 1170, 452, 22230],85,12]
%
% END OF LATTICE L86.12
%
\hbox{}\par\smallskip%
%
% LATTICE L86.13
%
\leavevmode%
\llap{\spacer}%
${L_{86.13}}=\pdualwitharg{3}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{*}{2}{260}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95193540\\128700\\86580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
128700\\-174\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
86580\\-117\\-58
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89\\65928\\-162
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
361\\267410\\-650
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1481\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-741\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-740\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\4446\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\17040\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\46670\\-117
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-18\\-162\\-1260\\-468\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\260\\0\\-650\\-5460\\-2210\\-1170\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\6\\-3\\-60\\-39\\-45\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-162\\-650\\-3\\2\\0\\-13\\-45\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-5460\\-60\\0\\60\\0\\-180\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-2210\\-39\\-13\\0\\26\\0\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-1170\\-45\\-45\\-180\\0\\90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-390\\-39\\-117\\-780\\-234\\0\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-95193540, 128700, 86580; 128700, -174, -117; 86580, -117, -58],[2340, 3, 1],[89, 361, 2, -1, -1, 6, 23, 63; 65928, 267410, 1481, -741, -740, 4446, 17040, 46670; -162, -650, -3, 2, 0, -13, -45, -117],85,13]
%
% END OF LATTICE L86.13
%
\hbox{}\par\smallskip%
%
% LATTICE L86.14
%
\leavevmode%
\llap{\spacer}%
${L_{86.14}}=\pdualwitharg{3.13}(\pfillwitharg{2.3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${39}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{}{6}{78}\above{1pt}{1pt}{l}{2}{65}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27019792410\\-209890395\\-8660172300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-209890395\\1630434\\67272426
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8660172300\\67272426\\2775690617
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-873060\\161009\\-2727855
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-759617\\140085\\-2373405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4635\\-857\\14482
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-186628\\34430\\-583115
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85076\\15693\\-265818
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-723462\\133435\\-2260440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108808\\20067\\-339968
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\0\\-39\\-1053\\-1365\\-234\\-390\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-975\\-1365\\-255\\-585\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\0\\26\\-39\\-130\\-39\\-195\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1053\\-975\\-39\\78\\0\\-39\\-585\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1365\\-1365\\-130\\0\\65\\0\\-390\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-255\\-39\\-39\\0\\6\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390\\-585\\-195\\-585\\-390\\0\\390\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-13\\-117\\-130\\-18\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[27019792410, -209890395, -8660172300; -209890395, 1630434, 67272426; -8660172300, 67272426, 2775690617],[195, 39, 1],[-873060, -759617, 4635, 25, -186628, -85076, -723462, -108808; 161009, 140085, -857, 0, 34430, 15693, 133435, 20067; -2727855, -2373405, 14482, 78, -583115, -265818, -2260440, -339968],85,14]
%
% END OF LATTICE L86.14
%
\hbox{}\par\smallskip%
%
% LATTICE L86.15
%
\leavevmode%
\llap{\spacer}%
${L_{86.15}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{117}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}{90}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{1170}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50004045\\93618135\\-15381990
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93618135\\-175236315\\28790955
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15381990\\28790955\\-4730242
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64206\\-185485\\-920180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-497000\\-1435785\\-7122843
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17191\\-49663\\-246375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1730\\4998\\24795
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1719\\4966\\24636
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8653\\-24999\\-124020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4015\\-11599\\-57542
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-242517\\-700609\\-3475680
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-15\\-405\\-105\\-1170\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\117\\0\\-2925\\-819\\-9945\\-117\\-585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\-45\\-30\\-585\\-15\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-405\\-2925\\-45\\90\\0\\-585\\-45\\-1755
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-819\\-30\\0\\3\\0\\-6\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1170\\-9945\\-585\\-585\\0\\1170\\0\\-3510
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-117\\-15\\-45\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-585\\-195\\-1755\\-390\\-3510\\0\\390
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-50004045, 93618135, -15381990; 93618135, -175236315, 28790955; -15381990, 28790955, -4730242],[585, 15, 1],[-64206, -497000, -17191, 1730, 1719, -8653, -4015, -242517; -185485, -1435785, -49663, 4998, 4966, -24999, -11599, -700609; -920180, -7122843, -246375, 24795, 24636, -124020, -57542, -3475680],85,15]
%
% END OF LATTICE L86.15
%
\hbox{}\par\smallskip%
%
% LATTICE L86.16
%
\leavevmode%
\llap{\spacer}%
${L_{86.16}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43920045\\1812915\\339885
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1812915\\-74715\\-14010
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
339885\\-14010\\-2627
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
139\\-27081\\162405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
118\\-22984\\137839
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-2138\\12825
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\197\\-1180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\194\\-1164
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-1183\\7085
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-1953\\11709
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\-35087\\210405
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-45\\-405\\-315\\-1170\\-90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\13\\0\\-325\\-273\\-1105\\-117\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\30\\-15\\-30\\-195\\-45\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-405\\-325\\-15\\10\\0\\-65\\-45\\-585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-273\\-30\\0\\3\\0\\-18\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1170\\-1105\\-195\\-65\\0\\130\\0\\-1170
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-117\\-45\\-45\\-18\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-195\\-195\\-585\\-390\\-1170\\0\\390
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-43920045, 1812915, 339885; 1812915, -74715, -14010; 339885, -14010, -2627],[585, 15, 1],[139, 118, 11, -1, -1, 6, 10, 180; -27081, -22984, -2138, 197, 194, -1183, -1953, -35087; 162405, 137839, 12825, -1180, -1164, 7085, 11709, 210405],85,16]
%
% END OF LATTICE L86.16
%
\hbox{}\par\smallskip%
%
% LATTICE L86.17
%
\leavevmode%
\llap{\spacer}%
${L_{86.17}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{780}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{312}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{s}{2}{104}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16829752680\\574860\\-4229131920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
574860\\24\\-144456
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4229131920\\-144456\\1062734381
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54877\\71\\-218382
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-684549\\910\\-2724150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3509\\6\\-13964
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-926\\0\\-3685
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21639\\13\\-86112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24953\\25\\-99300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74469\\91\\-296348
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-780\\-52\\-468\\-260\\-936\\0\\104
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16829752680, 574860, -4229131920; 574860, 24, -144456; -4229131920, -144456, 1062734381],[780, 12, 1],[-54877, -684549, -3509, 0, -926, -21639, -24953, -74469; 71, 910, 6, 1, 0, 13, 25, 91; -218382, -2724150, -13964, 0, -3685, -86112, -99300, -296348],85,17]
%
% END OF LATTICE L86.17
%
\hbox{}\par\smallskip%
%
% LATTICE L86.18
%
\leavevmode%
\llap{\spacer}%
${L_{86.18}}=\pdualwitharg{13}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{}{6}{26}\above{1pt}{1pt}{l}{2}{780}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{130}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3556020\\-1580280\\25740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1580280\\-691886\\13169
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25740\\13169\\102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23061\\47272\\-283686
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21577\\44230\\-265430
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1544\\3165\\-18993
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
782\\-1603\\9620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
761\\-1560\\9360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-361\\740\\-4441
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5976\\12250\\-73515
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3770\\7728\\-46377
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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52\\0\\-78\\-702\\-5460\\-156\\-260\\0
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\kern3pt%
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0\\20\\0\\-650\\-5460\\-170\\-390\\-30
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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-156\\-170\\-39\\-13\\0\\2\\0\\-18
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\kern3pt%
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-260\\-390\\-195\\-195\\-780\\0\\130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-39\\-117\\-780\\-18\\0\\6
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3556020, -1580280, 25740; -1580280, -691886, 13169; 25740, 13169, 102],[780, 13, 1],[-23061, -21577, -1544, 782, 761, -361, -5976, -3770; 47272, 44230, 3165, -1603, -1560, 740, 12250, 7728; -283686, -265430, -18993, 9620, 9360, -4441, -73515, -46377],85,18]
%
% END OF LATTICE L86.18
%
\hbox{}\par\smallskip%
%
% LATTICE L86.19
%
\leavevmode%
\llap{\spacer}%
${L_{86.19}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{130}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-131642940\\113880\\225420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
113880\\-90\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
225420\\-195\\-386
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
119\\-54\\69510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
305\\-130\\178152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-1\\2920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-13\\2925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\2337
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\-39\\44395
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-30\\-810\\-420\\-2340\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\156\\0\\-1950\\-1092\\-6630\\-234\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\-15\\-20\\-195\\-15\\-65
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\kern3pt%
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-810\\-1950\\-15\\30\\0\\-195\\-45\\-585
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-1092\\-20\\0\\4\\0\\-12\\-260
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2340\\-6630\\-195\\-195\\0\\390\\0\\-1170
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-234\\-15\\-45\\-12\\0\\6\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-390\\-65\\-585\\-260\\-1170\\0\\130
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-131642940, 113880, 225420; 113880, -90, -195; 225420, -195, -386],[780, 15, 1],[119, 305, 5, -1, -1, 5, 4, 76; -54, -130, -1, 2, 0, -13, -3, -39; 69510, 178152, 2920, -585, -584, 2925, 2337, 44395],85,19]
%
% END OF LATTICE L86.19
%
\hbox{}\par\smallskip%
%
% LATTICE L86.20
%
\leavevmode%
\llap{\spacer}%
${L_{86.20}}=\pdualwitharg{5.13}(\pfillwitharg{2.3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${65}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{}{6}{130}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
184363459410\\707804565\\35747068110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
707804565\\2717390\\137239440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35747068110\\137239440\\6931161319
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1814910\\-5283721\\-9255675
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
315817\\-919434\\-1610603
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28907\\84150\\147420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\153\\260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
232778\\-677676\\-1187121
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
176856\\-514875\\-901930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
300786\\-875671\\-1533948
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
678568\\-1975503\\-3460560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\0\\-195\\-1755\\-1365\\-390\\-130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-325\\-273\\-85\\-39\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\0\\390\\-195\\-390\\-195\\-195\\-195
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1755\\-325\\-195\\130\\0\\-65\\-195\\-585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1365\\-273\\-390\\0\\39\\0\\-78\\-390
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390\\-85\\-195\\-65\\0\\10\\0\\-90
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-39\\-195\\-195\\-78\\0\\26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-195\\-585\\-390\\-90\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[184363459410, 707804565, 35747068110; 707804565, 2717390, 137239440; 35747068110, 137239440, 6931161319],[195, 65, 1],[1814910, 315817, -28907, -51, 232778, 176856, 300786, 678568; -5283721, -919434, 84150, 153, -677676, -514875, -875671, -1975503; -9255675, -1610603, 147420, 260, -1187121, -901930, -1533948, -3460560],85,20]
%
% END OF LATTICE L86.20
%
\hbox{}\par\smallskip%
%
% LATTICE L86.21
%
\leavevmode%
\llap{\spacer}%
${L_{86.21}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{}{6}{40}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{520}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{1560}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
130405080\\-53820\\-32128980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53820\\40\\13260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32128980\\13260\\7915883
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12967\\240\\52630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32843\\611\\133302
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2927\\57\\11880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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17\\0\\69
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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3267\\52\\13260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1021\\18\\4144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51119\\936\\207480
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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20\\0\\-60\\-540\\-210\\-1560\\-40\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-40\\-156\\-60\\-60\\-12\\0\\8\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-780\\-780\\-2340\\-780\\-4680\\0\\1560
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[130405080, -53820, -32128980; -53820, 40, 13260; -32128980, 13260, 7915883],[780, 20, 1],[12967, 32843, 2927, 0, 17, 3267, 1021, 51119; 240, 611, 57, 1, 0, 52, 18, 936; 52630, 133302, 11880, 0, 69, 13260, 4144, 207480],85,21]
%
% END OF LATTICE L86.21
%
\hbox{}\par\smallskip%
%
% LATTICE L86.22
%
\leavevmode%
\llap{\spacer}%
${L_{86.22}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${13}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{}{6}{234}\above{1pt}{1pt}{l}{2}{195}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}{130}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3263985270\\-582811515\\2964780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-582811515\\-104065806\\529386
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2964780\\529386\\-2693
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
366\\-1839\\41431
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1063\\-5340\\120555
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\-386\\8892
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\207\\-4446
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\190\\-4485
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-81\\1692
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
146\\-735\\16250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\-563\\12630
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\0\\-39\\-1053\\-1365\\-234\\-130\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\45\\0\\-2925\\-4095\\-765\\-585\\-45
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-39\\0\\78\\-117\\-390\\-117\\-195\\-39
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1365\\-4095\\-390\\0\\195\\0\\-390\\-390
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-765\\-117\\-117\\0\\18\\0\\-54
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-585\\-195\\-585\\-390\\0\\130\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-39\\-351\\-390\\-54\\0\\6
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3263985270, -582811515, 2964780; -582811515, -104065806, 529386; 2964780, 529386, -2693],[585, 39, 1],[366, 1063, 77, -41, -38, 16, 146, 112; -1839, -5340, -386, 207, 190, -81, -735, -563; 41431, 120555, 8892, -4446, -4485, 1692, 16250, 12630],85,22]
%
% END OF LATTICE L86.22
%
\hbox{}\par\smallskip%
%
% LATTICE L86.23
%
\leavevmode%
\llap{\spacer}%
${L_{86.23}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${117}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{}{6}{26}\above{1pt}{1pt}{l}{2}{195}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{1170}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18823098645\\-2722961475\\17407845
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2722961475\\-393905343\\2518230
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17407845\\2518230\\-16099
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
796\\-2652\\445887
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
251\\-835\\140795
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\-137\\23985
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\51\\-8242
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\40\\-7800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-24\\3815
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
383\\-1290\\212355
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
85\\-284\\47487
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
117\\0\\-117\\-1053\\-4095\\-234\\-1170\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-325\\-1365\\-85\\-585\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\0\\78\\-39\\-390\\-39\\-585\\-39
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1053\\-325\\-39\\26\\0\\-13\\-585\\-117
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4095\\-1365\\-390\\0\\195\\0\\-1170\\-390
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-85\\-39\\-13\\0\\2\\0\\-18
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1170\\-585\\-585\\-585\\-1170\\0\\1170\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-39\\-117\\-390\\-18\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18823098645, -2722961475, 17407845; -2722961475, -393905343, 2518230; 17407845, 2518230, -16099],[585, 39, 1],[796, 251, 42, -15, -13, 7, 383, 85; -2652, -835, -137, 51, 40, -24, -1290, -284; 445887, 140795, 23985, -8242, -7800, 3815, 212355, 47487],85,23]
%
% END OF LATTICE L86.23
%
\hbox{}\par\smallskip%
%
% LATTICE L86.24
%
\leavevmode%
\llap{\spacer}%
${L_{86.24}}=\pdualwitharg{2}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{2340}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{72}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{936}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{312}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6402768840\\22791256020\\1608775740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22791256020\\81127612824\\5726588088
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1608775740\\5726588088\\404225023
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46397\\99\\-186058
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1731566\\3705\\-6943950
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8616\\19\\-34560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-199\\0\\792
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2579\\5\\-10335
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57791\\117\\-231660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21569\\45\\-86480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
189925\\403\\-761592
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\0\\-12\\-324\\-210\\-936\\-40\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2340\\0\\-11700\\-8190\\-39780\\-2340\\-2340
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\kern3pt%
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-12\\0\\24\\-36\\-60\\-468\\-60\\-156
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-324\\-11700\\-36\\72\\0\\-468\\-180\\-1404
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-8190\\-60\\0\\15\\0\\-60\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-39780\\-468\\-468\\0\\936\\0\\-2808
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-2340\\-60\\-180\\-60\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2340\\-156\\-1404\\-780\\-2808\\0\\312
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6402768840, 22791256020, 1608775740; 22791256020, 81127612824, 5726588088; 1608775740, 5726588088, 404225023],[2340, 12, 1],[46397, 1731566, 8616, -199, 2579, 57791, 21569, 189925; 99, 3705, 19, 0, 5, 117, 45, 403; -186058, -6943950, -34560, 792, -10335, -231660, -86480, -761592],85,24]
%
% END OF LATTICE L86.24
%
\hbox{}\par\smallskip%
%
% LATTICE L86.25
%
\leavevmode%
\llap{\spacer}%
${L_{86.25}}=\pdualwitharg{2.3}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{b}{2}{260}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{*}{2}{360}\above{1pt}{1pt}{s}{2}{312}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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24911640\\15862860\\6224400
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15862860\\10127544\\3963492
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6224400\\3963492\\1555223
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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163676\\297\\-655830
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680517\\1235\\-2726750
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10455\\19\\-41892
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-1\\0\\4
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2774\\5\\-11115
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21556\\39\\-86372
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74482\\135\\-298440
\end{tabular}\endgroup%
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222152\\403\\-890136
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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0\\-780\\-156\\-468\\-780\\-936\\0\\312
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24911640, 15862860, 6224400; 15862860, 10127544, 3963492; 6224400, 3963492, 1555223],[2340, 12, 1],[163676, 680517, 10455, -1, 2774, 21556, 74482, 222152; 297, 1235, 19, 0, 5, 39, 135, 403; -655830, -2726750, -41892, 4, -11115, -86372, -298440, -890136],85,25]
%
% END OF LATTICE L86.25
%
\hbox{}\par\smallskip%
%
% LATTICE L86.26
%
\leavevmode%
\llap{\spacer}%
${L_{86.26}}=\pdualwitharg{3.13}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{}{6}{78}\above{1pt}{1pt}{l}{2}{260}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{390}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-4783740\\40560\\1476540
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40560\\-234\\-12519
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1476540\\-12519\\-455746
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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11291\\-54\\36582
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10741\\-50\\34800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
321\\-1\\1040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-325\\2\\-1053
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\HardButStrongLineBreak\kern3pt%
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-321\\0\\-1040
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125\\-1\\405
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2588\\-15\\8385
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596\\-3\\1931
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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156\\0\\-78\\-2106\\-5460\\-468\\-780\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4783740, 40560, 1476540; 40560, -234, -12519; 1476540, -12519, -455746],[780, 39, 1],[11291, 10741, 321, -325, -321, 125, 2588, 596; -54, -50, -1, 2, 0, -1, -15, -3; 36582, 34800, 1040, -1053, -1040, 405, 8385, 1931],85,26]
%
% END OF LATTICE L86.26
%
\hbox{}\par\smallskip%
%
% LATTICE L86.27
%
\leavevmode%
\llap{\spacer}%
${L_{86.27}}=\pdualwitharg{5}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{468}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{90}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{1170}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-468910260\\870173460\\-142749360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
870173460\\-1614174630\\264777375
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\kern3pt%
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-142749360\\264777375\\-43431298
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-806803\\-2397358\\-11963620
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6203425\\-18433038\\-91987038
\end{tabular}\endgroup%
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-101676\\-302123\\-1507695
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20372\\60534\\302085
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\HardButStrongLineBreak\kern3pt%
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20335\\60424\\301536
\end{tabular}\endgroup%
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-101863\\-302679\\-1510470
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-27126\\-80603\\-402236
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-1545884\\-4593485\\-22923030
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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-20\\-234\\-15\\-45\\-12\\0\\2\\0
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0\\-1170\\-195\\-1755\\-780\\-3510\\0\\390
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-468910260, 870173460, -142749360; 870173460, -1614174630, 264777375; -142749360, 264777375, -43431298],[2340, 15, 1],[-806803, -6203425, -101676, 20372, 20335, -101863, -27126, -1545884; -2397358, -18433038, -302123, 60534, 60424, -302679, -80603, -4593485; -11963620, -91987038, -1507695, 302085, 301536, -1510470, -402236, -22923030],85,27]
%
% END OF LATTICE L86.27
%
\hbox{}\par\smallskip%
%
% LATTICE L86.28
%
\leavevmode%
\llap{\spacer}%
${L_{86.28}}=\pdualwitharg{3.5}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-394928820\\5431140\\1017900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5431140\\-74490\\-13965
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1017900\\-13965\\-2618
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
173\\-101262\\607410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
145\\-84864\\509054
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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6\\-3508\\21045
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\587\\-3520
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\HardButStrongLineBreak\kern3pt%
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-1\\584\\-3504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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6\\-3523\\21125
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\HardButStrongLineBreak\kern3pt%
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7\\-4101\\24597
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115\\-67327\\403845
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2340\\-2210\\-195\\-65\\0\\130\\0\\-1170
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-390\\-195\\-585\\-780\\-1170\\0\\390
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-394928820, 5431140, 1017900; 5431140, -74490, -13965; 1017900, -13965, -2618],[2340, 15, 1],[173, 145, 6, -1, -1, 6, 7, 115; -101262, -84864, -3508, 587, 584, -3523, -4101, -67327; 607410, 509054, 21045, -3520, -3504, 21125, 24597, 403845],85,28]
%
% END OF LATTICE L86.28
%
\hbox{}\par\smallskip%
%
% LATTICE L86.29
%
\leavevmode%
\llap{\spacer}%
${L_{86.29}}=\pdualwitharg{3.5.13}(\pfillwitharg{2.3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${195}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{}{6}{390}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38639589690\\107576880255\\-36206307735
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107576880255\\-299505750765\\100802303940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36206307735\\100802303940\\-33926241662
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21016\\-189095547\\-561821910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-3615\\-32525963\\-96637911
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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271\\2441038\\7252570
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-273\\-2461012\\-7311915
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1058\\-9522245\\-28291549
\end{tabular}\endgroup%
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-2274\\-20464552\\-60802245
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-3688\\-33186930\\-98601711
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2670\\-24024694\\-71379785
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
195\\0\\-195\\-5265\\-1365\\-1170\\-390\\0
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0\\3\\0\\-975\\-273\\-255\\-117\\-15
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-5265\\-975\\-195\\390\\0\\-195\\-585\\-585
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\kern3pt%
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-1365\\-273\\-130\\0\\13\\0\\-78\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1170\\-255\\-195\\-195\\0\\30\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390\\-117\\-195\\-585\\-78\\0\\78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-65\\-585\\-130\\-90\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38639589690, 107576880255, -36206307735; 107576880255, -299505750765, 100802303940; -36206307735, 100802303940, -33926241662],[195, 195, 1],[-21016, -3615, 271, -273, -1058, -2274, -3688, -2670; -189095547, -32525963, 2441038, -2461012, -9522245, -20464552, -33186930, -24024694; -561821910, -96637911, 7252570, -7311915, -28291549, -60802245, -98601711, -71379785],85,29]
%
% END OF LATTICE L86.29
%
\hbox{}\par\smallskip%
%
% LATTICE L86.30
%
\leavevmode%
\llap{\spacer}%
${L_{86.30}}=\pdualwitharg{2.13}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{}{6}{104}\above{1pt}{1pt}{l}{2}{195}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{520}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252237752520\\4152185700\\-63538633860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4152185700\\68350776\\-1045934656
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63538633860\\-1045934656\\16005367763
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37550\\5429\\149422
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36069\\5235\\143530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7291\\1101\\29016
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\6\\52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1814\\210\\7215
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1114\\147\\4432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16924\\2365\\67340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11734\\1683\\46692
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\0\\-156\\-1404\\-2730\\-312\\-520\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-1300\\-2730\\-340\\-780\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\0\\312\\-156\\-780\\-156\\-780\\-156
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1404\\-1300\\-156\\104\\0\\-52\\-780\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2730\\-2730\\-780\\0\\195\\0\\-780\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-340\\-156\\-52\\0\\8\\0\\-72
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-780\\-780\\-780\\-780\\0\\520\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-156\\-468\\-780\\-72\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[252237752520, 4152185700, -63538633860; 4152185700, 68350776, -1045934656; -63538633860, -1045934656, 16005367763],[780, 52, 1],[37550, 36069, 7291, 13, 1814, 1114, 16924, 11734; 5429, 5235, 1101, 6, 210, 147, 2365, 1683; 149422, 143530, 29016, 52, 7215, 4432, 67340, 46692],85,30]
%
% END OF LATTICE L86.30
%
\hbox{}\par\smallskip%
%
% LATTICE L86.31
%
\leavevmode%
\llap{\spacer}%
${L_{86.31}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{156}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{}{6}{120}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{1560}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{520}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14371395480\\453180\\-3611400780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
453180\\120\\-113880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3611400780\\-113880\\907512121
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-338813\\71\\-1348290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-858177\\182\\-3415074
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25501\\6\\-101480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-146\\0\\-581
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85263\\13\\-339300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26669\\5\\-106128
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-445197\\91\\-1771640
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-1620\\-210\\-4680\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\156\\0\\-3900\\-546\\-13260\\-468\\-780
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\-60\\-20\\-780\\-60\\-260
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1620\\-3900\\-60\\120\\0\\-780\\-180\\-2340
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-546\\-20\\0\\1\\0\\-12\\-260
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4680\\-13260\\-780\\-780\\0\\1560\\0\\-4680
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-468\\-60\\-180\\-12\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-780\\-260\\-2340\\-260\\-4680\\0\\520
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14371395480, 453180, -3611400780; 453180, 120, -113880; -3611400780, -113880, 907512121],[780, 60, 1],[-338813, -858177, -25501, 0, -146, -85263, -26669, -445197; 71, 182, 6, 1, 0, 13, 5, 91; -1348290, -3415074, -101480, 0, -581, -339300, -106128, -1771640],85,31]
%
% END OF LATTICE L86.31
%
\hbox{}\par\smallskip%
%
% LATTICE L86.32
%
\leavevmode%
\llap{\spacer}%
${L_{86.32}}=\pdualwitharg{5.13}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${260}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{390}\above{1pt}{1pt}{}{6}{130}\above{1pt}{1pt}{l}{2}{156}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080645540\\1082683680\\538358340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1082683680\\-1084725590\\-539373705
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
538358340\\-539373705\\-268200526
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
303299\\-54\\608920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58633\\-10\\117714
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31375\\-3\\62985
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5341\\2\\-10725
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5207\\0\\-10452
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2465\\-1\\4950
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12328\\-3\\24752
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46576\\-9\\93510
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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260\\0\\-390\\-3510\\-5460\\-780\\-260\\0
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0\\4\\0\\-650\\-1092\\-170\\-78\\-30
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5460\\-1092\\-780\\0\\156\\0\\-156\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\-170\\-195\\-65\\0\\10\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-260\\-78\\-195\\-195\\-156\\0\\26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-195\\-585\\-780\\-90\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1080645540, 1082683680, 538358340; 1082683680, -1084725590, -539373705; 538358340, -539373705, -268200526],[780, 65, 1],[303299, 58633, 31375, -5341, -5207, 2465, 12328, 46576; -54, -10, -3, 2, 0, -1, -3, -9; 608920, 117714, 62985, -10725, -10452, 4950, 24752, 93510],85,32]
%
% END OF LATTICE L86.32
%
\hbox{}\par\smallskip%
%
% LATTICE L86.33
%
\leavevmode%
\llap{\spacer}%
${L_{86.33}}=\pdualwitharg{13}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{*}{2}{180}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{}{6}{234}\above{1pt}{1pt}{l}{2}{780}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{130}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19499220\\210600\\-542880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210600\\-2262\\5889
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-542880\\5889\\-15062
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-419\\-17150\\8398
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1199\\-49080\\24030
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\-1516\\741
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\1431\\-702
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\1600\\-780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-531\\261
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-94\\-3845\\1885
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-2701\\1323
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\0\\-78\\-2106\\-5460\\-468\\-260\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\180\\0\\-5850\\-16380\\-1530\\-1170\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\0\\78\\-117\\-780\\-117\\-195\\-39
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2106\\-5850\\-117\\234\\0\\-117\\-585\\-351
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5460\\-16380\\-780\\0\\780\\0\\-780\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-1530\\-117\\-117\\0\\18\\0\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-260\\-1170\\-195\\-585\\-780\\0\\130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-39\\-351\\-780\\-54\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19499220, 210600, -542880; 210600, -2262, 5889; -542880, 5889, -15062],[2340, 39, 1],[-419, -1199, -37, 35, 39, -13, -94, -66; -17150, -49080, -1516, 1431, 1600, -531, -3845, -2701; 8398, 24030, 741, -702, -780, 261, 1885, 1323],85,33]
%
% END OF LATTICE L86.33
%
\hbox{}\par\smallskip%
%
% LATTICE L86.34
%
\leavevmode%
\llap{\spacer}%
${L_{86.34}}=\pdualwitharg{3.13}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${468}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{}{6}{26}\above{1pt}{1pt}{l}{2}{780}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{1170}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12670624980\\-1376256960\\11716380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1376256960\\-149486142\\1272609
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11716380\\1272609\\-10834
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1319\\-4464\\902070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
411\\-1390\\281200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-97\\19968
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\51\\-10231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\40\\-9360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-24\\4751
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
344\\-1170\\234585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-244\\49203
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
468\\0\\-234\\-2106\\-16380\\-468\\-2340\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-650\\-5460\\-170\\-1170\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\0\\78\\-39\\-780\\-39\\-585\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2106\\-650\\-39\\26\\0\\-13\\-585\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16380\\-5460\\-780\\0\\780\\0\\-2340\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-170\\-39\\-13\\0\\2\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2340\\-1170\\-585\\-585\\-2340\\0\\1170\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-39\\-117\\-780\\-18\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12670624980, -1376256960, 11716380; -1376256960, -149486142, 1272609; 11716380, 1272609, -10834],[2340, 39, 1],[1319, 411, 29, -15, -13, 7, 344, 72; -4464, -1390, -97, 51, 40, -24, -1170, -244; 902070, 281200, 19968, -10231, -9360, 4751, 234585, 49203],85,34]
%
% END OF LATTICE L86.34
%
\hbox{}\par\smallskip%
%
% LATTICE L86.35
%
\leavevmode%
\llap{\spacer}%
${L_{86.35}}=\pdualwitharg{5.13}(\pfillwitharg{2}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${65}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{}{6}{1170}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-550491435\\-112698495\\2815605
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112698495\\-23072010\\576420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2815605\\576420\\-14401
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\1421\\55120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\846\\32886
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\380\\15015
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-39\\-1170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-38\\-1521
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\15\\405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\89\\3367
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\413\\15945
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\0\\-195\\-5265\\-1365\\-1170\\-130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-2925\\-819\\-765\\-117\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\0\\390\\-585\\-390\\-585\\-195\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5265\\-2925\\-585\\1170\\0\\-585\\-585\\-1755
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1365\\-819\\-390\\0\\39\\0\\-78\\-390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1170\\-765\\-585\\-585\\0\\90\\0\\-270
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-117\\-195\\-585\\-78\\0\\26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-195\\-1755\\-390\\-270\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-550491435, -112698495, 2815605; -112698495, -23072010, 576420; 2815605, 576420, -14401],[585, 195, 1],[-9, -5, -1, 2, 0, -1, -1, -3; 1421, 846, 380, -39, -38, 15, 89, 413; 55120, 32886, 15015, -1170, -1521, 405, 3367, 15945],85,35]
%
% END OF LATTICE L86.35
%
\hbox{}\par\smallskip%
%
% LATTICE L86.36
%
\leavevmode%
\llap{\spacer}%
${L_{86.36}}=\pdualwitharg{3.5.13}(\pfillwitharg{2}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${585}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{}{6}{130}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{234}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29367585\\-289730610\\-4904055
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-289730610\\-2858197290\\-48378915
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4904055\\-48378915\\-818879
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92470\\-80031\\5281965
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6036\\-5224\\344779
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7289\\-6308\\416325
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
678\\587\\-38740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
661\\572\\-37752
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-313\\-271\\17885
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6679\\-5781\\381537
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9219\\-7979\\526605
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-195\\-585\\-390\\-90\\0\\30
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-29367585, -289730610, -4904055; -289730610, -2858197290, -48378915; -4904055, -48378915, -818879],[585, 195, 1],[-92470, -6036, -7289, 678, 661, -313, -6679, -9219; -80031, -5224, -6308, 587, 572, -271, -5781, -7979; 5281965, 344779, 416325, -38740, -37752, 17885, 381537, 526605],85,36]
%
% END OF LATTICE L86.36
%
\hbox{}\par\smallskip%
%
% LATTICE L86.37
%
\leavevmode%
\llap{\spacer}%
${L_{86.37}}=\pdualwitharg{2.3.13}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{}{6}{312}\above{1pt}{1pt}{l}{2}{65}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{1560}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144644760\\-192660\\-35370660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192660\\312\\47112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35370660\\47112\\8649353
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17567\\71\\71838
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16851\\70\\68910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1119\\6\\4576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
302\\0\\1235
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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537\\1\\2196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8011\\25\\32760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1835\\7\\7504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\-156\\-4212\\-2730\\-936\\-1560\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-3900\\-2730\\-1020\\-2340\\-60
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-156\\0\\104\\-156\\-260\\-156\\-780\\-52
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2730\\-2730\\-260\\0\\65\\0\\-780\\-260
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-1020\\-156\\-156\\0\\24\\0\\-72
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-2340\\-780\\-2340\\-780\\0\\1560\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-52\\-468\\-260\\-72\\0\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[144644760, -192660, -35370660; -192660, 312, 47112; -35370660, 47112, 8649353],[780, 156, 1],[17567, 16851, 1119, 0, 302, 537, 8011, 1835; 71, 70, 6, 1, 0, 1, 25, 7; 71838, 68910, 4576, 0, 1235, 2196, 32760, 7504],85,37]
%
% END OF LATTICE L86.37
%
\hbox{}\par\smallskip%
%
% LATTICE L86.38
%
\leavevmode%
\llap{\spacer}%
${L_{86.38}}=\pdualwitharg{2.5}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{468}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{}{6}{360}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{4680}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{1560}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6214312597254840\\-43724867502420\\1549941453637200
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43724867502420\\307654951080\\-10905628520640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1549941453637200\\-10905628520640\\386578317731267
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
816665\\182722\\-3269170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6208103\\1388673\\-24851502
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
185167\\41327\\-741240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1168\\105\\-4680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1020\\233\\-4083
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
610861\\137436\\-2445300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64053\\14362\\-256408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3216591\\720044\\-12876240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-60\\-1620\\-210\\-4680\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\468\\0\\-11700\\-1638\\-39780\\-468\\-2340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\-180\\-60\\-2340\\-60\\-780
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1620\\-11700\\-180\\360\\0\\-2340\\-180\\-7020
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-1638\\-60\\0\\3\\0\\-12\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4680\\-39780\\-2340\\-2340\\0\\4680\\0\\-14040
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-468\\-60\\-180\\-12\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2340\\-780\\-7020\\-780\\-14040\\0\\1560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6214312597254840, -43724867502420, 1549941453637200; -43724867502420, 307654951080, -10905628520640; 1549941453637200, -10905628520640, 386578317731267],[2340, 60, 1],[816665, 6208103, 185167, 1168, 1020, 610861, 64053, 3216591; 182722, 1388673, 41327, 105, 233, 137436, 14362, 720044; -3269170, -24851502, -741240, -4680, -4083, -2445300, -256408, -12876240],85,38]
%
% END OF LATTICE L86.38
%
\hbox{}\par\smallskip%
%
% LATTICE L86.39
%
\leavevmode%
\llap{\spacer}%
${L_{86.39}}=\pdualwitharg{2.3.5}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{}{6}{40}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{520}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{s}{2}{1560}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3075306291720\\13161430620\\767510841240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13161430620\\56327160\\3284726700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
767510841240\\3284726700\\191549340307
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1005877\\3765\\-4030470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
849244\\3185\\-3402854
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75694\\289\\-303300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\2\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
435\\1\\-1743
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84419\\299\\-338260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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79189\\291\\-317304
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1321763\\4927\\-5296200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\0\\-180\\-1620\\-630\\-4680\\-360\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\52\\0\\-1300\\-546\\-4420\\-468\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\120\\-60\\-60\\-780\\-180\\-780
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1620\\-1300\\-60\\40\\0\\-260\\-180\\-2340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-546\\-60\\0\\3\\0\\-36\\-780
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4680\\-4420\\-780\\-260\\0\\520\\0\\-4680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-468\\-180\\-180\\-36\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-780\\-780\\-2340\\-780\\-4680\\0\\1560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3075306291720, 13161430620, 767510841240; 13161430620, 56327160, 3284726700; 767510841240, 3284726700, 191549340307],[2340, 60, 1],[1005877, 849244, 75694, -5, 435, 84419, 79189, 1321763; 3765, 3185, 289, 2, 1, 299, 291, 4927; -4030470, -3402854, -303300, 20, -1743, -338260, -317304, -5296200],85,39]
%
% END OF LATTICE L86.39
%
\hbox{}\par\smallskip%
%
% LATTICE L86.40
%
\leavevmode%
\llap{\spacer}%
${L_{86.40}}=\pdualwitharg{3.5.13}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${780}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{130}\above{1pt}{1pt}{}{6}{390}\above{1pt}{1pt}{l}{2}{52}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9710220\\9598680\\-1510860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9598680\\-9488310\\1493505
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1510860\\1493505\\-235082
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10673\\-54\\68250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2103\\-10\\13452
\end{tabular}\endgroup%
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-446\\-1\\2860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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93\\2\\-585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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89\\0\\-572
\end{tabular}\endgroup%
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-36\\-1\\225
\end{tabular}\endgroup%
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-361\\-3\\2301
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\HardButStrongLineBreak\kern3pt%
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-525\\-3\\3355
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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780\\0\\-390\\-10530\\-5460\\-2340\\-780\\0
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-2340\\-510\\-195\\-195\\0\\30\\0\\-90
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-780\\-234\\-195\\-585\\-156\\0\\78\\0
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0\\-30\\-65\\-585\\-260\\-90\\0\\10
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9710220, 9598680, -1510860; 9598680, -9488310, 1493505; -1510860, 1493505, -235082],[780, 195, 1],[-10673, -2103, -446, 93, 89, -36, -361, -525; -54, -10, -1, 2, 0, -1, -3, -3; 68250, 13452, 2860, -585, -572, 225, 2301, 3355],85,40]
%
% END OF LATTICE L86.40
%
\hbox{}\par\smallskip%
%
% LATTICE L86.41
%
\leavevmode%
\llap{\spacer}%
${L_{86.41}}=\pdualwitharg{2.5.13}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${260}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{1560}\above{1pt}{1pt}{}{6}{520}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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236316386280\\8299980\\58767586020
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8299980\\520\\2063880
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58767586020\\2063880\\14614429631
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1995247\\6110307\\8022430
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-388727\\1190450\\1562982
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-450257\\1378884\\1810380
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\HardButStrongLineBreak\kern3pt%
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0\\1\\0
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-2648\\8109\\10647
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-38709\\118543\\155640
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-157147\\481251\\631852
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-605093\\1853055\\2432940
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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260\\0\\-780\\-7020\\-2730\\-1560\\-520\\0
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0\\4\\0\\-1300\\-546\\-340\\-156\\-60
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-780\\-2340\\-780\\-360\\0\\120
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[236316386280, 8299980, 58767586020; 8299980, 520, 2063880; 58767586020, 2063880, 14614429631],[780, 260, 1],[-1995247, -388727, -450257, 0, -2648, -38709, -157147, -605093; 6110307, 1190450, 1378884, 1, 8109, 118543, 481251, 1853055; 8022430, 1562982, 1810380, 0, 10647, 155640, 631852, 2432940],85,41]
%
% END OF LATTICE L86.41
%
\hbox{}\par\smallskip%
%
% LATTICE L86.42
%
\leavevmode%
\llap{\spacer}%
${L_{86.42}}=\pdualwitharg{2.13}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{b}{2}{180}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{}{6}{936}\above{1pt}{1pt}{l}{2}{195}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{*}{2}{520}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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5290744680\\600231060\\1320801300
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600231060\\68096184\\149843928
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1320801300\\149843928\\329729779
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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100381\\99\\-402142
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290186\\285\\-1162530
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20210\\19\\-80964
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701\\0\\-2808
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4332\\5\\-17355
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8393\\9\\-33624
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44067\\45\\-176540
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31167\\31\\-124860
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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52\\0\\-156\\-4212\\-2730\\-936\\-520\\0
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-520\\-2340\\-780\\-2340\\-780\\0\\520\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-156\\-1404\\-780\\-216\\0\\24
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5290744680, 600231060, 1320801300; 600231060, 68096184, 149843928; 1320801300, 149843928, 329729779],[2340, 156, 1],[100381, 290186, 20210, 701, 4332, 8393, 44067, 31167; 99, 285, 19, 0, 5, 9, 45, 31; -402142, -1162530, -80964, -2808, -17355, -33624, -176540, -124860],85,42]
%
% END OF LATTICE L86.42
%
\hbox{}\par\smallskip%
%
% LATTICE L86.43
%
\leavevmode%
\llap{\spacer}%
${L_{86.43}}=\pdualwitharg{2.3.13}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${468}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{}{6}{104}\above{1pt}{1pt}{l}{2}{195}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{4680}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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5558679963720\\1853244900\\1387280487060
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1853244900\\618072\\462514572
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\kern3pt%
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1387280487060\\462514572\\346223772971
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3491\\-39\\-13988
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
468\\0\\-468\\-4212\\-8190\\-936\\-4680\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5558679963720, 1853244900, 1387280487060; 1853244900, 618072, 462514572; 1387280487060, 462514572, 346223772971],[2340, 156, 1],[-6418528, -2034391, -367139, 3491, -157241, -77966, -3145988, -682494; 71901, 22790, 4114, -39, 1760, 873, 35235, 7645; 25718238, 8151550, 1471080, -13988, 630045, 312400, 12605580, 2734668],85,43]
%
% END OF LATTICE L86.43
%
\hbox{}\par\smallskip%
%
% LATTICE L86.44
%
\leavevmode%
\llap{\spacer}%
${L_{86.44}}=\pdualwitharg{5.13}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${260}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{b}{2}{390}\above{1pt}{1pt}{}{6}{1170}\above{1pt}{1pt}{l}{2}{156}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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91228880340\\-15091766430\\-117849225
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
712391940\\-117849225\\-920266
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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12149\\48578\\3183830
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7183\\28722\\1882332
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1526\\6103\\399750
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-314\\-1254\\-82485
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-305\\-1220\\-79872
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121\\483\\31815
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410\\1639\\107497
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1792\\7165\\469665
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-195\\-1755\\-780\\-270\\0\\30
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-551473455780, 91228880340, 712391940; 91228880340, -15091766430, -117849225; 712391940, -117849225, -920266],[2340, 195, 1],[12149, 7183, 1526, -314, -305, 121, 410, 1792; 48578, 28722, 6103, -1254, -1220, 483, 1639, 7165; 3183830, 1882332, 399750, -82485, -79872, 31815, 107497, 469665],85,44]
%
% END OF LATTICE L86.44
%
\hbox{}\par\smallskip%
%
% LATTICE L86.45
%
\leavevmode%
\llap{\spacer}%
${L_{86.45}}=\pdualwitharg{3.5.13}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2340}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{390}\above{1pt}{1pt}{}{6}{130}\above{1pt}{1pt}{l}{2}{156}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{234}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-29130660\\316702620\\4867200
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\kern3pt%
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316702620\\-3442804170\\-52910325
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4867200\\-52910325\\-813146
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15533\\-100002\\6599970
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1001\\-6444\\425294
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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536\\-3448\\227565
\end{tabular}\endgroup%
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-91\\587\\-38740
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-89\\572\\-37752
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42\\-271\\17885
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631\\-4065\\268281
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795\\-5119\\337845
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2340\\0\\-1170\\-10530\\-16380\\-2340\\-2340\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\-30\\-195\\-585\\-780\\-90\\0\\30
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-29130660, 316702620, 4867200; 316702620, -3442804170, -52910325; 4867200, -52910325, -813146],[2340, 195, 1],[15533, 1001, 536, -91, -89, 42, 631, 795; -100002, -6444, -3448, 587, 572, -271, -4065, -5119; 6599970, 425294, 227565, -38740, -37752, 17885, 268281, 337845],85,45]
%
% END OF LATTICE L86.45
%
\hbox{}\par\smallskip%
%
% LATTICE L86.46
%
\leavevmode%
\llap{\spacer}%
${L_{86.46}}=\pdualwitharg{2.3.5.13}(\pfillwitharg{3}({L_{86.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${780}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{520}\above{1pt}{1pt}{}{6}{1560}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1560\\746460\\-187200
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\kern3pt%
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746460\\2849126280\\-714530700
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187200\\-714530700\\179196733
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-6735\\539999\\2153190
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-1312\\105207\\419502
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-506\\40623\\161980
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\HardButStrongLineBreak\kern3pt%
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1\\0\\0
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-3\\238\\949
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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780\\0\\-780\\-21060\\-2730\\-4680\\-1560\\0
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-468\\-780\\-2340\\-156\\0\\312\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-260\\-2340\\-260\\-360\\0\\40
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1560, 746460, -187200; 746460, 2849126280, -714530700; -187200, -714530700, 179196733],[780, 780, 1],[-6735, -1312, -506, 1, -3, -131, -531, -681; 539999, 105207, 40623, 0, 238, 10473, 42527, 54587; 2153190, 419502, 161980, 0, 949, 41760, 169572, 217660],85,46]
%
% END OF LATTICE L86.46
%
\hbox{}\par\smallskip%
%
% LATTICE L86.47
%
\leavevmode%
\llap{\spacer}%
${L_{86.47}}=\pdualwitharg{2.5.13}({L_{86.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${260}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}{1560}\above{1pt}{1pt}{}{6}{4680}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{360}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63339649125480\\322303332780\\-15566116473180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
322303332780\\1640042040\\-79208064780
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15566116473180\\-79208064780\\3825470862911
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45338015\\-136168962\\181664210
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[63339649125480, 322303332780, -15566116473180; 322303332780, 1640042040, -79208064780; -15566116473180, -79208064780, 3825470862911],[2340, 780, 1],[45338015, 26499245, 10231589, -584, 59986, 2638219, 3570687, 13749409; -136168962, -79588281, -30729727, 1755, -180163, -7923672, -10724262, -41295208; 181664210, 106179426, 40996800, -2340, 240357, 10571040, 14307332, 55092300],85,47]
%
% END OF LATTICE L86.47
%
\hbox{}\par\smallskip%
%
% LATTICE L86.48
%
\leavevmode%
\llap{\spacer}%
${L_{86.48}}=\pdualwitharg{2.3.5.13}({L_{86.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26808239357640, -138808605780, -6794421290280; -138808605780, 718727880, 35180383680; -6794421290280, 35180383680, 1722013894831],[2340, 780, 1],[-327361, -21244, -24442, 65, -165, -2159, -25957, -33143; 1204428, 78177, 90115, -173, 586, 7900, 95322, 121888; -1316250, -85418, -98280, 260, -663, -8680, -104364, -133260],85,48]
%
% END OF LATTICE L86.48
%

\medskip%
%------------------------------WEYL-GROUP-W87--------------------------------
%
\leavevmode\llap{}%
$W_{87\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=84$%
\hfill%
$222222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L87.1
%
\leavevmode%
\llap{\spacer}%
${L_{87.1}}$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}41\above{1pt}{1pt}{1}{}$}\spacer%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[63140, -25420, -820; -25420, 10234, 329; -820, 329, -38],[820, 1, 1],[1125, 338, 495, 29, -412, -35, 2, 2573, 140, 243, 517, 6516, 3391, 70522, 2087, 5128, 81559, 2083; 2796, 840, 1230, 72, -1025, -87, 5, 6396, 348, 604, 1285, 16195, 8428, 175275, 5187, 12745, 202704, 5177; -80, -25, -41, -4, 0, 2, 0, -164, -9, -16, -35, -451, -236, -4920, -146, -360, -5740, -147],86,1]
%
% END OF LATTICE L87.1
%
\hbox{}\par\smallskip%
%
% LATTICE L87.2
%
\leavevmode%
\llap{\spacer}%
${L_{87.2}}=\pfillwitharg{2}({L_{87.1}})$%
{${}={}$}Nikulin $87$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-15785, -3075, 205; -3075, -599, 40; 205, 40, -2],[205, 1, 1],[162, 143, 423, 52, 1358, 16, 2, -8, -1, 6, 67, 1299, 368, 15862, 488, 1258, 10340, 547; -833, -735, -2173, -267, -6970, -82, -10, 41, 5, -31, -345, -6683, -1893, -81590, -2510, -6470, -53177, -2813; -162, -135, -369, -42, -1025, -9, 5, 0, -3, -10, -80, -1394, -388, -16605, -507, -1295, -10578, -556],86,2]
%
% END OF LATTICE L87.2
%
\hbox{}\par\smallskip%
%
% LATTICE L87.3
%
\leavevmode%
\llap{\spacer}%
${L_{87.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{87.1}}))$%
\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1151690, 13530, -227345; 13530, -155, 2670; -227345, 2670, -44878],[205, 5, 1],[-920, -123, -1159, -78, -168, 38, 8, -1198, -151, -152, -145, -9829, -2602, -21808, -3254, -1616, -64730, -3333; 993, 132, 1230, 81, 164, -42, -8, 1353, 168, 167, 158, 10660, 2819, 23616, 3522, 1748, 69987, 3602; 4720, 631, 5945, 400, 861, -195, -41, 6150, 775, 780, 744, 50430, 13350, 111889, 16695, 8291, 332100, 17100],86,3]
%
% END OF LATTICE L87.3
%
\hbox{}\par\smallskip%
%
% LATTICE L87.4
%
\leavevmode%
\llap{\spacer}%
${L_{87.4}}=\pdualwitharg{2}({L_{87.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}41\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1678000440, 5014300, 419471000; 5014300, 14984, 1253488; 419471000, 1253488, 104860473],[820, 4, 1],[-1585, -2394, -5632, -270, -11019, -1, 199, -623, -238, -169, -1991, -30325, -4115, -348551, -10523, -26509, -107248, -11163; -2389, -3625, -8569, -412, -16810, 0, 330, -738, -329, -247, -2965, -45469, -6178, -523570, -15816, -39870, -161376, -16805; 6369, 9620, 22632, 1085, 44280, 4, -800, 2501, 956, 679, 8000, 121852, 16535, 1400560, 42284, 106520, 430951, 44856],86,4]
%
% END OF LATTICE L87.4
%
\hbox{}\par\smallskip%
%
% LATTICE L87.5
%
\leavevmode%
\llap{\spacer}%
${L_{87.5}}=\pdualwitharg{5}({L_{87.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}41\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{410}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{82}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{820}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-491180, -246000, 3280; -246000, -123190, 1635; 3280, 1635, -18],[820, 5, 1],[-1303, -138, -2539, -747, -2239, -196, -39, -1199, 20, 39, -20, -3939, -2413, -10717, -1700, -909, -76537, -2074; 2672, 283, 5207, 1532, 4592, 402, 80, 2460, -41, -80, 41, 8077, 4948, 21976, 3486, 1864, 156948, 4253; 5240, 556, 10250, 3020, 9061, 795, 159, 4920, -80, -160, 79, 15785, 9680, 43009, 6825, 3651, 307500, 8335],86,5]
%
% END OF LATTICE L87.5
%
\hbox{}\par\smallskip%
%
% LATTICE L87.6
%
\leavevmode%
\llap{\spacer}%
${L_{87.6}}=\pdualwitharg{41}(\pfillwitharg{2}({L_{87.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}41\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${41}\above{1pt}{1pt}{r}{2}{410}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{41}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{82}\above{1pt}{1pt}{s}{2}{410}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{82}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2536465\\-66215\\-496305
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\kern3pt%
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-66215\\-1558\\-12956
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\kern3pt%
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{$\left.\llap{\phantom{%
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6022169\\27720\\-30780750
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2536465, -66215, -496305; -66215, -1558, -12956; -496305, -12956, -97111],[205, 41, 1],[-8928, -5936, -270, -730, -181, 377, 361, -296, -1508, -1500, -7099, -2337, -25340, -25889, -31661, -78571, -15346, -32383; -40, -25, -1, -2, 0, 2, 0, -2, -9, -8, -35, -11, -118, -120, -146, -360, -70, -147; 45633, 30340, 1380, 3731, 925, -1927, -1845, 1513, 7708, 7667, 36285, 11945, 129519, 132325, 161827, 401595, 78437, 165517],86,6]
%
% END OF LATTICE L87.6
%
\hbox{}\par\smallskip%
%
% LATTICE L87.7
%
\leavevmode%
\llap{\spacer}%
${L_{87.7}}=\pdualwitharg{2.5}({L_{87.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}41\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{1640}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{328}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{205}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1215825480\\-7539900\\303208940
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7539900\\46760\\-1880340
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\kern3pt%
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303208940\\-1880340\\75615837
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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187113093\\-76206249\\-752192560
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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\GramMatrix{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1215825480, -7539900, 303208940; -7539900, 46760, -1880340; 303208940, -1880340, 75615837],[820, 20, 1],[7153, 2799, 47731, 3342, 38878, 3184, 594, 5864, -5, 51, 803, 99951, 14668, 256486, 40040, 21018, 437182, 46831; -2914, -1142, -19516, -1369, -15949, -1311, -247, -2460, -2, -20, -325, -40631, -5966, -104345, -16293, -8555, -177981, -19069; -28755, -11252, -191880, -13435, -156292, -12800, -2388, -23575, 20, -205, -3228, -401800, -58965, -1031068, -160960, -84492, -1757465, -188260],86,7]
%
% END OF LATTICE L87.7
%
\hbox{}\par\smallskip%
%
% LATTICE L87.8
%
\leavevmode%
\llap{\spacer}%
${L_{87.8}}=\pdualwitharg{41}({L_{87.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}41\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[555140, 521520, -7380; 521520, 489786, -6929; -7380, -6929, 98],[820, 41, 1],[-11, -7, 2, 65, 38, 53, 152, 65, 75, 51, 32, 3, 35, 12, 7, -2, -5, -10; 28, 25, -1, -68, -45, -69, -215, -96, -115, -84, -60, -6, -72, -25, -15, 5, 12, 24; 1148, 1230, 76, 0, -365, -943, -3895, -1948, -2542, -2132, -1845, -199, -2460, -865, -533, 205, 472, 943],86,8]
%
% END OF LATTICE L87.8
%
\hbox{}\par\smallskip%
%
% LATTICE L87.9
%
\leavevmode%
\llap{\spacer}%
${L_{87.9}}=\pdualwitharg{5.41}(\pfillwitharg{2}({L_{87.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}41\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${205}\above{1pt}{1pt}{r}{2}{82}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{205}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{410}\above{1pt}{1pt}{s}{2}{82}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{410}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
410\\781460\\154365
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\kern3pt%
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781460\\1692972205\\334419780
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
154365\\334419780\\66059318
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[410, 781460, 154365; 781460, 1692972205, 334419780; 154365, 334419780, 66059318],[205, 205, 1],[212, 30, 8, 30, 3, 1, 1, 8, 34, 32, 31, 53, 580, 119, 731, 365, 358, 759; -125857, -17032, -4246, -14821, -1410, 0, -1474, -6489, -25876, -22191, -19980, -32766, -355299, -72614, -444144, -220598, -215583, -455280; 637140, 86223, 21495, 75030, 7138, 0, 7462, 32850, 130995, 112340, 101147, 165875, 1798670, 367602, 2248440, 1116758, 1091370, 2304815],86,9]
%
% END OF LATTICE L87.9
%
\hbox{}\par\smallskip%
%
% LATTICE L87.10
%
\leavevmode%
\llap{\spacer}%
${L_{87.10}}=\pdualwitharg{2.41}({L_{87.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[457856840, -2360780, 113931620; -2360780, 12136, -587448; 113931620, -587448, 28350377],[820, 164, 1],[-1397, -17539, -6553, -36475, -75359, -93283, -234941, -23176, -98871, -14021, -21006, -1156, -2070, -1731, 775, 3671, 49, -1346; 139, 1740, 650, 3618, 7475, 9253, 23305, 2299, 9808, 1391, 2085, 115, 207, 175, -73, -355, -4, 137; 5617, 70520, 26348, 146657, 303000, 375068, 944640, 93185, 397536, 56375, 84460, 4648, 8323, 6960, -3116, -14760, -197, 5412],86,10]
%
% END OF LATTICE L87.10
%
\hbox{}\par\smallskip%
%
% LATTICE L87.11
%
\leavevmode%
\llap{\spacer}%
${L_{87.11}}=\pdualwitharg{5.41}({L_{87.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}41\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${820}\above{1pt}{1pt}{r}{2}{82}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{820}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{410}\above{1pt}{1pt}{b}{2}{82}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{410}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
164820\\8200\\-4100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8200\\410\\-205
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4100\\-205\\102
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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503\\14112\\51660
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-2\\-37\\-164
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-1\\-13\\-70
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-1\\-8\\-59
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\-615\\-1845\\-44280\\-4715\\-30135\\-15785\\-31980\\-35055
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[164820, 8200, -4100; 8200, 410, -205; -4100, -205, 102],[820, 205, 1],[-17, -2, -1, -13, -1, -4, -1, -1, 0, 1, 0, -1, -27, -3, -20, -11, -23, -26; -428, -37, -13, -128, -8, -18, 0, 0, -1, -20, -19, -43, -992, -104, -654, -336, -672, -727; -1640, -164, -70, -820, -59, -205, -41, -40, 0, 0, -41, -135, -3280, -351, -2255, -1189, -2420, -2665],86,11]
%
% END OF LATTICE L87.11
%
\hbox{}\par\smallskip%
%
% LATTICE L87.12
%
\leavevmode%
\llap{\spacer}%
${L_{87.12}}=\pdualwitharg{2.5.41}({L_{87.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}41\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${205}\above{1pt}{1pt}{r}{2}{328}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{205}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{1640}\above{1pt}{1pt}{*}{2}{328}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{1640}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1640\\151700\\37720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
151700\\-332920\\-82820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37720\\-82820\\-20603
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
727\\-650832\\2617440
\end{tabular}\endgroup%
\kern3pt%
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-2561\\2289533\\-9207780
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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26\\-23193\\93275
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\HardButStrongLineBreak\kern3pt%
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10\\-8849\\35588
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4\\-3501\\14080
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11\\-9532\\38335
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3\\-2578\\10368
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9\\-7544\\30340
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
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\kern3pt%
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-9020\\-2788\\-820\\-1640\\-328\\0\\328\\0\\-820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4715\\-1640\\-580\\-1435\\-360\\-820\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21320\\-8200\\-3280\\-9020\\-2460\\-7380\\-820\\0\\1640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3485\\-1640\\-820\\-2665\\-820\\-3280\\-820\\-205\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1640\\-1148\\-820\\-3280\\-1148\\-5740\\-2132\\-820\\-2460
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-820\\-820\\-860\\-4100\\-1580\\-9020\\-4100\\-1880\\-7380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2665\\-3280\\-4100\\-20705\\-8200\\-48380\\-22960\\-10865\\-44280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-820\\-1148\\-1580\\-8200\\-3288\\-19680\\-9512\\-4560\\-18860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3280\\-5740\\-9020\\-48380\\-19680\\-119720\\-59040\\-28700\\-120540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-820\\-2132\\-4100\\-22960\\-9512\\-59040\\-29848\\-14760\\-63140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-205\\-820\\-1880\\-10865\\-4560\\-28700\\-14760\\-7385\\-31980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2460\\-7380\\-44280\\-18860\\-120540\\-63140\\-31980\\-140220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1640, 151700, 37720; 151700, -332920, -82820; 37720, -82820, -20603],[820, 820, 1],[26, 10, 4, 11, 3, 9, 1, 0, -2, 0, 3, 9, 54, 23, 147, 77, 39, 171; -23193, -8849, -3501, -9532, -2578, -7544, -734, 46, 1835, -51, -2773, -8121, -48578, -20666, -131920, -68998, -34912, -152921; 93275, 35588, 14080, 38335, 10368, 30340, 2952, -185, -7380, 205, 11152, 32660, 195365, 83112, 530540, 277488, 140405, 615000],86,12]
%
% END OF LATTICE L87.12
%

\medskip%
%------------------------------WEYL-GROUP-W88--------------------------------
%
\leavevmode\llap{}%
$W_{88\phantom{0}}$%
\qquad\llap{32} lattices, $\chi=18$%
\hfill%
$2222222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L88.1
%
\leavevmode%
\llap{\spacer}%
${L_{88.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{88}}%
\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-620760\\-248640\\2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-248640\\-99590\\1009
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2520\\1009\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-265\\672\\1008
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-33\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79\\-200\\-280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-63\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\28\\52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\33\\53
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-168\\-2520\\-1680\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-120\\-105\\-20\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\6\\0\\-21\\-12\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-120\\0\\40\\0\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-105\\-21\\0\\42\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-20\\-12\\-40\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-9\\-80\\-42\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-620760, -248640, 2520; -248640, -99590, 1009; 2520, 1009, -10],[840, 1, 1],[-265, -2, 13, 79, 25, -11, -13; 672, 5, -33, -200, -63, 28, 33; 1008, 0, -54, -280, -63, 52, 53],87,1]
%
% END OF LATTICE L88.1
%
\hbox{}\par\smallskip%
%
% LATTICE L88.2
%
\leavevmode%
\llap{\spacer}%
${L_{88.2}}=\pfillwitharg{2}({L_{88.1}})$%
{${}={}$}Nikulin $88$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16590\\6720\\210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6720\\-2722\\-85
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210\\-85\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\42\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-10\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-84\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-570\\-100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-298\\-735\\-147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-32\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-84\\-630\\-840\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-60\\-105\\-10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\6\\0\\-21\\-6\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-60\\0\\10\\0\\-10\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-105\\-21\\0\\42\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-10\\-6\\-10\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-9\\-40\\-42\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-16590, 6720, 210; 6720, -2722, -85; 210, -85, -2],[210, 1, 1],[17, -4, -34, -231, -298, -13, 2; 42, -10, -84, -570, -735, -32, 5; 0, 5, -9, -100, -147, -10, -3],87,2]
%
% END OF LATTICE L88.2
%
\hbox{}\par\smallskip%
%
% LATTICE L88.3
%
\leavevmode%
\llap{\spacer}%
${L_{88.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1060080\\114450\\-527730
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
114450\\-12280\\56974
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-527730\\56974\\-262715
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3944\\168\\7959
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\5\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
782\\-33\\-1578
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1162\\-50\\-2345
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1415\\-63\\-2856
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-169\\7\\341
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-781\\33\\1576
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-84\\-315\\-840\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-60\\-210\\-10\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\12\\0\\-42\\-6\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-60\\0\\5\\0\\-5\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-210\\-42\\0\\84\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-10\\-6\\-5\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-18\\-40\\-84\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1060080, 114450, -527730; 114450, -12280, 56974; -527730, 56974, -262715],[210, 2, 1],[-3944, -104, 782, 1162, 1415, -169, -781; 168, 5, -33, -50, -63, 7, 33; 7959, 210, -1578, -2345, -2856, 341, 1576],87,3]
%
% END OF LATTICE L88.3
%
\hbox{}\par\smallskip%
%
% LATTICE L88.4
%
\leavevmode%
\llap{\spacer}%
${L_{88.4}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{30}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1141770\\145320\\-378630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
145320\\-18420\\48189
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-378630\\48189\\-125560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1801\\112\\5474
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\5\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
179\\-11\\-544
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1589\\-100\\-4830
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
320\\-21\\-973
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-233\\14\\708
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-536\\33\\1629
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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14\\0\\-28\\-630\\-280\\-42\\0
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0\\30\\0\\-180\\-105\\-30\\-15
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-28\\0\\2\\0\\-7\\-6\\-9
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-630\\-180\\0\\30\\0\\-30\\-120
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-105\\-7\\0\\14\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-30\\-6\\-30\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-9\\-120\\-42\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1141770, 145320, -378630; 145320, -18420, 48189; -378630, 48189, -125560],[210, 3, 1],[-1801, -69, 179, 1589, 320, -233, -536; 112, 5, -11, -100, -21, 14, 33; 5474, 210, -544, -4830, -973, 708, 1629],87,4]
%
% END OF LATTICE L88.4
%
\hbox{}\par\smallskip%
%
% LATTICE L88.5
%
\leavevmode%
\llap{\spacer}%
${L_{88.5}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${210}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1393350\\207060\\559020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
207060\\-30700\\-83075
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
559020\\-83075\\-224282
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7043\\336\\17430
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\1\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-703\\-33\\-1740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-413\\-20\\-1022
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1231\\-63\\-3045
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
307\\14\\760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
701\\33\\1735
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210\\0\\-420\\-630\\-4200\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-12\\-105\\-10\\-5
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\30\\0\\-105\\-30\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-12\\0\\2\\0\\-10\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4200\\-105\\-105\\0\\210\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-10\\-30\\-10\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-45\\-40\\-210\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1393350, 207060, 559020; 207060, -30700, -83075; 559020, -83075, -224282],[210, 5, 1],[7043, 17, -703, -413, -1231, 307, 701; 336, 1, -33, -20, -63, 14, 33; 17430, 42, -1740, -1022, -3045, 760, 1735],87,5]
%
% END OF LATTICE L88.5
%
\hbox{}\par\smallskip%
%
% LATTICE L88.6
%
\leavevmode%
\llap{\spacer}%
${L_{88.6}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8323021140\\-2913330\\4137607110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2913330\\1020\\-1448298
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4137607110\\-1448298\\2056920475
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7033\\371\\-14147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2953\\170\\-5940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
521\\28\\-1048
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8941\\435\\-17985
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9201\\441\\-18508
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1302\\62\\-2619
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3299\\167\\-6636
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-28\\-315\\-280\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-180\\-210\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\4\\0\\-14\\-6\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-180\\0\\15\\0\\-15\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-210\\-14\\0\\28\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-30\\-6\\-15\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-18\\-120\\-84\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8323021140, -2913330, 4137607110; -2913330, 1020, -1448298; 4137607110, -1448298, 2056920475],[210, 6, 1],[7033, 2953, 521, 8941, 9201, 1302, 3299; 371, 170, 28, 435, 441, 62, 167; -14147, -5940, -1048, -17985, -18508, -2619, -6636],87,6]
%
% END OF LATTICE L88.6
%
\hbox{}\par\smallskip%
%
% LATTICE L88.7
%
\leavevmode%
\llap{\spacer}%
${L_{88.7}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{70}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1678530\\268800\\720930
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
268800\\-42980\\-115451
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720930\\-115451\\-309640
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1139\\48\\2634
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91\\5\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-799\\-33\\-1848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2331\\-100\\-5390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-196\\-9\\-453
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
351\\14\\812
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
796\\33\\1841
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-84\\-630\\-120\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-420\\-105\\-70\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\42\\0\\-21\\-42\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-420\\0\\70\\0\\-70\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-105\\-21\\0\\6\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-70\\-42\\-70\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-63\\-280\\-42\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1678530, 268800, 720930; 268800, -42980, -115451; 720930, -115451, -309640],[210, 7, 1],[1139, 91, -799, -2331, -196, 351, 796; 48, 5, -33, -100, -9, 14, 33; 2634, 210, -1848, -5390, -453, 812, 1841],87,7]
%
% END OF LATTICE L88.7
%
\hbox{}\par\smallskip%
%
% LATTICE L88.8
%
\leavevmode%
\llap{\spacer}%
${L_{88.8}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{420}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75799500\\-358890\\36018990
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-358890\\1700\\-170540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36018990\\-170540\\17115781
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3448\\1113\\-7245
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99\\34\\-208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
257\\84\\-540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
286\\87\\-601
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4397\\1323\\-9240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
207\\62\\-435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
533\\167\\-1120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\0\\-420\\-315\\-4200\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12\\-210\\-10\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\60\\0\\-210\\-30\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-12\\0\\1\\0\\-5\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4200\\-210\\-210\\0\\420\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-10\\-30\\-5\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-90\\-40\\-420\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[75799500, -358890, 36018990; -358890, 1700, -170540; 36018990, -170540, 17115781],[210, 10, 1],[3448, 99, 257, 286, 4397, 207, 533; 1113, 34, 84, 87, 1323, 62, 167; -7245, -208, -540, -601, -9240, -435, -1120],87,8]
%
% END OF LATTICE L88.8
%
\hbox{}\par\smallskip%
%
% LATTICE L88.9
%
\leavevmode%
\llap{\spacer}%
${L_{88.9}}=\pdualwitharg{3}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19320\\7560\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7560\\2958\\-333
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-333\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-331\\840\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-33\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79\\-200\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\56\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\160\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\132\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-56\\-2520\\-560\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-360\\-105\\-60\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\2\\0\\-7\\-12\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-360\\0\\120\\0\\-120\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\-105\\-7\\0\\14\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-60\\-12\\-120\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-9\\-240\\-42\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19320, 7560, -840; 7560, 2958, -333; -840, -333, -34],[840, 3, 1],[-331, -2, 13, 79, -22, -63, -52; 840, 5, -33, -200, 56, 160, 132; -56, 0, 2, 0, -7, -12, -9],87,9]
%
% END OF LATTICE L88.9
%
\hbox{}\par\smallskip%
%
% LATTICE L88.10
%
\leavevmode%
\llap{\spacer}%
${L_{88.10}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30237060\\-268170\\14053830
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-268170\\2380\\-124642
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14053830\\-124642\\6532055
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
449\\159\\-963
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
457\\170\\-980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
235\\84\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1289\\435\\-2765
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
565\\189\\-1212
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
186\\62\\-399
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
483\\167\\-1036
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-84\\-315\\-120\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\140\\0\\-420\\-210\\-70\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\84\\0\\-42\\-42\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-420\\0\\35\\0\\-35\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-210\\-42\\0\\12\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-70\\-42\\-35\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-126\\-280\\-84\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30237060, -268170, 14053830; -268170, 2380, -124642; 14053830, -124642, 6532055],[210, 14, 1],[449, 457, 235, 1289, 565, 186, 483; 159, 170, 84, 435, 189, 62, 167; -963, -980, -504, -2765, -1212, -399, -1036],87,10]
%
% END OF LATTICE L88.10
%
\hbox{}\par\smallskip%
%
% LATTICE L88.11
%
\leavevmode%
\llap{\spacer}%
${L_{88.11}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}{30}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61734750\\396690\\21964110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
396690\\2550\\141135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21964110\\141135\\7814434
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1773\\742\\4970
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76\\34\\213
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\28\\185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-443\\174\\1242
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1136\\441\\3185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-321\\124\\900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-412\\167\\1155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\0\\-140\\-630\\-1400\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-36\\-105\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\10\\0\\-35\\-30\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-36\\0\\6\\0\\-30\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1400\\-105\\-35\\0\\70\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-30\\-30\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-45\\-120\\-210\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[61734750, 396690, 21964110; 396690, 2550, 141135; 21964110, 141135, 7814434],[210, 15, 1],[-1773, -76, -66, -443, -1136, -321, -412; 742, 34, 28, 174, 441, 124, 167; 4970, 213, 185, 1242, 3185, 900, 1155],87,11]
%
% END OF LATTICE L88.11
%
\hbox{}\par\smallskip%
%
% LATTICE L88.12
%
\leavevmode%
\llap{\spacer}%
${L_{88.12}}=\pdualwitharg{5}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${840}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7740600\\-3911880\\25200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3911880\\-1976950\\12735
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25200\\12735\\-82
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-241\\504\\4200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\21\\188
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\99\\930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\112\\1104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89\\189\\1995
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-44\\-380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-19\\-185
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\0\\-840\\-2520\\-8400\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-24\\-105\\-20\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\0\\30\\0\\-105\\-60\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-24\\0\\8\\0\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8400\\-105\\-105\\0\\210\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-20\\-60\\-40\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-45\\-80\\-210\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7740600, -3911880, 25200; -3911880, -1976950, 12735; 25200, 12735, -82],[840, 5, 1],[-241, -10, -47, -53, -89, 21, 9; 504, 21, 99, 112, 189, -44, -19; 4200, 188, 930, 1104, 1995, -380, -185],87,12]
%
% END OF LATTICE L88.12
%
\hbox{}\par\smallskip%
%
% LATTICE L88.13
%
\leavevmode%
\llap{\spacer}%
${L_{88.13}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{210}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{42}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
97320090\\589260\\33999630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
589260\\3570\\205863
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33999630\\205863\\11878070
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-459\\106\\1312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-698\\170\\1995
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\28\\343
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3967\\870\\11340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-290\\63\\829
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-573\\124\\1638
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-742\\167\\2121
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-28\\-630\\-40\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\210\\0\\-1260\\-105\\-210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\14\\0\\-7\\-42\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-1260\\0\\210\\0\\-210\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-105\\-7\\0\\2\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-210\\-42\\-210\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-63\\-840\\-42\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[97320090, 589260, 33999630; 589260, 3570, 205863; 33999630, 205863, 11878070],[210, 21, 1],[-459, -698, -120, -3967, -290, -573, -742; 106, 170, 28, 870, 63, 124, 167; 1312, 1995, 343, 11340, 829, 1638, 2121],87,13]
%
% END OF LATTICE L88.13
%
\hbox{}\par\smallskip%
%
% LATTICE L88.14
%
\leavevmode%
\llap{\spacer}%
${L_{88.14}}=\pdualwitharg{7}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{280}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
271320\\-57120\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57120\\11998\\357
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\357\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\72\\624
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\15\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-27\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-83\\-320\\-2520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-27\\-213
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\19\\161
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-168\\-2520\\-240\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-840\\-105\\-140\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\42\\0\\-21\\-84\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-840\\0\\280\\0\\-280\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-105\\-21\\0\\6\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-140\\-84\\-280\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-63\\-560\\-42\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[271320, -57120, -1680; -57120, 11998, 357; -1680, 357, 10],[840, 7, 1],[19, 4, -7, -83, -7, -1, 5; 72, 15, -27, -320, -27, -4, 19; 624, 140, -210, -2520, -213, -28, 161],87,14]
%
% END OF LATTICE L88.14
%
\hbox{}\par\smallskip%
%
% LATTICE L88.15
%
\leavevmode%
\llap{\spacer}%
${L_{88.15}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1246557900\\15171870\\-619814580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15171870\\-184200\\7543770
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-619814580\\7543770\\-308184733
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110176\\56\\221585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1608\\1\\3234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21982\\-11\\-44210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19390\\-10\\-38997
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38599\\-21\\-77630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14387\\7\\28935
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65767\\33\\132270
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-140\\-315\\-1400\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-36\\-210\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\20\\0\\-70\\-30\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-36\\0\\3\\0\\-15\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1400\\-210\\-70\\0\\140\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-30\\-30\\-15\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-90\\-120\\-420\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1246557900, 15171870, -619814580; 15171870, -184200, 7543770; -619814580, 7543770, -308184733],[210, 30, 1],[-110176, -1608, 21982, 19390, 38599, -14387, -65767; 56, 1, -11, -10, -21, 7, 33; 221585, 3234, -44210, -38997, -77630, 28935, 132270],87,15]
%
% END OF LATTICE L88.15
%
\hbox{}\par\smallskip%
%
% LATTICE L88.16
%
\leavevmode%
\llap{\spacer}%
${L_{88.16}}=\pdualwitharg{2}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{336}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1664880\\-567000\\15120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-567000\\-193072\\5120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15120\\5120\\-107
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-425\\1281\\1239
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\-205\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
214\\-645\\-624
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\-235\\-225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-167\\504\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\208\\202
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-211\\636\\616
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-168\\-315\\-1680\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-120\\-840\\-40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\48\\0\\-168\\-24\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-120\\0\\5\\0\\-10\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-840\\-168\\0\\336\\0\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-40\\-24\\-10\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-72\\-80\\-336\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1664880, -567000, 15120; -567000, -193072, 5120; 15120, 5120, -107],[840, 8, 1],[-425, 68, 214, 78, -167, -69, -211; 1281, -205, -645, -235, 504, 208, 636; 1239, -200, -624, -225, 504, 202, 616],87,16]
%
% END OF LATTICE L88.16
%
\hbox{}\par\smallskip%
%
% LATTICE L88.17
%
\leavevmode%
\llap{\spacer}%
${L_{88.17}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{70}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
892809750\\2304120\\-353464860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2304120\\5950\\-912205
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-353464860\\-912205\\139937324
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6199\\318\\15660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
629\\34\\1589
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1621\\84\\4095
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3569\\174\\9016
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3913\\189\\9885
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2577\\124\\6510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3339\\167\\8435
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\0\\-420\\-630\\-600\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-84\\-105\\-70\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\210\\0\\-105\\-210\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-84\\0\\14\\0\\-70\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-600\\-105\\-105\\0\\30\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-70\\-210\\-70\\0\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-315\\-280\\-210\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[892809750, 2304120, -353464860; 2304120, 5950, -912205; -353464860, -912205, 139937324],[210, 35, 1],[6199, 629, 1621, 3569, 3913, 2577, 3339; 318, 34, 84, 174, 189, 124, 167; 15660, 1589, 4095, 9016, 9885, 6510, 8435],87,17]
%
% END OF LATTICE L88.17
%
\hbox{}\par\smallskip%
%
% LATTICE L88.18
%
\leavevmode%
\llap{\spacer}%
${L_{88.18}}=\pdualwitharg{2.3.7}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{420}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2966525520\\27681150\\-1474951590
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27681150\\-257880\\13763022
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1474951590\\13763022\\-733343495
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23146\\8\\46553
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11172\\5\\22470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32458\\-11\\-65282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
142156\\-50\\-285915
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7981\\-3\\-16052
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21373\\7\\42987
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-97019\\33\\195132
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-28\\-315\\-40\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-1260\\-210\\-210\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\28\\0\\-14\\-42\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-1260\\0\\105\\0\\-105\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-210\\-14\\0\\4\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-210\\-42\\-105\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-126\\-840\\-84\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2966525520, 27681150, -1474951590; 27681150, -257880, 13763022; -1474951590, 13763022, -733343495],[210, 42, 1],[-23146, -11172, 32458, 142156, 7981, -21373, -97019; 8, 5, -11, -50, -3, 7, 33; 46553, 22470, -65282, -285915, -16052, 42987, 195132],87,18]
%
% END OF LATTICE L88.18
%
\hbox{}\par\smallskip%
%
% LATTICE L88.19
%
\leavevmode%
\llap{\spacer}%
${L_{88.19}}=\pdualwitharg{3.5}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${280}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44520\\-21840\\4200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21840\\-10650\\1965
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4200\\1965\\-254
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-359\\840\\560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\14\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-96\\-65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
335\\-784\\-528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
356\\-833\\-560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\-180\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\89\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\0\\-280\\-2520\\-2800\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-72\\-105\\-60\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\0\\10\\0\\-35\\-60\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-72\\0\\24\\0\\-120\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2800\\-105\\-35\\0\\70\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-60\\-60\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-45\\-240\\-210\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-44520, -21840, 4200; -21840, -10650, 1965; 4200, 1965, -254],[840, 15, 1],[-359, -6, 41, 335, 356, 77, -38; 840, 14, -96, -784, -833, -180, 89; 560, 9, -65, -528, -560, -120, 60],87,19]
%
% END OF LATTICE L88.19
%
\hbox{}\par\smallskip%
%
% LATTICE L88.20
%
\leavevmode%
\llap{\spacer}%
${L_{88.20}}=\pdualwitharg{2.5.7}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}{420}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-422914800\\13492710\\-201789210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13492710\\-429800\\6437900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-201789210\\6437900\\-96281533
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31927\\24\\66915
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1022\\1\\2142
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44788\\-33\\-93870
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13069\\-10\\-27391
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10993\\-9\\-23040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9836\\7\\20615
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44621\\33\\93520
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-420\\-315\\-600\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-84\\-210\\-70\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\420\\0\\-210\\-210\\-630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-84\\0\\7\\0\\-35\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-600\\-210\\-210\\0\\60\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-70\\-210\\-35\\0\\35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-630\\-280\\-420\\0\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-422914800, 13492710, -201789210; 13492710, -429800, 6437900; -201789210, 6437900, -96281533],[210, 70, 1],[-31927, -1022, 44788, 13069, 10993, -9836, -44621; 24, 1, -33, -10, -9, 7, 33; 66915, 2142, -93870, -27391, -23040, 20615, 93520],87,20]
%
% END OF LATTICE L88.20
%
\hbox{}\par\smallskip%
%
% LATTICE L88.21
%
\leavevmode%
\llap{\spacer}%
${L_{88.21}}=\pdualwitharg{3.7}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{840}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6828360\\-1373400\\-18480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1373400\\276234\\3717
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18480\\3717\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\24\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\-200\\-2520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-73
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\16\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\24\\63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-56\\-2520\\-80\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\210\\0\\-2520\\-105\\-420\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\14\\0\\-7\\-84\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-2520\\0\\840\\0\\-840\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-105\\-7\\0\\2\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-420\\-84\\-840\\0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-63\\-1680\\-42\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6828360, -1373400, -18480; -1373400, 276234, 3717; -18480, 3717, 50],[840, 21, 1],[5, 1, -2, -47, -1, 3, 5; 24, 5, -9, -200, -4, 16, 24; 64, 0, -70, -2520, -73, -84, 63],87,21]
%
% END OF LATTICE L88.21
%
\hbox{}\par\smallskip%
%
% LATTICE L88.22
%
\leavevmode%
\llap{\spacer}%
${L_{88.22}}=\pdualwitharg{2.3}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-162960\\32760\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32760\\-6576\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\336\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\14\\77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\20\\96
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-56\\-315\\-560\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\240\\0\\-360\\-840\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\16\\0\\-56\\-24\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-360\\0\\15\\0\\-30\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\-840\\-56\\0\\112\\0\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-120\\-24\\-30\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-72\\-240\\-336\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-162960, 32760, -1680; 32760, -6576, 336; -1680, 336, -17],[840, 24, 1],[2, -1, -1, -1, 1, 1, 3; 14, -5, -7, -10, 0, 6, 20; 77, 0, -40, -105, -112, 18, 96],87,22]
%
% END OF LATTICE L88.22
%
\hbox{}\par\smallskip%
%
% LATTICE L88.23
%
\leavevmode%
\llap{\spacer}%
${L_{88.23}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{42}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{210}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-182555310\\10859310\\73277820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10859310\\-644700\\-4358025
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73277820\\-4358025\\-29413112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14427\\28870\\-40220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693\\1387\\-1932
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10119\\-20249\\28210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17717\\-35454\\49392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2484\\-4971\\6925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13333\\26680\\-37170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30244\\60521\\-84315
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-140\\-630\\-200\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-252\\-105\\-210\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\70\\0\\-35\\-210\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-252\\0\\42\\0\\-210\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-105\\-35\\0\\10\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\-210\\-210\\0\\210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-315\\-840\\-210\\0\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-182555310, 10859310, 73277820; 10859310, -644700, -4358025; 73277820, -4358025, -29413112],[210, 105, 1],[-14427, -693, 10119, 17717, 2484, -13333, -30244; 28870, 1387, -20249, -35454, -4971, 26680, 60521; -40220, -1932, 28210, 49392, 6925, -37170, -84315],87,23]
%
% END OF LATTICE L88.23
%
\hbox{}\par\smallskip%
%
% LATTICE L88.24
%
\leavevmode%
\llap{\spacer}%
${L_{88.24}}=\pdualwitharg{5.7}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{280}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4066440\\-1658160\\11760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1658160\\-676130\\4795
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11760\\4795\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\24\\960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-15\\-735
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-64\\-2464
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-33\\-1200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-28\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\-840\\-2520\\-1200\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-168\\-105\\-140\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\0\\210\\0\\-105\\-420\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-168\\0\\56\\0\\-280\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1200\\-105\\-105\\0\\30\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-140\\-420\\-280\\0\\280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-315\\-560\\-210\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4066440, -1658160, 11760; -1658160, -676130, 4795; 11760, 4795, -34],[840, 35, 1],[-7, -1, 4, 19, 10, 9, 0; 24, 3, -15, -64, -33, -28, 1; 960, 77, -735, -2464, -1200, -840, 140],87,24]
%
% END OF LATTICE L88.24
%
\hbox{}\par\smallskip%
%
% LATTICE L88.25
%
\leavevmode%
\llap{\spacer}%
${L_{88.25}}=\pdualwitharg{2.5}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{1680}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35449680\\-1848840\\52920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848840\\-96400\\2760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52920\\2760\\-79
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\5355
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\1304
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-9\\5040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\599
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-42\\5880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-670
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\0\\-840\\-315\\-8400\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-24\\-840\\-40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\0\\240\\0\\-840\\-120\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-24\\0\\1\\0\\-10\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8400\\-840\\-840\\0\\1680\\0\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-40\\-120\\-10\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-360\\-80\\-1680\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35449680, -1848840, 52920; -1848840, -96400, 2760; 52920, 2760, -79],[840, 40, 1],[8, 2, 8, 1, 11, -1, -1; 0, -1, -9, -2, -42, 0, 2; 5355, 1304, 5040, 599, 5880, -670, -600],87,25]
%
% END OF LATTICE L88.25
%
\hbox{}\par\smallskip%
%
% LATTICE L88.26
%
\leavevmode%
\llap{\spacer}%
${L_{88.26}}=\pdualwitharg{2.3.5.7}(\pfillwitharg{2}({L_{88.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{420}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
119124727680\\46899107010\\22659274260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46899107010\\18464061000\\8920899540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22659274260\\8920899540\\4310127041
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\8521\\-17915
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\5007\\-10542
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\4427\\-9310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
87\\15191\\-31899
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\11165\\-23440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62\\11054\\-23205
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
167\\27985\\-58800
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-140\\-315\\-200\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-252\\-210\\-210\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\140\\0\\-70\\-210\\-630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-252\\0\\21\\0\\-105\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-210\\-70\\0\\20\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-210\\-210\\-105\\0\\105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-630\\-840\\-420\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[119124727680, 46899107010, 22659274260; 46899107010, 18464061000, 8920899540; 22659274260, 8920899540, 4310127041],[210, 210, 1],[53, 34, 28, 87, 63, 62, 167; 8521, 5007, 4427, 15191, 11165, 11054, 27985; -17915, -10542, -9310, -31899, -23440, -23205, -58800],87,26]
%
% END OF LATTICE L88.26
%
\hbox{}\par\smallskip%
%
% LATTICE L88.27
%
\leavevmode%
\llap{\spacer}%
${L_{88.27}}=\pdualwitharg{2.7}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{560}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1886640\\189000\\-8400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
189000\\-18928\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8400\\840\\-37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-132\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-195\\-805
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-135\\-576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-13\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\35\\112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-168\\-315\\-240\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\560\\0\\-840\\-840\\-280\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\336\\0\\-168\\-168\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-840\\0\\35\\0\\-70\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-840\\-168\\0\\48\\0\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-280\\-168\\-70\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-504\\-560\\-336\\0\\112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1886640, 189000, -8400; 189000, -18928, 840; -8400, 840, -37],[840, 56, 1],[1, -1, -11, -16, -11, -1, 3; 12, -10, -132, -195, -135, -13, 35; 45, 0, -504, -805, -576, -70, 112],87,27]
%
% END OF LATTICE L88.27
%
\hbox{}\par\smallskip%
%
% LATTICE L88.28
%
\leavevmode%
\llap{\spacer}%
${L_{88.28}}=\pdualwitharg{3.5.7}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{840}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24360\\-2520\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-5250\\105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-11\\-595
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-96\\-5376
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-28\\-1680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-280\\-2520\\-400\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-504\\-105\\-420\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\0\\70\\0\\-35\\-420\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-504\\0\\168\\0\\-840\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-105\\-35\\0\\10\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-420\\-420\\-840\\0\\840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-315\\-1680\\-210\\0\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24360, -2520, 0; -2520, -5250, 105; 0, 105, -2],[840, 105, 1],[1, 0, -1, -7, -1, -1, 1; 8, 1, -11, -96, -15, -28, 5; 400, 63, -595, -5376, -850, -1680, 210],87,28]
%
% END OF LATTICE L88.28
%
\hbox{}\par\smallskip%
%
% LATTICE L88.29
%
\leavevmode%
\llap{\spacer}%
${L_{88.29}}=\pdualwitharg{2.3.5}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{560}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1953840\\-372120\\5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-372120\\-70800\\960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\960\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-153
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-14\\-2240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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35\\0\\-280\\-315\\-2800\\-210\\0
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0\\48\\0\\-72\\-840\\-120\\-120
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-280\\0\\80\\0\\-280\\-120\\-360
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-2800\\-840\\-280\\0\\560\\0\\-1680
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-210\\-120\\-120\\-30\\0\\60\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-360\\-240\\-1680\\0\\240
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1953840, -372120, 5040; -372120, -70800, 960; 5040, 960, -13],[840, 120, 1],[1, 1, 1, 0, -3, -1, -1; 0, -1, -3, -2, -14, 0, 2; 385, 312, 160, -153, -2240, -390, -240],87,29]
%
% END OF LATTICE L88.29
%
\hbox{}\par\smallskip%
%
% LATTICE L88.30
%
\leavevmode%
\llap{\spacer}%
${L_{88.30}}=\pdualwitharg{2.3.7}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{1680}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-777840\\6048
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6048\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\2\\257
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\-1288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-35\\-4515
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-7\\-904
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\17\\2184
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\0\\-56\\-315\\-80\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\1680\\0\\-2520\\-840\\-840\\-840
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\112\\0\\-56\\-168\\-504
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-2520\\0\\105\\0\\-210\\-1680
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-840\\-56\\0\\16\\0\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-840\\-168\\-210\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-504\\-1680\\-336\\0\\336
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1680, -840, 0; -840, -777840, 6048; 0, 6048, -47],[840, 168, 1],[1, -1, -5, -16, -3, 1, 9; 2, 0, -10, -35, -7, 1, 17; 257, 0, -1288, -4515, -904, 126, 2184],87,30]
%
% END OF LATTICE L88.30
%
\hbox{}\par\smallskip%
%
% LATTICE L88.31
%
\leavevmode%
\llap{\spacer}%
${L_{88.31}}=\pdualwitharg{2.5.7}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{*}{2}{1680}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{*}{2}{560}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38640\\-19320\\1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19320\\-9520\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\840\\-73
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-9\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-6\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-840\\-315\\-1200\\-210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\112\\0\\-168\\-840\\-280\\-280
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\0\\1680\\0\\-840\\-840\\-2520
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\kern3pt%
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-315\\-168\\0\\7\\0\\-70\\-560
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1200\\-840\\-840\\0\\240\\0\\-1680
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-280\\-840\\-70\\0\\140\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-2520\\-560\\-1680\\0\\560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38640, -19320, 1680; -19320, -9520, 840; 1680, 840, -73],[840, 280, 1],[2, 3, 5, -1, -7, -3, -1; 0, -1, -9, -2, -6, 0, 2; 45, 56, 0, -49, -240, -70, 0],87,31]
%
% END OF LATTICE L88.31
%
\hbox{}\par\smallskip%
%
% LATTICE L88.32
%
\leavevmode%
\llap{\spacer}%
${L_{88.32}}=\pdualwitharg{2.3.5.7}({L_{88.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{336}\above{1pt}{1pt}{*}{2}{560}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{*}{2}{1680}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\-5880\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5880\\-475440\\68880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\68880\\-9979
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\21\\145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\73\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-81\\-560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-155\\-1071
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-220\\-1520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-152\\-1050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-280\\-315\\-400\\-210\\0
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\kern3pt%
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0\\336\\0\\-504\\-840\\-840\\-840
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\kern3pt%
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-280\\0\\560\\0\\-280\\-840\\-2520
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-504\\0\\21\\0\\-210\\-1680
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-840\\-280\\0\\80\\0\\-1680
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-840\\-840\\-210\\0\\420\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-2520\\-1680\\-1680\\0\\1680
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1680, -5880, 840; -5880, -475440, 68880; 840, 68880, -9979],[840, 840, 1],[1, 4, -2, -6, -9, -7, -1; 21, 73, -81, -155, -220, -152, 0; 145, 504, -560, -1071, -1520, -1050, 0],87,32]
%
% END OF LATTICE L88.32
%

\medskip%
%------------------------------WEYL-GROUP-W89--------------------------------
%
\leavevmode\llap{}%
$W_{89\phantom{0}}$%
\qquad\llap{48} lattices, $\chi=32$%
\hfill%
$22262226\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L89.1
%
\leavevmode%
\llap{\spacer}%
${L_{89.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{23\rightarrow N_{89},3,2}%
\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{b}{2}{72}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{b}{2}{126}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22538902680\\-1942920\\19106640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1942920\\-138\\1629
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19106640\\1629\\-16186
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-397\\-305694\\-499401
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-60060\\-98119
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\26952\\44028
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\12320\\20127
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-13091\\-21385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-214\\-164787\\-269199
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-97\\-74692\\-122020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-160163\\-261651
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-144\\-180\\-153\\-441\\-36\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-42\\-49\\-189\\-28\\-21
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-22538902680, -1942920, 19106640; -1942920, -138, 1629; 19106640, 1629, -16186],[2520, 3, 1],[-397, -78, 35, 16, -17, -214, -97, -208; -305694, -60060, 26952, 12320, -13091, -164787, -74692, -160163; -499401, -98119, 44028, 20127, -21385, -269199, -122020, -261651],88,1]
%
% END OF LATTICE L89.1
%
\hbox{}\par\smallskip%
%
% LATTICE L89.2
%
\leavevmode%
\llap{\spacer}%
${L_{89.2}}=\pfillwitharg{2.3}({L_{89.1}})$%
{${}={}$}Nikulin $89$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23730\\-10290\\210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10290\\4462\\-91
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210\\-91\\2
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-6721\\-15540\\-2940
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2928\\6770\\1281
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-259\\-50
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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0\\0\\1
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6\\14\\7
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-13\\-30\\-4
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-122\\-282\\-51
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-51\\-49\\-6\\-3
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-49\\-63\\-14\\-21
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-14\\-6\\-18
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[23730, -10290, 210; -10290, 4462, -91; 210, -91, 2],[210, 1, 1],[0, 6, -13, -122, -112, -118, -19, -22; 0, 14, -30, -282, -259, -273, -44, -51; 1, 7, -4, -51, -50, -56, -10, -12],88,2]
%
% END OF LATTICE L89.2
%
\hbox{}\par\smallskip%
%
% LATTICE L89.3
%
\leavevmode%
\llap{\spacer}%
${L_{89.3}}=\pfillwitharg{3}({L_{89.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5985056280\\9843120\\-310800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9843120\\-16186\\509
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-310800\\509\\-14
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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3978239\\2498086080\\2506539840
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\kern3pt%
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-6512\\-4089130\\-4102967
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-34\\-21350\\-21427
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-65\\-40816\\-40956
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{$\left[\!\llap{\phantom{%
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-51\\-49\\-12\\-3
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-49\\-63\\-28\\-21
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-12\\-28\\-24\\-36
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-5985056280, 9843120, -310800; 9843120, -16186, 509; -310800, 509, -14],[840, 1, 1],[-1, -34, -65, -160, -111, -78, 1, 16; -628, -21350, -40816, -100470, -69701, -48979, 628, 10047; -633, -21427, -40956, -100809, -69934, -49140, 632, 10080],88,3]
%
% END OF LATTICE L89.3
%
\hbox{}\par\smallskip%
%
% LATTICE L89.4
%
\leavevmode%
\llap{\spacer}%
${L_{89.4}}=\pfillwitharg{2}({L_{89.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{126}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{18}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-4831470\\-2520\\9450
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\24\\-3
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9450\\-3\\-16
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\174\\557
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13\\2268\\7245
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3\\524\\1672
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25\\4371\\13932
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5\\875\\2786
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-1\\-174\\-558
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-1\\-175\\-558
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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-49\\-189\\-14\\-21\\0\\14\\0\\-42
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-3\\-63\\-18\\-87\\-180\\-42\\0\\6
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-4831470, -2520, 9450; -2520, 24, -3; 9450, -3, -16],[630, 3, 1],[1, 13, 3, 13, 25, 5, -1, -1; 174, 2268, 524, 2272, 4371, 875, -174, -175; 557, 7245, 1672, 7245, 13932, 2786, -558, -558],88,4]
%
% END OF LATTICE L89.4
%
\hbox{}\par\smallskip%
%
% LATTICE L89.5
%
\leavevmode%
\llap{\spacer}%
${L_{89.5}}=\pdualwitharg{2}(\pfillwitharg{2.3}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6958140\\31290\\-3471510
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31290\\-136\\15610
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6958140, 31290, -3471510; 31290, -136, 15610; -3471510, 15610, -1731983],[210, 2, 1],[-731, 26115, 20621, 250129, 195464, 168618, 7198, 242; 312, -11102, -8769, -106374, -83129, -71715, -3062, -105; 1468, -52444, -41411, -502308, -392530, -338618, -14455, -486],88,5]
%
% END OF LATTICE L89.5
%
\hbox{}\par\smallskip%
%
% LATTICE L89.6
%
\leavevmode%
\llap{\spacer}%
${L_{89.6}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-8908410\\43260\\-2962470
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8908410, 43260, -2962470; 43260, -204, 14385; -2962470, 14385, -985162],[210, 3, 1],[-626, 22370, 35327, 71418, 167429, 144433, 12331, 69; 312, -11102, -17538, -35458, -83129, -71715, -6124, -35; 1887, -67431, -106488, -215279, -504690, -435372, -37170, -208],88,6]
%
% END OF LATTICE L89.6
%
\hbox{}\par\smallskip%
%
% LATTICE L89.7
%
\leavevmode%
\llap{\spacer}%
${L_{89.7}}=\pdualwitharg{5}(\pfillwitharg{2.3}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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67200\\-340\\-26915
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\kern3pt%
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5186580\\-26915\\-2077256
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12950070, 67200, 5186580; 67200, -340, -26915; 5186580, -26915, -2077256],[210, 5, 1],[1093, -39069, -61697, -374183, -292405, -252243, -21535, -361; 312, -11102, -17538, -106374, -83129, -71715, -6124, -105; 2725, -97405, -153820, -932895, -729010, -628880, -53690, -900],88,7]
%
% END OF LATTICE L89.7
%
\hbox{}\par\smallskip%
%
% LATTICE L89.8
%
\leavevmode%
\llap{\spacer}%
${L_{89.8}}=\pdualwitharg{2.3}(\pfillwitharg{2.3}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5240814180, -14928690, 2605404690; -14928690, 42528, -7421610; 2605404690, -7421610, 1295244091],[210, 6, 1],[15818, -753492, -584110, -2351216, -5500773, -4731463, -199455, 521; 29, -1309, -1018, -4101, -9598, -8260, -349, 0; -31818, 1515654, 1174941, 4729486, 11064840, 9517368, 401205, -1048],88,8]
%
% END OF LATTICE L89.8
%
\hbox{}\par\smallskip%
%
% LATTICE L89.9
%
\leavevmode%
\llap{\spacer}%
${L_{89.9}}=\pdualwitharg{7}(\pfillwitharg{2.3}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17045490, 91140, 7311570; 91140, -476, -39095; 7311570, -39095, -3136258],[210, 7, 1],[1530, -7814, -86377, -523862, -409371, -50449, -30149, -505; 312, -1586, -17538, -106374, -83129, -10245, -6124, -105; 3563, -18197, -201152, -1219953, -953330, -117484, -70210, -1176],88,9]
%
% END OF LATTICE L89.9
%
\hbox{}\par\smallskip%
%
% LATTICE L89.10
%
\leavevmode%
\llap{\spacer}%
${L_{89.10}}=\pdualwitharg{2.5}(\pfillwitharg{2.3}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[92688540, -2563050, 44112810; -2563050, 70880, -1219820; 44112810, -1219820, 20994397],[210, 10, 1],[2804, -133062, -103173, -1245974, -971695, -835829, -35240, 257; 29, -1309, -1018, -12303, -9598, -8260, -349, 0; -5890, 279510, 216725, 2617290, 2041140, 1755740, 74025, -540],88,10]
%
% END OF LATTICE L89.10
%
\hbox{}\par\smallskip%
%
% LATTICE L89.11
%
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${L_{89.11}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
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0\\42\\0\\-42
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-48\\0\\24\\0
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-60\\-42\\0\\2
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-153\\-147\\-36\\-3
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5746440, 21840, 14280; 21840, -78, -57; 14280, -57, -34],[840, 3, 1],[-1, -2, 5, 8, 22, 23, 7, 1; -120, -238, 600, 958, 2633, 2751, 836, 119; -219, -441, 1092, 1751, 4818, 5040, 1536, 220],88,11]
%
% END OF LATTICE L89.11
%
\hbox{}\par\smallskip%
%
% LATTICE L89.12
%
\leavevmode%
\llap{\spacer}%
${L_{89.12}}=\pdualwitharg{2.7}(\pfillwitharg{2.3}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57649620\\-2391690\\26880210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2391690\\99232\\-1115170
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26880210\\-1115170\\12533399
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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37638713024\\392773275\\-80688330450
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\kern3pt%
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-1549825507\\-16172978\\3322452406
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17549799087\\183138357\\-37622540047
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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2658\\29\\-5698
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-17988\\-187\\38562
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-97642\\-1018\\209321
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\HardButStrongLineBreak\kern3pt%
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-1179208\\-12303\\2527938
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-714\\-98\\-42\\-42
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-98\\-18\\-14\\-42
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-42\\-14\\-21\\-126
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[57649620, -2391690, 26880210; -2391690, 99232, -1115170; 26880210, -1115170, 12533399],[210, 14, 1],[2658, -17988, -97642, -1179208, -919637, -113009, -33355, 235; 29, -187, -1018, -12303, -9598, -1180, -349, 0; -5698, 38562, 209321, 2527938, 1971480, 242264, 71505, -504],88,12]
%
% END OF LATTICE L89.12
%
\hbox{}\par\smallskip%
%
% LATTICE L89.13
%
\leavevmode%
\llap{\spacer}%
${L_{89.13}}=\pdualwitharg{3.5}(\pfillwitharg{2.3}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{l}{2}{30}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65282910\\2634450\\23290050
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2634450\\106320\\939855
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23290050\\939855\\8308858
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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31082602613\\-403677834\\-87080130270
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11088930207\\-144014817\\-31066429636
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0\\0\\0
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-2131\\29\\5970
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101193\\-1309\\-283500
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156919\\-2036\\-439620
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315837\\-4101\\-884840
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[65282910, 2634450, 23290050; 2634450, 106320, 939855; 23290050, 939855, 8308858],[210, 15, 1],[-2131, 101193, 156919, 315837, 738930, 635606, 53595, -66; 29, -1309, -2036, -4101, -9598, -8260, -698, 0; 5970, -283500, -439620, -884840, -2070165, -1780695, -150150, 185],88,13]
%
% END OF LATTICE L89.13
%
\hbox{}\par\smallskip%
%
% LATTICE L89.14
%
\leavevmode%
\llap{\spacer}%
${L_{89.14}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{252}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{}{6}{36}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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12750180660\\-38171070\\6354779130
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\kern3pt%
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-38171070\\114276\\-19024728
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6354779130\\-19024728\\3167266321
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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3124\\-5\\-6268
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29390\\-21\\-58968
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3598\\0\\-7219
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34426\\11\\-69072
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\HardButStrongLineBreak\kern3pt%
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71681\\36\\-143820
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\HardButStrongLineBreak\kern3pt%
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17221\\14\\-34552
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\HardButStrongLineBreak\kern3pt%
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785\\3\\-1575
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311\\-2\\-624
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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-98\\-378\\-14\\-42\\0\\28\\0\\-84
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-18\\-126\\-9\\-54\\-72\\0\\9\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-126\\-18\\-174\\-360\\-84\\0\\12
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12750180660, -38171070, 6354779130; -38171070, 114276, -19024728; 6354779130, -19024728, 3167266321],[630, 6, 1],[3124, 29390, 3598, 34426, 71681, 17221, 785, 311; -5, -21, 0, 11, 36, 14, 3, -2; -6268, -58968, -7219, -69072, -143820, -34552, -1575, -624],88,14]
%
% END OF LATTICE L89.14
%
\hbox{}\par\smallskip%
%
% LATTICE L89.15
%
\leavevmode%
\llap{\spacer}%
${L_{89.15}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1164743160, 9709560, 11670960; 9709560, -80930, -97015; 11670960, -97015, -110026],[840, 5, 1],[13879, 9750, -127, -2000, 128, 4257, 8131, 20009; 1748983, 1228661, -16004, -252033, 16130, 536452, 1024640, 2521464; -69950, -49140, 640, 10080, -645, -21455, -40980, -100845],88,15]
%
% END OF LATTICE L89.15
%
\hbox{}\par\smallskip%
%
% LATTICE L89.16
%
\leavevmode%
\llap{\spacer}%
${L_{89.16}}=\pdualwitharg{3.7}(\pfillwitharg{2.3}({L_{89.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[160224330, 4883340, 56055090; 4883340, 148848, 1708455; 56055090, 1708455, 19611086],[210, 21, 1],[-4027, 27267, 296011, 595809, 1393968, 171296, 101115, -120; 29, -187, -2036, -4101, -9598, -1180, -698, 0; 11508, -77922, -845922, -1702666, -3983595, -489519, -288960, 343],88,16]
%
% END OF LATTICE L89.16
%
\hbox{}\par\smallskip%
%
% LATTICE L89.17
%
\leavevmode%
\llap{\spacer}%
${L_{89.17}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{89.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-82545960, 3058440, -3959760; 3058440, -113302, 146489; -3959760, 146489, -187022],[840, 7, 1],[-10113, -1014, 97, 1456, -100, -445, -5933, -14587; -303161, -30397, 2908, 43647, -2998, -13340, -177856, -437280; -23338, -2340, 224, 3360, -231, -1027, -13692, -33663],88,17]
%
% END OF LATTICE L89.17
%
\hbox{}\par\smallskip%
%
% LATTICE L89.18
%
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${L_{89.18}}=\pdualwitharg{2.3.5}(\pfillwitharg{2.3}({L_{89.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11354558880, 4875150, -5645732400; 4875150, -2040, 2424030; -5645732400, 2424030, -2807180329],[210, 30, 1],[-99121, 3542953, 2797487, 11310897, 26516680, 22874606, 976450, 10914; 312, -11102, -8769, -35458, -83129, -71715, -3062, -35; 199350, -7125510, -5626245, -22748230, -53329770, -46004910, -1963815, -21950],88,18]
%
% END OF LATTICE L89.18
%
\hbox{}\par\smallskip%
%
% LATTICE L89.19
%
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${L_{89.19}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{89.1}}))$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4362041040, 50595720, -397717320; 50595720, -586864, 4613176; -397717320, 4613176, -36262091],[840, 8, 1],[3068, 8254, 2082, 14686, 7929, 2743, -511, -547; 214761, 577787, 145743, 1028049, 555050, 192024, -35770, -38292; -6328, -17024, -4294, -30288, -16352, -5656, 1054, 1128],88,19]
%
% END OF LATTICE L89.19
%
\hbox{}\par\smallskip%
%
% LATTICE L89.20
%
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${L_{89.20}}=\pdualwitharg{5.7}(\pfillwitharg{2.3}({L_{89.1}}))$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1522451910, 19433400, -602660520; 19433400, 248080, -7692685; -602660520, -7692685, 238562348],[210, 35, 1],[18177, -123063, -1335983, -8067187, -6291395, -773111, -456365, 1621; 29, -187, -2036, -12303, -9598, -1180, -698, 0; 45920, -310890, -3375050, -20379870, -15893745, -1953085, -1152900, 4095],88,20]
%
% END OF LATTICE L89.20
%
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%
% LATTICE L89.21
%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-29784351660, 9333030, -14808739890; 9333030, -2856, 4640370; -14808739890, 4640370, -7362885707],[210, 42, 1],[-182261, 930827, 5144749, 20801365, 48765572, 6009646, 1795724, 20054; 312, -1586, -8769, -35458, -83129, -10245, -3062, -35; 366576, -1872144, -10347477, -41837152, -98080710, -12087018, -3611685, -40334],88,21]
%
% END OF LATTICE L89.21
%
\hbox{}\par\smallskip%
%
% LATTICE L89.22
%
\leavevmode%
\llap{\spacer}%
${L_{89.22}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${90}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{90}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{s}{2}{630}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2565990\\-3150\\1047060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3150\\120\\585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1047060\\585\\-423296
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8989\\126717\\22410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1797\\25333\\4480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-361\\-5088\\-900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-361\\-5089\\-900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
359\\5060\\895
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4675\\65898\\11655
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1079\\15210\\2690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4675\\65902\\11655
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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90\\0\\-360\\-900\\-765\\-2205\\-90\\-45
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-360\\0\\90\\0\\-90\\-630\\-90\\-270
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-2205\\-945\\-630\\-315\\0\\630\\0\\-630
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-90\\-70\\-90\\-90\\-40\\0\\10\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-105\\-270\\-435\\-300\\-630\\0\\30
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2565990, -3150, 1047060; -3150, 120, 585; 1047060, 585, -423296],[630, 15, 1],[8989, 1797, -361, -361, 359, 4675, 1079, 4675; 126717, 25333, -5088, -5089, 5060, 65898, 15210, 65902; 22410, 4480, -900, -900, 895, 11655, 2690, 11655],88,22]
%
% END OF LATTICE L89.22
%
\hbox{}\par\smallskip%
%
% LATTICE L89.23
%
\leavevmode%
\llap{\spacer}%
${L_{89.23}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-696360\\17640\\224280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17640\\-390\\-5865
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
224280\\-5865\\-71642
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
259559\\2075640\\642600
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7107\\-56834\\-17595
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81885\\-654815\\-202726
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-103\\-824\\-255
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-212\\-1694\\-525
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509\\4072\\1260
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822\\6574\\2035
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\210\\0\\-210
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-300\\-210\\0\\10
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-765\\-735\\-180\\-15
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-735\\-945\\-420\\-105
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-420\\-360\\-180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-105\\-180\\-145
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-696360, 17640, 224280; 17640, -390, -5865; 224280, -5865, -71642],[840, 15, 1],[-103, -212, 509, 822, 2266, 2375, 727, 105; -824, -1694, 4072, 6574, 18121, 18991, 5812, 839; -255, -525, 1260, 2035, 5610, 5880, 1800, 260],88,23]
%
% END OF LATTICE L89.23
%
\hbox{}\par\smallskip%
%
% LATTICE L89.24
%
\leavevmode%
\llap{\spacer}%
${L_{89.24}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${126}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{126}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{}{6}{14}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1047690\\-6930\\-506520
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6930\\168\\-1533
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-506520\\-1533\\-229438
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-8021\\-159549\\18774
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-229\\-4555\\536
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
323\\6426\\-756
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\HardButStrongLineBreak\kern3pt%
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323\\6425\\-756
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\HardButStrongLineBreak\kern3pt%
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-320\\-6366\\749
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\HardButStrongLineBreak\kern3pt%
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-596\\-11856\\1395
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-963\\-19156\\2254
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4172\\-82988\\9765
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
126\\0\\-504\\-1260\\-1071\\-441\\-126\\-63
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-441\\-27\\-126\\-63\\0\\18\\0\\-126
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-14\\-126\\-126\\-56\\0\\14\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-21\\-378\\-609\\-420\\-126\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1047690, -6930, -506520; -6930, 168, -1533; -506520, -1533, -229438],[630, 21, 1],[-8021, -229, 323, 323, -320, -596, -963, -4172; -159549, -4555, 6426, 6425, -6366, -11856, -19156, -82988; 18774, 536, -756, -756, 749, 1395, 2254, 9765],88,24]
%
% END OF LATTICE L89.24
%
\hbox{}\par\smallskip%
%
% LATTICE L89.25
%
\leavevmode%
\llap{\spacer}%
${L_{89.25}}=\pdualwitharg{2.5.7}(\pfillwitharg{2.3}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4274738580, 4564350, -2039654610; 4564350, -4760, 2177840; -2039654610, 2177840, -973203589],[210, 70, 1],[-85303, 435657, 2407906, 29207093, 22823830, 2812704, 840455, 28156; 312, -1586, -8769, -106374, -83129, -10245, -3062, -105; 178780, -913060, -5046545, -61212900, -47834710, -5894930, -1761445, -59010],88,25]
%
% END OF LATTICE L89.25
%
\hbox{}\par\smallskip%
%
% LATTICE L89.26
%
\leavevmode%
\llap{\spacer}%
${L_{89.26}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-5885880\\-220920\\18480
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220920\\-8274\\693
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5885880, -220920, 18480; -220920, -8274, 693; 18480, 693, -58],[840, 21, 1],[-1, 0, 5, 6, 15, 2, 3, 0; -19, -1, 92, 117, 299, 41, 68, 3; -546, -12, 2688, 3304, 8337, 1125, 1764, 35],88,26]
%
% END OF LATTICE L89.26
%
\hbox{}\par\smallskip%
%
% LATTICE L89.27
%
\leavevmode%
\llap{\spacer}%
${L_{89.27}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{}{6}{20}\above{1pt}{1pt}{s}{2}{1260}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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618\\-5\\-1270
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7720\\-21\\-15750
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1134\\0\\-2305
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11660\\11\\-23670
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[261082080, -355430250, 128445660; -355430250, 483873360, -174862530; 128445660, -174862530, 63191957],[630, 30, 1],[25243, 6459, 469, -47, 618, 7720, 1134, 11660; 36, 14, 3, -2, -5, -21, 0, 11; -51210, -13090, -945, 90, -1270, -15750, -2305, -23670],88,27]
%
% END OF LATTICE L89.27
%
\hbox{}\par\smallskip%
%
% LATTICE L89.28
%
\leavevmode%
\llap{\spacer}%
${L_{89.28}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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61320\\-624\\-6552
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{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5223120, 61320, 542640; 61320, -624, -6552; 542640, -6552, -56033],[840, 24, 1],[2530, 2648, 202, 116, -115, -233, 143, 919; 37943, 39711, 3029, 1739, -1725, -3493, 2145, 13783; 20064, 21000, 1602, 920, -912, -1848, 1134, 7288],88,28]
%
% END OF LATTICE L89.28
%
\hbox{}\par\smallskip%
%
% LATTICE L89.29
%
\leavevmode%
\llap{\spacer}%
${L_{89.29}}=\pdualwitharg{3.5.7}(\pfillwitharg{2.3}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${210}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{210}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-2292755430\\4094370\\825411720
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\kern3pt%
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4094370\\-7140\\-1473885
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2292755430, 4094370, 825411720; 4094370, -7140, -1473885; 825411720, -1473885, -297155246],[210, 105, 1],[-57742, 294898, 3259843, 6590134, 15449545, 1903931, 1137815, 6353; 115796, -591382, -6537224, -13215726, -30982219, -3818107, -2281754, -12741; -160965, 822075, 9087330, 18371045, 43068060, 5307510, 3171840, 17710],88,29]
%
% END OF LATTICE L89.29
%
\hbox{}\par\smallskip%
%
% LATTICE L89.30
%
\leavevmode%
\llap{\spacer}%
${L_{89.30}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${252}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{63}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{}{6}{28}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{}{6}$}%
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{$\left.\llap{\phantom{%
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2672\\3\\-5355
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357\\-2\\-714
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\HardButStrongLineBreak\kern3pt%
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6724\\-5\\-13468
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9866\\-3\\-19764
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9039\\0\\-18109
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89010\\11\\-178332
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{$\left.\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11557435680, 3531095190, 5768825580; 3531095190, 1078840812, 1762525254; 5768825580, 1762525254, 2879475127],[630, 42, 1],[188351, 6641, 2672, 357, 6724, 9866, 9039, 89010; 36, 2, 3, -2, -5, -3, 0, 11; -377370, -13306, -5355, -714, -13468, -19764, -18109, -178332],88,30]
%
% END OF LATTICE L89.30
%
\hbox{}\par\smallskip%
%
% LATTICE L89.31
%
\leavevmode%
\llap{\spacer}%
${L_{89.31}}=\pdualwitharg{5.7}(\pfillwitharg{3}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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78960\\-490\\-11515
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{$\left.\llap{\phantom{%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1785\\-245\\-420\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-245\\-45\\-140\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-140\\-840\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-105\\-1260\\-3045
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11465160, 78960, 1648920; 78960, -490, -11515; 1648920, -11515, -236678],[840, 35, 1],[-2511, -252, 23, 362, -23, -110, -1471, -3620; -45202, -4536, 416, 6516, -417, -1981, -26484, -65169; -15295, -1535, 140, 2205, -140, -670, -8960, -22050],88,31]
%
% END OF LATTICE L89.31
%
\hbox{}\par\smallskip%
%
% LATTICE L89.32
%
\leavevmode%
\llap{\spacer}%
${L_{89.32}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{*}{2}{560}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-151576728240\\-36019405800\\-917158200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36019405800\\-8559345520\\-217945680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-917158200\\-217945680\\-5549527
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-344767501\\2412620280\\-37771473600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81923875\\573288381\\-8975281840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2085875\\14596574\\-228520881
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6941\\-48567\\760240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2403\\-16807\\262920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-447\\3129\\-49010
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-479\\3351\\-52440
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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80\\0\\-160\\-2400
\end{tabular}\endgroup%
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0\\560\\0\\-1680
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-160\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2400\\-1680\\0\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2040\\-1960\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1960\\-2520\\-280\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-280\\-60\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-840\\-360\\-3480
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-151576728240, -36019405800, -917158200; -36019405800, -8559345520, -217945680; -917158200, -217945680, -5549527],[840, 40, 1],[6941, 2403, -447, -479, 2684, 7222, 1822, 12854; -48567, -16807, 3129, 3351, -18787, -50547, -12751, -89949; 760240, 262920, -49010, -52440, 294240, 791560, 199650, 1408200],88,32]
%
% END OF LATTICE L89.32
%
\hbox{}\par\smallskip%
%
% LATTICE L89.33
%
\leavevmode%
\llap{\spacer}%
${L_{89.33}}=\pdualwitharg{5}({L_{89.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${90}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{b}{2}{360}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{b}{2}{630}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4663202040\\-1968120\\1946316960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1968120\\-690\\815925
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1946316960\\815925\\-812132474
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198847\\-19487010\\-496125
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39068\\-3828664\\-97475
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17531\\1718040\\43740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8014\\785372\\19995
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8515\\-834471\\-21245
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-107188\\-10504431\\-267435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48585\\-4761332\\-121220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104182\\-10209839\\-259935
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\0\\-720\\-900\\-765\\-2205\\-180\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-210\\-245\\-945\\-140\\-105
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\kern3pt%
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-720\\0\\360\\0\\-180\\-1260\\-360\\-540
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\kern3pt%
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-900\\-210\\0\\30\\-15\\-315\\-180\\-435
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-765\\-245\\-180\\-15\\10\\0\\-80\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2205\\-945\\-1260\\-315\\0\\630\\0\\-630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-140\\-360\\-180\\-80\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-105\\-540\\-435\\-300\\-630\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4663202040, -1968120, 1946316960; -1968120, -690, 815925; 1946316960, 815925, -812132474],[2520, 15, 1],[-198847, -39068, 17531, 8014, -8515, -107188, -48585, -104182; -19487010, -3828664, 1718040, 785372, -834471, -10504431, -4761332, -10209839; -496125, -97475, 43740, 19995, -21245, -267435, -121220, -259935],88,33]
%
% END OF LATTICE L89.33
%
\hbox{}\par\smallskip%
%
% LATTICE L89.34
%
\leavevmode%
\llap{\spacer}%
${L_{89.34}}=\pdualwitharg{2.3.5.7}(\pfillwitharg{2.3}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
203693872620\\80003805210\\38653583220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80003805210\\31422687240\\15181771080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38653583220\\15181771080\\7335024257
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6027771464\\-53039340009495\\109810695129030
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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29\\243838\\-504840
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-187\\-1651828\\3419880
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\HardButStrongLineBreak\kern3pt%
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-1018\\-8965886\\18562635
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4101\\-36092704\\74725000
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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0\\60\\0\\-420\\-1470\\-270\\-210\\-210
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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420\\0\\-840\\-4200
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0\\60\\0\\-420
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-840\\0\\105\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4200\\-420\\0\\140
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10710\\-1470\\-630\\-210
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1470\\-270\\-210\\-210
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-210\\-315\\-630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-210\\-630\\-2030
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[203693872620, 80003805210, 38653583220; 80003805210, 31422687240, 15181771080; 38653583220, 15181771080, 7335024257],[210, 210, 1],[29, -187, -1018, -4101, -9598, -1180, -349, 0; 243838, -1651828, -8965886, -36092704, -84443023, -10376627, -3062569, 7339; -504840, 3419880, 18562635, 74725000, 174827730, 21483390, 6340635, -15190],88,34]
%
% END OF LATTICE L89.34
%
\hbox{}\par\smallskip%
%
% LATTICE L89.35
%
\leavevmode%
\llap{\spacer}%
${L_{89.35}}=\pdualwitharg{2.7}(\pfillwitharg{3}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-201658848720, -47393939880, 830067000; -47393939880, -11138541904, 195082664; 830067000, 195082664, -3416717],[840, 56, 1],[6447, 319, -415, -445, 2492, 958, 1692, 11938; -32215, -1593, 2075, 2223, -12459, -4789, -8457, -59661; -273112, -13456, 17654, 18816, -105952, -40696, -71806, -506184],88,35]
%
% END OF LATTICE L89.35
%
\hbox{}\par\smallskip%
%
% LATTICE L89.36
%
\leavevmode%
\llap{\spacer}%
${L_{89.36}}=\pdualwitharg{7}({L_{89.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${126}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{504}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{}{6}{14}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{}{6}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1980720\\-966\\-867951
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-1491147000\\-867951\\-656429938
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{$\left.\llap{\phantom{%
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5937\\296850\\-13879
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-8525\\-426250\\19929
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9058\\452899\\-21175
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51683\\2584148\\-120820
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3386837160, -1980720, -1491147000; -1980720, -966, -867951; -1491147000, -867951, -656429938],[2520, 21, 1],[211526, 5937, -18649, -8525, 9058, 16289, 51683, 110825; 10576296, 296850, -932448, -426250, 452899, 814449, 2584148, 5541247; -494487, -13879, 43596, 19929, -21175, -38079, -120820, -259077],88,36]
%
% END OF LATTICE L89.36
%
\hbox{}\par\smallskip%
%
% LATTICE L89.37
%
\leavevmode%
\llap{\spacer}%
${L_{89.37}}=\pdualwitharg{2}({L_{89.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${144}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{}{6}{16}\above{1pt}{1pt}{*}{2}{1008}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{}{6}$}%
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{$\left[\!\llap{\phantom{%
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171360\\25248\\-2771
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-72\\-168\\-216\\-696\\-480\\-1008\\0\\48
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10589040, -1559880, 171360; -1559880, -229776, 25248; 171360, 25248, -2771],[2520, 24, 1],[46, 4, -56, -280, -237, -677, -13, -5; -249, -21, 303, 1513, 1280, 3654, 70, 26; 576, 56, -702, -3528, -2992, -8568, -166, -72],88,37]
%
% END OF LATTICE L89.37
%
\hbox{}\par\smallskip%
%
% LATTICE L89.38
%
\leavevmode%
\llap{\spacer}%
${L_{89.38}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${630}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{630}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{}{6}{70}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{l}{2}{70}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-2312730\\-2227050\\-1125180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2227050\\-2144310\\-1083495
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1125180\\-1083495\\-547418
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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-380\\-2\\785
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\HardButStrongLineBreak\kern3pt%
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613\\0\\-1260
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\HardButStrongLineBreak\kern3pt%
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560\\2\\-1155
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\HardButStrongLineBreak\kern3pt%
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-578\\-1\\1190
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\HardButStrongLineBreak\kern3pt%
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-1048\\-3\\2160
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\HardButStrongLineBreak\kern3pt%
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-1663\\-6\\3430
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-7124\\-29\\14700
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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630\\0\\-2520\\-6300\\-5355\\-2205\\-630\\-315
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\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2312730, -2227050, -1125180; -2227050, -2144310, -1083495; -1125180, -1083495, -547418],[630, 105, 1],[-13582, -380, 613, 560, -578, -1048, -1663, -7124; -60, -2, 0, 2, -1, -3, -6, -29; 28035, 785, -1260, -1155, 1190, 2160, 3430, 14700],88,38]
%
% END OF LATTICE L89.38
%
\hbox{}\par\smallskip%
%
% LATTICE L89.39
%
\leavevmode%
\llap{\spacer}%
${L_{89.39}}=\pdualwitharg{3.5.7}(\pfillwitharg{3}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-6567960\\-522480\\2635920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-522480\\-41370\\209685
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\kern3pt%
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2635920\\209685\\-1057874
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{$\left.\llap{\phantom{%
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0\\0\\0
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110476\\2839\\275835
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6567960, -522480, 2635920; -522480, -41370, 209685; 2635920, 209685, -1057874],[840, 105, 1],[-757, -24, 3701, 4598, 11649, 1580, 2523, 70; -19, -1, 92, 117, 299, 41, 68, 3; -1890, -60, 9240, 11480, 29085, 3945, 6300, 175],88,39]
%
% END OF LATTICE L89.39
%
\hbox{}\par\smallskip%
%
% LATTICE L89.40
%
\leavevmode%
\llap{\spacer}%
${L_{89.40}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-17742480\\6001800\\-73080
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6001800\\-2030160\\24720
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17742480, 6001800, -73080; 6001800, -2030160, 24720; -73080, 24720, -301],[840, 120, 1],[0, 2, 0, -2, -7, -9, -1, -1; -19, -35, 23, 145, 397, 413, 31, 17; -1560, -3360, 1890, 12400, 34320, 36120, 2790, 1640],88,40]
%
% END OF LATTICE L89.40
%
\hbox{}\par\smallskip%
%
% LATTICE L89.41
%
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\llap{\spacer}%
${L_{89.41}}=\pdualwitharg{2.5.7}(\pfillwitharg{2}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6308957340, 1433250, 3149396460; 1433250, 420, 715470; 3149396460, 715470, 1572161219],[630, 210, 1],[-689998, -24820, -11479, 0, -20477, -32887, -31956, -322041; 6375, 229, 105, 1, 192, 306, 296, 2978; 1382220, 49720, 22995, 0, 41020, 65880, 64015, 645120],88,41]
%
% END OF LATTICE L89.41
%
\hbox{}\par\smallskip%
%
% LATTICE L89.42
%
\leavevmode%
\llap{\spacer}%
${L_{89.42}}=\pdualwitharg{2.3.7}(\pfillwitharg{3}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15310238160, 15278293800, 36797040; 15278293800, -15246415632, -36720264; 36797040, -36720264, -88439],[840, 168, 1],[319, 11, -389, -1939, -4918, -668, -268, -32; -19, -1, 23, 117, 299, 41, 17, 3; 140616, 4992, -171402, -855344, -2170392, -294960, -118566, -14560],88,42]
%
% END OF LATTICE L89.42
%
\hbox{}\par\smallskip%
%
% LATTICE L89.43
%
\leavevmode%
\llap{\spacer}%
${L_{89.43}}=\pdualwitharg{2.5.7}(\pfillwitharg{3}({L_{89.1}}))$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5290320, -21330120, 151200; -21330120, -85989680, 609560; 151200, 609560, -4321],[840, 280, 1],[198, 76, 134, 944, 509, 25, -33, -35; 197, 75, 131, 915, 489, 23, -33, -33; 34720, 13240, 23170, 162120, 86800, 4120, -5810, -5880],88,43]
%
% END OF LATTICE L89.43
%
\hbox{}\par\smallskip%
%
% LATTICE L89.44
%
\leavevmode%
\llap{\spacer}%
${L_{89.44}}=\pdualwitharg{5.7}({L_{89.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-829450440, -827270640, 22954680; -827270640, -825095670, 22894305; 22954680, 22894305, -635258],[2520, 105, 1],[14710, 413, -1295, -593, 629, 1132, 3593, 7706; -29424, -826, 2592, 1186, -1259, -2265, -7188, -15415; -528885, -14845, 46620, 21315, -22645, -40725, -129220, -277095],88,44]
%
% END OF LATTICE L89.44
%
\hbox{}\par\smallskip%
%
% LATTICE L89.45
%
\leavevmode%
\llap{\spacer}%
${L_{89.45}}=\pdualwitharg{2.5}({L_{89.1}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-275693040, 778680, -32760; 778680, 211920, -5400; -32760, -5400, 137],[2520, 120, 1],[-19, -1, 23, 113, 95, 269, 5, 1; 10509, 553, -12723, -62509, -52552, -148806, -2766, -554; 409680, 21560, -495990, -2436840, -2048680, -5801040, -107830, -21600],88,45]
%
% END OF LATTICE L89.45
%
\hbox{}\par\smallskip%
%
% LATTICE L89.46
%
\leavevmode%
\llap{\spacer}%
${L_{89.46}}=\pdualwitharg{2.7}({L_{89.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1008}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{252}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{}{6}{112}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-155771280\\26109720\\-200309760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26109720\\-4372368\\33549264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200309760\\33549264\\-257417621
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
94015\\-2822388\\-441000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2357\\-70758\\-11056
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3089\\92736\\14490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3295\\98918\\15456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6184\\-185651\\-29008
\end{tabular}\endgroup%
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9010\\-270489\\-42264
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6390\\-191833\\-29974
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51538\\-1547207\\-241752
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1008\\0\\-2016\\-10080\\-8568\\-3528\\-504\\-504
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0\\16\\0\\-336\\-392\\-216\\-56\\-168
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-8568\\-392\\-504\\-168\\112\\0\\-224\\-3360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3528\\-216\\-504\\-504\\0\\144\\0\\-1008
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\kern3pt%
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-504\\-56\\-252\\-504\\-224\\0\\28\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-168\\-1512\\-4872\\-3360\\-1008\\0\\336
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-155771280, 26109720, -200309760; 26109720, -4372368, 33549264; -200309760, 33549264, -257417621],[2520, 168, 1],[94015, 2357, -3089, -3295, 6184, 9010, 6390, 51538; -2822388, -70758, 92736, 98918, -185651, -270489, -191833, -1547207; -441000, -11056, 14490, 15456, -29008, -42264, -29974, -241752],88,46]
%
% END OF LATTICE L89.46
%
\hbox{}\par\smallskip%
%
% LATTICE L89.47
%
\leavevmode%
\llap{\spacer}%
${L_{89.47}}=\pdualwitharg{2.3.5.7}(\pfillwitharg{3}({L_{89.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1680}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{*}{2}{560}\above{1pt}{1pt}{}{6}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14211120\\-8209320\\-3493560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8209320\\-4739280\\-2016840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3493560\\-2016840\\-858283
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-19721\\4792453\\-11181240
\end{tabular}\endgroup%
\kern3pt%
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-11280\\2741321\\-6395760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4800\\1166520\\-2721601
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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299\\-72727\\169680
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41\\-9927\\23160
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\HardButStrongLineBreak\kern3pt%
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17\\-4051\\9450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-601\\1400
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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1680\\0\\-3360\\-16800
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0\\240\\0\\-1680
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-16800\\-1680\\0\\560
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-42840\\-5880\\-2520\\-840
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-5880\\-1080\\-840\\-840
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-2520\\-840\\-1260\\-2520
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-840\\-2520\\-8120
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14211120, -8209320, -3493560; -8209320, -4739280, -2016840; -3493560, -2016840, -858283],[840, 840, 1],[299, 41, 17, 3, -19, -1, 23, 117; -72727, -9927, -4051, -601, 4680, 206, -5670, -28562; 169680, 23160, 9450, 1400, -10920, -480, 13230, 66640],88,47]
%
% END OF LATTICE L89.47
%
\hbox{}\par\smallskip%
%
% LATTICE L89.48
%
\leavevmode%
\llap{\spacer}%
${L_{89.48}}=\pdualwitharg{2.5.7}({L_{89.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5040}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{*}{2}{1260}\above{1pt}{1pt}{*}{2}{1680}\above{1pt}{1pt}{}{6}{560}\above{1pt}{1pt}{*}{2}{720}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{*}{2}{1680}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-248930640\\35025480\\48285720
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35025480\\-4927440\\-6794760
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48285720\\-6794760\\-9365329
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80113\\239340\\239400
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1981\\5918\\5920
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2741\\-8190\\-8190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2811\\-8398\\-8400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5435\\16239\\16240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7831\\23397\\23400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5505\\16447\\16450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44133\\131851\\131880
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\0\\-10080\\-50400\\-42840\\-17640\\-2520\\-2520
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-1680\\-1960\\-1080\\-280\\-840
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10080\\0\\1260\\0\\-2520\\-2520\\-1260\\-7560
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-50400\\-1680\\0\\1680\\-840\\-2520\\-2520\\-24360
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42840\\-1960\\-2520\\-840\\560\\0\\-1120\\-16800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17640\\-1080\\-2520\\-2520\\0\\720\\0\\-5040
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-280\\-1260\\-2520\\-1120\\0\\140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-840\\-7560\\-24360\\-16800\\-5040\\0\\1680
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-248930640, 35025480, 48285720; 35025480, -4927440, -6794760; 48285720, -6794760, -9365329],[2520, 840, 1],[80113, 1981, -2741, -2811, 5435, 7831, 5505, 44133; 239340, 5918, -8190, -8398, 16239, 23397, 16447, 131851; 239400, 5920, -8190, -8400, 16240, 23400, 16450, 131880],88,48]
%
% END OF LATTICE L89.48
%

\medskip%
%------------------------------WEYL-GROUP-W90--------------------------------
%
\leavevmode\llap{}%
$W_{90\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=37$%
\hfill%
$222462222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L90.1
%
\leavevmode%
\llap{\spacer}%
${L_{90.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}73\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{90}}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{292}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{4}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{s}{2}{146}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{438}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-263993988\\88065156\\275064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88065156\\-29377454\\-91759
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
275064\\-91759\\-286
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4225\\12598\\21556
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79077\\235790\\403398
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2470\\7365\\12597
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1263\\3766\\6438
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
330\\984\\1679
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-329\\-981\\-1680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-661\\-1971\\-3358
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
274\\817\\1400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49870\\148701\\254478
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-24\\-46\\-72\\-292\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\292\\0\\-292\\-730\\-1314\\-5986\\-146\\-2190
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-15\\-39\\-219\\-9\\-219
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-292\\0\\4\\-2\\-18\\-146\\-10\\-438
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\-730\\-15\\-2\\2\\-3\\-73\\-9\\-657
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-1314\\-39\\-18\\-3\\6\\0\\-6\\-876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-292\\-5986\\-219\\-146\\-73\\0\\146\\0\\-3066
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-146\\-9\\-10\\-9\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2190\\-219\\-438\\-657\\-876\\-3066\\0\\438
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-263993988, 88065156, 275064; 88065156, -29377454, -91759; 275064, -91759, -286],[876, 1, 1],[4225, 79077, 2470, 1263, 330, -329, -661, 274, 49870; 12598, 235790, 7365, 3766, 984, -981, -1971, 817, 148701; 21556, 403398, 12597, 6438, 1679, -1680, -3358, 1400, 254478],89,1]
%
% END OF LATTICE L90.1
%
\hbox{}\par\smallskip%
%
% LATTICE L90.2
%
\leavevmode%
\llap{\spacer}%
${L_{90.2}}=\pfillwitharg{2}({L_{90.1}})$%
{${}={}$}Nikulin $90$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}73\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{73}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{s}{2}{146}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{438}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15330\\-5037\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5037\\-1655\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\73\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-91\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
148\\-449\\-23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
265\\-804\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1203\\-3650\\-146
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-88\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
433\\-1314\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-6\\-23\\-36\\-146\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\73\\0\\-73\\-365\\-657\\-2993\\-73\\-1095
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\0\\-15\\-39\\-219\\-9\\-219
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-73\\0\\1\\-1\\-9\\-73\\-5\\-219
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-365\\-15\\-1\\2\\-3\\-73\\-9\\-657
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-657\\-39\\-9\\-3\\6\\0\\-6\\-876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-146\\-2993\\-219\\-73\\-73\\0\\146\\0\\-3066
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-73\\-9\\-5\\-9\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1095\\-219\\-219\\-657\\-876\\-3066\\0\\438
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-15330, -5037, 0; -5037, -1655, 0; 0, 0, 1],[219, 1, 1],[0, -24, 1, 30, 148, 265, 1203, 29, 433; 0, 73, -3, -91, -449, -804, -3650, -88, -1314; 1, 0, -3, -6, -23, -36, -146, -2, 0],89,2]
%
% END OF LATTICE L90.2
%
\hbox{}\par\smallskip%
%
% LATTICE L90.3
%
\leavevmode%
\llap{\spacer}%
${L_{90.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{90.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}73\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{219}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{4}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{s}{2}{438}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{146}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4220787\\30441\\-1397220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30441\\-213\\10077
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1397220\\10077\\-462526
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25820\\-23\\-78000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
380814\\-365\\-1150407
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3919\\-5\\-11839
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-432\\-1\\1305
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
146\\2\\-441
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4929\\-1\\-14890
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
155429\\-73\\-469536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13631\\-9\\-41178
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
266877\\-219\\-806212
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-18\\-69\\-36\\-438\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\219\\0\\-219\\-1095\\-657\\-8979\\-219\\-1095
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0\\-15\\-13\\-219\\-9\\-73
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-219\\0\\3\\-3\\-9\\-219\\-15\\-219
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\-1095\\-15\\-3\\6\\-3\\-219\\-27\\-657
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-657\\-13\\-9\\-3\\2\\0\\-6\\-292
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-438\\-8979\\-219\\-219\\-219\\0\\438\\0\\-3066
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-219\\-9\\-15\\-27\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1095\\-73\\-219\\-657\\-292\\-3066\\0\\146
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4220787, 30441, -1397220; 30441, -213, 10077; -1397220, 10077, -462526],[219, 3, 1],[25820, 380814, 3919, -432, 146, 4929, 155429, 13631, 266877; -23, -365, -5, -1, 2, -1, -73, -9, -219; -78000, -1150407, -11839, 1305, -441, -14890, -469536, -41178, -806212],89,3]
%
% END OF LATTICE L90.3
%
\hbox{}\par\smallskip%
%
% LATTICE L90.4
%
\leavevmode%
\llap{\spacer}%
${L_{90.4}}=\pdualwitharg{3}({L_{90.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}73\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{876}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{4}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{s}{2}{438}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{146}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-151983372\\-312732\\208488
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312732\\-642\\429
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208488\\429\\-286
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-46\\21066
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
421\\-730\\305724
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\1450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-732
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\2185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93\\-73\\67671
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-9\\5817
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
152\\-219\\110449
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-72\\-138\\-72\\-876\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\876\\0\\-876\\-2190\\-1314\\-17958\\-438\\-2190
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-15\\-13\\-219\\-9\\-73
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-876\\0\\12\\-6\\-18\\-438\\-30\\-438
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138\\-2190\\-15\\-6\\6\\-3\\-219\\-27\\-657
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-1314\\-13\\-18\\-3\\2\\0\\-6\\-292
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-876\\-17958\\-219\\-438\\-219\\0\\438\\0\\-3066
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-438\\-9\\-30\\-27\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2190\\-73\\-438\\-657\\-292\\-3066\\0\\146
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-151983372, -312732, 208488; -312732, -642, 429; 208488, 429, -286],[876, 3, 1],[29, 421, 2, -1, 0, 3, 93, 8, 152; -46, -730, -5, -2, 2, -1, -73, -9, -219; 21066, 305724, 1450, -732, 3, 2185, 67671, 5817, 110449],89,4]
%
% END OF LATTICE L90.4
%
\hbox{}\par\smallskip%
%
% LATTICE L90.5
%
\leavevmode%
\llap{\spacer}%
${L_{90.5}}=\pdualwitharg{2}({L_{90.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}73\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{292}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{4}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{s}{2}{584}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{1752}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4733639448\\-58423068\\-1208406960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58423068\\721112\\14914300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1208406960\\14914300\\308483023
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60579\\80802\\-241210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1149045\\1532635\\-4575202
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73837\\98487\\-294000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20925\\27911\\-83318
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17530\\23383\\-69800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-877\\1170\\-3492
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15327\\20440\\-61028
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7667\\10226\\-30528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1421227\\1895664\\-5658960
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-24\\-92\\-144\\-584\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\292\\0\\-292\\-1460\\-2628\\-11972\\-292\\-4380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0\\-60\\-156\\-876\\-36\\-876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-292\\0\\4\\-4\\-36\\-292\\-20\\-876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\-1460\\-60\\-4\\8\\-12\\-292\\-36\\-2628
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-2628\\-156\\-36\\-12\\24\\0\\-24\\-3504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-584\\-11972\\-876\\-292\\-292\\0\\584\\0\\-12264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-292\\-36\\-20\\-36\\-24\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4380\\-876\\-876\\-2628\\-3504\\-12264\\0\\1752
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4733639448, -58423068, -1208406960; -58423068, 721112, 14914300; -1208406960, 14914300, 308483023],[876, 4, 1],[-60579, -1149045, -73837, -20925, -17530, -877, -15327, -7667, -1421227; 80802, 1532635, 98487, 27911, 23383, 1170, 20440, 10226, 1895664; -241210, -4575202, -294000, -83318, -69800, -3492, -61028, -30528, -5658960],89,5]
%
% END OF LATTICE L90.5
%
\hbox{}\par\smallskip%
%
% LATTICE L90.6
%
\leavevmode%
\llap{\spacer}%
${L_{90.6}}=\pdualwitharg{2.3}({L_{90.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}73\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{876}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{4}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{s}{2}{1752}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{584}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16460332584\\-2628\\-4133977152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2628\\24\\660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4133977152\\660\\1038239477
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140359\\-397\\-558870
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2113043\\-5986\\-8413542
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23849\\-68\\-94960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-547\\-2\\-2178
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22755\\-63\\-90604
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-757479\\-2117\\-3016068
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69359\\-195\\-276168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1421969\\-4015\\-5661880
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-72\\-276\\-144\\-1752\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\876\\0\\-876\\-4380\\-2628\\-35916\\-876\\-4380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-60\\-52\\-876\\-36\\-292
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-876\\0\\12\\-12\\-36\\-876\\-60\\-876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276\\-4380\\-60\\-12\\24\\-12\\-876\\-108\\-2628
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-2628\\-52\\-36\\-12\\8\\0\\-24\\-1168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1752\\-35916\\-876\\-876\\-876\\0\\1752\\0\\-12264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-876\\-36\\-60\\-108\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4380\\-292\\-876\\-2628\\-1168\\-12264\\0\\584
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16460332584, -2628, -4133977152; -2628, 24, 660; -4133977152, 660, 1038239477],[876, 12, 1],[-140359, -2113043, -23849, -547, 0, -22755, -757479, -69359, -1421969; -397, -5986, -68, -2, 1, -63, -2117, -195, -4015; -558870, -8413542, -94960, -2178, 0, -90604, -3016068, -276168, -5661880],89,6]
%
% END OF LATTICE L90.6
%
\hbox{}\par\smallskip%
%
% LATTICE L90.7
%
\leavevmode%
\llap{\spacer}%
${L_{90.7}}=\pdualwitharg{73}(\pfillwitharg{2}({L_{90.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}73\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${73}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{438}\above{1pt}{1pt}{l}{2}{73}\above{1pt}{1pt}{}{4}{146}\above{1pt}{1pt}{}{6}{438}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{146}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27544287\\-367044\\-9060906
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-367044\\-4891\\-120742
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9060906\\-120742\\-2980655
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13692\\-12860\\-41099
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2775\\-2598\\-8330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6346\\-5853\\-19053
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\216\\584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-74\\-73
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7591\\-7371\\-22776
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1107\\-1061\\-3322
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7151\\-6791\\-21462
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5797\\-5463\\-17400
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\0\\-219\\-438\\-1679\\-2628\\-146\\-146\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-73\\-365\\-657\\-41\\-73\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-219\\0\\438\\0\\-1095\\-2847\\-219\\-657\\-219
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-438\\-73\\0\\73\\-73\\-657\\-73\\-365\\-219
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1679\\-365\\-1095\\-73\\146\\-219\\-73\\-657\\-657
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2628\\-657\\-2847\\-657\\-219\\438\\0\\-438\\-876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-146\\-41\\-219\\-73\\-73\\0\\2\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-146\\-73\\-657\\-365\\-657\\-438\\0\\146\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-219\\-219\\-657\\-876\\-42\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-27544287, -367044, -9060906; -367044, -4891, -120742; -9060906, -120742, -2980655],[219, 73, 1],[13692, 2775, 6346, -195, 25, 7591, 1107, 7151, 5797; -12860, -2598, -5853, 216, -74, -7371, -1061, -6791, -5463; -41099, -8330, -19053, 584, -73, -22776, -3322, -21462, -17400],89,7]
%
% END OF LATTICE L90.7
%
\hbox{}\par\smallskip%
%
% LATTICE L90.8
%
\leavevmode%
\llap{\spacer}%
${L_{90.8}}=\pdualwitharg{3.73}(\pfillwitharg{2}({L_{90.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}73\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${219}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{146}\above{1pt}{1pt}{l}{2}{219}\above{1pt}{1pt}{}{4}{438}\above{1pt}{1pt}{}{6}{146}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{438}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
772851\\-2254386\\-746352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2254386\\8825262\\2921679
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-746352\\2921679\\967247
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4010\\14481147\\-43745031
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-814\\2939588\\-8879985
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-625\\2257137\\-6818419
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\-187694\\566991
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-145\\438
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-729\\2632380\\-7951963
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-321\\1159154\\-3501603
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2083\\7522042\\-22722783
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-565\\2040344\\-6163525
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
219\\0\\-219\\-1314\\-5037\\-2628\\-438\\-438\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-219\\-1095\\-657\\-123\\-219\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-219\\0\\146\\0\\-1095\\-949\\-219\\-657\\-73
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1314\\-219\\0\\219\\-219\\-657\\-219\\-1095\\-219
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5037\\-1095\\-1095\\-219\\438\\-219\\-219\\-1971\\-657
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2628\\-657\\-949\\-657\\-219\\146\\0\\-438\\-292
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-438\\-123\\-219\\-219\\-219\\0\\6\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-438\\-219\\-657\\-1095\\-1971\\-438\\0\\438\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-73\\-219\\-657\\-292\\-42\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[772851, -2254386, -746352; -2254386, 8825262, 2921679; -746352, 2921679, 967247],[219, 219, 1],[-4010, -814, -625, 52, 0, -729, -321, -2083, -565; 14481147, 2939588, 2257137, -187694, -145, 2632380, 1159154, 7522042, 2040344; -43745031, -8879985, -6818419, 566991, 438, -7951963, -3501603, -22722783, -6163525],89,8]
%
% END OF LATTICE L90.8
%
\hbox{}\par\smallskip%
%
% LATTICE L90.9
%
\leavevmode%
\llap{\spacer}%
${L_{90.9}}=\pdualwitharg{73}({L_{90.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}73\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${292}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{438}\above{1pt}{1pt}{b}{2}{292}\above{1pt}{1pt}{*}{4}{146}\above{1pt}{1pt}{}{6}{438}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{146}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9982020\\3433044\\-158556
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3433044\\-1180702\\54531
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-158556\\54531\\-2518
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-717\\-2098\\-292
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-181\\-530\\-82
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-1173\\-219
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-191\\-562\\-146
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-108\\-73
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\189\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-157\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-118\\-345\\-42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
292\\0\\-438\\-1752\\-3358\\-5256\\-292\\-292\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-292\\-730\\-1314\\-82\\-146\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-438\\0\\438\\0\\-1095\\-2847\\-219\\-657\\-219
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1752\\-292\\0\\292\\-146\\-1314\\-146\\-730\\-438
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3358\\-730\\-1095\\-146\\146\\-219\\-73\\-657\\-657
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5256\\-1314\\-2847\\-1314\\-219\\438\\0\\-438\\-876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-292\\-82\\-219\\-146\\-73\\0\\2\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-292\\-146\\-657\\-730\\-657\\-438\\0\\146\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-219\\-438\\-657\\-876\\-42\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9982020, 3433044, -158556; 3433044, -1180702, 54531; -158556, 54531, -2518],[876, 73, 1],[-717, -181, -400, -191, -36, 65, 1, -54, -118; -2098, -530, -1173, -562, -108, 189, 3, -157, -345; -292, -82, -219, -146, -73, 0, 2, 0, -42],89,9]
%
% END OF LATTICE L90.9
%
\hbox{}\par\smallskip%
%
% LATTICE L90.10
%
\leavevmode%
\llap{\spacer}%
${L_{90.10}}=\pdualwitharg{3.73}({L_{90.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}73\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${876}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{146}\above{1pt}{1pt}{b}{2}{876}\above{1pt}{1pt}{*}{4}{438}\above{1pt}{1pt}{}{6}{146}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{438}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190092\\-95484\\94608
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95484\\-47742\\47523
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
94608\\47523\\-47086
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1723\\-46\\-3504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-303\\-10\\-618
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-5\\-73
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
219\\-2\\438
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\2\\-219
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-363\\-1\\-730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-125\\-1\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-650\\-9\\-1314
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138\\-3\\-280
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
876\\0\\-438\\-5256\\-10074\\-5256\\-876\\-876\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-876\\-2190\\-1314\\-246\\-438\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-438\\0\\146\\0\\-1095\\-949\\-219\\-657\\-73
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5256\\-876\\0\\876\\-438\\-1314\\-438\\-2190\\-438
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10074\\-2190\\-1095\\-438\\438\\-219\\-219\\-1971\\-657
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5256\\-1314\\-949\\-1314\\-219\\146\\0\\-438\\-292
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-876\\-246\\-219\\-438\\-219\\0\\6\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-876\\-438\\-657\\-2190\\-1971\\-438\\0\\438\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-73\\-438\\-657\\-292\\-42\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-190092, -95484, 94608; -95484, -47742, 47523; 94608, 47523, -47086],[876, 219, 1],[-1723, -303, -34, 219, -110, -363, -125, -650, -138; -46, -10, -5, -2, 2, -1, -1, -9, -3; -3504, -618, -73, 438, -219, -730, -252, -1314, -280],89,10]
%
% END OF LATTICE L90.10
%
\hbox{}\par\smallskip%
%
% LATTICE L90.11
%
\leavevmode%
\llap{\spacer}%
${L_{90.11}}=\pdualwitharg{2.73}({L_{90.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}73\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${292}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{1752}\above{1pt}{1pt}{*}{2}{292}\above{1pt}{1pt}{*}{4}{584}\above{1pt}{1pt}{}{6}{1752}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{584}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2209272\\-1779156\\1886904
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1779156\\1759592\\-1764264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1886904\\-1764264\\1794815
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117485\\350381\\467930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31211\\93082\\124310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-152419\\454566\\607068
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45931\\136982\\182938
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42302\\126159\\168484
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3739\\11151\\14892
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10997\\32797\\43800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36709\\109479\\146208
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
292\\0\\-876\\-1752\\-6716\\-10512\\-584\\-584\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-292\\-1460\\-2628\\-164\\-292\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-876\\0\\1752\\0\\-4380\\-11388\\-876\\-2628\\-876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1752\\-292\\0\\292\\-292\\-2628\\-292\\-1460\\-876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6716\\-1460\\-4380\\-292\\584\\-876\\-292\\-2628\\-2628
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10512\\-2628\\-11388\\-2628\\-876\\1752\\0\\-1752\\-3504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-584\\-164\\-876\\-292\\-292\\0\\8\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-584\\-292\\-2628\\-1460\\-2628\\-1752\\0\\584\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-876\\-876\\-2628\\-3504\\-168\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2209272, -1779156, 1886904; -1779156, 1759592, -1764264; 1886904, -1764264, 1794815],[876, 292, 1],[-117485, -31211, -152419, -45931, -42302, -3739, -1, -10997, -36709; 350381, 93082, 454566, 136982, 126159, 11151, 3, 32797, 109479; 467930, 124310, 607068, 182938, 168484, 14892, 4, 43800, 146208],89,11]
%
% END OF LATTICE L90.11
%
\hbox{}\par\smallskip%
%
% LATTICE L90.12
%
\leavevmode%
\llap{\spacer}%
${L_{90.12}}=\pdualwitharg{2.3.73}({L_{90.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}73\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${876}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{584}\above{1pt}{1pt}{*}{2}{876}\above{1pt}{1pt}{*}{4}{1752}\above{1pt}{1pt}{}{6}{584}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{1752}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1752\\-193596\\48180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-193596\\41825496\\-10408632
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48180\\-10408632\\2590277
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
444\\-51119\\-205422
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91\\-10483\\-42126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\-8429\\-33872
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\109\\438
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\-8647\\-34748
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\-3891\\-15636
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
224\\-25723\\-103368
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62\\-7133\\-28664
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
876\\0\\-876\\-5256\\-20148\\-10512\\-1752\\-1752\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-876\\-4380\\-2628\\-492\\-876\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-876\\0\\584\\0\\-4380\\-3796\\-876\\-2628\\-292
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5256\\-876\\0\\876\\-876\\-2628\\-876\\-4380\\-876
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20148\\-4380\\-4380\\-876\\1752\\-876\\-876\\-7884\\-2628
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10512\\-2628\\-3796\\-2628\\-876\\584\\0\\-1752\\-1168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1752\\-492\\-876\\-876\\-876\\0\\24\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1752\\-876\\-2628\\-4380\\-7884\\-1752\\0\\1752\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-292\\-876\\-2628\\-1168\\-168\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1752, -193596, 48180; -193596, 41825496, -10408632; 48180, -10408632, 2590277],[876, 876, 1],[444, 91, 73, -1, 1, 76, 34, 224, 62; -51119, -10483, -8429, 109, 0, -8647, -3891, -25723, -7133; -205422, -42126, -33872, 438, 0, -34748, -15636, -103368, -28664],89,12]
%
% END OF LATTICE L90.12
%

\medskip%
%------------------------------WEYL-GROUP-W91--------------------------------
%
\leavevmode\llap{}%
$W_{91\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=48$%
\hfill%
$222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L91.1
%
\leavevmode%
\llap{\spacer}%
${L_{91.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{91}}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3898356\\4620\\7392
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4620\\-2\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7392\\-9\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11705\\-403788\\-5933928
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\551\\8112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\759\\11153
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\36\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\33\\506
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-42\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-36\\-506
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-594\\-8613
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-208\\-3041
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-84\\-36\\-528\\-180\\-1212\\-1320\\-1596\\-84\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\0\\-22\\-495\\-187\\-1320\\-1474\\-1848\\-110\\-231\\-11
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\kern3pt%
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-84\\0\\84\\0\\-462\\-210\\-1596\\-1848\\-2436\\-168\\-462\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-84\\-36\\-528\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\0\\-22\\-495\\-187
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\84\\0\\-462\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-22\\0\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-495\\-462\\0\\66\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-187\\-210\\-8\\0\\2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1212\\-1320\\-1596\\-84\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\-1474\\-1848\\-110\\-231\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1596\\-1848\\-2436\\-168\\-462\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-110\\-168\\-16\\-66\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-231\\-462\\-66\\-429\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-42\\-10\\-99\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3898356, 4620, 7392; 4620, -2, -9; 7392, -9, -14],[924, 1, 1],[1, 1, -1, -1, -17, -6, -41, -45, -55, -3, -5, 0; 36, 33, -42, -36, -594, -208, -1416, -1551, -1890, -102, -165, 1; 504, 506, -504, -506, -8613, -3041, -20784, -22814, -27888, -1522, -2541, -1],90,1]
%
% END OF LATTICE L91.1
%
\hbox{}\par\smallskip%
%
% LATTICE L91.2
%
\leavevmode%
\llap{\spacer}%
${L_{91.2}}=\pfillwitharg{2}({L_{91.1}})$%
{${}={}$}Nikulin $91$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16401\\-5544\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5544\\1874\\77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\77\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1240625\\-3673593\\109725
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
416368\\1232900\\-36825
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87344\\-258633\\7724
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-385\\11
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
227\\672\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
179\\530\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6698\\19833\\-594
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2440\\7225\\-216
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-21\\-9\\-264\\-90\\-303\\-660\\-399\\-21\\-66\\0
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-399\\-924\\-609\\-42\\-231\\-21
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-21\\-55\\-42\\-4\\-33\\-5
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[16401, -5544, -231; -5544, 1874, 77; -231, 77, -38],[231, 1, 1],[1, -130, 227, 179, 6698, 2440, 8479, 18786, 11645, 669, 2630, 104; 3, -385, 672, 530, 19833, 7225, 25107, 55627, 34482, 1981, 7788, 308; 0, 11, -21, -16, -594, -216, -750, -1661, -1029, -59, -231, -9],90,2]
%
% END OF LATTICE L91.2
%
\hbox{}\par\smallskip%
%
% LATTICE L91.3
%
\leavevmode%
\llap{\spacer}%
${L_{91.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{91.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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426426\\-104412\\-142989
\end{tabular}\endgroup%
\kern3pt%
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-104412\\25401\\35013
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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746129\\19635\\2210901
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-198360\\-5221\\-587772
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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102\\3\\302
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-390\\-11\\-1155
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-494\\-14\\-1463
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-393\\-11\\-1164
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\HardButStrongLineBreak\kern3pt%
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-2421\\-66\\-7172
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-2015\\-54\\-5970
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{$\left.\llap{\phantom{%
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0\\0\\0
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[426426, -104412, -142989; -104412, 25401, 35013; -142989, 35013, 47947],[231, 3, 1],[102, -390, -494, -393, -2421, -2015, -2002, -12150, -2166, -177, 331, 305; 3, -11, -14, -11, -66, -54, -53, -319, -56, -4, 11, 9; 302, -1155, -1463, -1164, -7172, -5970, -5932, -36003, -6419, -525, 979, 903],90,3]
%
% END OF LATTICE L91.3
%
\hbox{}\par\smallskip%
%
% LATTICE L91.4
%
\leavevmode%
\llap{\spacer}%
${L_{91.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{91.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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541002\\-179256\\231231
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-179256\\59269\\-76615
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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266\\-6\\-627
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499\\-11\\-1176
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9413\\-198\\-22176
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2645\\-54\\-6230
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[541002, -179256, 231231; -179256, 59269, -76615; 231231, -76615, 98831],[231, 7, 1],[-374, 490, 266, 499, 9413, 2645, 7954, 16186, 1250, 259, -1075, -371; 9, -11, -6, -11, -198, -54, -159, -319, -24, -4, 33, 9; 882, -1155, -627, -1176, -22176, -6230, -18732, -38115, -2943, -609, 2541, 875],90,4]
%
% END OF LATTICE L91.4
%
\hbox{}\par\smallskip%
%
% LATTICE L91.5
%
\leavevmode%
\llap{\spacer}%
${L_{91.5}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{91.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[692538, -254100, -63525; -254100, 93137, 23309; -63525, 23309, 5827],[231, 11, 1],[82, -10, -418, -113, -197, -615, -1862, -346, -2074, -65, 19, 81; 9, -1, -42, -11, -18, -54, -159, -29, -168, -4, 3, 9; 858, -105, -4389, -1188, -2076, -6490, -19668, -3657, -21945, -693, 195, 847],90,5]
%
% END OF LATTICE L91.5
%
\hbox{}\par\smallskip%
%
% LATTICE L91.6
%
\leavevmode%
\llap{\spacer}%
${L_{91.6}}=\pdualwitharg{3}({L_{91.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1368444, 3696, 4620; 3696, -6, -15; 4620, -15, -14],[924, 3, 1],[1, 1, -1, -1, -2, -1, -1, -1, 1, 1, 2, 1; 124, 121, -126, -124, -242, -116, -104, -55, 154, 130, 253, 125; 196, 198, -196, -198, -407, -213, -236, -330, 140, 186, 385, 195],90,6]
%
% END OF LATTICE L91.6
%
\hbox{}\par\smallskip%
%
% LATTICE L91.7
%
\leavevmode%
\llap{\spacer}%
${L_{91.7}}=\pdualwitharg{2}({L_{91.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3080192808, -926772, -773408328; -926772, 280, 232704; -773408328, 232704, 194195779],[924, 4, 1],[-403, -1381, -2647, -1036, -34768, -12434, -21472, -94869, -58603, -3339, -13357, -691; 3, 0, 0, 3, 165, 65, 117, 528, 336, 21, 99, 7; -1605, -5500, -10542, -4126, -138468, -49520, -85515, -377828, -233394, -13298, -53196, -2752],90,7]
%
% END OF LATTICE L91.7
%
\hbox{}\par\smallskip%
%
% LATTICE L91.8
%
\leavevmode%
\llap{\spacer}%
${L_{91.8}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{91.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3866158758, -1535919, -1296084867; -1535919, 609, 514899; -1296084867, 514899, 434497414],[231, 21, 1],[-589, -6892, 59, 3182, 42307, 46667, 54209, 360784, 10665, 12904, 16675, 1591; 35, 418, -5, -209, -2750, -3028, -3515, -23386, -691, -835, -1078, -104; -1757, -20559, 176, 9492, 126203, 139209, 161707, 1076229, 31814, 38493, 49742, 4746],90,8]
%
% END OF LATTICE L91.8
%
\hbox{}\par\smallskip%
%
% LATTICE L91.9
%
\leavevmode%
\llap{\spacer}%
${L_{91.9}}=\pdualwitharg{7}({L_{91.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2471700, 17556, -5544; 17556, -14, -21; -5544, -21, 10],[924, 7, 1],[7, 1, -1, -1, 16, 9, 73, 87, 17, 9, 28, 3; 708, 99, -102, -102, 1617, 911, 7392, 8811, 1722, 912, 2838, 304; 5376, 770, -768, -770, 12243, 6895, 55944, 66682, 13032, 6902, 21483, 2303],90,9]
%
% END OF LATTICE L91.9
%
\hbox{}\par\smallskip%
%
% LATTICE L91.10
%
\leavevmode%
\llap{\spacer}%
${L_{91.10}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{91.1}}))$%
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{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[234962574, -475167, -79746051; -475167, 957, 161271; -79746051, 161271, 27065726],[231, 33, 1],[-265, -284, 209, 1478, 1781, 21599, 25085, 15176, 34539, 5968, 701, 739; 35, 38, -35, -209, -250, -3028, -3515, -2126, -4837, -835, -98, -104; -781, -837, 616, 4356, 5249, 63657, 73931, 44727, 101794, 17589, 2066, 2178],90,10]
%
% END OF LATTICE L91.10
%
\hbox{}\par\smallskip%
%
% LATTICE L91.11
%
\leavevmode%
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${L_{91.11}}=\pdualwitharg{11}({L_{91.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14940156, -158004, 32340; -158004, -1430, 341; 32340, 341, -70],[924, 11, 1],[-89, -7, -55, 5, 5, 12, 49, 3, -1, -9, -7, -18; -216, -18, -168, 6, 9, 23, 96, 6, 0, -18, -15, -41; -42240, -3329, -26334, 2332, 2352, 5654, 23100, 1415, -462, -4246, -3309, -8525],90,11]
%
% END OF LATTICE L91.11
%
\hbox{}\par\smallskip%
%
% LATTICE L91.12
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[402729096, -3380916, -101127180; -3380916, 28392, 848964; -101127180, 848964, 25393513],[924, 12, 1],[-286, -1591, -341, -574, -9424, -11034, -6609, -89423, -18965, -3563, -5745, -1377; 1, 0, 0, 3, 55, 65, 39, 528, 112, 21, 33, 7; -1139, -6336, -1358, -2286, -37532, -43944, -26321, -356136, -75530, -14190, -22880, -5484],90,12]
%
% END OF LATTICE L91.12
%
\hbox{}\par\smallskip%
%
% LATTICE L91.13
%
\leavevmode%
\llap{\spacer}%
${L_{91.13}}=\pdualwitharg{7.11}(\pfillwitharg{2}({L_{91.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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23131185\\227766\\-9327780
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227766\\2233\\-91861
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[23131185, 227766, -9327780; 227766, 2233, -91861; -9327780, -91861, 3761462],[231, 77, 1],[-481, -172, 55, 898, 3245, 13117, 45701, 9216, 8989, 3624, 1277, 449; 1548, 554, -180, -2903, -10485, -42379, -147648, -29774, -29040, -11707, -4125, -1451; -1155, -413, 132, 2156, 7791, 31493, 109725, 22127, 21582, 8701, 3066, 1078],90,13]
%
% END OF LATTICE L91.13
%
\hbox{}\par\smallskip%
%
% LATTICE L91.14
%
\leavevmode%
\llap{\spacer}%
${L_{91.14}}=\pdualwitharg{3.7}({L_{91.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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3837372\\29568\\-9240
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29568\\-42\\-63
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3837372, 29568, -9240; 29568, -42, -63; -9240, -63, 22],[924, 21, 1],[7, 1, -1, -1, 31, 44, 113, 395, 25, 37, 35, 10; 96, 11, -14, -14, 429, 609, 1564, 5467, 346, 512, 484, 138; 3220, 462, -460, -462, 14245, 20223, 51940, 181566, 11492, 17010, 16093, 4599],90,14]
%
% END OF LATTICE L91.14
%
\hbox{}\par\smallskip%
%
% LATTICE L91.15
%
\leavevmode%
\llap{\spacer}%
${L_{91.15}}=\pdualwitharg{2.7}({L_{91.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11195444568, -45188220, -2811040848; -45188220, 182392, 11346216; -2811040848, 11346216, 705818389],[924, 28, 1],[-4561, -14771, -577, 805, 14849, 3171, 3296, 9205, 71, -1058, -17632, -3930; 3, 0, 0, 3, 165, 65, 117, 528, 48, 21, 99, 7; -18165, -58828, -2298, 3206, 59136, 12628, 13125, 36652, 282, -4214, -70224, -15652],90,15]
%
% END OF LATTICE L91.15
%
\hbox{}\par\smallskip%
%
% LATTICE L91.16
%
\leavevmode%
\llap{\spacer}%
${L_{91.16}}=\pdualwitharg{3.11}({L_{91.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1961652, -43428, 17556; -43428, 66, 33; 17556, 33, -34],[924, 33, 1],[5, 1, -1, -5, -2, -21, -45, -13, -55, -7, 0, 3; 1340, 269, -266, -1348, -542, -5704, -12240, -3539, -14994, -1922, -3, 799; 3872, 777, -770, -3894, -1565, -16467, -35332, -10215, -43274, -5544, -8, 2310],90,16]
%
% END OF LATTICE L91.16
%
\hbox{}\par\smallskip%
%
% LATTICE L91.17
%
\leavevmode%
\llap{\spacer}%
${L_{91.17}}=\pdualwitharg{2.11}({L_{91.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[74340724920, -15126804, -18666324600; -15126804, 3080, 3798212; -18666324600, 3798212, 4686955561],[924, 44, 1],[-139313, -58416, -421676, -28863, -14719, -16329, -11327, -1840, -116, -1265, -15409, -74055; -741, -311, -2247, -154, -78, -84, -54, -7, 21, -5, -81, -393; -554829, -232648, -1679370, -114950, -58620, -65032, -45111, -7328, -462, -5038, -61368, -294932],90,17]
%
% END OF LATTICE L91.17
%
\hbox{}\par\smallskip%
%
% LATTICE L91.18
%
\leavevmode%
\llap{\spacer}%
${L_{91.18}}=\pdualwitharg{3.7.11}(\pfillwitharg{2}({L_{91.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[406335930, -75488259, 23852598; -75488259, 14024010, -4431273; 23852598, -4431273, 1400183],[231, 231, 1],[3, -1, -2, -11, -6, -54, -53, -29, -8, -4, 1, 9; -1468, 552, 1098, 6291, 3719, 35183, 35748, 20016, 5758, 3993, -291, -4331; -4697, 1764, 3509, 20097, 11872, 112266, 114037, 63840, 18359, 12705, -938, -13860],90,18]
%
% END OF LATTICE L91.18
%
\hbox{}\par\smallskip%
%
% LATTICE L91.19
%
\leavevmode%
\llap{\spacer}%
${L_{91.19}}=\pdualwitharg{7.11}({L_{91.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[924, 0, 0; 0, -2926, 77; 0, 77, -2],[924, 77, 1],[1, 0, -1, -3, -4, -15, -101, -10, -19, -7, -1, 0; 0, -1, -6, -10, -9, -29, -180, -17, -30, -8, 0, 2; 0, -42, -264, -462, -441, -1463, -9240, -882, -1584, -462, -21, 77],90,19]
%
% END OF LATTICE L91.19
%
\hbox{}\par\smallskip%
%
% LATTICE L91.20
%
\leavevmode%
\llap{\spacer}%
${L_{91.20}}=\pdualwitharg{2.3.7}({L_{91.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11192583864, -31180380, -2810365404; -31180380, 86856, 7829136; -2810365404, 7829136, 705659551],[924, 84, 1],[-3278, -26449, -115, 2183, 4795, -2341, -3647, -64961, -2655, -6338, -20030, -10124; 1, 0, 0, 3, 55, 65, 39, 528, 16, 21, 33, 7; -13055, -105336, -458, 8694, 19096, -9324, -14525, -258720, -10574, -25242, -79772, -40320],90,20]
%
% END OF LATTICE L91.20
%
\hbox{}\par\smallskip%
%
% LATTICE L91.21
%
\leavevmode%
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${L_{91.21}}=\pdualwitharg{2.3.11}({L_{91.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13963452888, -694098636, -3506053320; -694098636, 34502424, 174279732; -3506053320, 174279732, 880327379],[924, 132, 1],[-1478, -681, -657, -1674, -3268, -43894, -26757, -33201, -78401, -15267, -2379, -6929; -7, -1, 7, -5, -17, -239, -148, -185, -441, -88, -14, -40; -5885, -2712, -2618, -6666, -13012, -174768, -106535, -132192, -312158, -60786, -9472, -27588],90,21]
%
% END OF LATTICE L91.21
%
\hbox{}\par\smallskip%
%
% LATTICE L91.22
%
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${L_{91.22}}=\pdualwitharg{3.7.11}({L_{91.1}})$%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\-147\\-154\\-462\\-91\\-693
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-231\\-462\\-2310\\-693\\-6699
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28838964, 71148, 93324; 71148, -8778, 231; 93324, 231, 302],[924, 231, 1],[2979, 852, 509, 441, 6, -77, 1, 42, 87, 453, 143, 1418; -60, -17, -10, -8, 0, 2, 0, -1, -2, -10, -3, -29; -920612, -263298, -157300, -136290, -1855, 23793, -308, -12978, -26884, -139986, -44191, -438207],90,22]
%
% END OF LATTICE L91.22
%
\hbox{}\par\smallskip%
%
% LATTICE L91.23
%
\leavevmode%
\llap{\spacer}%
${L_{91.23}}=\pdualwitharg{2.7.11}({L_{91.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${231}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}{308}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{*}{2}{616}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8547000\\72996\\2091012
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72996\\616\\17864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2091012\\17864\\511559
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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418978\\-1311057\\-1666896
\end{tabular}\endgroup%
\kern3pt%
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2787\\-8722\\-11088
\end{tabular}\endgroup%
\kern3pt%
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103119\\-322677\\-410257
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-58\\174\\231
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\HardButStrongLineBreak\kern3pt%
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-7\\17\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\-267\\-330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
271\\-854\\-1078
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
739\\-2316\\-2940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2787\\-8722\\-11088
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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231\\0\\-462\\-1386\\-3696\\-13860
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0\\56\\0\\-308\\-1260\\-5236
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-1386\\-308\\0\\308\\0\\-1232
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-3696\\-1260\\-924\\0\\168\\0
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8547000, 72996, 2091012; 72996, 616, 17864; 2091012, 17864, 511559],[924, 308, 1],[-58, -7, 83, 271, 739, 2787, 4703, 1865, 1775, 658, 190, 0; 174, 17, -267, -854, -2316, -8722, -14709, -5831, -5547, -2053, -591, 1; 231, 28, -330, -1078, -2940, -11088, -18711, -7420, -7062, -2618, -756, 0],90,23]
%
% END OF LATTICE L91.23
%
\hbox{}\par\smallskip%
%
% LATTICE L91.24
%
\leavevmode%
\llap{\spacer}%
${L_{91.24}}=\pdualwitharg{2.3.7.11}({L_{91.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${77}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}{924}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{*}{2}{1848}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1848\\-22269324\\5591124
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22269324\\264938799048\\-66517765776
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5591124\\-66517765776\\16700510381
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-361\\-926325\\-3689532
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\kern3pt%
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5597040\\14401883549\\57362383848
\end{tabular}\endgroup%
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-1405240\\-3615858175\\-14401883189
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-200\\-520676\\-2073841
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-236\\-619195\\-2466240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74\\-196333\\-781990
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-79\\-217952\\-868098
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-20914\\-83300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-924\\-924\\-4620\\-2772\\-26796
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1848, -22269324, 5591124; -22269324, 264938799048, -66517765776; 5591124, -66517765776, 16700510381],[924, 924, 1],[-200, -236, -74, -79, -7, 1, 0, -4, -6, -41, -33, -361; -520676, -619195, -196333, -217952, -20914, 0, 6051, 1645, -9517, -90823, -82011, -926325; -2073841, -2466240, -781990, -868098, -83300, 0, 24101, 6552, -37906, -361746, -326648, -3689532],90,24]
%
% END OF LATTICE L91.24
%

\medskip%
%------------------------------WEYL-GROUP-W92--------------------------------
%
\leavevmode\llap{}%
$W_{92\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=15$%
\hfill%
$222242$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L92.1
%
\leavevmode%
\llap{\spacer}%
${L_{92.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{92}}%
\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{4}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3694273044\\3152688\\6305376
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3152688\\-2690\\-5381
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6305376\\-5381\\-10762
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\0\\11132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-7\\2933
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-17\\-24\\-9948
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-101\\-77\\-59136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-2\\-5272
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\-132\\-616\\-44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-77\\-14\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\12\\0\\-12\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-616\\-77\\0\\154\\0\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-14\\-12\\0\\4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-24\\-77\\-2\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3694273044, 3152688, 6305376; 3152688, -2690, -5381; 6305376, -5381, -10762],[924, 1, 1],[19, 5, -17, -101, -9, 0; 0, -7, -24, -77, -2, 2; 11132, 2933, -9948, -59136, -5272, -1],91,1]
%
% END OF LATTICE L92.1
%
\hbox{}\par\smallskip%
%
% LATTICE L92.2
%
\leavevmode%
\llap{\spacer}%
${L_{92.2}}=\pfillwitharg{2}({L_{92.1}})$%
{${}={}$}Nikulin $92$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\0\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\21\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\11\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-63\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\51\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82\\847\\-154
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\52\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\42\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-33\\-308\\-11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-77\\-7\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\3\\0\\-3\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\-77\\0\\154\\0\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-7\\-3\\0\\1\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-12\\-77\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-231, 0, -231; 0, 2, 21; -231, 21, -10],[231, 1, 1],[1, -6, 5, 82, 5, 4; 11, -63, 51, 847, 52, 42; 0, 7, -12, -154, -8, -5],91,2]
%
% END OF LATTICE L92.2
%
\hbox{}\par\smallskip%
%
% LATTICE L92.3
%
\leavevmode%
\llap{\spacer}%
${L_{92.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{92.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{4}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6254094\\24255\\2089857
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24255\\-93\\-8106
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2089857\\-8106\\-698342
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-209\\-231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
227\\630\\672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
178\\489\\527
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3823\\10472\\11319
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\209\\228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76\\-210\\-225
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\0\\-33\\-924\\-33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-231\\-21\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\1\\0\\-3\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\-231\\0\\462\\0\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-21\\-3\\0\\3\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-12\\-231\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6254094, 24255, 2089857; 24255, -93, -8106; 2089857, -8106, -698342],[231, 3, 1],[-78, 227, 178, 3823, 77, -76; -209, 630, 489, 10472, 209, -210; -231, 672, 527, 11319, 228, -225],91,3]
%
% END OF LATTICE L92.3
%
\hbox{}\par\smallskip%
%
% LATTICE L92.4
%
\leavevmode%
\llap{\spacer}%
${L_{92.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{92.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${77}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{4}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11550462\\50127\\-4945479
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50127\\-217\\21462
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4945479\\21462\\-2117470
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
98\\-209\\-231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\90\\92
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-650\\1467\\1533
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-667\\1496\\1573
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\209\\224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
92\\-210\\-217
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\0\\-231\\-308\\-77\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-11\\-7\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\0\\21\\0\\-21\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\-11\\0\\22\\0\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\-7\\-21\\0\\7\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-84\\-77\\-7\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11550462, 50127, -4945479; 50127, -217, 21462; -4945479, 21462, -2117470],[231, 7, 1],[98, -39, -650, -667, -95, 92; -209, 90, 1467, 1496, 209, -210; -231, 92, 1533, 1573, 224, -217],91,4]
%
% END OF LATTICE L92.4
%
\hbox{}\par\smallskip%
%
% LATTICE L92.5
%
\leavevmode%
\llap{\spacer}%
${L_{92.5}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{92.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{4}{22}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16920750\\75999\\1542849
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75999\\-341\\-6930
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1542849\\-6930\\-140678
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-19\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\630\\616
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
142\\1467\\1485
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93\\952\\973
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\209\\220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-210\\-209
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-33\\-28\\-11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\154\\0\\-77\\-77\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\33\\0\\-33\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-77\\0\\14\\0\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-77\\-33\\0\\11\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-77\\-132\\-77\\-11\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16920750, 75999, 1542849; 75999, -341, -6930; 1542849, -6930, -140678],[231, 11, 1],[-2, 59, 142, 93, 21, -20; -19, 630, 1467, 952, 209, -210; -21, 616, 1485, 973, 220, -209],91,5]
%
% END OF LATTICE L92.5
%
\hbox{}\par\smallskip%
%
% LATTICE L92.6
%
\leavevmode%
\llap{\spacer}%
${L_{92.6}}=\pdualwitharg{3}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{4}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2493263388\\4486020\\-828097116
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4486020\\-8070\\1489959
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-828097116\\1489959\\-275039066
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5261\\0\\15840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1388\\-7\\4179
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1565\\-8\\-4712
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27927\\-77\\-84084
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2491\\-2\\-7500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-132\\-1848\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-231\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\4\\0\\-12\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-231\\0\\462\\0\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-42\\-12\\0\\12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-24\\-231\\-6\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2493263388, 4486020, -828097116; 4486020, -8070, 1489959; -828097116, 1489959, -275039066],[924, 3, 1],[-5261, -1388, 1565, 27927, 2491, 1; 0, -7, -8, -77, -2, 2; 15840, 4179, -4712, -84084, -7500, -3],91,6]
%
% END OF LATTICE L92.6
%
\hbox{}\par\smallskip%
%
% LATTICE L92.7
%
\leavevmode%
\llap{\spacer}%
${L_{92.7}}=\pdualwitharg{2}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{616}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{4}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420882364056\\924\\-105679222572
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
924\\8\\-232
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105679222572\\-232\\26534963299
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16072\\-55\\64009
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58297\\-203\\232176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68378\\-237\\272325
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1023769\\-3542\\4077304
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26153\\-90\\104158
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-33\\-616\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-308\\-28\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\3\\0\\-6\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-616\\-308\\0\\616\\0\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-28\\-6\\0\\4\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-24\\-308\\-4\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[420882364056, 924, -105679222572; 924, 8, -232; -105679222572, -232, 26534963299],[924, 4, 1],[16072, 58297, 68378, 1023769, 26153, 0; -55, -203, -237, -3542, -90, 1; 64009, 232176, 272325, 4077304, 104158, 0],91,7]
%
% END OF LATTICE L92.7
%
\hbox{}\par\smallskip%
%
% LATTICE L92.8
%
\leavevmode%
\llap{\spacer}%
${L_{92.8}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{92.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${231}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{4}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
355787817\\-13655565\\-119273847
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13655565\\524391\\4577874
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119273847\\4577874\\39985210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12390\\11\\-36960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3539\\3\\-10557
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21110\\17\\-62972
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69163\\55\\-206316
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12911\\10\\-38514
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\0\\-525
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
231\\0\\-231\\-924\\-231\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-33\\-21\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\0\\7\\0\\-21\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\-33\\0\\66\\0\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-21\\-21\\0\\21\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-84\\-231\\-21\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[355787817, -13655565, -119273847; -13655565, 524391, 4577874; -119273847, 4577874, 39985210],[231, 21, 1],[-12390, -3539, -21110, -69163, -12911, -176; 11, 3, 17, 55, 10, 0; -36960, -10557, -62972, -206316, -38514, -525],91,8]
%
% END OF LATTICE L92.8
%
\hbox{}\par\smallskip%
%
% LATTICE L92.9
%
\leavevmode%
\llap{\spacer}%
${L_{92.9}}=\pdualwitharg{7}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${308}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{4}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252485772\\2180640\\-71515752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2180640\\-18830\\617659
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71515752\\617659\\-20256598
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2181\\0\\7700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-83\\-1\\293
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1927\\-24\\-6804
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1645\\-11\\-5808
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1031\\-2\\-3640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\2\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
308\\0\\-924\\-616\\-308\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-11\\-14\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\0\\84\\0\\-84\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-616\\-11\\0\\22\\0\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\-14\\-84\\0\\28\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-168\\-77\\-14\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-252485772, 2180640, -71515752; 2180640, -18830, 617659; -71515752, 617659, -20256598],[924, 7, 1],[-2181, -83, 1927, 1645, 1031, 2; 0, -1, -24, -11, -2, 2; 7700, 293, -6804, -5808, -3640, -7],91,9]
%
% END OF LATTICE L92.9
%
\hbox{}\par\smallskip%
%
% LATTICE L92.10
%
\leavevmode%
\llap{\spacer}%
${L_{92.10}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{92.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{}{4}{66}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20903421\\-4147605\\-7094703
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4147605\\824043\\1407714
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7094703\\1407714\\2407970
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-570\\1\\-1680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12305\\21\\-36267
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10286\\17\\-30316
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21347\\35\\-62916
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6203\\10\\-18282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\-165
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-33\\-84\\-33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\462\\0\\-231\\-231\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\11\\0\\-33\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-231\\0\\42\\0\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-231\\-33\\0\\33\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-231\\-132\\-231\\-33\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20903421, -4147605, -7094703; -4147605, 824043, 1407714; -7094703, 1407714, 2407970],[231, 33, 1],[-570, -12305, -10286, -21347, -6203, -56; 1, 21, 17, 35, 10, 0; -1680, -36267, -30316, -62916, -18282, -165],91,10]
%
% END OF LATTICE L92.10
%
\hbox{}\par\smallskip%
%
% LATTICE L92.11
%
\leavevmode%
\llap{\spacer}%
${L_{92.11}}=\pdualwitharg{11}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{l}{2}{132}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{4}{22}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-219314172\\-3326400\\316932
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3326400\\-50446\\4807
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
316932\\4807\\-458
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-692
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-7\\2002
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-24\\28116
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-7\\10997
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-2\\7590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-671
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-132\\-56\\-44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\154\\0\\-77\\-154\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\132\\0\\-132\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-77\\0\\14\\0\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-154\\-132\\0\\44\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-77\\-264\\-77\\-22\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-219314172, -3326400, 316932; -3326400, -50446, 4807; 316932, 4807, -458],[924, 11, 1],[-1, 3, 41, 16, 11, -1; 0, -7, -24, -7, -2, 2; -692, 2002, 28116, 10997, 7590, -671],91,11]
%
% END OF LATTICE L92.11
%
\hbox{}\par\smallskip%
%
% LATTICE L92.12
%
\leavevmode%
\llap{\spacer}%
${L_{92.12}}=\pdualwitharg{2.3}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{1848}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{4}{24}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1561407593592\\6468\\-391077605688
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6468\\24\\-1620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-391077605688\\-1620\\97951165537
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53485\\-55\\213543
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
194001\\-203\\774564
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75849\\-79\\302833
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3406873\\-3542\\13602204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
87031\\-90\\347478
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\0\\-33\\-1848\\-66\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-924\\-84\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\1\\0\\-6\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-924\\0\\1848\\0\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-84\\-6\\0\\12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-24\\-924\\-12\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1561407593592, 6468, -391077605688; 6468, 24, -1620; -391077605688, -1620, 97951165537],[924, 12, 1],[53485, 194001, 75849, 3406873, 87031, 0; -55, -203, -79, -3542, -90, 1; 213543, 774564, 302833, 13602204, 347478, 0],91,12]
%
% END OF LATTICE L92.12
%
\hbox{}\par\smallskip%
%
% LATTICE L92.13
%
\leavevmode%
\llap{\spacer}%
${L_{92.13}}=\pdualwitharg{7.11}(\pfillwitharg{2}({L_{92.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{231}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{77}\above{1pt}{1pt}{}{4}{154}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30765966\\7690914\\-3196347
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7690914\\1922767\\-798798
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3196347\\-798798\\332362
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-350\\1051\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1077\\3234\\-2585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18862\\56637\\-45276
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1863\\5594\\-4472
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3785\\11365\\-9086
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\96\\-77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-231\\-28\\-77\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\0\\-11\\-77\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\0\\231\\0\\-231\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-11\\0\\2\\0\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\-77\\-231\\0\\77\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-77\\-924\\-77\\-77\\154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30765966, 7690914, -3196347; 7690914, 1922767, -798798; -3196347, -798798, 332362],[231, 77, 1],[-350, -1077, -18862, -1863, -3785, -32; 1051, 3234, 56637, 5594, 11365, 96; -840, -2585, -45276, -4472, -9086, -77],91,13]
%
% END OF LATTICE L92.13
%
\hbox{}\par\smallskip%
%
% LATTICE L92.14
%
\leavevmode%
\llap{\spacer}%
${L_{92.14}}=\pdualwitharg{3.7}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${924}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{4}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43619268\\1569876\\-8158920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1569876\\-56490\\293643
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8158920\\293643\\-1526114
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1037\\0\\5544
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\-1\\219
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
293\\-8\\-1568
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
765\\-11\\-4092
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
487\\-2\\-2604
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\2\\-21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
924\\0\\-924\\-1848\\-924\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-33\\-42\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\0\\28\\0\\-84\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-33\\0\\66\\0\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\-42\\-84\\0\\84\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-168\\-231\\-42\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-43619268, 1569876, -8158920; 1569876, -56490, 293643; -8158920, 293643, -1526114],[924, 21, 1],[-1037, -41, 293, 765, 487, 4; 0, -1, -8, -11, -2, 2; 5544, 219, -1568, -4092, -2604, -21],91,14]
%
% END OF LATTICE L92.14
%
\hbox{}\par\smallskip%
%
% LATTICE L92.15
%
\leavevmode%
\llap{\spacer}%
${L_{92.15}}=\pdualwitharg{2.7}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${77}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{4}{56}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132680284968\\13860\\-33237391572
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13860\\56\\-3472
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33237391572\\-3472\\8326212133
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23822\\-55\\95095
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12343\\-29\\49272
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101336\\-237\\404523
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216743\\-506\\865216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38757\\-90\\154714
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\0\\-231\\-616\\-154\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-44\\-28\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\0\\21\\0\\-42\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-616\\-44\\0\\88\\0\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-154\\-28\\-42\\0\\28\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-168\\-308\\-28\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[132680284968, 13860, -33237391572; 13860, 56, -3472; -33237391572, -3472, 8326212133],[924, 28, 1],[23822, 12343, 101336, 216743, 38757, 0; -55, -29, -237, -506, -90, 1; 95095, 49272, 404523, 865216, 154714, 0],91,15]
%
% END OF LATTICE L92.15
%
\hbox{}\par\smallskip%
%
% LATTICE L92.16
%
\leavevmode%
\llap{\spacer}%
${L_{92.16}}=\pdualwitharg{3.11}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{*}{4}{66}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7934388\\-1095864\\2679600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1095864\\-151338\\370095
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2679600\\370095\\-904954
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\0\\-228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
157\\-7\\462
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
967\\-8\\2860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1157\\-7\\3423
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
825\\-2\\2442
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\2\\-165
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-132\\-168\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\462\\0\\-231\\-462\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\44\\0\\-132\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-231\\0\\42\\0\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-462\\-132\\0\\132\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-231\\-264\\-231\\-66\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7934388, -1095864, 2679600; -1095864, -151338, 370095; 2679600, 370095, -904954],[924, 33, 1],[-77, 157, 967, 1157, 825, -56; 0, -7, -8, -7, -2, 2; -228, 462, 2860, 3423, 2442, -165],91,16]
%
% END OF LATTICE L92.16
%
\hbox{}\par\smallskip%
%
% LATTICE L92.17
%
\leavevmode%
\llap{\spacer}%
${L_{92.17}}=\pdualwitharg{2.11}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{616}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{*}{4}{88}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29355101160\\616308\\-7370730444
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
616308\\88\\-154748
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7370730444\\-154748\\1850706185
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-173\\-1\\-689
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39905\\-224\\-158928
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74101\\-411\\-295119
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110027\\-609\\-438200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36099\\-199\\-143770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-33\\-56\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\616\\0\\-308\\-308\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\33\\0\\-66\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-308\\0\\56\\0\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-308\\-66\\0\\44\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-308\\-264\\-308\\-44\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[29355101160, 616308, -7370730444; 616308, 88, -154748; -7370730444, -154748, 1850706185],[924, 44, 1],[-173, -39905, -74101, -110027, -36099, 0; -1, -224, -411, -609, -199, 1; -689, -158928, -295119, -438200, -143770, 0],91,17]
%
% END OF LATTICE L92.17
%
\hbox{}\par\smallskip%
%
% LATTICE L92.18
%
\leavevmode%
\llap{\spacer}%
${L_{92.18}}=\pdualwitharg{3.7.11}(\pfillwitharg{2}({L_{92.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{77}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{231}\above{1pt}{1pt}{}{4}{462}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8803847283\\1758121596\\-555573942
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1758121596\\-351095514\\110947683
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-555573942\\110947683\\-35059942
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-1356\\-3990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\6499\\19140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
489\\35200\\103642
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
136\\9781\\28797
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
209\\14989\\44121
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-15140\\-44583
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-231\\-84\\-231\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\66\\0\\-33\\-231\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\0\\77\\0\\-231\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-33\\0\\6\\0\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-231\\-231\\0\\231\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-231\\-924\\-231\\-231\\462
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8803847283, 1758121596, -555573942; 1758121596, -351095514, 110947683; -555573942, 110947683, -35059942],[231, 231, 1],[-19, 90, 489, 136, 209, -210; -1356, 6499, 35200, 9781, 14989, -15140; -3990, 19140, 103642, 28797, 44121, -44583],91,18]
%
% END OF LATTICE L92.18
%
\hbox{}\par\smallskip%
%
% LATTICE L92.19
%
\leavevmode%
\llap{\spacer}%
${L_{92.19}}=\pdualwitharg{7.11}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{924}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{308}\above{1pt}{1pt}{*}{4}{154}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3281124\\-1076460\\-1391544
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1076460\\-353122\\-456533
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1391544\\-456533\\-590162
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
95\\0\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-1\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3127\\-24\\7392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-185\\-1\\437
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-979\\-2\\2310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\2\\-77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-924\\-56\\-308\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\0\\-11\\-154\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\0\\924\\0\\-924\\-1848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-11\\0\\2\\0\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\-154\\-924\\0\\308\\-154
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-77\\-1848\\-77\\-154\\154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3281124, -1076460, -1391544; -1076460, -353122, -456533; -1391544, -456533, -590162],[924, 77, 1],[95, -9, -3127, -185, -979, 32; 0, -1, -24, -1, -2, 2; -224, 22, 7392, 437, 2310, -77],91,19]
%
% END OF LATTICE L92.19
%
\hbox{}\par\smallskip%
%
% LATTICE L92.20
%
\leavevmode%
\llap{\spacer}%
${L_{92.20}}=\pdualwitharg{2.3.7}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${231}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{264}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{*}{4}{168}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15404083464\\34188\\-3860557008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34188\\168\\-8568
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3860557008\\-8568\\967529191
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14068\\-55\\56133
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7287\\-29\\29076
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19938\\-79\\79555
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127927\\-506\\510444
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22873\\-90\\91266
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
231\\0\\-231\\-1848\\-462\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-132\\-84\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\0\\7\\0\\-42\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-132\\0\\264\\0\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\-84\\-42\\0\\84\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-168\\-924\\-84\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15404083464, 34188, -3860557008; 34188, 168, -8568; -3860557008, -8568, 967529191],[924, 84, 1],[14068, 7287, 19938, 127927, 22873, 0; -55, -29, -79, -506, -90, 1; 56133, 29076, 79555, 510444, 91266, 0],91,20]
%
% END OF LATTICE L92.20
%
\hbox{}\par\smallskip%
%
% LATTICE L92.21
%
\leavevmode%
\llap{\spacer}%
${L_{92.21}}=\pdualwitharg{2.3.11}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{1848}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{*}{4}{264}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
124083960\\-54516\\-30638916
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54516\\264\\13464
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30638916\\13464\\7565387
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-1\\81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4563\\-224\\18480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2822\\-137\\11429
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12569\\-609\\50904
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4123\\-199\\16698
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-33\\-168\\-66\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1848\\0\\-924\\-924\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\11\\0\\-66\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-924\\0\\168\\0\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-924\\-66\\0\\132\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-924\\-264\\-924\\-132\\264
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[124083960, -54516, -30638916; -54516, 264, 13464; -30638916, 13464, 7565387],[924, 132, 1],[20, 4563, 2822, 12569, 4123, 0; -1, -224, -137, -609, -199, 1; 81, 18480, 11429, 50904, 16698, 0],91,21]
%
% END OF LATTICE L92.21
%
\hbox{}\par\smallskip%
%
% LATTICE L92.22
%
\leavevmode%
\llap{\spacer}%
${L_{92.22}}=\pdualwitharg{3.7.11}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{308}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{924}\above{1pt}{1pt}{*}{4}{462}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7511666316\\7422455964\\-24682812
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7422455964\\-7334304978\\24389673
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24682812\\24389673\\-81106
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\0\\18564
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\-1518
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
655\\-8\\-201740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
117\\-1\\-35907
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
625\\-2\\-190806
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\2\\5775
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-924\\-168\\-924\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\66\\0\\-33\\-462\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\0\\308\\0\\-924\\-1848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-33\\0\\6\\0\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\-462\\-924\\0\\924\\-462
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-231\\-1848\\-231\\-462\\462
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7511666316, 7422455964, -24682812; 7422455964, -7334304978, 24389673; -24682812, 24389673, -81106],[924, 231, 1],[-61, 4, 655, 117, 625, -17; 0, -1, -8, -1, -2, 2; 18564, -1518, -201740, -35907, -190806, 5775],91,22]
%
% END OF LATTICE L92.22
%
\hbox{}\par\smallskip%
%
% LATTICE L92.23
%
\leavevmode%
\llap{\spacer}%
${L_{92.23}}=\pdualwitharg{2.7.11}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{l}{2}{231}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{308}\above{1pt}{1pt}{*}{4}{616}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2812491528\\-187572\\704392920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187572\\616\\-47432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
704392920\\-47432\\176416655
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\-469\\-623
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4991\\-15005\\-19932
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64726\\-194589\\-258489
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13725\\-41262\\-54812
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31505\\-94714\\-125818
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-231\\-56\\-154\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\88\\0\\-44\\-308\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\0\\231\\0\\-462\\-1848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-44\\0\\8\\0\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-154\\-308\\-462\\0\\308\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-308\\-1848\\-308\\-308\\616
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2812491528, -187572, 704392920; -187572, 616, -47432; 704392920, -47432, 176416655],[924, 308, 1],[156, 4991, 64726, 13725, 31505, 0; -469, -15005, -194589, -41262, -94714, 1; -623, -19932, -258489, -54812, -125818, 0],91,23]
%
% END OF LATTICE L92.23
%
\hbox{}\par\smallskip%
%
% LATTICE L92.24
%
\leavevmode%
\llap{\spacer}%
${L_{92.24}}=\pdualwitharg{2.3.7.11}({L_{92.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{264}\above{1pt}{1pt}{l}{2}{77}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{924}\above{1pt}{1pt}{*}{4}{1848}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1848\\-5398932\\1355508
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5398932\\906327241512\\-227549981736
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1355508\\-227549981736\\57130572509
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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82\\-4877\\-19425
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0\\0\\0
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}}\!\right]$}%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1848, -5398932, 1355508; -5398932, 906327241512, -227549981736; 1355508, -227549981736, 57130572509],[924, 924, 1],[82, 2620, 11326, 7205, 16539, 1; -4877, -155829, -673555, -428471, -983509, 0; -19425, -620664, -2682757, -1706592, -3917298, 0],91,24]
%
% END OF LATTICE L92.24
%

\medskip%
%------------------------------WEYL-GROUP-W93--------------------------------
%
\leavevmode\llap{}%
$W_{93\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=54$%
\hfill%
$224222224222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L93.1
%
\leavevmode%
\llap{\spacer}%
${L_{93.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{93}}%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3813780\\10200\\15300
\end{tabular}\endgroup%
\kern3pt%
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10200\\-26\\-43
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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220319\\42769620\\26355780
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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\HardButStrongLineBreak\kern3pt%
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-1\\-194\\-120
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\HardButStrongLineBreak\kern3pt%
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-2\\-391\\-238
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2\\387\\240
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\68\\0\\-136\\-374\\-102
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-80\\-136\\-2\\4\\0\\-12
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\kern3pt%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3813780, 10200, 15300; 10200, -26, -43; 15300, -43, -58],[1020, 1, 1],[21, 39, 1, -1, -2, 2, 59, 505, 79, 273, 410, 22; 4080, 7582, 195, -194, -391, 387, 11450, 98022, 15335, 52996, 79594, 4272; 2510, 4658, 119, -120, -238, 240, 7060, 60418, 9451, 32658, 49045, 2631],92,1]
%
% END OF LATTICE L93.1
%
\hbox{}\par\smallskip%
%
% LATTICE L93.2
%
\leavevmode%
\llap{\spacer}%
${L_{93.2}}=\pfillwitharg{2}({L_{93.1}})$%
{${}={}$}Nikulin $93$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4845\\-1020\\1275
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1020\\-214\\255
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-4845, -1020, 1275; -1020, -214, 255; 1275, 255, -91],[255, 1, 1],[-1, -60, 11, 194, 1017, 263, 856, 5874, 1699, 2713, 7704, 250; 5, 306, -56, -989, -5185, -1341, -4365, -29954, -8664, -13835, -39287, -1275; 0, 17, -3, -55, -289, -75, -245, -1683, -487, -778, -2210, -72],92,2]
%
% END OF LATTICE L93.2
%
\hbox{}\par\smallskip%
%
% LATTICE L93.3
%
\leavevmode%
\llap{\spacer}%
${L_{93.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{93.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{51}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{4}{3}\above{1pt}{1pt}{r}{2}{102}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2753490\\-1530\\914685
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1530\\-3093\\-240
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
914685\\-240\\303827
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2753490, -1530, 914685; -1530, -3093, -240; 914685, -240, 303827],[255, 3, 1],[39344, 72551, 3628, -1674, -5861, 2697, 112726, 961525, 300512, 518712, 1556975, 27717; -10270, -18938, -947, 437, 1530, -704, -29425, -250988, -78443, -135400, -406419, -7235; -118455, -218433, -10923, 5040, 17646, -8120, -339390, -2894913, -904767, -1561713, -4687665, -83449],92,3]
%
% END OF LATTICE L93.3
%
\hbox{}\par\smallskip%
%
% LATTICE L93.4
%
\leavevmode%
\llap{\spacer}%
${L_{93.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{93.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3178830, 43350, 630615; 43350, -5155, 8940; 630615, 8940, 125081],[255, 5, 1],[6910, 63711, 3186, -1470, -5147, 7105, 19798, 844361, 263894, 455506, 1367255, 73019; -2054, -18938, -947, 437, 1530, -2112, -5885, -250988, -78443, -135400, -406419, -21705; -34691, -319855, -15995, 7380, 25840, -35670, -99394, -4239035, -1324855, -2286825, -6864175, -366585],92,4]
%
% END OF LATTICE L93.4
%
\hbox{}\par\smallskip%
%
% LATTICE L93.5
%
\leavevmode%
\llap{\spacer}%
${L_{93.5}}=\pdualwitharg{3}({L_{93.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35759356860, -1822740, 105814800; -1822740, -78, 5391; 105814800, 5391, -313114],[1020, 3, 1],[-2447, -4541, -116, 117, 232, -78, -6883, -58903, -9214, -31839, -47815, -855; -146520, -271898, -6945, 7006, 13889, -4671, -412150, -3527058, -551725, -1906484, -2863106, -51196; -829470, -1539282, -39321, 39660, 78642, -26440, -2333160, -19966602, -3123309, -10792602, -16208055, -289823],92,5]
%
% END OF LATTICE L93.5
%
\hbox{}\par\smallskip%
%
% LATTICE L93.6
%
\leavevmode%
\llap{\spacer}%
${L_{93.6}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{93.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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262227517530\\-571721910\\86765476867
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{$\left.\llap{\phantom{%
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11053\\0\\-33405
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[792518792415, -1727890200, 262227517530; -1727890200, 3767235, -571721910; 262227517530, -571721910, 86765476867],[255, 15, 1],[-5027, -61174, -6899, -1350, 11053, 885, -7660, -370184, -119971, -214329, -658090, -13572; -134, -1292, -83, -7, 0, -46, -377, -16082, -5027, -8680, -26061, -465; 15192, 184875, 20850, 4080, -33405, -2675, 23148, 1118685, 362550, 647700, 1988745, 41015],92,6]
%
% END OF LATTICE L93.6
%
\hbox{}\par\smallskip%
%
% LATTICE L93.7
%
\leavevmode%
\llap{\spacer}%
${L_{93.7}}=\pdualwitharg{2}({L_{93.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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1162997880\\-7667340\\-291929100
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1162997880, -7667340, -291929100; -7667340, 50552, 1924612; -291929100, 1924612, 73278379],[1020, 4, 1],[-8692, -16513, -1011, -127, -495, -2009, -25293, -214585, -66959, -115422, -346152, -18382; 95, 187, 13, 2, 0, 18, 260, 2227, 697, 1205, 3621, 195; -34630, -65790, -4028, -506, -1972, -8004, -100770, -854930, -266772, -459854, -1379108, -73236],92,7]
%
% END OF LATTICE L93.7
%
\hbox{}\par\smallskip%
%
% LATTICE L93.8
%
\leavevmode%
\llap{\spacer}%
${L_{93.8}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{93.1}}))$%
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{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-16120590\\207570\\7548255
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\kern3pt%
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207570\\-1003\\-97410
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7548255\\-97410\\-3534343
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16120590, 207570, 7548255; 207570, -1003, -97410; 7548255, -97410, -3534343],[255, 17, 1],[-98939, -10732, -9123, 4210, 867, -20347, -283476, -142234, -755707, -1304421, -230316, -209102; -27500, -2983, -2536, 1170, 241, -5655, -78790, -39533, -210044, -362556, -64015, -58119; -210545, -22838, -19414, 8959, 1845, -43299, -603245, -302678, -1608166, -2775845, -490119, -444975],92,8]
%
% END OF LATTICE L93.8
%
\hbox{}\par\smallskip%
%
% LATTICE L93.9
%
\leavevmode%
\llap{\spacer}%
${L_{93.9}}=\pdualwitharg{5}({L_{93.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7730268900, 1092420, -37509480; 1092420, -130, 5305; -37509480, 5305, -182006],[1020, 5, 1],[-481, -4463, -114, 115, 228, -230, -1353, -57893, -9056, -31293, -46995, -2521; -17256, -160106, -4089, 4126, 8177, -8253, -48542, -2077026, -324901, -1122692, -1686026, -90444; 98626, 915110, 23375, -23580, -46750, 47160, 277424, 11870590, 1856875, 6416430, 9636025, 516915],92,9]
%
% END OF LATTICE L93.9
%
\hbox{}\par\smallskip%
%
% LATTICE L93.10
%
\leavevmode%
\llap{\spacer}%
${L_{93.10}}=\pdualwitharg{2.3}({L_{93.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[44254639430040, -93064417500, -11107564851240; -93064417500, 195707976, 23358433512; -11107564851240, 23358433512, 2787911019353],[1020, 12, 1],[1481338, 2714367, 142427, 12269, 167227, 134335, 4562427, 38383235, 11945103, 20536532, 61479176, 1074418; 95, 187, 13, 2, 0, 6, 260, 2227, 697, 1205, 3621, 65; 5901930, 10814550, 567456, 48882, 666264, 535216, 18177570, 152926050, 47591544, 81821418, 244944636, 4280684],92,10]
%
% END OF LATTICE L93.10
%
\hbox{}\par\smallskip%
%
% LATTICE L93.11
%
\leavevmode%
\llap{\spacer}%
${L_{93.11}}=\pdualwitharg{3.17}(\pfillwitharg{2}({L_{93.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22596240795, 1440447570, 7748337690; 1440447570, 91824531, 493935000; 7748337690, 493935000, 2656934731],[255, 51, 1],[-65678, -7988, -10861, -1469, 468, -2630, -161447, -82862, -443389, -770756, -136743, -42795; 160, 17, 14, 0, 1, 15, 500, 247, 1306, 2244, 395, 117; 191505, 23292, 31671, 4284, -1365, 7667, 470730, 241602, 1292799, 2247315, 398706, 124780],92,11]
%
% END OF LATTICE L93.11
%
\hbox{}\par\smallskip%
%
% LATTICE L93.12
%
\leavevmode%
\llap{\spacer}%
${L_{93.12}}=\pdualwitharg{3.5}({L_{93.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7544940, 56100, 682380; 56100, -390, -5235; 682380, -5235, -60758],[1020, 15, 1],[1231, 52661, 8237, 28461, 42740, 764, 437, 4051, 103, -105, -206, 70; 54330, 2324206, 363543, 1256140, 1886354, 33720, 19288, 178806, 4547, -4634, -9095, 3089; 9144, 391170, 61185, 211410, 317475, 5675, 3246, 30090, 765, -780, -1530, 520],92,12]
%
% END OF LATTICE L93.12
%
\hbox{}\par\smallskip%
%
% LATTICE L93.13
%
\leavevmode%
\llap{\spacer}%
${L_{93.13}}=\pdualwitharg{17}({L_{93.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-84660, 6120, 3060; 6120, -442, -221; 3060, -221, -110],[1020, 17, 1],[39, 19, 50, 171, 15, 13, 11, 1, 0, -1, 0, 2; 690, 344, 912, 3142, 277, 249, 230, 24, 8, -14, -1, 27; -340, -182, -493, -1734, -155, -153, -170, -22, -17, 0, 2, 0],92,13]
%
% END OF LATTICE L93.13
%
\hbox{}\par\smallskip%
%
% LATTICE L93.14
%
\leavevmode%
\llap{\spacer}%
${L_{93.14}}=\pdualwitharg{2.5}({L_{93.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9989403706920, 33613560420, -2506957062840; 33613560420, 113107000, -8435714000; -2506957062840, -8435714000, 629150037311],[1020, 20, 1],[-183318, -1680643, -88459, -7697, -102563, -248693, -564051, -23729903, -7385231, -12697576, -38013256, -1993414; 19, 187, 13, 2, 0, 18, 52, 2227, 697, 1205, 3621, 195; -730462, -6696810, -352480, -30670, -408680, -990960, -2247558, -94555870, -29427720, -50595670, -151470340, -7943100],92,14]
%
% END OF LATTICE L93.14
%
\hbox{}\par\smallskip%
%
% LATTICE L93.15
%
\leavevmode%
\llap{\spacer}%
${L_{93.15}}=\pdualwitharg{5.17}(\pfillwitharg{2}({L_{93.1}}))$%
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\kern3pt%
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-85\\0\\170\\-85\\-85\\-765
\end{tabular}\endgroup%
\kern3pt%
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-340\\-170\\-85\\85\\0\\-510
\end{tabular}\endgroup%
\kern3pt%
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-85\\-55\\-85\\0\\10\\0
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-255\\-255\\-765\\-510\\0\\510
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-102\\-170\\-765\\-1105\\-170\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
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\kern3pt%
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0\\-255\\-1785\\-3825\\-765\\-1275
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[119987762730, -4285210995, -26954687790; -4285210995, 153040885, 962652540; -26954687790, 962652540, 6055244113],[255, 85, 1],[25668, 15604, 21197, 2863, -910, 15472, 63141, 162014, 866893, 1506890, 267337, 250955; -76972, -46795, -63577, -8589, 2731, -46371, -189323, -485795, -2599373, -4518426, -801616, -752514; 126497, 76900, 104465, 14110, -4485, 76245, 311168, 798430, 4272185, 7426195, 1317480, 1236750],92,15]
%
% END OF LATTICE L93.15
%
\hbox{}\par\smallskip%
%
% LATTICE L93.16
%
\leavevmode%
\llap{\spacer}%
${L_{93.16}}=\pdualwitharg{3.17}({L_{93.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1020}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{102}\above{1pt}{1pt}{*}{4}{204}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{34}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2575827420\\-1849260\\7601040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1849260\\-1326\\5457
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7601040\\5457\\-22430
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-6396721\\-8916640\\-2169941880
\end{tabular}\endgroup%
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-4686\\-6533\\-1589619
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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457\\690\\155040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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239\\344\\81078
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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643\\912\\218127
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2247\\3142\\762246
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\277\\67845
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\83\\21709
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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1020\\0\\-510\\-4080\\-510\\-510
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0\\12\\0\\-408\\-66\\-102
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-510\\0\\102\\-102\\-51\\-153
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-4080\\-408\\-102\\204\\0\\-204
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-510\\-66\\-51\\0\\6\\0
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-510\\-102\\-153\\-204\\0\\34
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-102\\-357\\-1530\\-153\\-85
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2575827420, -1849260, 7601040; -1849260, -1326, 5457; 7601040, 5457, -22430],[1020, 51, 1],[457, 239, 643, 2247, 200, 64, 203, 25, 17, -3, -2, 2; 690, 344, 912, 3142, 277, 83, 230, 24, 8, -14, -1, 9; 155040, 81078, 218127, 762246, 67845, 21709, 68850, 8478, 5763, -1020, -678, 680],92,16]
%
% END OF LATTICE L93.16
%
\hbox{}\par\smallskip%
%
% LATTICE L93.17
%
\leavevmode%
\llap{\spacer}%
${L_{93.17}}=\pdualwitharg{2.3.5}({L_{93.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{1020}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{*}{4}{60}\above{1pt}{1pt}{*}{2}{2040}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
741109560\\-561300900\\-186851760
\end{tabular}\endgroup%
\kern3pt%
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-561300900\\425117640\\141517620
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-186851760\\141517620\\47109877
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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879477437\\-2613440\\3496118820
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292784822\\-870034\\1163884919
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-235279\\697\\-935280
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-405706\\1205\\-1612770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1217000\\3621\\-4837860
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21578\\65\\-85780
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\-1020\\-420\\-900\\-3060\\-100
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[741109560, -561300900, -186851760; -561300900, 425117640, 141517620; -186851760, 141517620, 47109877],[1020, 60, 1],[-17749, -753741, -235279, -405706, -1217000, -21578, -6144, -58621, -3651, -475, -1543, -2315; 52, 2227, 697, 1205, 3621, 65, 19, 187, 13, 2, 0, 6; -70554, -2996250, -935280, -1612770, -4837860, -85780, -24426, -233070, -14520, -1890, -6120, -9200],92,17]
%
% END OF LATTICE L93.17
%
\hbox{}\par\smallskip%
%
% LATTICE L93.18
%
\leavevmode%
\llap{\spacer}%
${L_{93.18}}=\pdualwitharg{3.5.17}(\pfillwitharg{2}({L_{93.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${51}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{510}\above{1pt}{1pt}{}{4}{255}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{170}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46498224390\\-29909990655\\-9670186755
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29909990655\\-19239600570\\-6220347345
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9670186755\\-6220347345\\-2011097941
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-46498224390, -29909990655, -9670186755; -29909990655, -19239600570, -6220347345; -9670186755, -6220347345, -2011097941],[255, 255, 1],[-5500, -2983, -2536, 1170, 241, -1885, -15758, -39533, -210044, -362556, -64015, -19373; -1371281, -743716, -632195, 291764, 60081, -470023, -3928970, -9856786, -52370315, -90396031, -15960834, -4830228; 4267833, 2314665, 1967580, -908055, -186990, 1462850, 12228117, 30677235, 162991920, 281339205, 49674840, 15033100],92,18]
%
% END OF LATTICE L93.18
%
\hbox{}\par\smallskip%
%
% LATTICE L93.19
%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[573240, 29580, -143820; 29580, 3400, -7412; -143820, -7412, 36083],[1020, 68, 1],[-12851, -6484, -34514, -59695, -10555, -9683, -4739, -552, -666, -111, 1, -871; 260, 131, 697, 1205, 213, 195, 95, 11, 13, 2, 0, 18; -51170, -25818, -137428, -237694, -42028, -38556, -18870, -2198, -2652, -442, 4, -3468],92,19]
%
% END OF LATTICE L93.19
%
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%
% LATTICE L93.20
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-478517700, 1029180, -2331720; 1029180, -2210, 5015; -2331720, 5015, -11362],[1020, 85, 1],[45, 27, 17, -5, -2, 10, 107, 277, 743, 2589, 230, 218; 46, 24, 8, -14, -1, 27, 138, 344, 912, 3142, 277, 249; -9214, -5530, -3485, 1020, 410, -2040, -21896, -56690, -152065, -529890, -47075, -44625],92,20]
%
% END OF LATTICE L93.20
%
\hbox{}\par\smallskip%
%
% LATTICE L93.21
%
\leavevmode%
\llap{\spacer}%
${L_{93.21}}=\pdualwitharg{2.3.17}({L_{93.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20582565720, -8893380, -5166061320; -8893380, 10200, 2232168; -5166061320, 2232168, 1296640561],[1020, 204, 1],[4479569, 2259834, 12028414, 20803249, 3678223, 1124541, 1650381, 192146, 231486, 38581, -253, 101449; 260, 131, 697, 1205, 213, 65, 95, 11, 13, 2, 0, 6; 17847450, 9003606, 47923476, 82884078, 14654736, 4480384, 6575430, 765546, 922284, 153714, -1008, 404192],92,21]
%
% END OF LATTICE L93.21
%
\hbox{}\par\smallskip%
%
% LATTICE L93.22
%
\leavevmode%
\llap{\spacer}%
${L_{93.22}}=\pdualwitharg{3.5.17}({L_{93.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21017100, 21392460, -157080; 21392460, -21774450, 159885; -157080, 159885, -1174],[1020, 255, 1],[-47, -123, -331, -1157, -103, -33, -21, -13, -9, 1, 1, -1; 138, 344, 912, 3142, 277, 83, 46, 24, 8, -14, -1, 9; 25092, 63330, 168555, 582930, 51525, 15725, 9078, 5010, 2295, -2040, -270, 1360],92,22]
%
% END OF LATTICE L93.22
%
\hbox{}\par\smallskip%
%
% LATTICE L93.23
%
\leavevmode%
\llap{\spacer}%
${L_{93.23}}=\pdualwitharg{2.5.17}({L_{93.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5220986280, -6755460, 1315353240; -6755460, 17000, -1708160; 1315353240, -1708160, 331389223],[1020, 340, 1],[190117, 110782, 133878, 22313, -261, 174067, 515025, 1299502, 6917542, 11965101, 2115687, 1941359; -570332, -332335, -401621, -66937, 783, -522183, -1545023, -3898375, -20751929, -35894098, -6346848, -5823882; -757554, -441430, -533460, -88910, 1040, -693600, -2052206, -5178090, -27564140, -47677010, -8430320, -7735680],92,23]
%
% END OF LATTICE L93.23
%
\hbox{}\par\smallskip%
%
% LATTICE L93.24
%
\leavevmode%
\llap{\spacer}%
${L_{93.24}}=\pdualwitharg{2.3.5.17}({L_{93.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[51000, -8632260, 2166480; -8632260, 1574788200, -395229600; 2166480, -395229600, 99192029],[1020, 1020, 1],[2970, 7517, 40053, 69343, 12269, 3769, 1117, 657, 817, 136, -8, 318; -192311, -486774, -2593754, -4490627, -794549, -244111, -72363, -42574, -52986, -8831, 527, -20563; -766326, -1939710, -10335660, -17894370, -3166140, -972740, -288354, -169650, -211140, -35190, 2100, -81940],92,24]
%
% END OF LATTICE L93.24
%

\medskip%
%------------------------------WEYL-GROUP-W94--------------------------------
%
\leavevmode\llap{}%
$W_{94\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=16$%
\hfill%
$226222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L94.1
%
\leavevmode%
\llap{\spacer}%
${L_{94.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{94}}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-16365070740\\-63936660\\1970640
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\kern3pt%
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-63936660\\-249794\\7699
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-252\\64517\\509
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-4651\\1190748\\9384
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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0\\340\\0\\-170\\-5100\\-170
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-6\\0\\6\\-3\\-204\\-15
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-204\\-5100\\-204\\0\\204\\0
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0\\-170\\-15\\-5\\0\\10
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-16365070740, -63936660, 1970640; -63936660, -249794, 7699; 1970640, 7699, -226],[1020, 1, 1],[101, -583, -200, -252, -4651, 49; -25858, 149260, 51204, 64517, 1190748, -12545; -204, 1190, 405, 509, 9384, -100],93,1]
%
% END OF LATTICE L94.1
%
\hbox{}\par\smallskip%
%
% LATTICE L94.2
%
\leavevmode%
\llap{\spacer}%
${L_{94.2}}=\pfillwitharg{2}({L_{94.1}})$%
{${}={}$}Nikulin $94$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{85}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}{51}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20910\\-6885\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6885\\-2267\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\85\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\-103\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
505\\-1530\\-51
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\-100\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-5\\-51\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\85\\0\\-85\\-1275\\-85
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\-3\\-102\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-85\\-3\\2\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-1275\\-102\\0\\51\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-85\\-15\\-5\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-20910, -6885, 0; -6885, -2267, 0; 0, 0, 1],[255, 1, 1],[0, -28, 1, 34, 505, 33; 0, 85, -3, -103, -1530, -100; 1, 0, -3, -5, -51, 0],93,2]
%
% END OF LATTICE L94.2
%
\hbox{}\par\smallskip%
%
% LATTICE L94.3
%
\leavevmode%
\llap{\spacer}%
${L_{94.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{94.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{255}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{17}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2355945\\28305\\-781065
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28305\\-303\\9384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-781065\\9384\\-258946
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
186\\1\\-561
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1099\\0\\3315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-247\\-1\\745
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-929\\-5\\2802
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2852\\-17\\8602
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
184\\0\\-555
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-15\\-51\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\255\\0\\-255\\-1275\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\-3\\-34\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-255\\-3\\6\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-1275\\-34\\0\\17\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-255\\-15\\-15\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2355945, 28305, -781065; 28305, -303, 9384; -781065, 9384, -258946],[255, 3, 1],[186, -1099, -247, -929, -2852, 184; 1, 0, -1, -5, -17, 0; -561, 3315, 745, 2802, 8602, -555],93,3]
%
% END OF LATTICE L94.3
%
\hbox{}\par\smallskip%
%
% LATTICE L94.4
%
\leavevmode%
\llap{\spacer}%
${L_{94.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{94.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{17}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{l}{2}{255}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2614515\\38505\\-519435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38505\\-505\\7650
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-519435\\7650\\-103198
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
152\\1\\-765
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-179\\0\\901
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-605\\-3\\3045
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-759\\-5\\3820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6992\\-51\\35190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\0\\-151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-15\\-25\\-255\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\17\\0\\-85\\-1275\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\-15\\-510\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-85\\-15\\10\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-255\\-1275\\-510\\0\\255\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-17\\-15\\-5\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2614515, 38505, -519435; 38505, -505, 7650; -519435, 7650, -103198],[255, 5, 1],[152, -179, -605, -759, -6992, 30; 1, 0, -3, -5, -51, 0; -765, 901, 3045, 3820, 35190, -151],93,4]
%
% END OF LATTICE L94.4
%
\hbox{}\par\smallskip%
%
% LATTICE L94.5
%
\leavevmode%
\llap{\spacer}%
${L_{94.5}}=\pdualwitharg{3}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{1020}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{68}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2253180\\-1162800\\-103020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1162800\\-600078\\-53157
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103020\\-53157\\-4702
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89\\190\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2311\\4930\\-5100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\49\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
163\\-348\\363
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1593\\-3400\\3536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143\\-305\\315
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-30\\-204\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1020\\0\\-510\\-5100\\-510
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\-3\\-68\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-510\\-3\\6\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\-5100\\-68\\0\\68\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-510\\-15\\-15\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2253180, -1162800, -103020; -1162800, -600078, -53157; -103020, -53157, -4702],[1020, 3, 1],[-89, -2311, -23, 163, 1593, 143; 190, 4930, 49, -348, -3400, -305; -198, -5100, -50, 363, 3536, 315],93,5]
%
% END OF LATTICE L94.5
%
\hbox{}\par\smallskip%
%
% LATTICE L94.6
%
\leavevmode%
\llap{\spacer}%
${L_{94.6}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{94.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{51}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{l}{2}{85}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7028887830\\311355\\2325692565
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
311355\\15\\103020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2325692565\\103020\\769516606
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6733\\51\\-20349
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2379\\16\\-7190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9157\\46\\-27675
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33890\\153\\-102425
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
404\\0\\-1221
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\-75\\-255\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\51\\0\\-255\\-1275\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\10\\-15\\-170\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-255\\-15\\30\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-255\\-1275\\-170\\0\\85\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-51\\-15\\-15\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7028887830, 311355, 2325692565; 311355, 15, 103020; 2325692565, 103020, 769516606],[255, 15, 1],[0, 6733, 2379, 9157, 33890, 404; 1, 51, 16, 46, 153, 0; 0, -20349, -7190, -27675, -102425, -1221],93,6]
%
% END OF LATTICE L94.6
%
\hbox{}\par\smallskip%
%
% LATTICE L94.7
%
\leavevmode%
\llap{\spacer}%
${L_{94.7}}=\pdualwitharg{2}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{340}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{l}{2}{51}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26083723560\\-4349123940\\-6529515720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4349123940\\725160872\\1088712392
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6529515720\\1088712392\\1634527963
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-127\\-2\\-506
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-572972\\-8925\\-2282930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120766\\-1881\\-481176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-153275\\-2387\\-610704
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-848219\\-13209\\-3379617
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32763\\-510\\-130540
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-20\\-102\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\340\\0\\-340\\-2550\\-340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\-12\\-204\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-340\\-12\\8\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\-2550\\-204\\0\\51\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-340\\-60\\-20\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26083723560, -4349123940, -6529515720; -4349123940, 725160872, 1088712392; -6529515720, 1088712392, 1634527963],[1020, 4, 1],[-127, -572972, -120766, -153275, -848219, -32763; -2, -8925, -1881, -2387, -13209, -510; -506, -2282930, -481176, -610704, -3379617, -130540],93,7]
%
% END OF LATTICE L94.7
%
\hbox{}\par\smallskip%
%
% LATTICE L94.8
%
\leavevmode%
\llap{\spacer}%
${L_{94.8}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{94.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${17}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{102}\above{1pt}{1pt}{}{6}{34}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{170}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6880155\\-52020\\3231105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52020\\-391\\24429
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3231105\\24429\\-1517413
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111\\-102\\-238
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\35\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
452\\405\\969
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
555\\509\\1190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
298\\276\\639
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119\\-100\\-255
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\0\\-51\\-85\\-51\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-85\\-75\\-85
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\0\\102\\-51\\-102\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85\\-85\\-51\\34\\0\\-85
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-75\\-102\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-85\\-255\\-85\\0\\170
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6880155, -52020, 3231105; -52020, -391, 24429; 3231105, 24429, -1517413],[255, 17, 1],[-111, 42, 452, 555, 298, -119; -102, 35, 405, 509, 276, -100; -238, 90, 969, 1190, 639, -255],93,8]
%
% END OF LATTICE L94.8
%
\hbox{}\par\smallskip%
%
% LATTICE L94.9
%
\leavevmode%
\llap{\spacer}%
${L_{94.9}}=\pdualwitharg{5}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{68}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{l}{2}{1020}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-684420\\2040\\5100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2040\\10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5100\\-15\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-10\\940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-34\\2822
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\0\\3060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\134
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-30\\-50\\-1020\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\68\\0\\-170\\-5100\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\-15\\-1020\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-170\\-15\\10\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1020\\-5100\\-1020\\0\\1020\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-15\\-5\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-684420, 2040, 5100; 2040, 10, -15; 5100, -15, -38],[1020, 5, 1],[7, 21, 1, -1, 23, 1; -10, -34, -3, 2, 0, -1; 940, 2822, 135, -135, 3060, 134],93,9]
%
% END OF LATTICE L94.9
%
\hbox{}\par\smallskip%
%
% LATTICE L94.10
%
\leavevmode%
\llap{\spacer}%
${L_{94.10}}=\pdualwitharg{2.3}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{1020}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{l}{2}{17}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1112018280\\-15627420\\-276653580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15627420\\219624\\3887868
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276653580\\3887868\\68827289
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-155\\-89\\-618
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15223\\-8840\\-60690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-901\\-524\\-3592
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3010\\-1749\\-12000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5505\\-3196\\-21947
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-617\\-355\\-2460
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-60\\-102\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1020\\0\\-1020\\-2550\\-1020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\-12\\-68\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-1020\\-12\\24\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\-2550\\-68\\0\\17\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1020\\-60\\-60\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1112018280, -15627420, -276653580; -15627420, 219624, 3887868; -276653580, 3887868, 68827289],[1020, 12, 1],[-155, -15223, -901, -3010, -5505, -617; -89, -8840, -524, -1749, -3196, -355; -618, -60690, -3592, -12000, -21947, -2460],93,10]
%
% END OF LATTICE L94.10
%
\hbox{}\par\smallskip%
%
% LATTICE L94.11
%
\leavevmode%
\llap{\spacer}%
${L_{94.11}}=\pdualwitharg{3.17}(\pfillwitharg{2}({L_{94.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${51}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{34}\above{1pt}{1pt}{}{6}{102}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{510}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12703335\\2705040\\4359225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2705040\\576402\\928251
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4359225\\928251\\1495894
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1713\\5\\-4995
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1988\\6\\-5797
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7118\\23\\-20757
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1515\\5\\-4418
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1224\\5\\-3570
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\0\\-51\\-255\\-51\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-255\\-75\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\0\\34\\-51\\-34\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-255\\-255\\-51\\102\\0\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-75\\-34\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-255\\-255\\-255\\0\\510
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12703335, 2705040, 4359225; 2705040, 576402, 928251; 4359225, 928251, 1495894],[255, 51, 1],[35, 1713, 1988, 7118, 1515, 1224; 0, 5, 6, 23, 5, 5; -102, -4995, -5797, -20757, -4418, -3570],93,11]
%
% END OF LATTICE L94.11
%
\hbox{}\par\smallskip%
%
% LATTICE L94.12
%
\leavevmode%
\llap{\spacer}%
${L_{94.12}}=\pdualwitharg{3.5}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{204}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{l}{2}{340}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128843340\\-794580\\158100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-794580\\-4890\\975
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
158100\\975\\-194
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-10\\8910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-34\\16932
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\825
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\0\\43180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\2439
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-30\\-150\\-1020\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\204\\0\\-510\\-5100\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\-15\\-340\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-150\\-510\\-15\\30\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1020\\-5100\\-340\\0\\340\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-102\\-15\\-15\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-128843340, -794580, 158100; -794580, -4890, 975; 158100, 975, -194],[1020, 15, 1],[11, 21, -1, 1, 53, 3; -10, -34, -1, 2, 0, -1; 8910, 16932, -820, 825, 43180, 2439],93,12]
%
% END OF LATTICE L94.12
%
\hbox{}\par\smallskip%
%
% LATTICE L94.13
%
\leavevmode%
\llap{\spacer}%
${L_{94.13}}=\pdualwitharg{17}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${68}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{102}\above{1pt}{1pt}{}{6}{34}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{170}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8875020\\389640\\14280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
389640\\17102\\629
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14280\\629\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-190\\-408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-190\\-410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-21\\-51
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\254\\527
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\360\\744
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\85\\170
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\0\\-102\\-170\\-204\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-170\\-300\\-170
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\0\\102\\-51\\-204\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-170\\-170\\-51\\34\\0\\-85
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\-300\\-204\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-170\\-255\\-85\\0\\170
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8875020, 389640, 14280; 389640, 17102, 629; 14280, 629, 22],[1020, 17, 1],[9, 9, 1, -12, -17, -4; -190, -190, -21, 254, 360, 85; -408, -410, -51, 527, 744, 170],93,13]
%
% END OF LATTICE L94.13
%
\hbox{}\par\smallskip%
%
% LATTICE L94.14
%
\leavevmode%
\llap{\spacer}%
${L_{94.14}}=\pdualwitharg{2.5}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{68}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{}{6}{40}\above{1pt}{1pt}{l}{2}{255}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528496680\\133620\\-132658140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
133620\\40\\-33540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132658140\\-33540\\33298567
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2111\\8\\-8410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6085\\17\\-24242
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-497\\-3\\-1980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8257\\51\\-32895
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-745\\4\\-2968
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-60\\-100\\-510\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\68\\0\\-340\\-2550\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\-60\\-1020\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-340\\-60\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-510\\-2550\\-1020\\0\\255\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-68\\-60\\-20\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[528496680, 133620, -132658140; 133620, 40, -33540; -132658140, -33540, 33298567],[1020, 20, 1],[-2111, -6085, -497, 0, -8257, -745; 8, 17, -3, 1, 51, 4; -8410, -24242, -1980, 0, -32895, -2968],93,14]
%
% END OF LATTICE L94.14
%
\hbox{}\par\smallskip%
%
% LATTICE L94.15
%
\leavevmode%
\llap{\spacer}%
${L_{94.15}}=\pdualwitharg{5.17}(\pfillwitharg{2}({L_{94.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${85}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{510}\above{1pt}{1pt}{}{6}{170}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{34}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39343695\\-6143460\\-11724900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6143460\\960670\\1831665
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11724900\\1831665\\3494674
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\207\\-340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-669\\2008\\-3297
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11642\\34944\\-57375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13884\\41675\\-68425
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8863\\26604\\-43680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-476\\1429\\-2346
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
85\\0\\-255\\-425\\-255\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-85\\-75\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-255\\0\\510\\-255\\-510\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-425\\-85\\-255\\170\\0\\-85
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-255\\-75\\-510\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-17\\-255\\-85\\0\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[39343695, -6143460, -11724900; -6143460, 960670, 1831665; -11724900, 1831665, 3494674],[255, 85, 1],[-69, -669, -11642, -13884, -8863, -476; 207, 2008, 34944, 41675, 26604, 1429; -340, -3297, -57375, -68425, -43680, -2346],93,15]
%
% END OF LATTICE L94.15
%
\hbox{}\par\smallskip%
%
% LATTICE L94.16
%
\leavevmode%
\llap{\spacer}%
${L_{94.16}}=\pdualwitharg{3.17}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${204}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{34}\above{1pt}{1pt}{}{6}{102}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{510}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20464260\\357000\\-43860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
357000\\6222\\-765
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43860\\-765\\94
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-612
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\17\\1071
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\532
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\510
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
204\\0\\-102\\-510\\-204\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-510\\-300\\-510
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\0\\34\\-51\\-68\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-510\\-510\\-51\\102\\0\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\-300\\-68\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-510\\-255\\-255\\0\\510
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20464260, 357000, -43860; 357000, 6222, -765; -43860, -765, 94],[1020, 51, 1],[-1, -1, 0, 2, 1, 1; -18, -20, -2, 17, 8, 5; -612, -630, -17, 1071, 532, 510],93,16]
%
% END OF LATTICE L94.16
%
\hbox{}\par\smallskip%
%
% LATTICE L94.17
%
\leavevmode%
\llap{\spacer}%
${L_{94.17}}=\pdualwitharg{2.3.5}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{204}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{}{6}{120}\above{1pt}{1pt}{l}{2}{85}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22110595080\\-837420\\-5549432400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-837420\\120\\210180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5549432400\\210180\\1392825469
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82411\\-141\\-328350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-217937\\-374\\-868326
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1019\\-2\\-4060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121069\\-204\\-482375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32353\\-55\\-128904
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-300\\-510\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\204\\0\\-1020\\-2550\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\-60\\-340\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\-1020\\-60\\120\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-510\\-2550\\-340\\0\\85\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-204\\-60\\-60\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22110595080, -837420, -5549432400; -837420, 120, 210180; -5549432400, 210180, 1392825469],[1020, 60, 1],[-82411, -217937, -1019, 0, -121069, -32353; -141, -374, -2, 1, -204, -55; -328350, -868326, -4060, 0, -482375, -128904],93,17]
%
% END OF LATTICE L94.17
%
\hbox{}\par\smallskip%
%
% LATTICE L94.18
%
\leavevmode%
\llap{\spacer}%
${L_{94.18}}=\pdualwitharg{3.5.17}(\pfillwitharg{2}({L_{94.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${255}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{170}\above{1pt}{1pt}{}{6}{510}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{102}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20107970430\\-12917868795\\-4176485880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12917868795\\-8298765645\\-2683080165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4176485880\\-2683080165\\-867468667
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\-1655\\5610
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\186\\-609
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
135\\2538\\-8500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
509\\8359\\-28305
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
92\\1420\\-4835
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-513\\1683
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
255\\0\\-255\\-1275\\-255\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-255\\-75\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-255\\0\\170\\-255\\-170\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1275\\-255\\-255\\510\\0\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-255\\-75\\-170\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-51\\-255\\-255\\0\\102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-20107970430, -12917868795, -4176485880; -12917868795, -8298765645, -2683080165; -4176485880, -2683080165, -867468667],[255, 255, 1],[-102, 7, 135, 509, 92, -20; -1655, 186, 2538, 8359, 1420, -513; 5610, -609, -8500, -28305, -4835, 1683],93,18]
%
% END OF LATTICE L94.18
%
\hbox{}\par\smallskip%
%
% LATTICE L94.19
%
\leavevmode%
\llap{\spacer}%
${L_{94.19}}=\pdualwitharg{2.17}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${68}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{408}\above{1pt}{1pt}{}{6}{136}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{680}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
689758680\\-19968540\\-173060340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19968540\\577864\\5010104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-173060340\\5010104\\43420811
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2193\\22\\8738
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1413\\-15\\-5630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9011\\-93\\-35904
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1536\\-17\\-6120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1228\\12\\4893
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11861\\120\\47260
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\0\\-204\\-340\\-102\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-340\\-150\\-340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\0\\408\\-204\\-204\\-1020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-340\\-340\\-204\\136\\0\\-340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\-150\\-204\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-340\\-1020\\-340\\0\\680
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[689758680, -19968540, -173060340; -19968540, 577864, 5010104; -173060340, 5010104, 43420811],[1020, 68, 1],[2193, -1413, -9011, -1536, 1228, 11861; 22, -15, -93, -17, 12, 120; 8738, -5630, -35904, -6120, 4893, 47260],93,19]
%
% END OF LATTICE L94.19
%
\hbox{}\par\smallskip%
%
% LATTICE L94.20
%
\leavevmode%
\llap{\spacer}%
${L_{94.20}}=\pdualwitharg{5.17}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${340}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{510}\above{1pt}{1pt}{}{6}{170}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{34}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-251940\\-141780\\59160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-141780\\-79730\\33235
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59160\\33235\\-13834
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
173\\-520\\-510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\-154\\-152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
169\\-513\\-510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\87\\85
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59\\180\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\16\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
340\\0\\-510\\-850\\-1020\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-170\\-300\\-34
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-510\\0\\510\\-255\\-1020\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-850\\-170\\-255\\170\\0\\-85
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1020\\-300\\-1020\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-255\\-85\\0\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-251940, -141780, 59160; -141780, -79730, 33235; 59160, 33235, -13834],[1020, 85, 1],[173, 51, 169, -29, -59, -5; -520, -154, -513, 87, 180, 16; -510, -152, -510, 85, 180, 17],93,20]
%
% END OF LATTICE L94.20
%
\hbox{}\par\smallskip%
%
% LATTICE L94.21
%
\leavevmode%
\llap{\spacer}%
${L_{94.21}}=\pdualwitharg{2.3.17}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${204}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{136}\above{1pt}{1pt}{}{6}{408}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{2040}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
593576760\\-2840700\\-148980180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2840700\\11832\\712980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-148980180\\712980\\37392121
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5965\\-8\\-23766
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3426\\5\\13650
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7868\\11\\31348
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5837\\9\\23256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-825\\-1\\-3287
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29697\\-40\\-118320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
204\\0\\-204\\-1020\\-102\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-1020\\-150\\-1020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\0\\136\\-204\\-68\\-1020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1020\\-1020\\-204\\408\\0\\-1020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\-150\\-68\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1020\\-1020\\-1020\\0\\2040
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[593576760, -2840700, -148980180; -2840700, 11832, 712980; -148980180, 712980, 37392121],[1020, 204, 1],[-5965, 3426, 7868, 5837, -825, -29697; -8, 5, 11, 9, -1, -40; -23766, 13650, 31348, 23256, -3287, -118320],93,21]
%
% END OF LATTICE L94.21
%
\hbox{}\par\smallskip%
%
% LATTICE L94.22
%
\leavevmode%
\llap{\spacer}%
${L_{94.22}}=\pdualwitharg{3.5.17}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1020}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{170}\above{1pt}{1pt}{}{6}{510}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{102}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12106380\\-2080800\\16320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2080800\\-357510\\2805
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16320\\2805\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-2040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-1002
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-1615
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\255
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\612
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1020\\0\\-510\\-2550\\-1020\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-510\\-300\\-102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-510\\0\\170\\-255\\-340\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2550\\-510\\-255\\510\\0\\-255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1020\\-300\\-340\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-102\\-255\\-255\\0\\102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12106380, -2080800, 16320; -2080800, -357510, 2805; 16320, 2805, -22],[1020, 255, 1],[-1, -1, -2, 0, 1, 1; -10, -2, -1, 2, 0, -1; -2040, -1002, -1615, 255, 740, 612],93,22]
%
% END OF LATTICE L94.22
%
\hbox{}\par\smallskip%
%
% LATTICE L94.23
%
\leavevmode%
\llap{\spacer}%
${L_{94.23}}=\pdualwitharg{2.5.17}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${340}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{2040}\above{1pt}{1pt}{}{6}{680}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{136}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
479210280\\48238860\\83764440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48238860\\4855880\\8432000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83764440\\8432000\\14641759
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\-281\\-170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-105\\-94
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
271\\-921\\-1020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
85\\-254\\-340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-51\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\21\\68
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
340\\0\\-1020\\-1700\\-510\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-340\\-150\\-68
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1020\\0\\2040\\-1020\\-1020\\-1020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1700\\-340\\-1020\\680\\0\\-340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-510\\-150\\-1020\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-68\\-1020\\-340\\0\\136
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[479210280, 48238860, 83764440; 48238860, 4855880, 8432000; 83764440, 8432000, 14641759],[1020, 340, 1],[58, 27, 271, 85, 13, -14; -281, -105, -921, -254, -51, 21; -170, -94, -1020, -340, -45, 68],93,23]
%
% END OF LATTICE L94.23
%
\hbox{}\par\smallskip%
%
% LATTICE L94.24
%
\leavevmode%
\llap{\spacer}%
${L_{94.24}}=\pdualwitharg{2.3.5.17}({L_{94.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1020}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{680}\above{1pt}{1pt}{}{6}{2040}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{408}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2040\\-260100\\65280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-260100\\1560124680\\-391568820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65280\\-391568820\\98278133
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-47233\\-188190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-10567\\-42102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\-15787\\-62900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-1846\\-7355
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-9421\\-37536
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1020\\0\\-1020\\-5100\\-510\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-1020\\-150\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1020\\0\\680\\-1020\\-340\\-1020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5100\\-1020\\-1020\\2040\\0\\-1020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-510\\-150\\-340\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-204\\-1020\\-1020\\0\\408
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2040, -260100, 65280; -260100, 1560124680, -391568820; 65280, -391568820, 98278133],[1020, 1020, 1],[-130, -29, -43, 1, -5, -26; -47233, -10567, -15787, 0, -1846, -9421; -188190, -42102, -62900, 0, -7355, -37536],93,24]
%
% END OF LATTICE L94.24
%

\medskip%
%------------------------------WEYL-GROUP-W95--------------------------------
%
\leavevmode\llap{}%
$W_{95\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=36$%
\hfill%
$2222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L95.1
%
\leavevmode%
\llap{\spacer}%
${L_{95.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{95}}%
\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{182}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{52}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-9\\-47\\2
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-331\\-1729\\0
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\HardButStrongLineBreak\kern3pt%
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-58\\-303\\-6
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-209\\-1092\\-52
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{$\left.\llap{\phantom{%
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0\\0\\0
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\kern3pt%
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-21\\-3\\-91\\-15\\-52
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\kern3pt%
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-273\\-91\\-8463\\-1911\\-9100
\end{tabular}\endgroup%
\kern3pt%
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-21\\-15\\-1911\\-447\\-2184
\end{tabular}\endgroup%
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0\\-52\\-9100\\-2184\\-10868
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[5460, -1092, -1092; -1092, 218, 209; -1092, 209, -2],[1092, 1, 1],[4, -9, -331, -58, -209, -4, 9, 331, 58, 209; 21, -47, -1729, -303, -1092, -21, 47, 1729, 303, 1092; 0, 2, 0, -6, -52, -21, -3, -91, -15, -52],94,1]
%
% END OF LATTICE L95.1
%
\hbox{}\par\smallskip%
%
% LATTICE L95.2
%
\leavevmode%
\llap{\spacer}%
${L_{95.2}}=\pfillwitharg{2}({L_{95.1}})$%
{${}={}$}Nikulin $95$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4914\\1882\\105
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\kern3pt%
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273\\105\\10
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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589679\\-1549548\\137592
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226080\\-594089\\52752
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[12831, 4914, 273; 4914, 1882, 105; 273, 105, 10],[273, 1, 1],[-3676, -70, -346, 16, -5, -104, -110, -16034, -3796, -9355; 9660, 184, 910, -42, 13, 273, 289, 42133, 9975, 24583; -861, -17, -91, 3, 0, -21, -25, -3731, -885, -2184],94,2]
%
% END OF LATTICE L95.2
%
\hbox{}\par\smallskip%
%
% LATTICE L95.3
%
\leavevmode%
\llap{\spacer}%
${L_{95.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{95.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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-21\\-9\\-273\\-15\\-78
\end{tabular}\endgroup%
\kern3pt%
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-273\\-273\\-25389\\-1911\\-13650
\end{tabular}\endgroup%
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-7\\-15\\-1911\\-149\\-1092
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1330329, -11193, 442806; -11193, 87, -3723; 442806, -3723, 147389],[273, 3, 1],[-30, -92, -2790, -148, -694, 37, 93, 2881, 155, 746; 35, 103, 3185, 171, 819, -35, -103, -3185, -171, -819; 91, 279, 8463, 449, 2106, -112, -282, -8736, -470, -2262],94,3]
%
% END OF LATTICE L95.3
%
\hbox{}\par\smallskip%
%
% LATTICE L95.4
%
\leavevmode%
\llap{\spacer}%
${L_{95.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{95.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{91}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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2602509\\-23205\\742833
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\kern3pt%
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-23205\\203\\-6622
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-39\\-91\\-1209\\-1911\\-4550
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2602509, -23205, 742833; -23205, 203, -6622; 742833, -6622, 212026],[273, 7, 1],[-11, -81, -341, -373, -564, 17, 83, 367, 415, 668; 15, 103, 455, 513, 819, -15, -103, -455, -513, -819; 39, 287, 1209, 1323, 2002, -60, -294, -1300, -1470, -2366],94,4]
%
% END OF LATTICE L95.4
%
\hbox{}\par\smallskip%
%
% LATTICE L95.5
%
\leavevmode%
\llap{\spacer}%
${L_{95.5}}=\pdualwitharg{3}({L_{95.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{546}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{156}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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-5033028\\-2130492\\-13104
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\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5033028, -2130492, -13104; -2130492, -901842, -5547; -13104, -5547, -34],[1092, 3, 1],[24, -11, -463, -28, -309, -3, 14, 736, 49, 621; -56, 26, 1092, 66, 728, 7, -33, -1729, -115, -1456; -119, -3, 273, 23, 312, 14, -12, -1638, -128, -1872],94,5]
%
% END OF LATTICE L95.5
%
\hbox{}\par\smallskip%
%
% LATTICE L95.6
%
\leavevmode%
\llap{\spacer}%
${L_{95.6}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{95.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4478019, -41223, 1721538; -41223, 377, -15847; 1721538, -15847, 661831],[273, 13, 1],[-104, -114, -248, -490, -54, 209, 119, 283, 595, 74; 105, 103, 245, 513, 63, -105, -103, -245, -513, -63; 273, 299, 651, 1287, 142, -546, -312, -742, -1560, -194],94,6]
%
% END OF LATTICE L95.6
%
\hbox{}\par\smallskip%
%
% LATTICE L95.7
%
\leavevmode%
\llap{\spacer}%
${L_{95.7}}=\pdualwitharg{2}({L_{95.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[214763640, 972972, 53680536; 972972, 4408, 243196; 53680536, 243196, 13417541],[1092, 4, 1],[251, -1, -20219, -5189, -6758, -6005, -273, -4715, -565, -366; 210, 0, -15470, -3972, -5174, -4599, -209, -3549, -417, -260; -1008, 4, 81172, 20832, 27131, 24108, 1096, 18928, 2268, 1469],94,7]
%
% END OF LATTICE L95.7
%
\hbox{}\par\smallskip%
%
% LATTICE L95.8
%
\leavevmode%
\llap{\spacer}%
${L_{95.8}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{95.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[450575034, -6694779, 149297967; -6694779, 99435, -2218314; 149297967, -2218314, 49469858],[273, 21, 1],[1063, 167, -43861, -26271, -204172, -4257, -3361, 2339, 3913, 38084; 2, 0, -104, -62, -481, -10, -8, 0, 6, 65; -3208, -504, 132366, 79282, 616161, 12847, 10143, -7059, -11809, -114933],94,8]
%
% END OF LATTICE L95.8
%
\hbox{}\par\smallskip%
%
% LATTICE L95.9
%
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\llap{\spacer}%
${L_{95.9}}=\pdualwitharg{7}({L_{95.1}})$%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-21\\-21\\-91\\-105\\-364
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-39\\-91\\-1209\\-1911\\-9100
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-21\\-105\\-1911\\-3129\\-15288
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[116844, 38220, -1092; 38220, 12502, -357; -1092, -357, 10],[1092, 7, 1],[-1, 2, 74, 127, 641, 19, 4, 4, -1, -17; 3, -7, -247, -423, -2132, -63, -13, -13, 3, 52; -3, -35, -793, -1323, -6552, -186, -28, -26, 0, 0],94,9]
%
% END OF LATTICE L95.9
%
\hbox{}\par\smallskip%
%
% LATTICE L95.10
%
\leavevmode%
\llap{\spacer}%
${L_{95.10}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{95.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${182}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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-8133762\\184665\\-2697357
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{$\left.\llap{\phantom{%
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-273\\-273\\-1953\\-1911\\-1050
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[357860958, -8133762, 118675557; -8133762, 184665, -2697357; 118675557, -2697357, 39355754],[273, 39, 1],[14878, 194, -52072, -57792, -34519, -65397, -7411, 1553, 7273, 5651; 14, 0, -56, -62, -37, -70, -8, 0, 6, 5; -44863, -585, 157017, 174265, 104088, 197197, 22347, -4683, -21931, -17040],94,10]
%
% END OF LATTICE L95.10
%
\hbox{}\par\smallskip%
%
% LATTICE L95.11
%
\leavevmode%
\llap{\spacer}%
${L_{95.11}}=\pdualwitharg{2.3}({L_{95.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{2184}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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-1243788\\6504\\-310728
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240443112, -1243788, 60066552; -1243788, 6504, -310728; 60066552, -310728, 15005591],[1092, 12, 1],[-18678, -2472, -54674, -2748, -7490, -551, -275, -195303, -16481, -63932; 10479, 1387, 30667, 1541, 4199, 308, 154, 109564, 9246, 35867; 74984, 9924, 219492, 11032, 30069, 2212, 1104, 784056, 66164, 256659],94,11]
%
% END OF LATTICE L95.11
%
\hbox{}\par\smallskip%
%
% LATTICE L95.12
%
\leavevmode%
\llap{\spacer}%
${L_{95.12}}=\pdualwitharg{13}({L_{95.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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546\\0\\-1638\\-5460\\-2184\\-6279\\-273\\-273\\-273\\0
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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546\\0\\-1638\\-5460\\-2184
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0\\26\\0\\-78\\-52
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-1638\\0\\14\\0\\-28
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-5460\\-78\\0\\78\\0
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\end{tabular}\endgroup%
\kern3pt%
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-273\\-39\\-91\\-195\\-52
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\kern3pt%
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-273\\-91\\-651\\-1911\\-700
\end{tabular}\endgroup%
\kern3pt%
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-273\\-195\\-1911\\-5811\\-2184
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3556644, 1975428, -21840; 1975428, 1097174, -12129; -21840, -12129, 134],[1092, 13, 1],[-10, 13, 65, 178, 61, 115, -8, -30, -73, -21; 21, -27, -133, -363, -124, -231, 17, 63, 153, 44; 273, -325, -1449, -3861, -1288, -2184, 234, 812, 1950, 560],94,12]
%
% END OF LATTICE L95.12
%
\hbox{}\par\smallskip%
%
% LATTICE L95.13
%
\leavevmode%
\llap{\spacer}%
${L_{95.13}}=\pdualwitharg{3.7}({L_{95.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{1092}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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568932\\12012\\-1092
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12012\\210\\-21
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1092\\-21\\2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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-209\\-8632\\-203112
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-42\\-1744\\-41013
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2\\83\\1952
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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0\\0\\-1
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-1\\-42\\-987
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-25\\-1040\\-24453
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-14\\-582\\-13685
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\HardButStrongLineBreak\kern3pt%
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-209\\-8684\\-204204
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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-234\\0\\78\\0\\-1092
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-140\\-42\\0\\14\\0
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-23\\-21\\-39\\-7\\0
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-21\\-63\\-273\\-105\\-1092
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\kern3pt%
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-39\\-273\\-3627\\-1911\\-27300
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\kern3pt%
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-7\\-105\\-1911\\-1043\\-15288
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[568932, 12012, -1092; 12012, 210, -21; -1092, -21, 2],[1092, 21, 1],[0, -1, -25, -14, -209, -2, -1, -1, 0, 1; 0, -42, -1040, -582, -8684, -83, -41, -39, 1, 52; -1, -987, -24453, -13685, -204204, -1952, -966, -936, 14, 1092],94,13]
%
% END OF LATTICE L95.13
%
\hbox{}\par\smallskip%
%
% LATTICE L95.14
%
\leavevmode%
\llap{\spacer}%
${L_{95.14}}=\pdualwitharg{7.13}(\pfillwitharg{2}({L_{95.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${78}\above{1pt}{1pt}{s}{2}{182}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{546}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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536608527\\15212925\\-139786920
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15212925\\430885\\-3963323
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\kern3pt%
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{$\left.\llap{\phantom{%
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1529711\\-4590288\\5372640
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-285\\855\\-1001
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10949\\-32855\\38455
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255185\\-765741\\896259
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50807\\-152458\\178444
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[536608527, 15212925, -139786920; 15212925, 430885, -3963323; -139786920, -3963323, 36414302],[273, 91, 1],[-9383, -285, 10949, 255185, 50807, 41252, 10908, -326, -32102, -8315; 28155, 855, -32855, -765741, -152458, -123786, -32732, 978, 96324, 24950; -32955, -1001, 38455, 896259, 178444, 144885, 38311, -1145, -112749, -29204],94,14]
%
% END OF LATTICE L95.14
%
\hbox{}\par\smallskip%
%
% LATTICE L95.15
%
\leavevmode%
\llap{\spacer}%
${L_{95.15}}=\pdualwitharg{2.7}({L_{95.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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3304392\\-53508\\825552
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4471973\\-858429\\-17955756
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3304392, -53508, 825552; -53508, 616, -13356; 825552, -13356, 206251],[1092, 28, 1],[-263, -1053, -17393, -28201, -34200, -3691, -265, 259, 523, -68; 51, 203, 3341, 5415, 6565, 708, 50, -52, -102, 13; 1056, 4228, 69836, 113232, 137319, 14820, 1064, -1040, -2100, 273],94,15]
%
% END OF LATTICE L95.15
%
\hbox{}\par\smallskip%
%
% LATTICE L95.16
%
\leavevmode%
\llap{\spacer}%
${L_{95.16}}=\pdualwitharg{3.13}({L_{95.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22874124, 11709516, -29484; 11709516, 5994222, -15093; -29484, -15093, 38],[1092, 39, 1],[59, 2, -6, -7, -7, -3, 6, 62, 63, 71; -126, -4, 14, 16, 16, 7, -13, -133, -135, -152; -4277, -39, 903, 923, 924, 455, -507, -4725, -4745, -5292],94,16]
%
% END OF LATTICE L95.16
%
\hbox{}\par\smallskip%
%
% LATTICE L95.17
%
\leavevmode%
\llap{\spacer}%
${L_{95.17}}=\pdualwitharg{2.13}({L_{95.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2184}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1647070152, -5572476, 410465328; -5572476, 18824, -1388712; 410465328, -1388712, 102291809],[1092, 52, 1],[10879, 531, -23327, -79261, -8019, -94375, -4507, -4505, -4235, 67; -1911, -93, 4123, 14007, 1417, 16674, 796, 798, 756, -11; -43680, -2132, 93660, 318240, 32197, 378924, 18096, 18088, 17004, -269],94,17]
%
% END OF LATTICE L95.17
%
\hbox{}\par\smallskip%
%
% LATTICE L95.18
%
\leavevmode%
\llap{\spacer}%
${L_{95.18}}=\pdualwitharg{3.7.13}(\pfillwitharg{2}({L_{95.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[158045133246, 26977468791, 8314881120; 26977468791, 4604911311, 1419306252; 8314881120, 1419306252, 437452559],[273, 273, 1],[5, 103, 35, 171, 63, -5, -103, -35, -171, -63; -1035, -22733, -7335, -34687, -12052, 1624, 23322, 7924, 38810, 14408; 3263, 71799, 23133, 109291, 37905, -5174, -73710, -25044, -122668, -45549],94,18]
%
% END OF LATTICE L95.18
%
\hbox{}\par\smallskip%
%
% LATTICE L95.19
%
\leavevmode%
\llap{\spacer}%
${L_{95.19}}=\pdualwitharg{2.3.7}({L_{95.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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3030\\-247\\-12168
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[320180952, -1165164, 79750944; -1165164, 4200, -290220; 79750944, -290220, 19864433],[1092, 84, 1],[-270, -1088, 3030, 2852, 13666, 793, 1611, 3769, 809, -68; 1, -5, -247, -143, -546, -22, -16, -26, -4, 0; 1084, 4368, -12168, -11452, -54873, -3184, -6468, -15132, -3248, 273],94,19]
%
% END OF LATTICE L95.19
%
\hbox{}\par\smallskip%
%
% LATTICE L95.20
%
\leavevmode%
\llap{\spacer}%
${L_{95.20}}=\pdualwitharg{7.13}({L_{95.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1092, 0, -1092; 0, 182, 455; -1092, 455, 46],[1092, 91, 1],[37, 84, 84, 1721, 629, 209, -2, -2, 1, 27; 93, 211, 211, 4323, 1580, 525, -5, -5, 3, 68; -39, -91, -93, -1911, -700, -234, 0, 2, 0, -28],94,20]
%
% END OF LATTICE L95.20
%
\hbox{}\par\smallskip%
%
% LATTICE L95.21
%
\leavevmode%
\llap{\spacer}%
${L_{95.21}}=\pdualwitharg{2.3.13}({L_{95.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${728}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1422860712, -4377828, 354463200; -4377828, 12792, -1090596; 354463200, -1090596, 88303907],[1092, 156, 1],[-68724, -14882, -15964, -4870, 609, 17589, 7577, -35171, -46265, -15219; 3647, 789, 847, 259, -32, -931, -401, 1869, 2457, 808; 275912, 59748, 64092, 19552, -2445, -70616, -30420, 141204, 185744, 61101],94,21]
%
% END OF LATTICE L95.21
%
\hbox{}\par\smallskip%
%
% LATTICE L95.22
%
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${L_{95.22}}=\pdualwitharg{3.7.13}({L_{95.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30474444, 15137304, -82992; 15137304, -7518966, 41223; -82992, 41223, -226],[1092, 273, 1],[6, 41, 41, 280, 307, 34, -1, -1, 0, 13; 18, 120, 118, 804, 880, 97, -5, -3, 1, 40; 1079, 6825, 6459, 43771, 47712, 5200, -546, -180, 182, 2520],94,22]
%
% END OF LATTICE L95.22
%
\hbox{}\par\smallskip%
%
% LATTICE L95.23
%
\leavevmode%
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${L_{95.23}}=\pdualwitharg{2.7.13}({L_{95.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3723720, -1241604, -1860768; -1241604, 414232, 620620; -1860768, 620620, 929975],[1092, 364, 1],[661, 1995, 2447, 51407, 4786, 6683, 453, 1, 1, 110; -1992, -6018, -7384, -155130, -14443, -20169, -1369, -3, 3, -331; 2652, 8008, 9824, 206388, 19215, 26832, 1820, 4, 0, 441],94,23]
%
% END OF LATTICE L95.23
%
\hbox{}\par\smallskip%
%
% LATTICE L95.24
%
\leavevmode%
\llap{\spacer}%
${L_{95.24}}=\pdualwitharg{2.3.7.13}({L_{95.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[447720, -26534508, -6608784; -26534508, 1593326280, 396840444; -6608784, 396840444, 98838725],[1092, 1092, 1],[-139, -1097, -1279, -8907, -2478, -1142, -184, -2, -60, -84; -19248, -152054, -177348, -1235110, -343628, -158375, -25569, -275, -8251, -11618; 77272, 610428, 711972, 4958408, 1379511, 635804, 102648, 1104, 33124, 46641],94,24]
%
% END OF LATTICE L95.24
%

\medskip%
%------------------------------WEYL-GROUP-W96--------------------------------
%
\leavevmode\llap{}%
$W_{96\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=56$%
\hfill%
$222262222262\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L96.1
%
\leavevmode%
\llap{\spacer}%
${L_{96.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{96}}%
\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-156692172, 269724, -125580; 269724, -434, 177; -125580, 177, -50],[1092, 1, 1],[13, -29, -54, -290, -149, -1214, -1315, -1180, -163, -113, -6, 5; 11802, -26325, -49021, -263263, -135263, -1102077, -1193766, -1071213, -147973, -102583, -5447, 4539; 9128, -20358, -37912, -203606, -104612, -852345, -923258, -828477, -114443, -79339, -4213, 3510],95,1]
%
% END OF LATTICE L96.1
%
\hbox{}\par\smallskip%
%
% LATTICE L96.2
%
\leavevmode%
\llap{\spacer}%
${L_{96.2}}=\pfillwitharg{2}({L_{96.1}})$%
{${}={}$}Nikulin $96$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[20202, 5733, 819; 5733, 1627, 234; 819, 234, 70],[273, 1, 1],[-9343, -16541, -2199, -1415, -53, 22, -2, -814, -916, -4370, -2172, -17377; 33131, 58656, 7798, 5018, 188, -78, 7, 2886, 3248, 15496, 7702, 61620; -1365, -2418, -322, -208, -8, 3, 0, -117, -133, -637, -317, -2538],95,2]
%
% END OF LATTICE L96.2
%
\hbox{}\par\smallskip%
%
% LATTICE L96.3
%
\leavevmode%
\llap{\spacer}%
${L_{96.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{96.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4940705679, 54529566, -1646769033; 54529566, -601815, 18175050; -1646769033, 18175050, -548878726],[273, 3, 1],[6232441, 3581577, 1316281, 692961, -10021, 4014, 101978, 339730, 795192, 3228346, 1518216, 3917293; -6377070, -3664687, -1346821, -709033, 10255, -4108, -104349, -347620, -813652, -3303274, -1553450, -4008199; -18909996, -10866947, -3993759, -2102529, 30405, -12179, -309414, -1030783, -2412711, -9795201, -4606455, -11885551],95,3]
%
% END OF LATTICE L96.3
%
\hbox{}\par\smallskip%
%
% LATTICE L96.4
%
\leavevmode%
\llap{\spacer}%
${L_{96.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{96.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11431685811, 126699846, -3266063346; 126699846, -1404235, 36198484; -3266063346, 36198484, -933123071],[273, 7, 1],[737587, 8901214, 155778, 574076, -8300, 9973, 12068, 844301, 94107, 2674429, 1257725, 9735542; -911010, -10994061, -192403, -709033, 10255, -12324, -14907, -1042860, -116236, -3303274, -1553450, -12024597; -2617000, -31582005, -552709, -2036853, 29449, -35385, -42818, -2995629, -333897, -9489025, -4462479, -34542249],95,4]
%
% END OF LATTICE L96.4
%
\hbox{}\par\smallskip%
%
% LATTICE L96.5
%
\leavevmode%
\llap{\spacer}%
${L_{96.5}}=\pdualwitharg{3}({L_{96.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13760292, -279552, 60060; -279552, -1878, 579; 60060, 579, -154],[1092, 3, 1],[13, -53, -194, -940, -469, -1253, -4045, -1195, -478, -308, -11, 5; 2338, -9529, -34881, -169013, -84327, -225292, -727300, -214864, -85946, -55380, -1978, 899; 13860, -56498, -206808, -1002066, -499968, -1335737, -4312098, -1273909, -509565, -328341, -11727, 5330],95,5]
%
% END OF LATTICE L96.5
%
\hbox{}\par\smallskip%
%
% LATTICE L96.6
%
\leavevmode%
\llap{\spacer}%
${L_{96.6}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{96.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21168237909, 234955266, -8141497455; 234955266, -2607865, 90365938; -8141497455, 90365938, -3131294210],[273, 13, 1],[6633919, 879763, 1401093, 56741, -10661, 12808, 108530, 83444, 846390, 264328, 1616006, 12508883; -6377070, -845697, -1346821, -54541, 10255, -12324, -104349, -80220, -813652, -254098, -1553450, -12024597; -17432506, -2311827, -3681769, -149103, 28015, -33657, -285194, -219273, -2224131, -694597, -4246515, -32870643],95,6]
%
% END OF LATTICE L96.6
%
\hbox{}\par\smallskip%
%
% LATTICE L96.7
%
\leavevmode%
\llap{\spacer}%
${L_{96.7}}=\pdualwitharg{2}({L_{96.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1281107923368, -6102459636, 319097083032; -6102459636, 29068600, -1519994556; 319097083032, -1519994556, 79480382989],[1092, 4, 1],[1217, -36643, -56507, -305317, -157685, -1290301, -700097, -1261277, -176453, -127323, -8715, -7619; 0, -312, -406, -2080, -1058, -8583, -4641, -8307, -1141, -793, -47, -36; -4886, 147108, 226856, 1225744, 633052, 5180124, 2810654, 5063604, 708400, 511160, 34988, 30588],95,7]
%
% END OF LATTICE L96.7
%
\hbox{}\par\smallskip%
%
% LATTICE L96.8
%
\leavevmode%
\llap{\spacer}%
${L_{96.8}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{96.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3438235140522, -49906672998, 1139254175241; -49906672998, 724405395, -16536502962; 1139254175241, -16536502962, 377490201443],[273, 21, 1],[-66151448, -267396629, -14257039, -54734015, 199579, -617, -847702, -22933021, -8076011, -235595635, -111864829, -290329033; -169864, -686621, -36609, -140543, 513, -2, -2177, -58890, -20738, -604968, -287248, -745509; 199635657, 806964977, 43025715, 165179469, -602301, 1862, 2558244, 69208594, 24372252, 710994102, 337592136, 876171709],95,8]
%
% END OF LATTICE L96.8
%
\hbox{}\par\smallskip%
%
% LATTICE L96.9
%
\leavevmode%
\llap{\spacer}%
${L_{96.9}}=\pdualwitharg{7}({L_{96.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[221676, -28392, 2184; -28392, -3038, 329; 2184, 329, -34],[1092, 7, 1],[1, -29, -6, -212, -107, -863, -133, -829, -16, -74, -3, 5; 50, -1443, -299, -10569, -5335, -43032, -6632, -41340, -798, -3692, -150, 249; 548, -15834, -3280, -115934, -58520, -472017, -72746, -453453, -8753, -40495, -1645, 2730],95,9]
%
% END OF LATTICE L96.9
%
\hbox{}\par\smallskip%
%
% LATTICE L96.10
%
\leavevmode%
\llap{\spacer}%
${L_{96.10}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{96.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2255997630705, -55091270598, 748140607305; -55091270598, 1345324305, -18269530110; 748140607305, -18269530110, 248100601118],[273, 39, 1],[-995761325, -44231333, -214606837, -9053717, 429385, -1496, -12761008, -3793540, -121567274, -38971156, -240553750, -624321853; -1189048, -52817, -256263, -10811, 513, -2, -15239, -4530, -145166, -46536, -287248, -745509; 3002603604, 133374491, 647122203, 27300441, -1294761, 4511, 38479350, 11438983, 366572115, 117513033, 725362131, 1882570651],95,10]
%
% END OF LATTICE L96.10
%
\hbox{}\par\smallskip%
%
% LATTICE L96.11
%
\leavevmode%
\llap{\spacer}%
${L_{96.11}}=\pdualwitharg{2.3}({L_{96.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4724327368488, -8789578980, 1176739240404; -8789578980, 16352952, -2189315364; 1176739240404, -2189315364, 293103151391],[1092, 12, 1],[5555, 30515, 83627, 367227, 177649, 466310, 747274, 435522, 167316, 98810, 1596, -273; 0, -208, -742, -3640, -1826, -4897, -7917, -4693, -1897, -1261, -59, -4; -22302, -122512, -335748, -1474356, -713232, -1872160, -3000186, -1748552, -671748, -396708, -6408, 1096],95,11]
%
% END OF LATTICE L96.11
%
\hbox{}\par\smallskip%
%
% LATTICE L96.12
%
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${L_{96.12}}=\pdualwitharg{13}({L_{96.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17147676, 313404, -1116024; 313404, -5642, 18993; -1116024, 18993, -49706],[1092, 13, 1],[1189, -203, -4926, -2036, -13601, -110825, -120049, -8287, -14884, -794, -549, 455; 83202, -14205, -344701, -142471, -951743, -7755087, -8400546, -579891, -1041523, -55561, -38417, 31839; 5096, -870, -21112, -8726, -58292, -474981, -514514, -35517, -63791, -3403, -2353, 1950],95,12]
%
% END OF LATTICE L96.12
%
\hbox{}\par\smallskip%
%
% LATTICE L96.13
%
\leavevmode%
\llap{\spacer}%
${L_{96.13}}=\pdualwitharg{3.7}({L_{96.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-113016540, -1232868, 121212; -1232868, -13146, 1323; 121212, 1323, -130],[1092, 21, 1],[1, -53, -26, -862, -427, -1136, -523, -1078, -61, -269, -8, 5; -2, 117, 57, 1885, 933, 2481, 1142, 2353, 133, 585, 17, -11; 912, -48230, -23664, -784602, -388668, -1034033, -476058, -981253, -55527, -244881, -7287, 4550],95,13]
%
% END OF LATTICE L96.13
%
\hbox{}\par\smallskip%
%
% LATTICE L96.14
%
\leavevmode%
\llap{\spacer}%
${L_{96.14}}=\pdualwitharg{7.13}(\pfillwitharg{2}({L_{96.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3423424976790, 103665029013, -886153946496; 103665029013, 3139090045, -26833704352; -886153946496, -26833704352, 229381050545],[273, 91, 1],[209364103, 195297322, 45122226, 13325144, -631964, 6607, 2683070, 16749851, 25560165, 57357247, 354043919, 2756606642; -628262173, -586050417, -135403287, -39986243, 1896405, -19827, -8051387, -50263143, -76701233, -172118277, -1062419005, -8272056453; 735327632, 685922349, 158478073, 46800509, -2219581, 23205, 9423466, 58828749, 89772293, 201449857, 1243471411, 9681741069],95,14]
%
% END OF LATTICE L96.14
%
\hbox{}\par\smallskip%
%
% LATTICE L96.15
%
\leavevmode%
\llap{\spacer}%
${L_{96.15}}=\pdualwitharg{2.7}({L_{96.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[56861560392, -133952364, 14162935332; -133952364, 315560, -33364520; 14162935332, -33364520, 3527668531],[1092, 28, 1],[135, 9791, 1043, 28373, 13095, 100094, 7542, 90010, 1516, 4894, -300, -293; 0, -312, -58, -2080, -1058, -8583, -663, -8307, -163, -793, -47, -36; -542, -39312, -4188, -113932, -52584, -401940, -30286, -361452, -6088, -19656, 1204, 1176],95,15]
%
% END OF LATTICE L96.15
%
\hbox{}\par\smallskip%
%
% LATTICE L96.16
%
\leavevmode%
\llap{\spacer}%
${L_{96.16}}=\pdualwitharg{3.13}({L_{96.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[169260, -148512, 295932; -148512, -24414, 76011; 295932, 76011, -210826],[1092, 39, 1],[-473521, -10761, -55960, -2774, -1289, 585, 1525, -477, -22706, -8464, -54901, -146679; -6620404, -150452, -782390, -38784, -18022, 8179, 21322, -6669, -317457, -118337, -767583, -2050752; -3051594, -69349, -360633, -17877, -8307, 3770, 9828, -3074, -146328, -54546, -353808, -945269],95,16]
%
% END OF LATTICE L96.16
%
\hbox{}\par\smallskip%
%
% LATTICE L96.17
%
\leavevmode%
\llap{\spacer}%
${L_{96.17}}=\pdualwitharg{2.13}({L_{96.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[137803848, -1606236996, 32559072; -1606236996, 18722246920, -379507440; 32559072, -379507440, 7692769],[1092, 52, 1],[43, -5335, -90563, -35733, -236369, -1917931, -1037149, -142823, -255139, -13653, -10561, -8123; 0, -24, -406, -160, -1058, -8583, -4641, -639, -1141, -61, -47, -36; -182, 21396, 363272, 143344, 948220, 7694076, 4160702, 572964, 1023568, 54776, 42380, 32604],95,17]
%
% END OF LATTICE L96.17
%
\hbox{}\par\smallskip%
%
% LATTICE L96.18
%
\leavevmode%
\llap{\spacer}%
${L_{96.18}}=\pdualwitharg{3.7.13}(\pfillwitharg{2}({L_{96.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-747646811172534, -127612505363889, -39332164909947; -127612505363889, -21781610356506, -6713432105289; -39332164909947, -6713432105289, -2069184504481],[273, 273, 1],[-911010, -281899, -192403, -54541, 10255, -4108, -14907, -26740, -116236, -254098, -1553450, -4008199; 193972582, 60022191, 40966957, 11613273, -2182669, 874191, 3173636, 5693215, 24748449, 54102133, 330759699, 853426087; -612023607, -189382438, -129259260, -36642312, 6886698, -2758210, -10013445, -17963246, -78086424, -170703372, -1043615118, -2692735682],95,18]
%
% END OF LATTICE L96.18
%
\hbox{}\par\smallskip%
%
% LATTICE L96.19
%
\leavevmode%
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${L_{96.19}}=\pdualwitharg{2.3.7}({L_{96.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[441319851336, 82123860, 109924039680; 82123860, 15288, 20455428; 109924039680, 20455428, 27379902497],[1092, 84, 1],[677, 105081, 50645, 1705413, 850405, 2272803, 524191, 2169347, 124311, 569015, 24375, 1625; 0, -208, -106, -3640, -1826, -4897, -1131, -4693, -271, -1261, -59, -4; -2718, -421876, -203328, -6846840, -3414180, -9124780, -2104506, -8709428, -499080, -2284464, -97860, -6524],95,19]
%
% END OF LATTICE L96.19
%
\hbox{}\par\smallskip%
%
% LATTICE L96.20
%
\leavevmode%
\llap{\spacer}%
${L_{96.20}}=\pdualwitharg{7.13}({L_{96.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1547364, -616980, -68796; -616980, -245882, -27391; -68796, -27391, -3046],[1092, 91, 1],[-13, 41, 100, 264, 1717, 13778, 2119, 1012, 251, 87, 38, -89; 50, -159, -387, -1021, -6639, -53268, -8192, -3912, -970, -336, -146, 345; -156, 504, 1222, 3220, 20930, 167895, 25818, 12327, 3055, 1057, 455, -1092],95,20]
%
% END OF LATTICE L96.20
%
\hbox{}\par\smallskip%
%
% LATTICE L96.21
%
\leavevmode%
\llap{\spacer}%
${L_{96.21}}=\pdualwitharg{2.3.13}({L_{96.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26375244168, -21550660404, 6530134884; -21550660404, 17608593912, -5335636644; 6530134884, -5335636644, 1616768411],[1092, 156, 1],[-527054, -24186, -129432, -6874, -4992, -441, 1487, -841, -45757, -18153, -120143, -324910; -7917, -361, -1897, -97, -59, -4, 0, -16, -742, -280, -1826, -4897; 2102646, 96496, 516516, 27444, 19968, 1768, -6006, 3344, 182364, 72396, 479232, 1296152],95,21]
%
% END OF LATTICE L96.21
%
\hbox{}\par\smallskip%
%
% LATTICE L96.22
%
\leavevmode%
\llap{\spacer}%
${L_{96.22}}=\pdualwitharg{3.7.13}({L_{96.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14408940, 15989064, -5553912; 15989064, -17740086, 6162975; -5553912, 6162975, -2140750],[1092, 273, 1],[-363, 1440, 9203, 23489, 151300, 402582, 185355, 29392, 21629, 7343, 2860, -1766; -2, 9, 57, 145, 933, 2481, 1142, 181, 133, 45, 17, -11; 936, -3710, -23712, -60522, -389844, -1037309, -477594, -75733, -55731, -18921, -7371, 4550],95,22]
%
% END OF LATTICE L96.22
%
\hbox{}\par\smallskip%
%
% LATTICE L96.23
%
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${L_{96.23}}=\pdualwitharg{2.7.13}({L_{96.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[74105304, -22009260, -34842444; -22009260, 6539624, 10350340; -34842444, 10350340, 16383631],[1092, 364, 1],[-149, -4459, -9927, -26629, -174631, -1410022, -108712, -104402, -26360, -9658, -6804, -5171; 445, 13311, 29633, 79489, 521281, 4208967, 324509, 311643, 78685, 28829, 20309, 15435; -598, -17892, -39832, -106848, -700700, -5657652, -436202, -418908, -105768, -38752, -27300, -20748],95,23]
%
% END OF LATTICE L96.23
%
\hbox{}\par\smallskip%
%
% LATTICE L96.24
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[198744, 210778932, 52619112; 210778932, 223682277000, 55840318716; 52619112, 55840318716, 13940045837],[1092, 1092, 1],[-10, -168, -1070, -2788, -18106, -48441, -11177, -3561, -2659, -941, -539, -36; -1733, -26331, -166991, -434451, -2820115, -7542933, -1740223, -554321, -413661, -146201, -83145, -5543; 6942, 105476, 668928, 1740312, 11296740, 30215276, 6970938, 2220484, 1657032, 585648, 333060, 22204],95,24]
%
% END OF LATTICE L96.24
%

\medskip%
%------------------------------WEYL-GROUP-W97--------------------------------
%
\leavevmode\llap{}%
$W_{97\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=60$%
\hfill%
$22222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L97.1
%
\leavevmode%
\llap{\spacer}%
${L_{97.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\spacer%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[22003140, 6840, -30780; 6840, -2, -5; -30780, -5, 38],[1140, 1, 1],[13, 21, 3, 85, 2, 1, 1, -1, -2, 0, 29, 3, 9, 94; 13578, 21945, 3137, 88920, 2095, 1048, 1045, -1050, -2109, -5, 30096, 3125, 9392, 98135; 12300, 19874, 2840, 80484, 1895, 948, 950, -948, -1900, -2, 27360, 2835, 8512, 88920],96,1]
%
% END OF LATTICE L97.1
%
\hbox{}\par\smallskip%
%
% LATTICE L97.2
%
\leavevmode%
\llap{\spacer}%
${L_{97.2}}=\pfillwitharg{2}({L_{97.1}})$%
{${}={}$}Nikulin $97$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[21090, -7125, -855; -7125, 2407, 285; -855, 285, -122],[285, 1, 1],[1, -256, 62, 6625, 1233, 1837, 38404, 2663, 8692, 1270, 18683, 987, 383, 3776; 3, -760, 184, 19665, 3660, 5453, 114000, 7905, 25802, 3770, 55461, 2930, 1137, 11210; 0, 19, -5, -513, -95, -141, -2945, -204, -665, -97, -1425, -75, -29, -285],96,2]
%
% END OF LATTICE L97.2
%
\hbox{}\par\smallskip%
%
% LATTICE L97.3
%
\leavevmode%
\llap{\spacer}%
${L_{97.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{97.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11480370, 78375, -3840660; 78375, 525, -26223; -3840660, -26223, 1284859],[285, 3, 1],[-1717, -44519, -14387, -127900, -45983, -45012, -825779, -16171, -125417, -12445, -42036, 1263, 292, -23901; 1685, 43681, 14115, 125476, 45110, 44156, 810065, 15863, 123025, 12207, 41230, -1240, -286, 23465; -5098, -132183, -42717, -379753, -136530, -133647, -2451855, -48014, -372381, -36951, -124811, 3750, 867, -70965],96,3]
%
% END OF LATTICE L97.3
%
\hbox{}\par\smallskip%
%
% LATTICE L97.4
%
\leavevmode%
\llap{\spacer}%
${L_{97.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{97.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14202690, 112005, -5693160; 112005, 875, -44900; -5693160, -44900, 2282107],[285, 5, 1],[-6221, -53772, -17378, -463481, -11109, -54373, -199504, -58603, -151504, -15034, -152347, 305, 353, -5772; 5055, 43681, 14115, 376428, 9022, 44156, 162013, 47589, 123025, 12207, 123690, -248, -286, 4693; -15420, -133285, -43075, -1148835, -27536, -134775, -494513, -145260, -375535, -37265, -377625, 756, 875, -14307],96,4]
%
% END OF LATTICE L97.4
%
\hbox{}\par\smallskip%
%
% LATTICE L97.5
%
\leavevmode%
\llap{\spacer}%
${L_{97.5}}=\pdualwitharg{3}({L_{97.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{114}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{76}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{570}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40883820, 5700, -72960; 5700, -6, -9; -72960, -9, 130],[1140, 3, 1],[-1, -2, 2, 73, 18, 47, 474, 21, 97, 13, 117, 7, 3, 1; -98, -209, 189, 6992, 1730, 4528, 45695, 2026, 9367, 1257, 11324, 680, 292, 95; -568, -1140, 1134, 41420, 10215, 26676, 269040, 11920, 55062, 7380, 66424, 3975, 1704, 570],96,5]
%
% END OF LATTICE L97.5
%
\hbox{}\par\smallskip%
%
% LATTICE L97.6
%
\leavevmode%
\llap{\spacer}%
${L_{97.6}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{97.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[98926350, -8073195, -31324635; -8073195, 658830, 2556345; -31324635, 2556345, 9918821],[285, 15, 1],[4712, 109845, 33683, 290125, 20384, 97451, 354305, 34240, 260199, 24745, 79919, -908, -71, 15739; -132, -3097, -953, -8227, -579, -2773, -10089, -976, -7429, -709, -2299, 25, 3, -437; 14915, 347700, 106620, 918365, 64524, 308475, 1121532, 108385, 823650, 78330, 252985, -2874, -225, 49818],96,6]
%
% END OF LATTICE L97.6
%
\hbox{}\par\smallskip%
%
% LATTICE L97.7
%
\leavevmode%
\llap{\spacer}%
${L_{97.7}}=\pdualwitharg{2}({L_{97.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25657813560, -100095420, 6391851180; -100095420, 390488, -24935680; 6391851180, -24935680, 1592332153],[1140, 4, 1],[-39587, -110719, -12675, -73000, -1963, 496, 22341, 707, -12401, -6765, -111680, -30437, -16696, -637941; 132, 380, 46, 285, 15, 1, 0, 0, 38, 20, 342, 95, 54, 2090; 158910, 444448, 50880, 293037, 7880, -1991, -89680, -2838, 49780, 27156, 448305, 122180, 67021, 2560820],96,7]
%
% END OF LATTICE L97.7
%
\hbox{}\par\smallskip%
%
% LATTICE L97.8
%
\leavevmode%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L97.8
%
\hbox{}\par\smallskip%
%
% LATTICE L97.9
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[520980, -2280, -1140; -2280, 10, 5; -1140, 5, 2],[1140, 5, 1],[-5, -4, 0, 1, 0, -3, -8, -7, -15, -3, -115, -1, -7, -11; -1026, -817, 1, 228, 0, -620, -1653, -1446, -3097, -619, -23712, -206, -1440, -2261; -240, -190, 0, 0, -1, -160, -418, -360, -760, -150, -5700, -49, -340, -532],96,9]
%
% END OF LATTICE L97.9
%
\hbox{}\par\smallskip%
%
% LATTICE L97.10
%
\leavevmode%
\llap{\spacer}%
${L_{97.10}}=\pdualwitharg{2.3}({L_{97.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26019950520, -47580180, 6482001240; -47580180, 87000, -11853012; 6482001240, -11853012, 1614774019],[1140, 12, 1],[141, -53277, -26779, -146507, -119516, -66029, -2530373, -52569, -444789, -51863, -102526, -11554, 494, 40043; 0, 38, 20, 114, 95, 54, 2090, 44, 380, 46, 95, 15, 1, 0; -566, 213864, 107496, 588107, 479760, 265053, 10157400, 211022, 1785468, 208188, 411559, 46380, -1983, -160740],96,10]
%
% END OF LATTICE L97.10
%
\hbox{}\par\smallskip%
%
% LATTICE L97.11
%
\leavevmode%
\llap{\spacer}%
${L_{97.11}}=\pdualwitharg{3.19}(\pfillwitharg{2}({L_{97.1}}))$%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[51251089155, 5439795, -17184299295; 5439795, 570, -1823943; -17184299295, -1823943, 5761831546],[285, 57, 1],[121335, 149804, 875764, 397883, 2660088, 2547863, 2439396, 896717, 359222, 651314, 111143, -114958, -2733, 105734; 38, 45, 257, 115, 760, 719, 685, 250, 99, 175, 29, -40, 1, 35; 361874, 446781, 2611911, 1186661, 7933545, 7598841, 7275345, 2674402, 1071357, 1942503, 331477, -342855, -8151, 315345],96,11]
%
% END OF LATTICE L97.11
%
\hbox{}\par\smallskip%
%
% LATTICE L97.12
%
\leavevmode%
\llap{\spacer}%
${L_{97.12}}=\pdualwitharg{3.5}({L_{97.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[298619580, 11696400, 785460; 11696400, 458130, 30765; 785460, 30765, 2066],[1140, 15, 1],[75, 178, 0, 1, 1, 151, 394, 113, 715, 141, 1785, 46, 319, 499; -602, -1425, 1, 0, -8, -1216, -3173, -910, -5757, -1135, -14364, -370, -2564, -4009; -19550, -46455, -15, -380, -261, -39300, -102543, -29410, -186105, -36705, -464740, -11979, -83100, -130017],96,12]
%
% END OF LATTICE L97.12
%
\hbox{}\par\smallskip%
%
% LATTICE L97.13
%
\leavevmode%
\llap{\spacer}%
${L_{97.13}}=\pdualwitharg{19}({L_{97.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${228}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{38}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{190}\above{1pt}{1pt}{l}{2}{76}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1140, 0, -1140; 0, 38, 19; -1140, 19, -1130],[1140, 19, 1],[445, 39, 111, 175, 92, 73, 19, -1, -2, 0, 47, 93, 297, 166; 198, 17, 47, 72, 35, 24, 5, -6, -1, 1, 24, 45, 136, 75; -456, -40, -114, -180, -95, -76, -20, 0, 2, 0, -48, -95, -304, -170],96,13]
%
% END OF LATTICE L97.13
%
\hbox{}\par\smallskip%
%
% LATTICE L97.14
%
\leavevmode%
\llap{\spacer}%
${L_{97.14}}=\pdualwitharg{2.5}({L_{97.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-645240, -1140, -160740; -1140, 4520, -280; -160740, -280, -40043],[1140, 20, 1],[-1039, 1041, 563, -71, -282, -2059, -19331, -7541, -28211, -4853, -40684, -1148, -1516, -7839; -6, 0, 2, 0, -1, -8, -76, -30, -114, -20, -171, -5, -7, -38; 4170, -4180, -2260, 285, 1132, 8265, 77596, 30270, 113240, 19480, 163305, 4608, 6085, 31464],96,14]
%
% END OF LATTICE L97.14
%
\hbox{}\par\smallskip%
%
% LATTICE L97.15
%
\leavevmode%
\llap{\spacer}%
${L_{97.15}}=\pdualwitharg{5.19}(\pfillwitharg{2}({L_{97.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[51424525905, -7034655, 20656310370; -7034655, 950, -2825680; 20656310370, -2825680, 8297269633],[285, 95, 1],[-563923, -232079, -1356749, -1849222, -824211, -3947194, -755831, -4167629, -556513, -1009027, -516554, 35619, 4234, -32761; 1691883, 696282, 4070504, 5548011, 2472785, 11842301, 2267630, 12503637, 1669638, 3027256, 1549749, -106865, -12701, 98290; 1404480, 578005, 3379055, 4605585, 2052741, 9830695, 1882437, 10379700, 1386025, 2513035, 1286505, -88711, -10545, 81593],96,15]
%
% END OF LATTICE L97.15
%
\hbox{}\par\smallskip%
%
% LATTICE L97.16
%
\leavevmode%
\llap{\spacer}%
${L_{97.16}}=\pdualwitharg{3.19}({L_{97.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-641820, -160740, 1140; -160740, -40242, 285; 1140, 285, -2],[1140, 57, 1],[1, 0, -2, -3, -13, -31, -16, -13, -3, -7, -3, -2, 1, 1; -6, -1, 3, 8, 40, 104, 55, 46, 11, 27, 12, 10, -4, -5; -304, -156, -798, -652, -1995, -3420, -1560, -1064, -186, -228, -32, 285, 0, -150],96,16]
%
% END OF LATTICE L97.16
%
\hbox{}\par\smallskip%
%
% LATTICE L97.17
%
\leavevmode%
\llap{\spacer}%
${L_{97.17}}=\pdualwitharg{2.3.5}({L_{97.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15955392120, 23307300, -3975058020; 23307300, -33720, 5806680; -3975058020, 5806680, -990328921],[1140, 60, 1],[63855, 410111, 21525, -9112, -10906, -9832, -3181, 11333, 304175, 91257, 366255, 48500, 103817, 717467; -667, -4275, -223, 95, 113, 97, -19, -125, -3249, -965, -3857, -509, -1087, -7505; -256310, -1646160, -86400, 36575, 43776, 39465, 12768, -45490, -1220940, -366300, -1470125, -194676, -416715, -2879868],96,17]
%
% END OF LATTICE L97.17
%
\hbox{}\par\smallskip%
%
% LATTICE L97.18
%
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\llap{\spacer}%
${L_{97.18}}=\pdualwitharg{3.5.19}(\pfillwitharg{2}({L_{97.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[918791194035, -1681338121995, 563122063935; -1681338121995, 3076757692125, -1030482875010; 563122063935, -1030482875010, 345134411594],[285, 285, 1],[-21, -29, -179, -84, -115, -564, -109, -203, -83, -157, -28, 3, 4, -3; 13127214, 17910487, 109963221, 51448487, 70285807, 343996355, 66429367, 123580174, 50443207, 95097861, 16905207, -1932113, -2227885, 1924327; 39194530, 53476170, 328322280, 153612130, 209855589, 1027085850, 198341238, 368979050, 150610620, 283938090, 50474660, -5768799, -6651900, 5745552],96,18]
%
% END OF LATTICE L97.18
%
\hbox{}\par\smallskip%
%
% LATTICE L97.19
%
\leavevmode%
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${L_{97.19}}=\pdualwitharg{2.19}({L_{97.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L97.19
%
\hbox{}\par\smallskip%
%
% LATTICE L97.20
%
\leavevmode%
\llap{\spacer}%
${L_{97.20}}=\pdualwitharg{5.19}({L_{97.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36804900, -43320, 80940; -43320, 190, 95; 80940, 95, -178],[1140, 95, 1],[401, 35, 99, 155, 16, 61, 3, -5, -2, 0, 43, 17, 269, 30; 198, 17, 47, 72, 7, 24, 1, -6, -1, 1, 24, 9, 136, 15; 182400, 15920, 45030, 70500, 7277, 27740, 1364, -2280, -910, 0, 19560, 7733, 122360, 13646],96,20]
%
% END OF LATTICE L97.20
%
\hbox{}\par\smallskip%
%
% LATTICE L97.21
%
\leavevmode%
\llap{\spacer}%
${L_{97.21}}=\pdualwitharg{2.3.19}({L_{97.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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48794280\\-338580\\12156960
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-338580\\3192\\-84360
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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26809\\-468\\-107616
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89174\\-1555\\-357960
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44516\\-776\\-178695
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[48794280, -338580, 12156960; -338580, 3192, -84360; 12156960, -84360, 3028873],[1140, 228, 1],[2963, 4233, 26809, 6420, 89174, 44516, 86633, 32593, 13545, 26525, 2469, -284, -71, 2257; -52, -74, -468, -112, -1555, -776, -1510, -568, -236, -462, -43, 5, 1, -40; -11894, -16992, -107616, -25771, -357960, -178695, -347760, -130834, -54372, -106476, -9911, 1140, 285, -9060],96,21]
%
% END OF LATTICE L97.21
%
\hbox{}\par\smallskip%
%
% LATTICE L97.22
%
\leavevmode%
\llap{\spacer}%
${L_{97.22}}=\pdualwitharg{3.5.19}({L_{97.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${380}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{570}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{114}\above{1pt}{1pt}{l}{2}{1140}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\1\\0
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[438900, -21660, -5700; -21660, 570, 285; -5700, 285, 74],[1140, 285, 1],[-5, -2, 0, 11, 13, 199, 22, 97, 25, 69, 35, 10, 31, 1; -2, -1, 1, 8, 9, 136, 15, 66, 17, 47, 24, 7, 24, 1; -380, -150, 0, 820, 969, 14820, 1638, 7220, 1860, 5130, 2600, 741, 2280, 72],96,22]
%
% END OF LATTICE L97.22
%
\hbox{}\par\smallskip%
%
% LATTICE L97.23
%
\leavevmode%
\llap{\spacer}%
${L_{97.23}}=\pdualwitharg{2.5.19}({L_{97.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1140}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{760}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{152}\above{1pt}{1pt}{l}{2}{95}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26201506920, 9780060, -6519880020; 9780060, 3800, -2433520; -6519880020, -2433520, 1622381407],[1140, 380, 1],[-479803, -84729, -245179, -99296, -44757, -53272, -13793, -31949, -563, -10697, -28642, -40535, -159958, -71499; 1439379, 254181, 735517, 297879, 134266, 159809, 41377, 95841, 1689, 32093, 85926, 121604, 479866, 214493; -1926030, -340120, -984200, -398595, -179664, -213845, -55368, -128250, -2260, -42940, -114975, -162716, -642105, -287012],96,23]
%
% END OF LATTICE L97.23
%
\hbox{}\par\smallskip%
%
% LATTICE L97.24
%
\leavevmode%
\llap{\spacer}%
${L_{97.24}}=\pdualwitharg{2.3.5.19}({L_{97.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${380}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{s}{2}{2280}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{456}\above{1pt}{1pt}{l}{2}{285}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11400\\-1537860\\-383040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1537860\\196214520\\48871800
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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0\\0\\0
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47\\2272\\-9120
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-4854\\-229757\\922260
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2\\-1751\\7030
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0\\269\\-1080
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2\\5111\\-20520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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0\\1142\\-4585
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-1\\2953\\-11856
\end{tabular}\endgroup%
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-8\\4614\\-18525
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-4\\1333\\-5352
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-4560\\-2400\\-6840\\-900\\-1140\\-1140\\-240
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11400, -1537860, -383040; -1537860, 196214520, 48871800; -383040, 48871800, 12172661],[1140, 1140, 1],[-2, 0, 2, 0, -1, -8, -4, -10, -6, -20, -3, -5, -7, -2; -1751, 269, 5111, 1142, 2953, 4614, 1333, 1183, -553, -5963, -1284, -3237, -5324, -1617; 7030, -1080, -20520, -4585, -11856, -18525, -5352, -4750, 2220, 23940, 5155, 12996, 21375, 6492],96,24]
%
% END OF LATTICE L97.24
%

\medskip%
%------------------------------WEYL-GROUP-W98--------------------------------
%
\leavevmode\llap{}%
$W_{98\phantom{0}}$%
\qquad\llap{24} lattices, $\chi=18$%
\hfill%
$2222222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L98.1
%
\leavevmode%
\llap{\spacer}%
${L_{98.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{98}}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{380}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{570}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{76}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15644220\\9120\\5700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9120\\-2\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5700\\-7\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1098\\-990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-9880\\-8930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\14250\\12825
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1096\\988
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1095\\990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1102\\-988
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-570\\-36\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\380\\0\\-4560\\-380\\-570\\-760
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-6\\0\\2\\0\\-4\\-15\\-38
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-570\\-4560\\0\\570\\0\\-285\\-1710
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-380\\-4\\0\\4\\0\\-76
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-570\\-15\\-285\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-760\\-38\\-1710\\-76\\0\\76
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-15644220, 9120, 5700; 9120, -2, -7; 5700, -7, 2],[1140, 1, 1],[-1, -9, 0, 13, 1, 1, -1; -1098, -9880, 0, 14250, 1096, 1095, -1102; -990, -8930, -1, 12825, 988, 990, -988],97,1]
%
% END OF LATTICE L98.1
%
\hbox{}\par\smallskip%
%
% LATTICE L98.2
%
\leavevmode%
\llap{\spacer}%
${L_{98.2}}=\pfillwitharg{2}({L_{98.1}})$%
{${}={}$}Nikulin $98$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{95}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{570}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{19}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27930\\-9405\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9405\\3167\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\95\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-863\\-2565\\-285
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-104\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-101\\-300\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\-190\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-3\\-285\\-9\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\95\\0\\-2280\\-95\\-285\\-190
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0\\-2\\-15\\-19
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-285\\-2280\\0\\570\\0\\-285\\-855
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-95\\-2\\0\\1\\0\\-19
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-285\\-15\\-285\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-190\\-19\\-855\\-19\\0\\19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[27930, -9405, 0; -9405, 3167, 0; 0, 0, -1],[285, 1, 1],[1, 32, -1, -863, -35, -101, -64; 3, 95, -3, -2565, -104, -300, -190; 0, 0, -1, -285, -11, -30, -19],97,2]
%
% END OF LATTICE L98.2
%
\hbox{}\par\smallskip%
%
% LATTICE L98.3
%
\leavevmode%
\llap{\spacer}%
${L_{98.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{98.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{285}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{190}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{57}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-815333415\\-7547085\\272094915
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7547085\\-69774\\2518659
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
272094915\\2518659\\-90804122
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15523\\15529\\-46084
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2376379\\-2377280\\7054890
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
291802\\-291913\\866289
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5760788\\-5762985\\17102375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
460891\\-461067\\1368273
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
153566\\-153625\\455900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93331\\93366\\-277077
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-95\\-9\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\285\\0\\-2280\\-285\\-285\\-570
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\0\\-6\\-15\\-57
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\-2280\\0\\190\\0\\-95\\-855
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-285\\-6\\0\\3\\0\\-57
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-285\\-15\\-95\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-570\\-57\\-855\\-57\\0\\57
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-815333415, -7547085, 272094915; -7547085, -69774, 2518659; 272094915, 2518659, -90804122],[285, 3, 1],[-15523, 2376379, 291802, 5760788, 460891, 153566, -93331; 15529, -2377280, -291913, -5762985, -461067, -153625, 93366; -46084, 7054890, 866289, 17102375, 1368273, 455900, -277077],97,3]
%
% END OF LATTICE L98.3
%
\hbox{}\par\smallskip%
%
% LATTICE L98.4
%
\leavevmode%
\llap{\spacer}%
${L_{98.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{98.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{19}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{114}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{95}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1240849245\\-12015315\\496643850
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12015315\\-116290\\4809090
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
496643850\\4809090\\-198779269
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55793\\46587\\-138270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
569417\\-475456\\1411168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
349601\\-291913\\866405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4141115\\-3457791\\10262793
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
552182\\-461067\\1368455
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110390\\-92175\\273576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111818\\93366\\-277115
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\-285\\-45\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\19\\0\\-456\\-95\\-57\\-190
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\10\\0\\-10\\-15\\-95
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-285\\-456\\0\\114\\0\\-57\\-855
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-95\\-10\\0\\5\\0\\-95
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-57\\-15\\-57\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-190\\-95\\-855\\-95\\0\\95
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1240849245, -12015315, 496643850; -12015315, -116290, 4809090; 496643850, 4809090, -198779269],[285, 5, 1],[-55793, 569417, 349601, 4141115, 552182, 110390, -111818; 46587, -475456, -291913, -3457791, -461067, -92175, 93366; -138270, 1411168, 866405, 10262793, 1368455, 273576, -277115],97,4]
%
% END OF LATTICE L98.4
%
\hbox{}\par\smallskip%
%
% LATTICE L98.5
%
\leavevmode%
\llap{\spacer}%
${L_{98.5}}=\pdualwitharg{3}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{1140}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{190}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{228}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25306860\\12656280\\-34200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12656280\\6329562\\-17103
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34200\\-17103\\46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-6\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1223\\-2470\\-9120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-97\\-378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\95\\380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\192\\756
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-94\\190\\755
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-451\\912\\3762
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-190\\-36\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1140\\0\\-4560\\-1140\\-570\\-2280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-12\\-15\\-114
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190\\-4560\\0\\190\\0\\-95\\-1710
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-1140\\-12\\0\\12\\0\\-228
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-570\\-15\\-95\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2280\\-114\\-1710\\-228\\0\\228
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25306860, 12656280, -34200; 12656280, 6329562, -17103; -34200, -17103, 46],[1140, 3, 1],[3, 1223, 48, -47, -95, -94, -451; -6, -2470, -97, 95, 192, 190, 912; -2, -9120, -378, 380, 756, 755, 3762],97,5]
%
% END OF LATTICE L98.5
%
\hbox{}\par\smallskip%
%
% LATTICE L98.6
%
\leavevmode%
\llap{\spacer}%
${L_{98.6}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{98.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{57}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{38}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{285}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58470445815\\-4925200995\\-18515625780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4925200995\\414869505\\1559645685
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18515625780\\1559645685\\5863276622
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14646\\-449\\46370
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480742\\14801\\-1522071
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-291905\\8981\\-924195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1143711\\35169\\-3621077
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-455741\\14010\\-1442910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29838\\916\\-94469
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96047\\-2964\\304095
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-15\\-95\\-45\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\57\\0\\-456\\-285\\-57\\-570
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\0\\-30\\-15\\-285
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\-456\\0\\38\\0\\-19\\-855
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-285\\-30\\0\\15\\0\\-285
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-57\\-15\\-19\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-570\\-285\\-855\\-285\\0\\285
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[58470445815, -4925200995, -18515625780; -4925200995, 414869505, 1559645685; -18515625780, 1559645685, 5863276622],[285, 15, 1],[14646, -480742, -291905, -1143711, -455741, -29838, 96047; -449, 14801, 8981, 35169, 14010, 916, -2964; 46370, -1522071, -924195, -3621077, -1442910, -94469, 304095],97,6]
%
% END OF LATTICE L98.6
%
\hbox{}\par\smallskip%
%
% LATTICE L98.7
%
\leavevmode%
\llap{\spacer}%
${L_{98.7}}=\pdualwitharg{2}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{380}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{2280}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{76}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14668177080\\-57292980\\3654125220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57292980\\223784\\-14272784
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3654125220\\-14272784\\910312921
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1559\\3\\-6258
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25986\\95\\-104310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1422\\7\\-5708
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77816\\285\\-312360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1064\\3\\-4271
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3677\\0\\-14760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4667\\0\\-18734
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-1140\\-18\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\380\\0\\-9120\\-190\\-1140\\-760
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\kern3pt%
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-12\\0\\8\\0\\-4\\-60\\-76
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1140\\-9120\\0\\2280\\0\\-1140\\-3420
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-190\\-4\\0\\1\\0\\-38
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-1140\\-60\\-1140\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-760\\-76\\-3420\\-38\\0\\76
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14668177080, -57292980, 3654125220; -57292980, 223784, -14272784; 3654125220, -14272784, 910312921],[1140, 4, 1],[1559, 25986, 1422, 77816, 1064, 3677, 4667; 3, 95, 7, 285, 3, 0, 0; -6258, -104310, -5708, -312360, -4271, -14760, -18734],97,7]
%
% END OF LATTICE L98.7
%
\hbox{}\par\smallskip%
%
% LATTICE L98.8
%
\leavevmode%
\llap{\spacer}%
${L_{98.8}}=\pdualwitharg{19}(\pfillwitharg{2}({L_{98.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${57}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{38}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{19}\above{1pt}{1pt}{r}{2}{570}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2903466570\\-4564845\\-762184905
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4564845\\-6479\\-1198311
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-762184905\\-1198311\\-200080082
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-196075\\-9\\746928
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
526604\\25\\-2006045
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1228601\\58\\-4680232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3829769\\180\\-14589120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1940535\\91\\-7392273
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1939727\\90\\-7389195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20682\\-1\\78786
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57\\0\\-57\\-285\\-171\\-285\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\5\\0\\-120\\-95\\-285\\-10
\end{tabular}\endgroup%
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-57\\0\\38\\0\\-38\\-285\\-19
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-285\\-120\\0\\30\\0\\-285\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171\\-95\\-38\\0\\19\\0\\-19
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-285\\-285\\-285\\-285\\0\\570\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-19\\-45\\-19\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2903466570, -4564845, -762184905; -4564845, -6479, -1198311; -762184905, -1198311, -200080082],[285, 19, 1],[-196075, 526604, 1228601, 3829769, 1940535, 1939727, -20682; -9, 25, 58, 180, 91, 90, -1; 746928, -2006045, -4680232, -14589120, -7392273, -7389195, 78786],97,8]
%
% END OF LATTICE L98.8
%
\hbox{}\par\smallskip%
%
% LATTICE L98.9
%
\leavevmode%
\llap{\spacer}%
${L_{98.9}}=\pdualwitharg{5}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{76}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{114}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{380}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15979380\\-6840\\-14820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6840\\10\\5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14820\\5\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-186\\-990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-190\\-988
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-1311\\-6954
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-560\\-2980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-186\\-993
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-1672\\-8930
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-30\\-570\\-180\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\76\\0\\-912\\-380\\-114\\-760
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-30\\0\\10\\0\\-20\\-15\\-190
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-570\\-912\\0\\114\\0\\-57\\-1710
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-380\\-20\\0\\20\\0\\-380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-114\\-15\\-57\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-760\\-190\\-1710\\-380\\0\\380
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15979380, -6840, -14820; -6840, 10, 5; -14820, 5, 14],[1140, 5, 1],[-1, -1, 0, -7, -3, -1, -9; -186, -190, -1, -1311, -560, -186, -1672; -990, -988, 0, -6954, -2980, -993, -8930],97,9]
%
% END OF LATTICE L98.9
%
\hbox{}\par\smallskip%
%
% LATTICE L98.10
%
\leavevmode%
\llap{\spacer}%
${L_{98.10}}=\pdualwitharg{2.3}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{1140}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}{760}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{228}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18144588840\\-32484300\\4525589100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32484300\\58152\\-8102172
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4525589100\\-8102172\\1128763891
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6353\\4279\\25502
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-146399\\98610\\587670
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
565\\-380\\-2268
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17797\\-11970\\-71440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\-47\\-285
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6776\\4565\\27200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58588\\39463\\235182
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-380\\-18\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1140\\0\\-9120\\-570\\-1140\\-2280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0\\-12\\-60\\-228
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-380\\-9120\\0\\760\\0\\-380\\-3420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-570\\-12\\0\\3\\0\\-114
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-1140\\-60\\-380\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2280\\-228\\-3420\\-114\\0\\228
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18144588840, -32484300, 4525589100; -32484300, 58152, -8102172; 4525589100, -8102172, 1128763891],[1140, 12, 1],[-6353, -146399, 565, 17797, 71, -6776, -58588; 4279, 98610, -380, -11970, -47, 4565, 39463; 25502, 587670, -2268, -71440, -285, 27200, 235182],97,10]
%
% END OF LATTICE L98.10
%
\hbox{}\par\smallskip%
%
% LATTICE L98.11
%
\leavevmode%
\llap{\spacer}%
${L_{98.11}}=\pdualwitharg{3.19}(\pfillwitharg{2}({L_{98.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${19}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{114}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{57}\above{1pt}{1pt}{r}{2}{190}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32371857218910\\42366960\\-10854162474705
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42366960\\57\\-14205483
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10854162474705\\-14205483\\3639360022831
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
414454\\54\\1236083
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3595313\\-440\\-10722795
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8289991\\-1025\\-24724377
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8542911\\-1065\\-25478695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12932058\\-1619\\-38569050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4227613\\-540\\-12608590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143997\\17\\429462
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\0\\-57\\-95\\-171\\-95\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-120\\-285\\-285\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\0\\114\\0\\-114\\-285\\-57
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\-120\\0\\10\\0\\-95\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171\\-285\\-114\\0\\57\\0\\-57
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\-285\\-285\\-95\\0\\190\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-57\\-45\\-57\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[32371857218910, 42366960, -10854162474705; 42366960, 57, -14205483; -10854162474705, -14205483, 3639360022831],[285, 57, 1],[414454, -3595313, -8289991, -8542911, -12932058, -4227613, 143997; 54, -440, -1025, -1065, -1619, -540, 17; 1236083, -10722795, -24724377, -25478695, -38569050, -12608590, 429462],97,11]
%
% END OF LATTICE L98.11
%
\hbox{}\par\smallskip%
%
% LATTICE L98.12
%
\leavevmode%
\llap{\spacer}%
${L_{98.12}}=\pdualwitharg{3.5}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{228}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{b}{2}{38}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{1140}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
286140\\-14820\\1140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14820\\690\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1140\\-45\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-34\\-190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-38\\-228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\33\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\133\\722
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\100\\540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-304\\-1710
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-30\\-190\\-180\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\228\\0\\-912\\-1140\\-114\\-2280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\0\\-60\\-15\\-570
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190\\-912\\0\\38\\0\\-19\\-1710
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-1140\\-60\\0\\60\\0\\-1140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-114\\-15\\-19\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2280\\-570\\-1710\\-1140\\0\\1140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[286140, -14820, 1140; -14820, 690, -45; 1140, -45, 2],[1140, 15, 1],[-1, -1, 1, 4, 3, 0, -9; -34, -38, 33, 133, 100, 0, -304; -190, -228, 180, 722, 540, -1, -1710],97,12]
%
% END OF LATTICE L98.12
%
\hbox{}\par\smallskip%
%
% LATTICE L98.13
%
\leavevmode%
\llap{\spacer}%
${L_{98.13}}=\pdualwitharg{19}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${228}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{38}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{76}\above{1pt}{1pt}{r}{2}{570}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-107224980\\76380\\185820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76380\\-38\\-133
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
185820\\-133\\-322
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-570
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\200\\5110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\198\\5111
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\480\\12495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\412\\10792
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\15\\570
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\-568
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
228\\0\\-114\\-570\\-684\\-570\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-240\\-380\\-570\\-40
\end{tabular}\endgroup%
\kern3pt%
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-114\\0\\38\\0\\-76\\-285\\-38
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-570\\-240\\0\\30\\0\\-285\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-684\\-380\\-76\\0\\76\\0\\-76
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-570\\-570\\-285\\-285\\0\\570\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-38\\-90\\-76\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-107224980, 76380, 185820; 76380, -38, -133; 185820, -133, -322],[1140, 19, 1],[-1, 9, 9, 22, 19, 1, -1; -18, 200, 198, 480, 412, 15, -22; -570, 5110, 5111, 12495, 10792, 570, -568],97,13]
%
% END OF LATTICE L98.13
%
\hbox{}\par\smallskip%
%
% LATTICE L98.14
%
\leavevmode%
\llap{\spacer}%
${L_{98.14}}=\pdualwitharg{2.5}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{76}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{*}{2}{456}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{380}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6389588280\\21385260\\-1591769460
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21385260\\-71560\\5327480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1591769460\\5327480\\-396540419
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4387\\3\\-17610
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4667\\0\\-18734
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1699\\-2\\6820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42145\\0\\169176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3464\\1\\13905
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2260\\3\\9072
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4402\\19\\-17670
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-1140\\-90\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\76\\0\\-1824\\-190\\-228\\-760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\0\\-20\\-60\\-380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1140\\-1824\\0\\456\\0\\-228\\-3420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-190\\-20\\0\\5\\0\\-190
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-228\\-60\\-228\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-760\\-380\\-3420\\-190\\0\\380
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6389588280, 21385260, -1591769460; 21385260, -71560, 5327480; -1591769460, 5327480, -396540419],[1140, 20, 1],[4387, 4667, -1699, -42145, -3464, -2260, 4402; 3, 0, -2, 0, 1, 3, 19; -17610, -18734, 6820, 169176, 13905, 9072, -17670],97,14]
%
% END OF LATTICE L98.14
%
\hbox{}\par\smallskip%
%
% LATTICE L98.15
%
\leavevmode%
\llap{\spacer}%
${L_{98.15}}=\pdualwitharg{5.19}(\pfillwitharg{2}({L_{98.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${285}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{190}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{95}\above{1pt}{1pt}{r}{2}{114}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32507940465285\\-54810345\\13052565812760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54810345\\95\\-22007415
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13052565812760\\-22007415\\5240857213901
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1926241\\5778885\\4797405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1113986\\-3342046\\-2774441
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12843018\\-38530079\\-31986215
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7940908\\-23823363\\-19777251
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20034600\\-60105419\\-49897230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3929701\\-11789427\\-9787128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-223083\\669266\\555600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
285\\0\\-285\\-285\\-855\\-285\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-24\\-95\\-57\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-285\\0\\190\\0\\-190\\-285\\-95
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-285\\-24\\0\\6\\0\\-57\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-855\\-95\\-190\\0\\95\\0\\-95
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-285\\-57\\-285\\-57\\0\\114\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-95\\-45\\-95\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[32507940465285, -54810345, 13052565812760; -54810345, 95, -22007415; 13052565812760, -22007415, 5240857213901],[285, 95, 1],[-1926241, 1113986, 12843018, 7940908, 20034600, 3929701, -223083; 5778885, -3342046, -38530079, -23823363, -60105419, -11789427, 669266; 4797405, -2774441, -31986215, -19777251, -49897230, -9787128, 555600],97,15]
%
% END OF LATTICE L98.15
%
\hbox{}\par\smallskip%
%
% LATTICE L98.16
%
\leavevmode%
\llap{\spacer}%
${L_{98.16}}=\pdualwitharg{3.19}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${76}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{114}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{228}\above{1pt}{1pt}{r}{2}{190}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22037340\\5509620\\-23940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5509620\\-1377462\\5985
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23940\\5985\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-48\\-2774
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-50\\-3240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\9\\228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-35\\-1615
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-26\\-1386
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\0\\-114\\-190\\-684\\-190\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-240\\-1140\\-570\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114\\0\\114\\0\\-228\\-285\\-114
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190\\-240\\0\\10\\0\\-95\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-684\\-1140\\-228\\0\\228\\0\\-228
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190\\-570\\-285\\-95\\0\\190\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-114\\-90\\-228\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-22037340, 5509620, -23940; 5509620, -1377462, 5985; -23940, 5985, -26],[1140, 57, 1],[-9, -9, 2, 1, -1, -7, -5; -48, -50, 9, 5, -4, -35, -26; -2774, -3240, 228, 230, 0, -1615, -1386],97,16]
%
% END OF LATTICE L98.16
%
\hbox{}\par\smallskip%
%
% LATTICE L98.17
%
\leavevmode%
\llap{\spacer}%
${L_{98.17}}=\pdualwitharg{2.3.5}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{228}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{*}{2}{152}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{1140}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
403721880\\-3931860\\100568520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3931860\\38280\\-979440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100568520\\-979440\\25051967
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-147\\-3\\590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2187\\-38\\8778
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-583\\-10\\2340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
833\\0\\-3344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
497\\4\\-1995
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
286\\3\\-1148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2414\\19\\-9690
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-60\\-380\\-90\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\228\\0\\-1824\\-570\\-228\\-2280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\0\\-60\\-60\\-1140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-380\\-1824\\0\\152\\0\\-76\\-3420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-570\\-60\\0\\15\\0\\-570
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-228\\-60\\-76\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2280\\-1140\\-3420\\-570\\0\\1140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[403721880, -3931860, 100568520; -3931860, 38280, -979440; 100568520, -979440, 25051967],[1140, 60, 1],[-147, -2187, -583, 833, 497, 286, 2414; -3, -38, -10, 0, 4, 3, 19; 590, 8778, 2340, -3344, -1995, -1148, -9690],97,17]
%
% END OF LATTICE L98.17
%
\hbox{}\par\smallskip%
%
% LATTICE L98.18
%
\leavevmode%
\llap{\spacer}%
${L_{98.18}}=\pdualwitharg{3.5.19}(\pfillwitharg{2}({L_{98.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${95}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{570}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{285}\above{1pt}{1pt}{r}{2}{38}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-122003331109185\\223281946114740\\-74782691649075
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
223281946114740\\-408634969252530\\136862041174335
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74782691649075\\136862041174335\\-45838510464898
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\121159898\\361752400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-195241459\\-582940953
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\-2277554434\\-6800193765
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-473302966\\-1413161341
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91\\-3597322617\\-10740683280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-239721657\\-715747423
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\38339850\\114472965
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
95\\0\\-285\\-95\\-855\\-95\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-24\\-285\\-57\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-285\\0\\570\\0\\-570\\-285\\-285
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\-24\\0\\2\\0\\-19\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-855\\-285\\-570\\0\\285\\0\\-285
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\-57\\-285\\-19\\0\\38\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-285\\-45\\-285\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-122003331109185, 223281946114740, -74782691649075; 223281946114740, -408634969252530, 136862041174335; -74782691649075, 136862041174335, -45838510464898],[285, 285, 1],[-3, 5, 58, 12, 91, 6, -1; 121159898, -195241459, -2277554434, -473302966, -3597322617, -239721657, 38339850; 361752400, -582940953, -6800193765, -1413161341, -10740683280, -715747423, 114472965],97,18]
%
% END OF LATTICE L98.18
%
\hbox{}\par\smallskip%
%
% LATTICE L98.19
%
\leavevmode%
\llap{\spacer}%
${L_{98.19}}=\pdualwitharg{2.19}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${228}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{152}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{19}\above{1pt}{1pt}{r}{2}{2280}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26019950520\\-67511940\\6482001240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67511940\\175256\\-16818344
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6482001240\\-16818344\\1614774019
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8321\\3\\-33402
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25806\\5\\-103590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43432\\7\\-174344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115346\\15\\-463020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27003\\3\\-108395
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40043\\0\\-160740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
141\\0\\-566
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
228\\0\\-228\\-1140\\-342\\-1140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-480\\-190\\-1140\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-228\\0\\152\\0\\-76\\-1140\\-76
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1140\\-480\\0\\120\\0\\-1140\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-342\\-190\\-76\\0\\19\\0\\-38
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1140\\-1140\\-1140\\-1140\\0\\2280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-76\\-180\\-38\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26019950520, -67511940, 6482001240; -67511940, 175256, -16818344; 6482001240, -16818344, 1614774019],[1140, 76, 1],[8321, 25806, 43432, 115346, 27003, 40043, 141; 3, 5, 7, 15, 3, 0, 0; -33402, -103590, -174344, -463020, -108395, -160740, -566],97,19]
%
% END OF LATTICE L98.19
%
\hbox{}\par\smallskip%
%
% LATTICE L98.20
%
\leavevmode%
\llap{\spacer}%
${L_{98.20}}=\pdualwitharg{5.19}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1140}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{190}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{380}\above{1pt}{1pt}{r}{2}{114}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26451420\\2772480\\-75240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2772480\\290510\\-7885
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75240\\-7885\\214
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-570
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\794
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\58\\3895
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\24\\1587
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\88\\5700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\-684
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-720
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1140\\0\\-570\\-570\\-3420\\-570\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-48\\-380\\-114\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-570\\0\\190\\0\\-380\\-285\\-190
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-570\\-48\\0\\6\\0\\-57\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3420\\-380\\-380\\0\\380\\0\\-380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-570\\-114\\-285\\-57\\0\\114\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-190\\-90\\-380\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26451420, 2772480, -75240; 2772480, 290510, -7885; -75240, -7885, 214],[1140, 95, 1],[-1, 1, 5, 2, 7, -1, -1; -6, 12, 58, 24, 88, -9, -10; -570, 794, 3895, 1587, 5700, -684, -720],97,20]
%
% END OF LATTICE L98.20
%
\hbox{}\par\smallskip%
%
% LATTICE L98.21
%
\leavevmode%
\llap{\spacer}%
${L_{98.21}}=\pdualwitharg{2.3.19}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${76}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{456}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{57}\above{1pt}{1pt}{r}{2}{760}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1516480440\\-6699780\\377766360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6699780\\30552\\-1668960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
377766360\\-1668960\\94104361
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35510\\-621\\142538
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52623\\-920\\211230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13405\\-234\\53808
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1141\\-20\\4580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\1\\-285
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21963\\-385\\88160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14689\\-257\\58962
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\0\\-228\\-380\\-342\\-380\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-480\\-570\\-1140\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-228\\0\\456\\0\\-228\\-1140\\-228
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-380\\-480\\0\\40\\0\\-380\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-342\\-570\\-228\\0\\57\\0\\-114
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-380\\-1140\\-1140\\-380\\0\\760\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-228\\-180\\-114\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1516480440, -6699780, 377766360; -6699780, 30552, -1668960; 377766360, -1668960, 94104361],[1140, 228, 1],[-35510, -52623, -13405, -1141, 71, -21963, -14689; -621, -920, -234, -20, 1, -385, -257; 142538, 211230, 53808, 4580, -285, 88160, 58962],97,21]
%
% END OF LATTICE L98.21
%
\hbox{}\par\smallskip%
%
% LATTICE L98.22
%
\leavevmode%
\llap{\spacer}%
${L_{98.22}}=\pdualwitharg{3.5.19}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${380}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{570}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{1140}\above{1pt}{1pt}{r}{2}{38}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-337247340\\-112361820\\592800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112361820\\-37435890\\197505
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
592800\\197505\\-1042
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-948
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\7\\3602
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93\\172\\85500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\18\\8531
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\20\\8910
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
380\\0\\-570\\-190\\-3420\\-190\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-48\\-1140\\-114\\-120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-570\\0\\570\\0\\-1140\\-285\\-570
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190\\-48\\0\\2\\0\\-19\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3420\\-1140\\-1140\\0\\1140\\0\\-1140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190\\-114\\-285\\-19\\0\\38\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-570\\-90\\-1140\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-337247340, -112361820, 592800; -112361820, -37435890, 197505; 592800, 197505, -1042],[1140, 285, 1],[-1, -1, 1, 4, 93, 9, 9; 2, -2, -3, 7, 172, 18, 20; -190, -948, 0, 3602, 85500, 8531, 8910],97,22]
%
% END OF LATTICE L98.22
%
\hbox{}\par\smallskip%
%
% LATTICE L98.23
%
\leavevmode%
\llap{\spacer}%
${L_{98.23}}=\pdualwitharg{2.5.19}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1140}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{760}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{95}\above{1pt}{1pt}{r}{2}{456}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6849409560\\32228940\\-1682225040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32228940\\151240\\-7915780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1682225040\\-7915780\\413156711
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24565\\73584\\-98610
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2386\\7147\\-9578
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
284\\-853\\1140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8986\\-26919\\36072
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16637\\-49838\\66785
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14881\\-44577\\59736
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-421\\1261\\-1690
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1140\\0\\-1140\\-1140\\-1710\\-1140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-96\\-190\\-228\\-40
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\kern3pt%
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-1140\\0\\760\\0\\-380\\-1140\\-380
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\kern3pt%
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-1140\\-96\\0\\24\\0\\-228\\-180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1710\\-190\\-380\\0\\95\\0\\-190
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1140\\-228\\-1140\\-228\\0\\456\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-380\\-180\\-190\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6849409560, 32228940, -1682225040; 32228940, 151240, -7915780; -1682225040, -7915780, 413156711],[1140, 380, 1],[-24565, -2386, 284, 8986, 16637, 14881, -421; 73584, 7147, -853, -26919, -49838, -44577, 1261; -98610, -9578, 1140, 36072, 66785, 59736, -1690],97,23]
%
% END OF LATTICE L98.23
%
\hbox{}\par\smallskip%
%
% LATTICE L98.24
%
\leavevmode%
\llap{\spacer}%
${L_{98.24}}=\pdualwitharg{2.3.5.19}({L_{98.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${380}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{2280}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{285}\above{1pt}{1pt}{r}{2}{152}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
111720\\-50799540\\-12655140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50799540\\24065785320\\5995246320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12655140\\5995246320\\1493530253
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\13395\\-53770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\423\\-1698
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190\\40043\\-160740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\63449\\-254696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1706\\360884\\-1448655
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-657\\139024\\-558068
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-359\\75998\\-305070
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
380\\0\\-1140\\-380\\-1710\\-380\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-96\\-570\\-228\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1140\\0\\2280\\0\\-1140\\-1140\\-1140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-380\\-96\\0\\8\\0\\-76\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1710\\-570\\-1140\\0\\285\\0\\-570
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-380\\-228\\-1140\\-76\\0\\152\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-1140\\-180\\-570\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[111720, -50799540, -12655140; -50799540, 24065785320, 5995246320; -12655140, 5995246320, 1493530253],[1140, 1140, 1],[-63, -2, -190, -300, -1706, -657, -359; 13395, 423, 40043, 63449, 360884, 139024, 75998; -53770, -1698, -160740, -254696, -1448655, -558068, -305070],97,24]
%
% END OF LATTICE L98.24
%

\medskip%
%------------------------------WEYL-GROUP-W99--------------------------------
%
\leavevmode\llap{}%
$W_{99\phantom{0}}$%
\qquad\llap{12} lattices, $\chi=98$%
\hfill%
$222242262222242262\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L99.1
%
\leavevmode%
\llap{\spacer}%
${L_{99.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}97\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{99}}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{388}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{194}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{b}{2}{582}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57207108\\148992\\147828
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
148992\\-386\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
147828\\-385\\-382
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6104015\\-5307840\\2367031248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16583\\14419\\-6430599
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15778\\13720\\-6118435
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
125\\-106\\48470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3029\\-2522\\1174476
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\-81\\38772
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
622\\-485\\241142
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-15\\8140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-4\\3486
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
85\\0\\32883
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-389
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\390
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-194\\-22\\-72\\-2910\\-106\\-90\\-40\\-1164\\-48\\-388\\-20\\-24\\-582\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\388\\0\\-2716\\-388\\-1552\\-66930\\-2522\\-2328\\-1164\\-38024\\-1746\\-16102\\-970\\-1552\\-45978\\-1358\\-582
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-6\\-42\\-2037\\-81\\-84\\-48\\-1746\\-87\\-873\\-57\\-102\\-3201\\-99\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-194\\-2716\\0\\194\\0\\-194\\-11349\\-485\\-582\\-388\\-16102\\-873\\-9603\\-679\\-1358\\-45105\\-1455\\-873
\end{tabular}\endgroup%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-57207108, 148992, 147828; 148992, -386, -385; 147828, -385, -382],[1164, 1, 1],[125, 3029, 100, 622, 21, 9, 85, -1, 1, 13, 1433, 107, 1609, 140, 359, 13301, 461, 344; -106, -2522, -81, -485, -15, -4, 0, 2, -3, -14, -1358, -99, -1455, -125, -316, -11640, -402, -297; 48470, 1174476, 38772, 241142, 8140, 3486, 32883, -389, 390, 5044, 555810, 41499, 624001, 54293, 139218, 5157975, 178769, 133395],98,1]
%
% END OF LATTICE L99.1
%
\hbox{}\par\smallskip%
%
% LATTICE L99.2
%
\leavevmode%
\llap{\spacer}%
${L_{99.2}}=\pfillwitharg{2}({L_{99.1}})$%
{${}={}$}Nikulin $99$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}97\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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\GramMatrix{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-27354, -9021, 0; -9021, -2975, 0; 0, 0, 1],[291, 1, 1],[0, -32, 1, 545, 77, 153, 13174, 496, 457, 114, 3718, 341, 3141, 189, 151, 8942, 264, 113; 0, 97, -3, -1649, -233, -463, -39867, -1501, -1383, -345, -11252, -1032, -9506, -572, -457, -27063, -799, -342; 1, 0, -3, -97, -11, -18, -1455, -53, -45, -10, -291, -24, -194, -10, -6, -291, -7, 0],98,2]
%
% END OF LATTICE L99.2
%
\hbox{}\par\smallskip%
%
% LATTICE L99.3
%
\leavevmode%
\llap{\spacer}%
${L_{99.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{99.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}97\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{291}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{582}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{4}{3}\above{1pt}{1pt}{r}{2}{194}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10162011, 54417, -3369780; 54417, -285, 18045; -3369780, 18045, -1117438],[291, 3, 1],[24378, 1828816, 83749, 3496867, 291163, 355457, 8605848, 883838, 212337, 108630, 2471776, 49259, 803725, 19189, -769, -4664, 2882, 9343; 31, 2231, 101, 4171, 345, 418, 10088, 1034, 247, 125, 2813, 55, 873, 19, -2, 0, 6, 13; -73515, -5515032, -252556, -10545258, -878040, -1071927, -25952059, -2665329, -640330, -327588, -7453965, -148547, -2423739, -57867, 2319, 14065, -8691, -28175],98,3]
%
% END OF LATTICE L99.3
%
\hbox{}\par\smallskip%
%
% LATTICE L99.4
%
\leavevmode%
\llap{\spacer}%
${L_{99.4}}=\pdualwitharg{3}({L_{99.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}97\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{1164}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{582}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{*}{4}{12}\above{1pt}{1pt}{b}{2}{194}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1164\\-1164\\-1164
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1164\\-1158\\-1161
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1164\\-1161\\-1162
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25609\\15520\\11640
\end{tabular}\endgroup%
\kern3pt%
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-23793\\14419\\10815
\end{tabular}\endgroup%
\kern3pt%
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-24618\\14920\\11189
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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185\\-106\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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4581\\-2522\\-2328
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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52\\-27\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1010\\-485\\-582
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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37\\-15\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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21\\-4\\-18
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\HardButStrongLineBreak\kern3pt%
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93\\0\\-97
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\2\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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0\\-582\\-17\\-873\\-81\\-222\\-2813\\-297\\-77
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1164, -1164, -1164; -1164, -1158, -1161; -1164, -1161, -1162],[1164, 3, 1],[185, 4581, 52, 1010, 37, 21, 93, 1, -1, 13, 1821, 47, 2191, 194, 507, 6309, 659, 166; -106, -2522, -27, -485, -15, -4, 0, 2, -1, -14, -1358, -33, -1455, -125, -316, -3880, -402, -99; -90, -2328, -28, -582, -24, -18, -97, -3, 2, 0, -582, -17, -873, -81, -222, -2813, -297, -77],98,4]
%
% END OF LATTICE L99.4
%
\hbox{}\par\smallskip%
%
% LATTICE L99.5
%
\leavevmode%
\llap{\spacer}%
${L_{99.5}}=\pdualwitharg{2}({L_{99.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}97\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{388}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{776}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{*}{2}{2328}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[33256134168, 448140, -8342901324; 448140, 8, -112424; -8342901324, -112424, 2092967335],[1164, 4, 1],[-181053, -4425467, -295927, -1872663, -65601, -16328, -391294, 0, -289, -16617, -1965863, -297083, -4518667, -395629, -510792, -37956686, -1317800, -988061; 1983, 48500, 3246, 20564, 722, 181, 4365, 1, 0, 180, 21437, 3243, 49373, 4325, 5587, 415257, 14419, 10815; -721706, -17640614, -1179612, -7464732, -261496, -65086, -1559760, 0, -1152, -66238, -7836242, -1184220, -18012124, -1577040, -2036098, -151301376, -5252960, -3938568],98,5]
%
% END OF LATTICE L99.5
%
\hbox{}\par\smallskip%
%
% LATTICE L99.6
%
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${L_{99.6}}=\pdualwitharg{2.3}({L_{99.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6995640, 8148, -1750656; 8148, 24, -2040; -1750656, -2040, 438101],[1164, 12, 1],[7759, 189721, 4231, 80373, 2819, 704, 5630, 0, 1, 707, 84013, 4235, 193361, 16935, 21872, 541838, 56440, 14109; 1983, 48500, 1082, 20564, 722, 181, 1455, 1, 0, 180, 21437, 1081, 49373, 4325, 5587, 138419, 14419, 3605; 31014, 758346, 16912, 321264, 11268, 2814, 22504, 0, 4, 2826, 335814, 16928, 772896, 67692, 87426, 2165816, 225600, 56396],98,6]
%
% END OF LATTICE L99.6
%
\hbox{}\par\smallskip%
%
% LATTICE L99.7
%
\leavevmode%
\llap{\spacer}%
${L_{99.7}}=\pdualwitharg{97}(\pfillwitharg{2}({L_{99.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}97\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-86570463, -874164, -28568634; -874164, -8827, -288478; -28568634, -288478, -9427775],[291, 97, 1],[11327, 8792, 117277, 16843, 136109, 166268, 124531, 413597, 298231, 50902, 11951, 69410, 3900, 9092, -324, -73, 1263, 12914; -12186, -9424, -125583, -18019, -145531, -177665, -133032, -441758, -318387, -54294, -12736, -73857, -4141, -9589, 385, 72, -1442, -14013; -33950, -26353, -351528, -50486, -407982, -498386, -373281, -1239757, -893952, -152581, -35824, -208065, -11691, -27257, 970, 219, -3783, -38703],98,7]
%
% END OF LATTICE L99.7
%
\hbox{}\par\smallskip%
%
% LATTICE L99.8
%
\leavevmode%
\llap{\spacer}%
${L_{99.8}}=\pdualwitharg{3.97}(\pfillwitharg{2}({L_{99.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}97\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${291}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{194}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{582}\above{1pt}{1pt}{}{4}{291}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{582}\above{1pt}{1pt}{}{6}{194}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1816131, -5335194, -1769280; -5335194, 21001470, 6964503; -1769280, 6964503, 2309567],[291, 291, 1],[-2698, -2098, -9333, -4023, -32519, -39737, -9922, -98868, -23769, -12176, -2860, -5541, -935, -2187, 73, 6, -292, -1021; 17123192, 13315329, 59233672, 25532794, 206388758, 252199692, 62972213, 627488315, 150855540, 77277997, 18151734, 35167517, 5934269, 13880683, -463188, -38087, 1852945, 6479775; -51637077, -40154001, -178626373, -76997259, -622390509, -760538976, -189900400, -1892267658, -454923307, -233041239, -54738771, -106051943, -17895513, -41858895, 1396800, 114856, -5587782, -19540553],98,8]
%
% END OF LATTICE L99.8
%
\hbox{}\par\smallskip%
%
% LATTICE L99.9
%
\leavevmode%
\llap{\spacer}%
${L_{99.9}}=\pdualwitharg{97}({L_{99.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}97\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1167492, -75660, 10476; -75660, -4850, 679; 10476, 679, -94],[1164, 97, 1],[1, 5, 41, 7, 62, 167, 65, 221, 170, 65, 17, 58, 4, 15, 9, 1, -1, -5; -14, -14, -99, -15, -125, -316, -120, -402, -297, -106, -26, -81, -5, -15, -4, 0, 2, -3; 0, 446, 3783, 661, 5917, 16102, 6291, 21437, 16587, 6402, 1688, 5820, 406, 1552, 970, 111, -97, -582],98,9]
%
% END OF LATTICE L99.9
%
\hbox{}\par\smallskip%
%
% LATTICE L99.10
%
\leavevmode%
\llap{\spacer}%
${L_{99.10}}=\pdualwitharg{3.97}({L_{99.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}97\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1164}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{194}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{582}\above{1pt}{1pt}{*}{4}{1164}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{582}\above{1pt}{1pt}{}{6}{194}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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191351\\-182240\\124624824
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-88055436, -26257512, 96612; -26257512, -7829646, 28809; 96612, 28809, -106],[1164, 291, 1],[125, 33, 38, 8, 31, 21, 1, 1, -3, 1, 9, 25, 13, 116, 315, 41, 419, 108; -106, -26, -27, -5, -15, -4, 0, 2, -1, -14, -14, -33, -15, -125, -316, -40, -402, -99; 84972, 22974, 27257, 5925, 24153, 18042, 911, 1455, -3007, -2910, 4380, 13774, 7752, 71586, 200790, 26443, 272085, 71392],98,10]
%
% END OF LATTICE L99.10
%
\hbox{}\par\smallskip%
%
% LATTICE L99.11
%
\leavevmode%
\llap{\spacer}%
${L_{99.11}}=\pdualwitharg{2.97}({L_{99.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}97\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${388}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{2328}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{776}\above{1pt}{1pt}{*}{4}{388}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{776}\above{1pt}{1pt}{}{6}{2328}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
172353480\\-105924\\43321752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105924\\776\\-27160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43321752\\-27160\\10889503
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-110135419\\330740253\\438976992
\end{tabular}\endgroup%
\kern3pt%
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732040\\-2198341\\-2917760
\end{tabular}\endgroup%
\kern3pt%
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-28183540\\84636090\\112333759
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13093\\-39319\\-52186
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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12811\\-38472\\-51062
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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180479\\-541986\\-719352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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27311\\-82016\\-108856
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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227497\\-683182\\-906756
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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287608\\-863697\\-1146346
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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218486\\-656121\\-870840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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732040\\-2198341\\-2917760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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541145\\-1625076\\-2156892
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[172353480, -105924, 43321752; -105924, 776, -27160; 43321752, -27160, 10889503],[1164, 388, 1],[13093, 12811, 180479, 27311, 227497, 287608, 218486, 732040, 541145, 96713, 23791, 148939, 9291, 28717, 5208, 1126, 0, 7885; -39319, -38472, -541986, -82016, -683182, -863697, -656121, -2198341, -1625076, -290432, -71445, -447267, -27901, -86237, -15639, -3381, 1, -23679; -52186, -51062, -719352, -108856, -906756, -1146346, -870840, -2917760, -2156892, -385478, -94826, -593640, -37032, -114460, -20758, -4488, 0, -31428],98,11]
%
% END OF LATTICE L99.11
%
\hbox{}\par\smallskip%
%
% LATTICE L99.12
%
\leavevmode%
\llap{\spacer}%
${L_{99.12}}=\pdualwitharg{2.3.97}({L_{99.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}97\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1164}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{776}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{2328}\above{1pt}{1pt}{*}{4}{1164}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{2328}\above{1pt}{1pt}{}{6}{776}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2328\\-1688964\\423696
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1688964\\13925199720\\-3493314156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
423696\\-3493314156\\876342461
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-54581\\-13245320\\-52799040
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\kern3pt%
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-37095297\\-9002181739\\-35884867536
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9305890\\2258327060\\9002236319
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2328, -1688964, 423696; -1688964, 13925199720, -3493314156; 423696, -3493314156, 876342461],[1164, 1164, 1],[-7368, -1837, -3911, -751, -2455, -545, -43, 1, -91, -819, -892, -4276, -1978, -16648, -21287, -5415, -54581, -13554; -1788089, -445823, -949209, -182283, -595981, -132424, -10466, 0, -22095, -198709, -216443, -1037589, -479983, -4039881, -5165704, -1314066, -13245320, -3289235; -7127754, -1777158, -3783776, -726624, -2375724, -527874, -41720, 0, -88076, -792102, -862794, -4136080, -1913328, -16103940, -20591742, -5238184, -52799040, -13111684],98,12]
%
% END OF LATTICE L99.12
%

\medskip%
%------------------------------WEYL-GROUP-W100--------------------------------
%
\leavevmode\llap{}%
$W_{100}$%
\qquad\llap{32} lattices, $\chi=24$%
\hfill%
$22222222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L100.1
%
\leavevmode%
\llap{\spacer}%
${L_{100.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{100}}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{264}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
703560\\-141240\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-141240\\28354\\265
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\265\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-299\\-1488\\-176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-641\\-3190\\-385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\264\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-209\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-213\\-1060\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-683\\-3399\\-396
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-8\\-1848\\-160\\-88\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\110\\0\\-3960\\-385\\-275\\-220\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\2\\0\\-5\\-11\\-20\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-3960\\0\\264\\0\\-264\\-1320\\-4224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-385\\-5\\0\\10\\0\\-80\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-275\\-11\\-264\\0\\22\\0\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-220\\-20\\-1320\\-80\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-33\\-4224\\-330\\-132\\0\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[703560, -141240, -1320; -141240, 28354, 265; -1320, 265, 2],[1320, 1, 1],[-299, -641, 0, 53, 1, -42, -213, -683; -1488, -3190, 0, 264, 5, -209, -1060, -3399; -176, -385, -1, 0, 0, -22, -120, -396],99,1]
%
% END OF LATTICE L100.1
%
\hbox{}\par\smallskip%
%
% LATTICE L100.2
%
\leavevmode%
\llap{\spacer}%
${L_{100.2}}=\pfillwitharg{2}({L_{100.1}})$%
{${}={}$}Nikulin $100$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{66}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37290\\18480\\-2970
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18480\\9158\\-1485
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2970\\-1485\\-514
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1558\\-3135\\55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\161\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40441\\81378\\-1452
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7668\\15430\\-275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5226\\10516\\-187
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1963\\3950\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2788\\5610\\-99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-4\\-462\\-80\\-44\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\110\\0\\-1980\\-385\\-275\\-110\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\2\\0\\-5\\-11\\-10\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\-1980\\0\\66\\0\\-132\\-330\\-2112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-385\\-5\\0\\10\\0\\-40\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-275\\-11\\-132\\0\\22\\0\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-110\\-10\\-330\\-40\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-33\\-2112\\-330\\-132\\0\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[37290, 18480, -2970; 18480, 9158, -1485; -2970, -1485, -514],[330, 1, 1],[-1, 1558, -80, -40441, -7668, -5226, -1963, -2788; 2, -3135, 161, 81378, 15430, 10516, 3950, 5610; 0, 55, -3, -1452, -275, -187, -70, -99],99,2]
%
% END OF LATTICE L100.2
%
\hbox{}\par\smallskip%
%
% LATTICE L100.3
%
\leavevmode%
\llap{\spacer}%
${L_{100.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{220}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{44}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{132}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
417120\\39270\\207900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39270\\3512\\19570
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
207900\\19570\\103621
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4466\\-141\\8987
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38319\\-1210\\77110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1066\\33\\-2145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
164\\5\\-330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2438\\-77\\4906
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3168\\-100\\6375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40768\\-1287\\82038
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\-231\\-80\\-44\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\220\\0\\-1980\\-770\\-550\\-110\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\4\\0\\-10\\-22\\-10\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-1980\\0\\33\\0\\-132\\-165\\-2112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-770\\-10\\0\\20\\0\\-40\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-550\\-22\\-132\\0\\44\\0\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-110\\-10\\-165\\-40\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-330\\-66\\-2112\\-660\\-264\\0\\132
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[417120, 39270, 207900; 39270, 3512, 19570; 207900, 19570, 103621],[330, 2, 1],[-4466, -38319, 1, 1066, 164, -2438, -3168, -40768; -141, -1210, 0, 33, 5, -77, -100, -1287; 8987, 77110, -2, -2145, -330, 4906, 6375, 82038],99,3]
%
% END OF LATTICE L100.3
%
\hbox{}\par\smallskip%
%
% LATTICE L100.4
%
\leavevmode%
\llap{\spacer}%
${L_{100.4}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{22}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{l}{2}{30}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
426030\\48180\\-142560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48180\\5268\\-16125
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-142560\\-16125\\47704
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6017\\-282\\17886
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25811\\-1210\\76725
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-481\\22\\-1430
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111\\5\\-330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1643\\-77\\4884
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4269\\-200\\12690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9155\\-429\\27214
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-12\\-462\\-240\\-132\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\330\\0\\-1980\\-1155\\-825\\-330\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\6\\0\\-15\\-33\\-30\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\-1980\\0\\22\\0\\-132\\-330\\-704
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1155\\-15\\0\\30\\0\\-120\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-825\\-33\\-132\\0\\66\\0\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-330\\-30\\-330\\-120\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-33\\-704\\-330\\-132\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[426030, 48180, -142560; 48180, 5268, -16125; -142560, -16125, 47704],[330, 3, 1],[6017, 25811, -1, -481, -111, 1643, 4269, 9155; -282, -1210, 0, 22, 5, -77, -200, -429; 17886, 76725, -3, -1430, -330, 4884, 12690, 27214],99,4]
%
% END OF LATTICE L100.4
%
\hbox{}\par\smallskip%
%
% LATTICE L100.5
%
\leavevmode%
\llap{\spacer}%
${L_{100.5}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{330}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
487410\\66000\\97020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66000\\8780\\13135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
97020\\13135\\19312
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3487\\-282\\17710
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2991\\-242\\15191
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
845\\66\\-4290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\1\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-953\\-77\\4840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-495\\-40\\2514
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15919\\-1287\\80850
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-20\\-2310\\-80\\-220\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\0\\-1980\\-77\\-275\\-22\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\10\\0\\-5\\-55\\-10\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-1980\\0\\330\\0\\-660\\-330\\-10560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-77\\-5\\0\\2\\0\\-8\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-275\\-55\\-660\\0\\110\\0\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-22\\-10\\-330\\-8\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-165\\-10560\\-330\\-660\\0\\330
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[487410, 66000, 97020; 66000, 8780, 13135; 97020, 13135, 19312],[330, 5, 1],[-3487, -2991, 1, 845, 13, -953, -495, -15919; -282, -242, 0, 66, 1, -77, -40, -1287; 17710, 15191, -5, -4290, -66, 4840, 2514, 80850],99,5]
%
% END OF LATTICE L100.5
%
\hbox{}\par\smallskip%
%
% LATTICE L100.6
%
\leavevmode%
\llap{\spacer}%
${L_{100.6}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{660}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{s}{2}{132}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{44}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
453447060\\390390\\222524610
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
390390\\336\\191580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
222524610\\191580\\109201727
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-623\\289\\1269
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7614\\3025\\15510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-268\\65\\546
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\11\\55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
589\\-130\\-1200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
907\\-132\\-1848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\95\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1253\\726\\2552
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-231\\-240\\-132\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\660\\0\\-1980\\-2310\\-1650\\-330\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-30\\-66\\-30\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-1980\\0\\11\\0\\-132\\-165\\-704
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-2310\\-30\\0\\60\\0\\-120\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-1650\\-66\\-132\\0\\132\\0\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-330\\-30\\-165\\-120\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-330\\-66\\-704\\-660\\-264\\0\\44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[453447060, 390390, 222524610; 390390, 336, 191580; 222524610, 191580, 109201727],[330, 6, 1],[-623, -7614, -268, -27, 589, 907, 22, -1253; 289, 3025, 65, 11, -130, -132, 95, 726; 1269, 15510, 546, 55, -1200, -1848, -45, 2552],99,6]
%
% END OF LATTICE L100.6
%
\hbox{}\par\smallskip%
%
% LATTICE L100.7
%
\leavevmode%
\llap{\spacer}%
${L_{100.7}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{44}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{220}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{660}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4907537580\\1658250\\2443099890
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1658250\\560\\825520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2443099890\\825520\\1216238689
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4717\\289\\9475
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10810\\605\\21714
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1608\\65\\3230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-575\\33\\1155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
689\\-26\\-1384
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4819\\-132\\-9680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116\\19\\233
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31543\\2178\\63360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-20\\-1155\\-80\\-220\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\44\\0\\-1980\\-154\\-550\\-22\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0\\-10\\-110\\-10\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-1980\\0\\165\\0\\-660\\-165\\-10560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-154\\-10\\0\\4\\0\\-8\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-550\\-110\\-660\\0\\220\\0\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-22\\-10\\-165\\-8\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-330\\-330\\-10560\\-660\\-1320\\0\\660
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4907537580, 1658250, 2443099890; 1658250, 560, 825520; 2443099890, 825520, 1216238689],[330, 10, 1],[-4717, -10810, -1608, -575, 689, 4819, -116, -31543; 289, 605, 65, 33, -26, -132, 19, 2178; 9475, 21714, 3230, 1155, -1384, -9680, 233, 63360],99,7]
%
% END OF LATTICE L100.7
%
\hbox{}\par\smallskip%
%
% LATTICE L100.8
%
\leavevmode%
\llap{\spacer}%
${L_{100.8}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{110}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
734910\\119460\\-267630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
119460\\19316\\-43505
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-267630\\-43505\\97462
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6303\\-282\\17182
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2456\\-110\\6695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-143\\6\\-390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\5\\-330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
157\\-7\\428
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4479\\-200\\12210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2617\\-117\\7134
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\0\\-44\\-462\\-880\\-44\\-110\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-180\\-385\\-25\\-110\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\0\\22\\0\\-55\\-11\\-110\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\-180\\0\\6\\0\\-12\\-330\\-192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-880\\-385\\-55\\0\\110\\0\\-440\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-25\\-11\\-12\\0\\2\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\-110\\-110\\-330\\-440\\0\\110\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-33\\-192\\-330\\-12\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[734910, 119460, -267630; 119460, 19316, -43505; -267630, -43505, 97462],[330, 11, 1],[6303, 2456, -4, -143, -121, 157, 4479, 2617; -282, -110, 0, 6, 5, -7, -200, -117; 17182, 6695, -11, -390, -330, 428, 12210, 7134],99,8]
%
% END OF LATTICE L100.8
%
\hbox{}\par\smallskip%
%
% LATTICE L100.9
%
\leavevmode%
\llap{\spacer}%
${L_{100.9}}=\pdualwitharg{3}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{88}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
147865237080\\-844231080\\8193240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-844231080\\4820106\\-46779
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8193240\\-46779\\454
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
787\\137880\\3984
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1726\\302390\\8745
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\876\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\176\\88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-175\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\16819\\495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
533\\93380\\2700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
589\\103191\\2981
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-1848\\-480\\-264\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\330\\0\\-3960\\-1155\\-825\\-660\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\6\\0\\-15\\-33\\-60\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-3960\\0\\88\\0\\-264\\-1320\\-1408
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-1155\\-15\\0\\30\\0\\-240\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-825\\-33\\-264\\0\\66\\0\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-660\\-60\\-1320\\-240\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-33\\-1408\\-330\\-132\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[147865237080, -844231080, 8193240; -844231080, 4820106, -46779; 8193240, -46779, 454],[1320, 3, 1],[787, 1726, 5, 1, -1, 96, 533, 589; 137880, 302390, 876, 176, -175, 16819, 93380, 103191; 3984, 8745, 27, 88, 15, 495, 2700, 2981],99,9]
%
% END OF LATTICE L100.9
%
\hbox{}\par\smallskip%
%
% LATTICE L100.10
%
\leavevmode%
\llap{\spacer}%
${L_{100.10}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{110}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{330}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4261540470\\1892550\\-1428631050
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1892550\\840\\-634455
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1428631050\\-634455\\478931666
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7261\\578\\21660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8319\\605\\24816
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1237\\65\\3690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
295\\22\\880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-530\\-26\\-1581
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3706\\-132\\-11055
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
179\\38\\534
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8094\\726\\24145
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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30\\0\\-60\\-2310\\-240\\-660\\-30\\0
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0\\66\\0\\-1980\\-231\\-825\\-66\\-165
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-240\\-231\\-15\\0\\6\\0\\-24\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-825\\-165\\-660\\0\\330\\0\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-66\\-30\\-330\\-24\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-165\\-3520\\-330\\-660\\0\\110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4261540470, 1892550, -1428631050; 1892550, 840, -634455; -1428631050, -634455, 478931666],[330, 15, 1],[7261, 8319, 1237, 295, -530, -3706, 179, 8094; 578, 605, 65, 22, -26, -132, 38, 726; 21660, 24816, 3690, 880, -1581, -11055, 534, 24145],99,10]
%
% END OF LATTICE L100.10
%
\hbox{}\par\smallskip%
%
% LATTICE L100.11
%
\leavevmode%
\llap{\spacer}%
${L_{100.11}}=\pdualwitharg{5}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{1320}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}{330}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6305640\\5280\\-19800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5280\\-10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19800\\-15\\62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\176\\1640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\176\\1639
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\70\\655
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\3432\\33000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\33\\326
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-77\\-660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-36\\-328
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\33\\330
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-40\\-9240\\-160\\-440\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\0\\-3960\\-77\\-275\\-44\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\10\\0\\-5\\-55\\-20\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9240\\-3960\\0\\1320\\0\\-1320\\-1320\\-21120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-77\\-5\\0\\2\\0\\-16\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\-275\\-55\\-1320\\0\\110\\0\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-44\\-20\\-1320\\-16\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-165\\-21120\\-330\\-660\\0\\330
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6305640, 5280, -19800; 5280, -10, -15; -19800, -15, 62],[1320, 5, 1],[5, 5, 2, 101, 1, -2, -1, 1; 176, 176, 70, 3432, 33, -77, -36, 33; 1640, 1639, 655, 33000, 326, -660, -328, 330],99,11]
%
% END OF LATTICE L100.11
%
\hbox{}\par\smallskip%
%
% LATTICE L100.12
%
\leavevmode%
\llap{\spacer}%
${L_{100.12}}=\pdualwitharg{2.11}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{44}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{220}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80119380\\314490\\39148890
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
314490\\1232\\153670
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39148890\\153670\\19129399
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2092\\289\\4279
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2068\\275\\4230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-570\\65\\1166
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\3\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1183\\-130\\-2420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
131\\-12\\-268
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-511\\95\\1045
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1367\\198\\2796
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-44\\-231\\-880\\-44\\-55\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-180\\-770\\-50\\-110\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\0\\44\\0\\-110\\-22\\-110\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-180\\0\\3\\0\\-12\\-165\\-192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-880\\-770\\-110\\0\\220\\0\\-440\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-50\\-22\\-12\\0\\4\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-110\\-110\\-165\\-440\\0\\55\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-66\\-192\\-660\\-24\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[80119380, 314490, 39148890; 314490, 1232, 153670; 39148890, 153670, 19129399],[330, 22, 1],[-2092, -2068, -570, -22, 1183, 131, -511, -1367; 289, 275, 65, 3, -130, -12, 95, 198; 4279, 4230, 1166, 45, -2420, -268, 1045, 2796],99,12]
%
% END OF LATTICE L100.12
%
\hbox{}\par\smallskip%
%
% LATTICE L100.13
%
\leavevmode%
\llap{\spacer}%
${L_{100.13}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{132}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{660}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{220}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2450635440\\-11462550\\1202782680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11462550\\52680\\-5625870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1202782680\\-5625870\\590331043
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
751762\\-141\\-1531695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1289665\\-242\\-2627658
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-427\\0\\870
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60710\\11\\123695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5604\\1\\11418
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
410906\\-77\\-837210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
106716\\-20\\-217431
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2287976\\-429\\-4661690
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-60\\-1155\\-240\\-660\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\132\\0\\-1980\\-462\\-1650\\-66\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\60\\0\\-30\\-330\\-30\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-1980\\0\\55\\0\\-660\\-165\\-3520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-462\\-30\\0\\12\\0\\-24\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-1650\\-330\\-660\\0\\660\\0\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-66\\-30\\-165\\-24\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-330\\-330\\-3520\\-660\\-1320\\0\\220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2450635440, -11462550, 1202782680; -11462550, 52680, -5625870; 1202782680, -5625870, 590331043],[330, 30, 1],[751762, 1289665, -427, -60710, -5604, 410906, 106716, 2287976; -141, -242, 0, 11, 1, -77, -20, -429; -1531695, -2627658, 870, 123695, 11418, -837210, -217431, -4661690],99,13]
%
% END OF LATTICE L100.13
%
\hbox{}\par\smallskip%
%
% LATTICE L100.14
%
\leavevmode%
\llap{\spacer}%
${L_{100.14}}=\pdualwitharg{2}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{880}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}{176}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{528}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-984720\\-6600\\6600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6600\\16\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6600\\16\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-136\\-287
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-1265\\-2640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\131\\280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\264\\561
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-130\\-280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-660\\-1408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-400\\-850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-3366\\-7128
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-8\\-231\\-160\\-88\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\880\\0\\-3960\\-3080\\-2200\\-440\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\16\\0\\-40\\-88\\-40\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-3960\\0\\33\\0\\-264\\-330\\-4224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-3080\\-40\\0\\80\\0\\-160\\-2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-2200\\-88\\-264\\0\\176\\0\\-1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-440\\-40\\-330\\-160\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1320\\-264\\-4224\\-2640\\-1056\\0\\528
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-984720, -6600, 6600; -6600, 16, 16; 6600, 16, -31],[1320, 8, 1],[-1, -9, 1, 2, -1, -5, -3, -25; -136, -1265, 131, 264, -130, -660, -400, -3366; -287, -2640, 280, 561, -280, -1408, -850, -7128],99,14]
%
% END OF LATTICE L100.14
%
\hbox{}\par\smallskip%
%
% LATTICE L100.15
%
\leavevmode%
\llap{\spacer}%
${L_{100.15}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{330}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7540170\\-118140\\-2285250
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-118140\\1848\\35805
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2285250\\35805\\692606
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-851\\578\\-2838
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-423\\275\\-1410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119\\65\\-396
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
248\\-130\\825
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-12\\93
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-197\\190\\-660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\66\\-307
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\0\\-132\\-462\\-2640\\-132\\-330\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-180\\-1155\\-75\\-330\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\66\\0\\-165\\-33\\-330\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\-180\\0\\2\\0\\-12\\-330\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2640\\-1155\\-165\\0\\330\\0\\-1320\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-75\\-33\\-12\\0\\6\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-330\\-330\\-330\\-1320\\0\\330\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-33\\-64\\-330\\-12\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7540170, -118140, -2285250; -118140, 1848, 35805; -2285250, 35805, 692606],[330, 33, 1],[-851, -423, -119, -3, 248, 28, -197, -92; 578, 275, 65, 2, -130, -12, 190, 66; -2838, -1410, -396, -10, 825, 93, -660, -307],99,15]
%
% END OF LATTICE L100.15
%
\hbox{}\par\smallskip%
%
% LATTICE L100.16
%
\leavevmode%
\llap{\spacer}%
${L_{100.16}}=\pdualwitharg{11}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${88}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{440}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9240\\-3960\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3960\\1694\\539
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\539\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
221\\536\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\165\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\85\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79\\192\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-17\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\-100\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\63\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\0\\-88\\-1848\\-1760\\-88\\-440\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-360\\-385\\-25\\-220\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\0\\22\\0\\-55\\-11\\-220\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-360\\0\\24\\0\\-24\\-1320\\-384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1760\\-385\\-55\\0\\110\\0\\-880\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-25\\-11\\-24\\0\\2\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\-220\\-220\\-1320\\-880\\0\\440\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-33\\-384\\-330\\-12\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9240, -3960, -1320; -3960, 1694, 539; -1320, 539, -38],[1320, 11, 1],[221, 68, 35, 79, 2, -7, -41, 26; 536, 165, 85, 192, 5, -17, -100, 63; -88, -25, -11, -24, 0, 2, 0, -12],99,16]
%
% END OF LATTICE L100.16
%
\hbox{}\par\smallskip%
%
% LATTICE L100.17
%
\leavevmode%
\llap{\spacer}%
${L_{100.17}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{l}{2}{30}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{22}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
313401990\\-983400\\63819690
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-983400\\3080\\-200255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63819690\\-200255\\12995938
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5221\\578\\-25630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
517\\55\\-2538
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
717\\65\\-3520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\6\\-270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-298\\-26\\1463
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-166\\-12\\815
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
251\\38\\-1232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1702\\198\\-8355
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\0\\-220\\-2310\\-880\\-220\\-110\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-180\\-77\\-25\\-22\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\0\\110\\0\\-55\\-55\\-110\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-180\\0\\30\\0\\-60\\-330\\-960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-880\\-77\\-55\\0\\22\\0\\-88\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-25\\-55\\-60\\0\\10\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\-22\\-110\\-330\\-88\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-165\\-960\\-330\\-60\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[313401990, -983400, 63819690; -983400, 3080, -200255; 63819690, -200255, 12995938],[330, 55, 1],[5221, 517, 717, 55, -298, -166, 251, 1702; 578, 55, 65, 6, -26, -12, 38, 198; -25630, -2538, -3520, -270, 1463, 815, -1232, -8355],99,17]
%
% END OF LATTICE L100.17
%
\hbox{}\par\smallskip%
%
% LATTICE L100.18
%
\leavevmode%
\llap{\spacer}%
${L_{100.18}}=\pdualwitharg{3.5}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{440}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{330}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2234760\\-69960\\15840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69960\\-2190\\495
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15840\\495\\-106
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-32\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\231\\33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-67\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
651\\-21560\\-3520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-1225\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
126\\-4169\\-660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-628\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-759\\-110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\-120\\-9240\\-480\\-1320\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\66\\0\\-3960\\-231\\-825\\-132\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\30\\0\\-15\\-165\\-60\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9240\\-3960\\0\\440\\0\\-1320\\-1320\\-7040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-231\\-15\\0\\6\\0\\-48\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\-825\\-165\\-1320\\0\\330\\0\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-132\\-60\\-1320\\-48\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-165\\-7040\\-330\\-660\\0\\110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2234760, -69960, 15840; -69960, -2190, 495; 15840, 495, -106],[1320, 15, 1],[1, -7, 2, 651, 37, 126, 19, 23; -32, 231, -67, -21560, -1225, -4169, -628, -759; 0, 33, -15, -3520, -198, -660, -96, -110],99,18]
%
% END OF LATTICE L100.18
%
\hbox{}\par\smallskip%
%
% LATTICE L100.19
%
\leavevmode%
\llap{\spacer}%
${L_{100.19}}=\pdualwitharg{2.3.11}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{s}{2}{132}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{660}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18155940\\1454970\\8940030
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1454970\\115896\\716430
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8940030\\716430\\4402093
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55220\\-141\\112167
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43037\\-110\\87420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\0\\-132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
416\\1\\-845
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2112\\5\\-4290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2750\\-7\\5586
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39234\\-100\\79695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15284\\-39\\31046
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\0\\-132\\-231\\-2640\\-132\\-165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-180\\-2310\\-150\\-330\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\132\\0\\-330\\-66\\-330\\-66
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-180\\0\\1\\0\\-12\\-165\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2640\\-2310\\-330\\0\\660\\0\\-1320\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-150\\-66\\-12\\0\\12\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-330\\-330\\-165\\-1320\\0\\165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-66\\-64\\-660\\-24\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18155940, 1454970, 8940030; 1454970, 115896, 716430; 8940030, 716430, 4402093],[330, 66, 1],[-55220, -43037, 65, 416, 2112, -2750, -39234, -15284; -141, -110, 0, 1, 5, -7, -100, -39; 112167, 87420, -132, -845, -4290, 5586, 79695, 31046],99,19]
%
% END OF LATTICE L100.19
%
\hbox{}\par\smallskip%
%
% LATTICE L100.20
%
\leavevmode%
\llap{\spacer}%
${L_{100.20}}=\pdualwitharg{2.3}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{2640}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{*}{2}{528}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{2}{176}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-393360\\-3960\\3960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3960\\48\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3960\\24\\-37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-180\\-981
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-129\\-2915\\-15840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\21\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\22\\121
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-286\\-1584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-290\\-1590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-1144\\-6248
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-24\\-231\\-480\\-264\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2640\\0\\-3960\\-9240\\-6600\\-1320\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\48\\0\\-120\\-264\\-120\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-3960\\0\\11\\0\\-264\\-330\\-1408
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-9240\\-120\\0\\240\\0\\-480\\-2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-6600\\-264\\-264\\0\\528\\0\\-1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-1320\\-120\\-330\\-480\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1320\\-264\\-1408\\-2640\\-1056\\0\\176
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-393360, -3960, 3960; -3960, 48, 24; 3960, 24, -37],[1320, 24, 1],[-8, -129, 1, 1, -1, -13, -13, -51; -180, -2915, 21, 22, -20, -286, -290, -1144; -981, -15840, 120, 121, -120, -1584, -1590, -6248],99,20]
%
% END OF LATTICE L100.20
%
\hbox{}\par\smallskip%
%
% LATTICE L100.21
%
\leavevmode%
\llap{\spacer}%
${L_{100.21}}=\pdualwitharg{2.5.11}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${55}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{220}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{44}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10680665820\\45544950\\5317335870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45544950\\193160\\22674410
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5317335870\\22674410\\2647218931
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1656886\\-141\\3328105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-258249\\-22\\518732
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2081\\0\\-4180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37570\\3\\-75465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12716\\1\\-25542
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82538\\-7\\165790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-235476\\-20\\472989
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1375864\\-117\\2763630
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\0\\-220\\-1155\\-880\\-220\\-55\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-180\\-154\\-50\\-22\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\0\\220\\0\\-110\\-110\\-110\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-180\\0\\15\\0\\-60\\-165\\-960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-880\\-154\\-110\\0\\44\\0\\-88\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-50\\-110\\-60\\0\\20\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-22\\-110\\-165\\-88\\0\\11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-330\\-960\\-660\\-120\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10680665820, 45544950, 5317335870; 45544950, 193160, 22674410; 5317335870, 22674410, 2647218931],[330, 110, 1],[-1656886, -258249, 2081, 37570, 12716, -82538, -235476, -1375864; -141, -22, 0, 3, 1, -7, -20, -117; 3328105, 518732, -4180, -75465, -25542, 165790, 472989, 2763630],99,21]
%
% END OF LATTICE L100.21
%
\hbox{}\par\smallskip%
%
% LATTICE L100.22
%
\leavevmode%
\llap{\spacer}%
${L_{100.22}}=\pdualwitharg{3.11}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${264}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{1320}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49564680\\-9942240\\-55440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9942240\\1994322\\11121
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55440\\11121\\62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\184\\1848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\65\\855
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\41\\693
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\32\\520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-174
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-100\\-2640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
264\\0\\-264\\-1848\\-5280\\-264\\-1320\\0
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0\\30\\0\\-360\\-1155\\-75\\-660\\-15
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\kern3pt%
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-264\\0\\66\\0\\-165\\-33\\-660\\-33
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5280\\-1155\\-165\\0\\330\\0\\-2640\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-75\\-33\\-24\\0\\6\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\-660\\-660\\-1320\\-2640\\0\\1320\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-33\\-128\\-330\\-12\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[49564680, -9942240, -55440; -9942240, 1994322, 11121; -55440, 11121, 62],[1320, 33, 1],[39, 14, 9, 7, 1, -2, -23, 1; 184, 65, 41, 32, 5, -9, -100, 5; 1848, 855, 693, 520, 0, -174, -2640, -4],99,22]
%
% END OF LATTICE L100.22
%
\hbox{}\par\smallskip%
%
% LATTICE L100.23
%
\leavevmode%
\llap{\spacer}%
${L_{100.23}}=\pdualwitharg{2.5}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{176}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{880}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{2640}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2640\\-1320\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\-6160\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\360\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\4\\75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\33\\616
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\13\\240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\66\\1155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\4\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-22\\-440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-40\\-1155\\-160\\-440\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\176\\0\\-3960\\-616\\-2200\\-88\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\80\\0\\-40\\-440\\-40\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-3960\\0\\165\\0\\-1320\\-330\\-21120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-616\\-40\\0\\16\\0\\-32\\-2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\-2200\\-440\\-1320\\0\\880\\0\\-5280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-88\\-40\\-330\\-32\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1320\\-1320\\-21120\\-2640\\-5280\\0\\2640
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2640, -1320, 0; -1320, -6160, 360; 0, 360, -19],[1320, 40, 1],[2, 17, 7, 41, 3, -9, -1, -1; 4, 33, 13, 66, 4, -22, -2, 0; 75, 616, 240, 1155, 64, -440, -38, 0],99,23]
%
% END OF LATTICE L100.23
%
\hbox{}\par\smallskip%
%
% LATTICE L100.24
%
\leavevmode%
\llap{\spacer}%
${L_{100.24}}=\pdualwitharg{3.5.11}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${330}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{330}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{66}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9123303090\\-51557220\\3092826660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51557220\\289740\\-17476965
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3092826660\\-17476965\\1048476634
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2554123\\5107964\\7619370
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-199048\\398074\\593793
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1604\\-3208\\-4785
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19305\\-38608\\-57590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9801\\-19601\\-29238
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63617\\127227\\189780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-362991\\725942\\1082862
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-353487\\706935\\1054510
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
330\\0\\-660\\-2310\\-2640\\-660\\-330\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\6\\0\\-180\\-231\\-75\\-66\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\0\\330\\0\\-165\\-165\\-330\\-165
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-180\\0\\10\\0\\-60\\-330\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2640\\-231\\-165\\0\\66\\0\\-264\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-75\\-165\\-60\\0\\30\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-66\\-330\\-330\\-264\\0\\66\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-165\\-320\\-330\\-60\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9123303090, -51557220, 3092826660; -51557220, 289740, -17476965; 3092826660, -17476965, 1048476634],[330, 165, 1],[-2554123, -199048, 1604, 19305, 9801, -63617, -362991, -353487; 5107964, 398074, -3208, -38608, -19601, 127227, 725942, 706935; 7619370, 593793, -4785, -57590, -29238, 189780, 1082862, 1054510],99,24]
%
% END OF LATTICE L100.24
%
\hbox{}\par\smallskip%
%
% LATTICE L100.25
%
\leavevmode%
\llap{\spacer}%
${L_{100.25}}=\pdualwitharg{5.11}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${440}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{88}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2044680\\15840\\21120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15840\\-110\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21120\\-165\\-218
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\56\\440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-23\\-275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-432\\-4440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-81\\-814
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-19\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
440\\0\\-440\\-9240\\-1760\\-440\\-440\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-360\\-77\\-25\\-44\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\0\\110\\0\\-55\\-55\\-220\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9240\\-360\\0\\120\\0\\-120\\-1320\\-1920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1760\\-77\\-55\\0\\22\\0\\-176\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\-25\\-55\\-120\\0\\10\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\-44\\-220\\-1320\\-176\\0\\88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-165\\-1920\\-330\\-60\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2044680, 15840, 21120; 15840, -110, -165; 21120, -165, -218],[1320, 55, 1],[5, 0, -3, -49, -9, -2, -1, 1; 56, 1, -23, -432, -81, -19, -12, 9; 440, -1, -275, -4440, -814, -180, -88, 90],99,25]
%
% END OF LATTICE L100.25
%
\hbox{}\par\smallskip%
%
% LATTICE L100.26
%
\leavevmode%
\llap{\spacer}%
${L_{100.26}}=\pdualwitharg{2.3.5.11}(\pfillwitharg{2}({L_{100.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${165}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{660}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{132}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6634041756540\\-16530135384510\\-8235080591760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16530135384510\\41188371411660\\20519466424710
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8235080591760\\20519466424710\\10222509119057
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
289\\5622560\\-11285835
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\1113904\\-2235876
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\1546714\\-3104640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\19746\\-39635
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-643017\\1290696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-358609\\719820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\269445\\-540837
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\1221399\\-2451640
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
165\\0\\-660\\-1155\\-2640\\-660\\-165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-180\\-462\\-150\\-66\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\0\\660\\0\\-330\\-330\\-330\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-180\\0\\5\\0\\-60\\-165\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2640\\-462\\-330\\0\\132\\0\\-264\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-150\\-330\\-60\\0\\60\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-66\\-330\\-165\\-264\\0\\33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-330\\-320\\-660\\-120\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6634041756540, -16530135384510, -8235080591760; -16530135384510, 41188371411660, 20519466424710; -8235080591760, 20519466424710, 10222509119057],[330, 330, 1],[289, 55, 65, 1, -26, -12, 19, 66; 5622560, 1113904, 1546714, 19746, -643017, -358609, 269445, 1221399; -11285835, -2235876, -3104640, -39635, 1290696, 719820, -540837, -2451640],99,26]
%
% END OF LATTICE L100.26
%
\hbox{}\par\smallskip%
%
% LATTICE L100.27
%
\leavevmode%
\llap{\spacer}%
${L_{100.27}}=\pdualwitharg{2.11}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}{176}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{880}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{220}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2911920\\698280\\-9240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
698280\\-166672\\2200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9240\\2200\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\20\\869
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\65\\3000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\47\\2288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\591
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-592
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-40\\-2090
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-88\\-231\\-1760\\-88\\-110\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-360\\-3080\\-200\\-440\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\0\\176\\0\\-440\\-88\\-440\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-360\\0\\3\\0\\-24\\-330\\-384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1760\\-3080\\-440\\0\\880\\0\\-1760\\-2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-200\\-88\\-24\\0\\16\\0\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\-440\\-440\\-330\\-1760\\0\\220\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-264\\-384\\-2640\\-96\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2911920, 698280, -9240; 698280, -166672, 2200; -9240, 2200, -29],[1320, 88, 1],[2, 6, 4, 1, -1, -1, -3, 1; 20, 65, 47, 12, -10, -12, -40, 6; 869, 3000, 2288, 591, -440, -592, -2090, 120],99,27]
%
% END OF LATTICE L100.27
%
\hbox{}\par\smallskip%
%
% LATTICE L100.28
%
\leavevmode%
\llap{\spacer}%
${L_{100.28}}=\pdualwitharg{2.3.5}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{528}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{2640}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{880}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2721840\\175560\\7920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
175560\\7440\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7920\\240\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-364\\9075
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-605\\15048
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\365\\-9120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-121\\3025
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-243\\6072
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-2431\\60720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-243\\6066
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-3157\\78760
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-120\\-1155\\-480\\-1320\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\528\\0\\-3960\\-1848\\-6600\\-264\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\240\\0\\-120\\-1320\\-120\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-3960\\0\\55\\0\\-1320\\-330\\-7040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-1848\\-120\\0\\48\\0\\-96\\-2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\-6600\\-1320\\-1320\\0\\2640\\0\\-5280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-264\\-120\\-330\\-96\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1320\\-1320\\-7040\\-2640\\-5280\\0\\880
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2721840, 175560, 7920; 175560, 7440, 240; 7920, 240, 7],[1320, 120, 1],[3, 5, -3, 1, 2, 20, 2, 26; -364, -605, 365, -121, -243, -2431, -243, -3157; 9075, 15048, -9120, 3025, 6072, 60720, 6066, 78760],99,28]
%
% END OF LATTICE L100.28
%
\hbox{}\par\smallskip%
%
% LATTICE L100.29
%
\leavevmode%
\llap{\spacer}%
${L_{100.29}}=\pdualwitharg{3.5.11}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1320}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{330}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{b}{2}{264}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1320\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-235290\\1815
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1815\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\129
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\1155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\24\\3040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\11\\1386
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-130
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1320\\0\\-1320\\-9240\\-5280\\-1320\\-1320\\0
\end{tabular}\endgroup%
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0\\6\\0\\-360\\-231\\-75\\-132\\-15
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-1320\\0\\330\\0\\-165\\-165\\-660\\-165
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-9240\\-360\\0\\40\\0\\-120\\-1320\\-640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5280\\-231\\-165\\0\\66\\0\\-528\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\-75\\-165\\-120\\0\\30\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\-132\\-660\\-1320\\-528\\0\\264\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-165\\-640\\-330\\-60\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1320, 0, 0; 0, -235290, 1815; 0, 1815, -14],[1320, 165, 1],[1, 0, -1, -7, -4, -1, -1, 0; 0, 1, 9, 24, 11, 1, -4, -1; 0, 129, 1155, 3040, 1386, 120, -528, -130],99,29]
%
% END OF LATTICE L100.29
%
\hbox{}\par\smallskip%
%
% LATTICE L100.30
%
\leavevmode%
\llap{\spacer}%
${L_{100.30}}=\pdualwitharg{2.3.11}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{*}{2}{528}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{2640}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{*}{2}{660}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2640\\-3960\\1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3960\\-494736\\167112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1320\\167112\\-56447
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-11\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\365\\1080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\535\\1584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\51\\151
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-154\\-456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-780\\-2310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-108\\-320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\0\\-264\\-231\\-5280\\-264\\-330\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\240\\0\\-360\\-9240\\-600\\-1320\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\528\\0\\-1320\\-264\\-1320\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-360\\0\\1\\0\\-24\\-330\\-128
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5280\\-9240\\-1320\\0\\2640\\0\\-5280\\-2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-600\\-264\\-24\\0\\48\\0\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-1320\\-1320\\-330\\-5280\\0\\660\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-264\\-128\\-2640\\-96\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2640, -3960, 1320; -3960, -494736, 167112; 1320, 167112, -56447],[1320, 264, 1],[2, 11, 11, 1, -1, -3, -13, -1; -11, 365, 535, 51, 0, -154, -780, -108; -33, 1080, 1584, 151, 0, -456, -2310, -320],99,30]
%
% END OF LATTICE L100.30
%
\hbox{}\par\smallskip%
%
% LATTICE L100.31
%
\leavevmode%
\llap{\spacer}%
${L_{100.31}}=\pdualwitharg{2.5.11}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${55}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{880}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{176}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1000560\\-130680\\-58080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130680\\-6160\\-2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58080\\-2640\\-1129
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-773\\1705
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-769\\1696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-4589\\10120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-4170\\9195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-4909\\10824
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-363\\800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\389\\-858
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\381\\-840
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\0\\-440\\-1155\\-1760\\-440\\-110\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-360\\-616\\-200\\-88\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\0\\880\\0\\-440\\-440\\-440\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-360\\0\\15\\0\\-120\\-330\\-1920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1760\\-616\\-440\\0\\176\\0\\-352\\-2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\-200\\-440\\-120\\0\\80\\0\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110\\-88\\-440\\-330\\-352\\0\\44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-1320\\-1920\\-2640\\-480\\0\\240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1000560, -130680, -58080; -130680, -6160, -2640; -58080, -2640, -1129],[1320, 440, 1],[2, 2, 12, 11, 13, 1, -1, -1; -773, -769, -4589, -4170, -4909, -363, 389, 381; 1705, 1696, 10120, 9195, 10824, 800, -858, -840],99,31]
%
% END OF LATTICE L100.31
%
\hbox{}\par\smallskip%
%
% LATTICE L100.32
%
\leavevmode%
\llap{\spacer}%
${L_{100.32}}=\pdualwitharg{2.3.5.11}({L_{100.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${165}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{2640}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{528}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60720\\4014120\\-9240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4014120\\-189179760\\435600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9240\\435600\\-1003
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-411\\-178695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-86\\-37392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\252\\109560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-20\\-8695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-235\\-102168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-191\\-83040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-223\\-96954
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-291\\-126520
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
165\\0\\-1320\\-1155\\-5280\\-1320\\-330\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-360\\-1848\\-600\\-264\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\0\\2640\\0\\-1320\\-1320\\-1320\\-1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-360\\0\\5\\0\\-120\\-330\\-640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5280\\-1848\\-1320\\0\\528\\0\\-1056\\-2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\-600\\-1320\\-120\\0\\240\\0\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-264\\-1320\\-330\\-1056\\0\\132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-1320\\-640\\-2640\\-480\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-60720, 4014120, -9240; 4014120, -189179760, 435600; -9240, 435600, -1003],[1320, 1320, 1],[23, 5, -13, 1, 12, 10, 12, 16; -411, -86, 252, -20, -235, -191, -223, -291; -178695, -37392, 109560, -8695, -102168, -83040, -96954, -126520],99,32]
%
% END OF LATTICE L100.32
%

\medskip%
%------------------------------WEYL-GROUP-W101--------------------------------
%
\leavevmode\llap{}%
$W_{101}$%
\qquad\llap{24} lattices, $\chi=22$%
\hfill%
$2262222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L101.1
%
\leavevmode%
\llap{\spacer}%
${L_{101.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{101}}%
\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{92}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}{276}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{46}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-579195660\\194669700\\-401580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
194669700\\-65428814\\134783
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-401580\\134783\\-178
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19003\\-56760\\-107070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10303\\30774\\58052
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4808\\14361\\27090
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5037\\-15045\\-28381
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-612451\\-1829328\\-3450828
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45867\\-137000\\-258435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61741\\-184414\\-347875
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-30\\-30\\-1380\\-60\\0
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\kern3pt%
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0\\92\\0\\-46\\-3588\\-230\\-230
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-30\\0\\6\\-3\\-552\\-45\\-69
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-30\\-46\\-3\\2\\0\\-5\\-23
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1380\\-3588\\-552\\0\\276\\0\\-552
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-230\\-45\\-5\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-230\\-69\\-23\\-552\\0\\46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-579195660, 194669700, -401580; 194669700, -65428814, 134783; -401580, 134783, -178],[1380, 1, 1],[-19003, 10303, 4808, -5037, -612451, -45867, -61741; -56760, 30774, 14361, -15045, -1829328, -137000, -184414; -107070, 58052, 27090, -28381, -3450828, -258435, -347875],100,1]
%
% END OF LATTICE L101.1
%
\hbox{}\par\smallskip%
%
% LATTICE L101.2
%
\leavevmode%
\llap{\spacer}%
${L_{101.2}}=\pfillwitharg{2}({L_{101.1}})$%
{${}={}$}Nikulin $101$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{23}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{l}{2}{69}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{46}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13110\\1725\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1725\\227\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\15\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\23\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-15\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-38\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
163\\-1242\\-207
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-145\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-115\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\-15\\-345\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\23\\0\\-23\\-897\\-115\\-115
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\6\\-3\\-276\\-45\\-69
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-23\\-3\\2\\0\\-5\\-23
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-345\\-897\\-276\\0\\69\\0\\-276
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-115\\-45\\-5\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-115\\-69\\-23\\-276\\0\\46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[13110, 1725, 0; 1725, 227, 0; 0, 0, -1],[345, 1, 1],[-2, -3, 2, 5, 163, 19, 15; 15, 23, -15, -38, -1242, -145, -115; 0, 0, -3, -6, -207, -25, -23],100,2]
%
% END OF LATTICE L101.2
%
\hbox{}\par\smallskip%
%
% LATTICE L101.3
%
\leavevmode%
\llap{\spacer}%
${L_{101.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{101.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{69}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{23}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{138}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1237759822695\\11185948110\\-412695155490
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11185948110\\-101090238\\3729630345
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-412695155490\\3729630345\\-137601243991
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6336\\-4870\\-19135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9344\\7245\\28221
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3187\\2454\\9625
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9230\\-7131\\-27876
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
193025\\-148879\\-582958
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
87429\\-67405\\-264045
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
119147\\-91793\\-359835
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-5\\-15\\-115\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\69\\0\\-69\\-897\\-345\\-345
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\2\\-3\\-92\\-45\\-69
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-69\\-3\\6\\0\\-15\\-69
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-115\\-897\\-92\\0\\23\\0\\-276
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-345\\-45\\-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-345\\-69\\-69\\-276\\0\\138
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1237759822695, 11185948110, -412695155490; 11185948110, -101090238, 3729630345; -412695155490, 3729630345, -137601243991],[345, 3, 1],[6336, -9344, -3187, 9230, 193025, 87429, 119147; -4870, 7245, 2454, -7131, -148879, -67405, -91793; -19135, 28221, 9625, -27876, -582958, -264045, -359835],100,3]
%
% END OF LATTICE L101.3
%
\hbox{}\par\smallskip%
%
% LATTICE L101.4
%
\leavevmode%
\llap{\spacer}%
${L_{101.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{101.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{115}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{l}{2}{345}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{230}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2113167226965\\18868871100\\422523583800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18868871100\\-168483730\\-3772793245
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
422523583800\\-3772793245\\-84482750153
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2684\\-2922\\-13293
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6572\\7245\\32545
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6745\\7362\\33405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6502\\-7131\\-32200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-408223\\-446637\\-2021700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12329\\-13481\\-61059
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84035\\-91793\\-416185
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-15\\-15\\-345\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\115\\0\\-115\\-4485\\-115\\-575
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\-15\\-1380\\-45\\-345
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-115\\-15\\10\\0\\-5\\-115
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-345\\-4485\\-1380\\0\\345\\0\\-1380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-115\\-45\\-5\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-575\\-345\\-115\\-1380\\0\\230
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2113167226965, 18868871100, 422523583800; 18868871100, -168483730, -3772793245; 422523583800, -3772793245, -84482750153],[345, 5, 1],[-2684, 6572, 6745, -6502, -408223, -12329, -84035; -2922, 7245, 7362, -7131, -446637, -13481, -91793; -13293, 32545, 33405, -32200, -2021700, -61059, -416185],100,4]
%
% END OF LATTICE L101.4
%
\hbox{}\par\smallskip%
%
% LATTICE L101.5
%
\leavevmode%
\llap{\spacer}%
${L_{101.5}}=\pdualwitharg{3}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{276}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{92}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{138}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4637088420\\25149120\\-1713960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25149120\\-135606\\9021
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1713960\\9021\\-538
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-293\\-67190\\-193180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
135\\30958\\89010
\end{tabular}\endgroup%
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63\\14447\\41537
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-129\\-29582\\-85053
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6539\\-1499508\\-4311304
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1546\\-354525\\-1019310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2258\\-517799\\-1488744
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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20\\0\\-10\\-30\\-460\\-60\\0
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-460\\-3588\\-184\\0\\92\\0\\-552
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\kern3pt%
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-60\\-690\\-45\\-15\\0\\30\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-690\\-69\\-69\\-552\\0\\138
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4637088420, 25149120, -1713960; 25149120, -135606, 9021; -1713960, 9021, -538],[1380, 3, 1],[-293, 135, 63, -129, -6539, -1546, -2258; -67190, 30958, 14447, -29582, -1499508, -354525, -517799; -193180, 89010, 41537, -85053, -4311304, -1019310, -1488744],100,5]
%
% END OF LATTICE L101.5
%
\hbox{}\par\smallskip%
%
% LATTICE L101.6
%
\leavevmode%
\llap{\spacer}%
${L_{101.6}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{101.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{345}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{l}{2}{115}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{690}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
270241525995\\-6578213325\\88093642395
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6578213325\\160128885\\-2144373510
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88093642395\\-2144373510\\28716866521
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
158043\\2163\\-484661
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1268786\\17365\\-3890910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75\\1\\-230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1102309\\15086\\-3380385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19860847\\271814\\-60906070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1734406\\23737\\-5318799
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11832740\\161943\\-36286755
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-5\\-15\\-115\\-6\\0
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0\\345\\0\\-345\\-4485\\-345\\-1725
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\kern3pt%
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-5\\0\\10\\-15\\-460\\-45\\-345
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-345\\-15\\30\\0\\-15\\-345
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-115\\-4485\\-460\\0\\115\\0\\-1380
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-345\\-45\\-15\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1725\\-345\\-345\\-1380\\0\\690
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[270241525995, -6578213325, 88093642395; -6578213325, 160128885, -2144373510; 88093642395, -2144373510, 28716866521],[345, 15, 1],[158043, 1268786, 75, 1102309, 19860847, 1734406, 11832740; 2163, 17365, 1, 15086, 271814, 23737, 161943; -484661, -3890910, -230, -3380385, -60906070, -5318799, -36286755],100,6]
%
% END OF LATTICE L101.6
%
\hbox{}\par\smallskip%
%
% LATTICE L101.7
%
\leavevmode%
\llap{\spacer}%
${L_{101.7}}=\pdualwitharg{2}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{92}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{l}{2}{69}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{s}{2}{184}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6008791084440\\-31829682060\\1487187136440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31829682060\\168607736\\-7877906396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1487187136440\\-7877906396\\368081623693
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15712\\-21135\\63030
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7373\\9890\\-29578
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6539\\-8784\\26232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36163\\-48593\\145072
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-730745\\-982008\\2931465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190866\\-256505\\765680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-207456\\-278829\\832232
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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60\\0\\-60\\-60\\-690\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\92\\0\\-92\\-1794\\-460\\-460
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-60\\0\\24\\-12\\-552\\-180\\-276
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\kern3pt%
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-60\\-92\\-12\\8\\0\\-20\\-92
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-690\\-1794\\-552\\0\\69\\0\\-552
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-460\\-180\\-20\\0\\40\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-460\\-276\\-92\\-552\\0\\184
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6008791084440, -31829682060, 1487187136440; -31829682060, 168607736, -7877906396; 1487187136440, -7877906396, 368081623693],[1380, 4, 1],[-15712, 7373, -6539, -36163, -730745, -190866, -207456; -21135, 9890, -8784, -48593, -982008, -256505, -278829; 63030, -29578, 26232, 145072, 2931465, 765680, 832232],100,7]
%
% END OF LATTICE L101.7
%
\hbox{}\par\smallskip%
%
% LATTICE L101.8
%
\leavevmode%
\llap{\spacer}%
${L_{101.8}}=\pdualwitharg{5}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{460}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{l}{2}{1380}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{230}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-159736380\\-51409140\\-12578700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51409140\\-16544710\\-4060785
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12578700\\-4060785\\-758962
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93341\\-286248\\-15438
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-253093\\776158\\41860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-118082\\362121\\19530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
123735\\-379457\\-20465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15043741\\-46134504\\-2488140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
225323\\-690996\\-37267
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1516473\\-4650554\\-250815
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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12\\0\\-30\\-30\\-1380\\-12\\0
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\kern3pt%
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0\\460\\0\\-230\\-17940\\-230\\-1150
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-30\\0\\30\\-15\\-2760\\-45\\-345
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-30\\-230\\-15\\10\\0\\-5\\-115
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-1380\\-17940\\-2760\\0\\1380\\0\\-2760
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-230\\-45\\-5\\0\\2\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1150\\-345\\-115\\-2760\\0\\230
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-159736380, -51409140, -12578700; -51409140, -16544710, -4060785; -12578700, -4060785, -758962],[1380, 5, 1],[93341, -253093, -118082, 123735, 15043741, 225323, 1516473; -286248, 776158, 362121, -379457, -46134504, -690996, -4650554; -15438, 41860, 19530, -20465, -2488140, -37267, -250815],100,8]
%
% END OF LATTICE L101.8
%
\hbox{}\par\smallskip%
%
% LATTICE L101.9
%
\leavevmode%
\llap{\spacer}%
${L_{101.9}}=\pdualwitharg{23}(\pfillwitharg{2}({L_{101.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${345}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{138}\above{1pt}{1pt}{}{6}{46}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{230}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12897386218995\\2910574560\\-1686614680380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2910574560\\-656834\\380621135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1686614680380\\380621135\\-220561672867
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49672\\-4530\\-379845
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1071\\94\\8190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25030\\2265\\191406
\end{tabular}\endgroup%
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24245\\-2162\\-185403
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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66023\\-5931\\-504882
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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229171\\-20625\\-1752485
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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13569\\-1225\\-103763
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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345\\0\\-345\\-345\\-345\\-690\\0
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-690\\-115\\-1035\\-115\\0\\230\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-69\\-23\\-12\\0\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12897386218995, 2910574560, -1686614680380; 2910574560, -656834, 380621135; -1686614680380, 380621135, -220561672867],[345, 23, 1],[49672, -1071, -25030, 24245, 66023, 229171, 13569; -4530, 94, 2265, -2162, -5931, -20625, -1225; -379845, 8190, 191406, -185403, -504882, -1752485, -103763],100,9]
%
% END OF LATTICE L101.9
%
\hbox{}\par\smallskip%
%
% LATTICE L101.10
%
\leavevmode%
\llap{\spacer}%
${L_{101.10}}=\pdualwitharg{2.3}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{276}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{l}{2}{23}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{s}{2}{552}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
415947509425320\\-1265685451980\\103683626331420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1265685451980\\3851350536\\-315498601344
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
103683626331420\\-315498601344\\25845314915063
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40829\\-300\\163790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7327\\-23\\-29394
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2409\\-15\\9664
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-144268\\-967\\578748
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1069803\\-7268\\4291639
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-872611\\-5960\\3500580
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-1031109\\-7130\\4136412
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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20\\0\\-20\\-60\\-230\\-120\\0
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0\\276\\0\\-276\\-1794\\-1380\\-1380
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-20\\0\\8\\-12\\-184\\-180\\-276
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-60\\-276\\-12\\24\\0\\-60\\-276
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-230\\-1794\\-184\\0\\23\\0\\-552
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-120\\-1380\\-180\\-60\\0\\120\\0
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0\\-1380\\-276\\-276\\-552\\0\\552
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[415947509425320, -1265685451980, 103683626331420; -1265685451980, 3851350536, -315498601344; 103683626331420, -315498601344, 25845314915063],[1380, 12, 1],[-40829, 7327, -2409, -144268, -1069803, -872611, -1031109; -300, -23, -15, -967, -7268, -5960, -7130; 163790, -29394, 9664, 578748, 4291639, 3500580, 4136412],100,10]
%
% END OF LATTICE L101.10
%
\hbox{}\par\smallskip%
%
% LATTICE L101.11
%
\leavevmode%
\llap{\spacer}%
${L_{101.11}}=\pdualwitharg{3.5}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{1380}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{l}{2}{460}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{690}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-984708660\\25886040\\82068600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25886040\\-678030\\-2112315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82068600\\-2112315\\-6013706
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12519\\575914\\-31444
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28845\\-1326962\\72450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13459\\-619157\\33805
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27561\\1267894\\-69225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1397013\\64267060\\-3508880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66058\\3038879\\-165918
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
482398\\22191849\\-1211640
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\0\\-10\\-30\\-460\\-12\\0
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0\\1380\\0\\-690\\-17940\\-690\\-3450
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-10\\0\\10\\-15\\-920\\-45\\-345
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-30\\-690\\-15\\30\\0\\-15\\-345
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-460\\-17940\\-920\\0\\460\\0\\-2760
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\kern3pt%
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-12\\-690\\-45\\-15\\0\\6\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3450\\-345\\-345\\-2760\\0\\690
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-984708660, 25886040, 82068600; 25886040, -678030, -2112315; 82068600, -2112315, -6013706],[1380, 15, 1],[12519, -28845, -13459, 27561, 1397013, 66058, 482398; 575914, -1326962, -619157, 1267894, 64267060, 3038879, 22191849; -31444, 72450, 33805, -69225, -3508880, -165918, -1211640],100,11]
%
% END OF LATTICE L101.11
%
\hbox{}\par\smallskip%
%
% LATTICE L101.12
%
\leavevmode%
\llap{\spacer}%
${L_{101.12}}=\pdualwitharg{3.23}(\pfillwitharg{2}({L_{101.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${115}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{46}\above{1pt}{1pt}{}{6}{138}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{690}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24130231850865\\-40827570135\\7993175520465
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40827570135\\69087147\\-13524193890
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7993175520465\\-13524193890\\2647751389039
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14799299\\-315\\44676925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1033142\\-22\\3118905
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1379\\0\\4163
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-20643905\\-439\\62320938
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-16171889\\-344\\48820574
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-162409826\\-3455\\490291575
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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115\\0\\-115\\-345\\-115\\-690\\0
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0\\3\\0\\-69\\-39\\-345\\-15
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-115\\0\\46\\-69\\-92\\-1035\\-69
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-345\\-69\\-69\\138\\0\\-345\\-69
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-115\\-39\\-92\\0\\1\\0\\-12
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-690\\-345\\-1035\\-345\\0\\690\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-69\\-69\\-12\\0\\6
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24130231850865, -40827570135, 7993175520465; -40827570135, 69087147, -13524193890; 7993175520465, -13524193890, 2647751389039],[345, 69, 1],[-14799299, -1033142, -1379, -20643905, -16171889, -162409826, -9634968; -315, -22, 0, -439, -344, -3455, -205; 44676925, 3118905, 4163, 62320938, 48820574, 490291575, 29086563],100,12]
%
% END OF LATTICE L101.12
%
\hbox{}\par\smallskip%
%
% LATTICE L101.13
%
\leavevmode%
\llap{\spacer}%
${L_{101.13}}=\pdualwitharg{2.5}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{460}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{}{6}{40}\above{1pt}{1pt}{l}{2}{345}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{920}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4708729545163320\\-50242196807580\\1177046902511760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50242196807580\\536084800760\\-12559103588460
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1177046902511760\\-12559103588460\\294227858581433
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5624\\1017\\22542
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2457\\-1426\\-9890
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7091\\1704\\28440
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-44759\\9919\\179480
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-939311\\203550\\3766365
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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-24\\-460\\-180\\-20\\0\\8\\0
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0\\-2300\\-1380\\-460\\-2760\\0\\920
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4708729545163320, -50242196807580, 1177046902511760; -50242196807580, 536084800760, -12559103588460; 1177046902511760, -12559103588460, 294227858581433],[1380, 20, 1],[-5624, 2457, -7091, -44759, -939311, -49894, -282236; 1017, -1426, 1704, 9919, 203550, 10707, 59179; 22542, -9890, 28440, 179480, 3766365, 200056, 1131600],100,13]
%
% END OF LATTICE L101.13
%
\hbox{}\par\smallskip%
%
% LATTICE L101.14
%
\leavevmode%
\llap{\spacer}%
${L_{101.14}}=\pdualwitharg{23}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1380}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{138}\above{1pt}{1pt}{}{6}{46}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{230}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18947329620\\-908287020\\3022200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-908287020\\-43540978\\144877
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3022200\\144877\\-482
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-419\\8550\\-57270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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5\\-102\\692
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\HardButStrongLineBreak\kern3pt%
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106\\-2163\\14490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\1122\\-7613
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-397\\8100\\-54588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-731\\14915\\-100395
\end{tabular}\endgroup%
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-47\\959\\-6445
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1380\\0\\-690\\-690\\-1380\\-1380\\0
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\kern3pt%
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0\\4\\0\\-46\\-156\\-230\\-10
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-690\\0\\138\\-69\\-552\\-1035\\-69
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-1380\\-156\\-552\\0\\12\\0\\-24
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-1380\\-230\\-1035\\-115\\0\\230\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-69\\-23\\-24\\0\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18947329620, -908287020, 3022200; -908287020, -43540978, 144877; 3022200, 144877, -482],[1380, 23, 1],[-419, 5, 106, -55, -397, -731, -47; 8550, -102, -2163, 1122, 8100, 14915, 959; -57270, 692, 14490, -7613, -54588, -100395, -6445],100,14]
%
% END OF LATTICE L101.14
%
\hbox{}\par\smallskip%
%
% LATTICE L101.15
%
\leavevmode%
\llap{\spacer}%
${L_{101.15}}=\pdualwitharg{5.23}(\pfillwitharg{2}({L_{101.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${69}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{690}\above{1pt}{1pt}{}{6}{230}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{46}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20720328456390\\48842270310\\4150818229515
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48842270310\\115145245\\9784363070
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4150818229515\\9784363070\\831516359081
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6424133\\19272210\\31841637
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-747450\\2242328\\3704785
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2993\\8979\\14835
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14935301\\44805464\\74027800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35099759\\105298245\\173974260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23499814\\70498751\\116478371
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6970636\\20911703\\34550415
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69\\0\\-345\\-345\\-345\\-138\\0
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\kern3pt%
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0\\5\\0\\-115\\-195\\-115\\-25
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-345\\0\\690\\-345\\-1380\\-1035\\-345
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-345\\-195\\-1380\\0\\15\\0\\-60
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-138\\-115\\-1035\\-115\\0\\46\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-25\\-345\\-115\\-60\\0\\10
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20720328456390, 48842270310, 4150818229515; 48842270310, 115145245, 9784363070; 4150818229515, 9784363070, 831516359081],[345, 115, 1],[-6424133, -747450, -2993, -14935301, -35099759, -23499814, -6970636; 19272210, 2242328, 8979, 44805464, 105298245, 70498751, 20911703; 31841637, 3704785, 14835, 74027800, 173974260, 116478371, 34550415],100,15]
%
% END OF LATTICE L101.15
%
\hbox{}\par\smallskip%
%
% LATTICE L101.16
%
\leavevmode%
\llap{\spacer}%
${L_{101.16}}=\pdualwitharg{2.3.5}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{1380}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{}{6}{120}\above{1pt}{1pt}{l}{2}{115}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{2760}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
374888445391560\\115150992175140\\93617326960500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115150992175140\\35369857785480\\28755562399440
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93617326960500\\28755562399440\\23378164931371
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5405\\-60\\21718
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20167\\-23\\-80730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2133\\-15\\8560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114040\\-967\\457860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-826249\\-7268\\3317635
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\HardButStrongLineBreak\kern3pt%
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-133527\\-1192\\536172
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-771469\\-7130\\3098100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\0\\-20\\-60\\-230\\-24\\0
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0\\1380\\0\\-1380\\-8970\\-1380\\-6900
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-20\\0\\40\\-60\\-920\\-180\\-1380
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-230\\-8970\\-920\\0\\115\\0\\-2760
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-24\\-1380\\-180\\-60\\0\\24\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6900\\-1380\\-1380\\-2760\\0\\2760
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[374888445391560, 115150992175140, 93617326960500; 115150992175140, 35369857785480, 28755562399440; 93617326960500, 28755562399440, 23378164931371],[1380, 60, 1],[-5405, 20167, -2133, -114040, -826249, -133527, -771469; -60, -23, -15, -967, -7268, -1192, -7130; 21718, -80730, 8560, 457860, 3317635, 536172, 3098100],100,16]
%
% END OF LATTICE L101.16
%
\hbox{}\par\smallskip%
%
% LATTICE L101.17
%
\leavevmode%
\llap{\spacer}%
${L_{101.17}}=\pdualwitharg{3.23}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${460}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{46}\above{1pt}{1pt}{}{6}{138}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{690}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-794448060\\-97343820\\1105380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-97343820\\-11923062\\135447
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1105380\\135447\\-1538
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-293\\200\\-192970
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\1980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\-43\\41492
\end{tabular}\endgroup%
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-96\\65\\-63273
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-247\\168\\-162728
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1381\\940\\-909765
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-91\\62\\-59943
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-794448060, -97343820, 1105380; -97343820, -11923062, 135447; 1105380, 135447, -1538],[1380, 69, 1],[-293, 3, 63, -96, -247, -1381, -91; 200, -2, -43, 65, 168, 940, 62; -192970, 1980, 41492, -63273, -162728, -909765, -59943],100,17]
%
% END OF LATTICE L101.17
%
\hbox{}\par\smallskip%
%
% LATTICE L101.18
%
\leavevmode%
\llap{\spacer}%
${L_{101.18}}=\pdualwitharg{3.5.23}(\pfillwitharg{2}({L_{101.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${23}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{230}\above{1pt}{1pt}{}{6}{690}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{138}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102545810471842365\\271973490454204530\\90166738340303880
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
271973490454204530\\-721332048276650805\\-239141535246016260
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90166738340303880\\-239141535246016260\\-79282036638261493
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-302\\7723852\\-23298103
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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94\\-2497641\\7533840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
755\\-19459788\\58698185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2162\\56544795\\-170560755
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-1977\\51328489\\-154826395
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-4125\\106899991\\-322451352
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-1225\\31647677\\-95461530
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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23\\0\\-115\\-345\\-115\\-138\\0
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0\\15\\0\\-345\\-195\\-345\\-75
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-115\\0\\230\\-345\\-460\\-1035\\-345
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-345\\-345\\-345\\690\\0\\-345\\-345
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-115\\-195\\-460\\0\\5\\0\\-60
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-138\\-345\\-1035\\-345\\0\\138\\0
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0\\-75\\-345\\-345\\-60\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-102545810471842365, 271973490454204530, 90166738340303880; 271973490454204530, -721332048276650805, -239141535246016260; 90166738340303880, -239141535246016260, -79282036638261493],[345, 345, 1],[-302, 94, 755, -2162, -1977, -4125, -1225; 7723852, -2497641, -19459788, 56544795, 51328489, 106899991, 31647677; -23298103, 7533840, 58698185, -170560755, -154826395, -322451352, -95461530],100,18]
%
% END OF LATTICE L101.18
%
\hbox{}\par\smallskip%
%
% LATTICE L101.19
%
\leavevmode%
\llap{\spacer}%
${L_{101.19}}=\pdualwitharg{2.23}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1380}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{552}\above{1pt}{1pt}{}{6}{184}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{920}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
686395983720\\7029809700\\171086628840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7029809700\\71997544\\1752204952
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
171086628840\\1752204952\\42643947899
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
536123\\-4485\\-2150730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1203\\-10\\-4826
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108979\\-912\\-437184
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
726412\\-6079\\-2914100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
676133\\-5658\\-2712399
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4165259\\-34855\\-16709500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
209019\\-1749\\-838508
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1380\\0\\-1380\\-1380\\-690\\-2760\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\4\\0\\-92\\-78\\-460\\-20
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-1380\\0\\552\\-276\\-552\\-4140\\-276
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\kern3pt%
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-1380\\-92\\-276\\184\\0\\-460\\-92
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-690\\-78\\-552\\0\\3\\0\\-24
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-2760\\-460\\-4140\\-460\\0\\920\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-276\\-92\\-24\\0\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[686395983720, 7029809700, 171086628840; 7029809700, 71997544, 1752204952; 171086628840, 1752204952, 42643947899],[1380, 92, 1],[536123, 1203, 108979, 726412, 676133, 4165259, 209019; -4485, -10, -912, -6079, -5658, -34855, -1749; -2150730, -4826, -437184, -2914100, -2712399, -16709500, -838508],100,19]
%
% END OF LATTICE L101.19
%
\hbox{}\par\smallskip%
%
% LATTICE L101.20
%
\leavevmode%
\llap{\spacer}%
${L_{101.20}}=\pdualwitharg{5.23}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${276}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{690}\above{1pt}{1pt}{}{6}{230}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{46}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-369355620\\-109940460\\-13892460
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109940460\\-32724170\\-4135055
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13892460\\-4135055\\-522454
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2207\\8862\\-11454
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
135\\-542\\700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2792\\-11211\\14490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1486\\5966\\-7705
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10583\\42492\\-54900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-3888\\15611\\-20171
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1246\\5003\\-6465
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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276\\0\\-690\\-690\\-1380\\-276\\0
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0\\20\\0\\-230\\-780\\-230\\-50
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-50\\-345\\-115\\-120\\0\\10
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-369355620, -109940460, -13892460; -109940460, -32724170, -4135055; -13892460, -4135055, -522454],[1380, 115, 1],[-2207, 135, 2792, -1486, -10583, -3888, -1246; 8862, -542, -11211, 5966, 42492, 15611, 5003; -11454, 700, 14490, -7705, -54900, -20171, -6465],100,20]
%
% END OF LATTICE L101.20
%
\hbox{}\par\smallskip%
%
% LATTICE L101.21
%
\leavevmode%
\llap{\spacer}%
${L_{101.21}}=\pdualwitharg{2.3.23}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${460}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{184}\above{1pt}{1pt}{}{6}{552}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{2760}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17613178240440\\3672192420\\4390453398600
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\kern3pt%
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3672192420\\765624\\915370836
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4390453398600\\915370836\\1094412421321
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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81928\\-585\\-328670
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\HardButStrongLineBreak\kern3pt%
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513\\-2\\-2058
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\HardButStrongLineBreak\kern3pt%
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4105\\-30\\-16468
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
258615\\-1889\\-1037484
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\HardButStrongLineBreak\kern3pt%
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84751\\-617\\-339995
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1600262\\-11635\\-6419760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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83468\\-605\\-334848
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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460\\0\\-460\\-1380\\-230\\-2760\\0
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-2760\\-1380\\-4140\\-1380\\0\\2760\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-276\\-276\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17613178240440, 3672192420, 4390453398600; 3672192420, 765624, 915370836; 4390453398600, 915370836, 1094412421321],[1380, 276, 1],[81928, 513, 4105, 258615, 84751, 1600262, 83468; -585, -2, -30, -1889, -617, -11635, -605; -328670, -2058, -16468, -1037484, -339995, -6419760, -334848],100,21]
%
% END OF LATTICE L101.21
%
\hbox{}\par\smallskip%
%
% LATTICE L101.22
%
\leavevmode%
\llap{\spacer}%
${L_{101.22}}=\pdualwitharg{3.5.23}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${92}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{230}\above{1pt}{1pt}{}{6}{690}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{138}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-213716460\\326601840\\85230180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
326601840\\-499102530\\-130248885
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
85230180\\-130248885\\-33989818
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12395\\40\\-31234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
643\\-2\\1620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13326\\-43\\33580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20401\\65\\-51405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52353\\168\\-131920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58517\\188\\-147453
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19269\\62\\-48555
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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92\\0\\-230\\-690\\-460\\-276\\0
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0\\60\\0\\-690\\-780\\-690\\-150
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-230\\0\\230\\-345\\-920\\-1035\\-345
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-460\\-780\\-920\\0\\20\\0\\-120
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-276\\-690\\-1035\\-345\\0\\138\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-150\\-345\\-345\\-120\\0\\30
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-213716460, 326601840, 85230180; 326601840, -499102530, -130248885; 85230180, -130248885, -33989818],[1380, 345, 1],[-12395, 643, 13326, -20401, -52353, -58517, -19269; 40, -2, -43, 65, 168, 188, 62; -31234, 1620, 33580, -51405, -131920, -147453, -48555],100,22]
%
% END OF LATTICE L101.22
%
\hbox{}\par\smallskip%
%
% LATTICE L101.23
%
\leavevmode%
\llap{\spacer}%
${L_{101.23}}=\pdualwitharg{2.5.23}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${276}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{2760}\above{1pt}{1pt}{}{6}{920}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{184}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43274511960\\20496473460\\4496140740
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20496473460\\9707920040\\2129546540
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4496140740\\2129546540\\467141599
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38233\\114354\\-153318
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-217\\649\\-870
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39919\\119397\\-160080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-266012\\795637\\-1066740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-246812\\738210\\-989745
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-303683\\908310\\-1217804
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75943\\227144\\-304540
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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276\\0\\-1380\\-1380\\-690\\-552\\0
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0\\20\\0\\-460\\-390\\-460\\-100
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-552\\-460\\-4140\\-460\\0\\184\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-100\\-1380\\-460\\-120\\0\\40
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[43274511960, 20496473460, 4496140740; 20496473460, 9707920040, 2129546540; 4496140740, 2129546540, 467141599],[1380, 460, 1],[-38233, -217, -39919, -266012, -246812, -303683, -75943; 114354, 649, 119397, 795637, 738210, 908310, 227144; -153318, -870, -160080, -1066740, -989745, -1217804, -304540],100,23]
%
% END OF LATTICE L101.23
%
\hbox{}\par\smallskip%
%
% LATTICE L101.24
%
\leavevmode%
\llap{\spacer}%
${L_{101.24}}=\pdualwitharg{2.3.5.23}({L_{101.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${92}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{920}\above{1pt}{1pt}{}{6}{2760}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{552}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3828120\\-1756479180\\-441286740
\end{tabular}\endgroup%
\kern3pt%
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-1756479180\\805945256760\\202480597980
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-441286740\\202480597980\\50869947077
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-339\\11452\\-45586
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\309\\-1230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-86\\2889\\-11500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5417\\182007\\-724500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1772\\59585\\-237185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6687\\224926\\-895344
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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92\\0\\-460\\-1380\\-230\\-552\\0
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0\\60\\0\\-1380\\-390\\-1380\\-300
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-460\\0\\920\\-1380\\-920\\-4140\\-1380
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-230\\-390\\-920\\0\\5\\0\\-120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-552\\-1380\\-4140\\-1380\\0\\552\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-300\\-1380\\-1380\\-120\\0\\120
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3828120, -1756479180, -441286740; -1756479180, 805945256760, 202480597980; -441286740, 202480597980, 50869947077],[1380, 1380, 1],[-339, -8, -86, -5417, -1772, -6687, -1741; 11452, 309, 2889, 182007, 59585, 224926, 58604; -45586, -1230, -11500, -724500, -237185, -895344, -233280],100,24]
%
% END OF LATTICE L101.24
%

\medskip%
%------------------------------WEYL-GROUP-W102--------------------------------
%
\leavevmode\llap{}%
$W_{102}$%
\qquad\llap{24} lattices, $\chi=72$%
\hfill%
$2222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L102.1
%
\leavevmode%
\llap{\spacer}%
${L_{102.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{102}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{68}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{34}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{714}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-506940\\-252756\\1428
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252756\\-126022\\713
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1428\\713\\178
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7608385\\15268176\\-359856
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3794276\\7614188\\-179459
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-122692\\246213\\-5804
\end{tabular}\endgroup%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-506940, -252756, 1428; -252756, -126022, 713; 1428, 713, 178],[1428, 1, 1],[179, 2679, 426, -197, -5155, -26252, -17403, -264002, -1215, -12743, -1462, -691, -1061, -1424, -357, -178; -359, -5372, -854, 396, 10346, 52683, 34924, 529788, 2438, 25568, 2933, 1386, 2128, 2856, 716, 357; 2, 0, -7, -30, -280, -1292, -836, -12495, -51, -476, -42, -12, -14, -17, -4, 0],101,1]
%
% END OF LATTICE L102.1
%
\hbox{}\par\smallskip%
%
% LATTICE L102.2
%
\leavevmode%
\llap{\spacer}%
${L_{102.2}}=\pfillwitharg{2}({L_{102.1}})$%
{${}={}$}Nikulin $102$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[53193, -22848, -1785; -22848, 9814, 765; -1785, 765, 91],[357, 1, 1],[-1886, -7889, -957, 22, 3, -1298, -553, -19249, -179, -2141, -1108, -907, -6198, -53867, -17147, -507326; -4373, -18292, -2219, 51, 7, -3009, -1282, -44625, -415, -4964, -2569, -2103, -14371, -124899, -39758, -1176315; -236, -986, -119, 3, 0, -170, -72, -2499, -23, -272, -140, -114, -777, -6749, -2148, -63546],101,2]
%
% END OF LATTICE L102.2
%
\hbox{}\par\smallskip%
%
% LATTICE L102.3
%
\leavevmode%
\llap{\spacer}%
${L_{102.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{102.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7181769, 714, -2414391; 714, -3, -240; -2414391, -240, 811678],[357, 3, 1],[-1297, -81012, -59197, -21973, -515561, -4609745, -1478878, -14657729, -171225, -756952, -113325, -2673, -2005, 943, 118, -6641; -25, -1615, -1183, -440, -10332, -92395, -29643, -293811, -3433, -15181, -2275, -54, -42, 17, 3, -119; -3858, -240975, -176085, -65360, -1533567, -13711962, -4399011, -43600291, -509319, -2251599, -337092, -7951, -5964, 2805, 351, -19754],101,3]
%
% END OF LATTICE L102.3
%
\hbox{}\par\smallskip%
%
% LATTICE L102.4
%
\leavevmode%
\llap{\spacer}%
${L_{102.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{102.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2936325, 714, -1266993; 714, -7, -308; -1266993, -308, 546694],[357, 7, 1],[-1625, -101514, -10597, -82603, -92293, -5776489, -1853188, -7871855, -214563, -948542, -20287, -10049, -359, 1181, 148, -3565; -25, -1615, -169, -1320, -1476, -92395, -29643, -125919, -3433, -15181, -325, -162, -6, 17, 3, -51; -3766, -235263, -24559, -191436, -213893, -13387262, -4294843, -18243363, -497259, -2198287, -47016, -23289, -832, 2737, 343, -8262],101,4]
%
% END OF LATTICE L102.4
%
\hbox{}\par\smallskip%
%
% LATTICE L102.5
%
\leavevmode%
\llap{\spacer}%
${L_{102.5}}=\pdualwitharg{3}({L_{102.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{l}{2}{204}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{102}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{238}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-297996468\\-2483292\\-813960
\end{tabular}\endgroup%
\kern3pt%
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-2483292\\-20694\\-6783
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-297996468, -2483292, -813960; -2483292, -20694, -6783; -813960, -6783, -2222],[1428, 3, 1],[2, 2199, 857, 671, 16067, 72115, 46321, 229709, 2700, 24049, 1845, 101, 145, 67, -1, -39; -239, -261868, -102053, -79902, -1913226, -8587295, -5515800, -27353221, -321509, -2863684, -219695, -12026, -17262, -7973, 120, 4641; -3, -6120, -2394, -1880, -45066, -202317, -129960, -644504, -7578, -67524, -5187, -286, -420, -204, 0, 119],101,5]
%
% END OF LATTICE L102.5
%
\hbox{}\par\smallskip%
%
% LATTICE L102.6
%
\leavevmode%
\llap{\spacer}%
${L_{102.6}}=\pdualwitharg{2}({L_{102.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[130305899640, -183522276, 32576729808; -183522276, 258472, -45880928; 32576729808, -45880928, 8144246177],[1428, 4, 1],[-1, 19045, 29944, 35440, 283707, 2548241, 409261, 24357038, 95516, 212920, 65571, 5495, 2838, 3374, 89, -713; 0, -35785, -56259, -66585, -533036, -4787710, -768932, -45762759, -179459, -400044, -123200, -10326, -5341, -6409, -178, 714; 4, -76381, -120092, -142134, -1137822, -10219856, -1641363, -97685196, -383072, -853927, -262976, -22038, -11382, -13532, -357, 2856],101,6]
%
% END OF LATTICE L102.6
%
\hbox{}\par\smallskip%
%
% LATTICE L102.7
%
\leavevmode%
\llap{\spacer}%
${L_{102.7}}=\pdualwitharg{17}(\pfillwitharg{2}({L_{102.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${34}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{238}\above{1pt}{1pt}{l}{2}{51}\above{1pt}{1pt}{}{2}{119}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{17}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1069215\\714\\-254898
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1069215, 714, -254898; 714, -17, -170; -254898, -170, 60767],[357, 17, 1],[-843, -3099, -38498, -42871, -335304, -176355, -961817, -1682284, -111360, -28959, -73705, -5216, -1305, 36, 77, -761; -25, -95, -1183, -1320, -10332, -5435, -29643, -51849, -3433, -893, -2275, -162, -42, 1, 3, -21; -3536, -12999, -161483, -179826, -1406461, -739736, -4034423, -7056483, -467109, -121471, -309162, -21879, -5474, 151, 323, -3192],101,7]
%
% END OF LATTICE L102.7
%
\hbox{}\par\smallskip%
%
% LATTICE L102.8
%
\leavevmode%
\llap{\spacer}%
${L_{102.8}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{102.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31048647, 72114, -10054191; 72114, 147, -23352; -10054191, -23352, 3255754],[357, 21, 1],[4801, 255140, 26287, 67545, 225415, 14096169, 4521206, 6400675, 522677, 2306896, 49067, 7809, 647, -4971, 34, 4415; -21, -1241, -129, -334, -1118, -69955, -22441, -31773, -2597, -11475, -245, -40, -4, 17, 1, -17; 14826, 787899, 81177, 208586, 696105, 43530438, 13961955, 19765951, 1614081, 7123935, 151524, 24115, 1998, -15351, 105, 13634],101,8]
%
% END OF LATTICE L102.8
%
\hbox{}\par\smallskip%
%
% LATTICE L102.9
%
\leavevmode%
\llap{\spacer}%
${L_{102.9}}=\pdualwitharg{7}({L_{102.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21164388, -406980, -55692; -406980, -7826, -1071; -55692, -1071, -146],[1428, 7, 1],[2, 1247, 69, 1129, 1285, 40359, 25921, 55087, 1510, 13441, 147, 167, 11, 33, -1, -7; -103, -63784, -3529, -57738, -65714, -2063919, -1325572, -2817087, -77219, -687344, -7517, -8538, -562, -1683, 52, 357; -7, -7616, -424, -6972, -7950, -249781, -160440, -340986, -9352, -83300, -913, -1050, -72, -238, 0, 51],101,9]
%
% END OF LATTICE L102.9
%
\hbox{}\par\smallskip%
%
% LATTICE L102.10
%
\leavevmode%
\llap{\spacer}%
${L_{102.10}}=\pdualwitharg{2.3}({L_{102.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[158505144, -22549548, 39578448; -22549548, 3216216, -5630604; 39578448, -5630604, 9882667],[1428, 12, 1],[11153, 1139552, 1752473, 678861, 16192109, 145243761, 23318549, 462493589, 5429821, 12074242, 3688501, 98421, 130813, 107973, -89, -10799; -123, -12563, -19320, -7484, -178507, -1601213, -257071, -5098674, -59860, -133110, -40663, -1085, -1442, -1190, 1, 119; -44736, -4570875, -7029372, -2722990, -64948422, -582589524, -93533397, -1855115276, -21779640, -48431181, -14795004, -394778, -524706, -433092, 357, 43316],101,10]
%
% END OF LATTICE L102.10
%
\hbox{}\par\smallskip%
%
% LATTICE L102.11
%
\leavevmode%
\llap{\spacer}%
${L_{102.11}}=\pdualwitharg{3.17}(\pfillwitharg{2}({L_{102.1}}))$%
\nopagebreak\par%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10612182, -65688, -3641757; -65688, 357, 22542; -3641757, 22542, 1249733],[357, 51, 1],[-4568, -14261, -174827, -64168, -1498955, -787692, -4294945, -2503673, -496513, -128905, -326254, -7415, -4288, 279, -35, -1732; -21, -73, -903, -334, -7826, -4115, -22441, -13083, -2597, -675, -1715, -40, -28, 1, 1, -7; -13311, -41556, -509439, -186983, -4367895, -2295303, -12515298, -7295603, -1446819, -375624, -950691, -21607, -12495, 813, -102, -5047],101,11]
%
% END OF LATTICE L102.11
%
\hbox{}\par\smallskip%
%
% LATTICE L102.12
%
\leavevmode%
\llap{\spacer}%
${L_{102.12}}=\pdualwitharg{17}({L_{102.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[704004, 371280, -1428; 371280, 195262, -731; -1428, -731, 2],[1428, 17, 1],[11, 151, 970, 2219, 17533, 4620, 50421, 44110, 2929, 1529, 1970, 301, 107, 0, -21, -10; -23, -316, -2030, -4644, -36694, -9669, -105524, -92316, -6130, -3200, -4123, -630, -224, 0, 44, 21; -544, -7644, -49147, -112506, -889168, -234310, -2557208, -2237151, -148563, -77560, -99960, -15300, -5474, -1, 1088, 546],101,12]
%
% END OF LATTICE L102.12
%
\hbox{}\par\smallskip%
%
% LATTICE L102.13
%
\leavevmode%
\llap{\spacer}%
${L_{102.13}}=\pdualwitharg{3.7}({L_{102.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7781172, -56301756, 8043924; -56301756, 405003018, -57862539; 8043924, -57862539, 8266786],[1428, 21, 1],[-704, -18485, -701, -2565, -7219, -217431, -138035, -97061, -7444, -60667, -463, 237, 235, -237, -1645, -1879; -36605, -961248, -36455, -133402, -375470, -11309029, -7179508, -5048371, -387189, -3155608, -24087, 12318, 12218, -12325, -85532, -97699; -255528, -6710172, -254481, -931238, -2621040, -78944838, -50117928, -35241119, -2702847, -22028328, -168144, 85988, 85290, -86037, -597072, -682006],101,13]
%
% END OF LATTICE L102.13
%
\hbox{}\par\smallskip%
%
% LATTICE L102.14
%
\leavevmode%
\llap{\spacer}%
${L_{102.14}}=\pdualwitharg{2.7}({L_{102.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[27077736, -2437596, 6757296; -2437596, 227416, -608328; 6757296, -608328, 1686295],[1428, 28, 1],[5033, 568578, 125173, 1019917, 1159123, 72790093, 11686649, 99340439, 2721831, 6053806, 264379, 148739, 9533, 56131, -89, -4679; -55, -6205, -1366, -11130, -12649, -794325, -127531, -1084056, -29702, -66062, -2885, -1623, -104, -612, 1, 51; -20188, -2280635, -502084, -4091010, -4649382, -291969832, -46876557, -398466468, -10917592, -24282545, -1060456, -596610, -38238, -225148, 357, 18768],101,14]
%
% END OF LATTICE L102.14
%
\hbox{}\par\smallskip%
%
% LATTICE L102.15
%
\leavevmode%
\llap{\spacer}%
${L_{102.15}}=\pdualwitharg{7.17}(\pfillwitharg{2}({L_{102.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24151407, -151725, 10442607; -151725, 833, -65450; 10442607, -65450, 4514989],[357, 119, 1],[-13282, -41471, -72629, -559814, -622725, -2290670, -12490039, -3120377, -1443901, -374867, -135540, -64693, -1782, 811, -101, -2158; 39825, 124340, 217758, 1678440, 1867057, 6867895, 37447676, 9355524, 4329106, 1123926, 406375, 193959, 5342, -2432, 304, 6471; 31297, 97720, 171139, 1319115, 1467355, 5397609, 29430842, 7352685, 3402329, 883316, 319379, 152439, 4199, -1911, 238, 5085],101,15]
%
% END OF LATTICE L102.15
%
\hbox{}\par\smallskip%
%
% LATTICE L102.16
%
\leavevmode%
\llap{\spacer}%
${L_{102.16}}=\pdualwitharg{3.17}({L_{102.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-50758260, -6344604, 54264; -6344604, -793050, 6783; 54264, 6783, -58],[1428, 51, 1],[636, 329, 411, 17, 1, -1, -1, 3, 8, 39, 233, 167, 3863, 1013, 11041, 3217; -4661, -2412, -3017, -126, -14, 7, 8, -21, -57, -284, -1701, -1222, -28294, -7421, -80888, -23569; 49521, 25512, 31416, 1156, -714, -117, 0, 350, 816, 3252, 18921, 13226, 302736, 79218, 862920, 251335],101,16]
%
% END OF LATTICE L102.16
%
\hbox{}\par\smallskip%
%
% LATTICE L102.17
%
\leavevmode%
\llap{\spacer}%
${L_{102.17}}=\pdualwitharg{2.17}({L_{102.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9352106232, -16766148, 2331948276; -16766148, 30056, -4180640; 2331948276, -4180640, 581471449],[1428, 68, 1],[4611, 6477, 157232, 171616, 1331607, 699261, 1906021, 6665710, 440148, 57060, 287527, 18995, 2670, -354, 89, 5675; -546, -761, -18459, -20133, -156170, -82004, -223520, -781683, -51611, -6690, -33698, -2220, -301, 43, -10, -672; -18496, -25981, -630700, -688398, -5341434, -2804924, -7645563, -26737956, -1765552, -228883, -1153348, -76194, -10710, 1420, -357, -22764],101,17]
%
% END OF LATTICE L102.17
%
\hbox{}\par\smallskip%
%
% LATTICE L102.18
%
\leavevmode%
\llap{\spacer}%
${L_{102.18}}=\pdualwitharg{2.3.7}({L_{102.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[307432709258808, -2619486467556, 76640489122680; -2619486467556, 22319386152, -653016800352; 76640489122680, -653016800352, 19105854374197],[1428, 84, 1],[-29931, -613280, -122883, -306845, -1011709, -63112679, -10114669, -28626627, -2328649, -5114700, -214057, -30095, 889, 104899, 6469, -13273; -8194, -167144, -33475, -83567, -275508, -17186524, -2754363, -7795401, -634105, -1392725, -58282, -8190, 237, 27965, 1653, -3944; 119784, 2454375, 491784, 1228010, 4048914, 252580356, 40479453, 114565312, 9319380, 20469309, 856668, 120442, -3558, -419832, -25893, 53108],101,18]
%
% END OF LATTICE L102.18
%
\hbox{}\par\smallskip%
%
% LATTICE L102.19
%
\leavevmode%
\llap{\spacer}%
${L_{102.19}}=\pdualwitharg{3.7.17}(\pfillwitharg{2}({L_{102.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4704016926, -8399192550, 2810080875; -8399192550, 14997061275, -5017500768; 2810080875, -5017500768, 1678683143],[357, 357, 1],[-25, -95, -169, -440, -1476, -5435, -29643, -2469, -3433, -893, -325, -54, -6, 1, 3, -1; 639109, 2349063, 4168768, 10831919, 36307970, 133674541, 729043225, 60721262, 84409134, 21950435, 7980981, 1317830, 141277, -27302, -58285, 27483; 1910307, 7021392, 12460524, 32376806, 108525195, 399555681, 2179123716, 181496978, 252300468, 65610258, 23855301, 3939019, 422280, -81606, -174216, 82147],101,19]
%
% END OF LATTICE L102.19
%
\hbox{}\par\smallskip%
%
% LATTICE L102.20
%
\leavevmode%
\llap{\spacer}%
${L_{102.20}}=\pdualwitharg{7.17}({L_{102.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-829668, 207060, -4284; 207060, -51646, 1071; -4284, 1071, -22],[1428, 119, 1],[-344, -177, -31, -23, 1, 1, 1, -1, -6, -23, -19, -277, -301, -551, -6001, -749; -1241, -640, -113, -90, 2, 3, 4, -3, -19, -80, -67, -990, -1082, -1983, -21604, -2697; 5593, 2800, 442, 0, -102, -49, 0, 48, 238, 532, 391, 4998, 5100, 9212, 99960, 12447],101,20]
%
% END OF LATTICE L102.20
%
\hbox{}\par\smallskip%
%
% LATTICE L102.21
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L102.21
%
\hbox{}\par\smallskip%
%
% LATTICE L102.22
%
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${L_{102.22}}=\pdualwitharg{3.7.17}({L_{102.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2477580, -278492844, -2340492; -278492844, -31297679490, -263030103; -2340492, -263030103, -2210542],[1428, 357, 1],[352, 2291, 2011, 10323, 34421, 63297, 690229, 28739, 39884, 20701, 3737, 1173, 67, -69, -469, 1; 1419, 9200, 8073, 41430, 138130, 254003, 2769788, 115325, 160043, 83064, 14993, 4702, 266, -277, -1868, 5; -169218, -1097124, -962727, -4940642, -16472388, -30290568, -330304968, -13752829, -19085577, -9905616, -1787958, -560728, -31722, 33033, 222768, -596],101,22]
%
% END OF LATTICE L102.22
%
\hbox{}\par\smallskip%
%
% LATTICE L102.23
%
\leavevmode%
\llap{\spacer}%
${L_{102.23}}=\pdualwitharg{2.7.17}({L_{102.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[77520408, -1560804, -20497512; -1560804, 33320, 414120; -20497512, 414120, 5420903],[1428, 476, 1],[1139379, 148500, 108857, 56785, 2721, 363, -89, 727, 6171, 15150, 54793, 434165, 488229, 1799787, 4909589, 2454023; -3427766, -446755, -327491, -170835, -8186, -1092, 268, -2187, -18565, -45578, -164842, -1306164, -1468813, -5414571, -14770258, -7382808; 4570076, 595637, 436628, 227766, 10914, 1456, -357, 2916, 24752, 60767, 219776, 1741446, 1958298, 7218988, 19692477, 9843144],101,23]
%
% END OF LATTICE L102.23
%
\hbox{}\par\smallskip%
%
% LATTICE L102.24
%
\leavevmode%
\llap{\spacer}%
${L_{102.24}}=\pdualwitharg{2.3.7.17}({L_{102.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}{}{\hbox{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1108153704, 133256971596, 33221700288; 133256971596, 16024329877416, 3994954096056; 33221700288, 3994954096056, 995964158981],[1428, 1428, 1],[-24392, -27794, -94289, -236349, -781290, -2868562, -7816539, -1301447, -1801355, -233027, -166532, -24472, -1173, -469, -8091, -2658; 92765, 105552, 358013, 897267, 2965867, 10889249, 29671987, 4940349, 6837879, 884540, 632079, 92825, 4409, 1763, 30773, 10111; -371280, -422457, -1432896, -3591182, -11870454, -43582644, -118757835, -19773032, -27367620, -3540243, -2529804, -371518, -17646, -7056, -123165, -40468],101,24]
%
% END OF LATTICE L102.24
%

\medskip%
%------------------------------WEYL-GROUP-W103--------------------------------
%
\leavevmode\llap{}%
$W_{103}$%
\qquad\llap{24} lattices, $\chi=48$%
\hfill%
$222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L103.1
%
\leavevmode%
\llap{\spacer}%
${L_{103.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{103}}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{476}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{102}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{238}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51517956\\8568\\-52836
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8568\\-2\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52836\\-5\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-52837\\-69056652\\-62663496
\end{tabular}\endgroup%
\kern3pt%
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-24\\-31369\\-28464
\end{tabular}\endgroup%
\kern3pt%
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71\\92797\\84205
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\-1314\\-1188
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\HardButStrongLineBreak\kern3pt%
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-11\\-14518\\-13090
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\57477\\52173
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\30056\\27276
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
237\\309757\\281078
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-1122\\-564\\-5712\\-588\\-5712\\-18\\-102\\-12\\0
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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\kern3pt%
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-1122\\-9996\\0\\102\\0\\-357
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-5712\\-54026\\-119\\-357\\0\\238
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\kern3pt%
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-588\\-5712\\-18\\-102\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2618\\-119\\-11781\\-5712\\-56882
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[51517956, 8568, -52836; 8568, -2, -5; -52836, -5, 50],[1428, 1, 1],[-1, -11, 0, 44, 23, 237, 25, 249, 1, 7, 1, 1; -1314, -14518, -3, 57477, 30056, 309757, 32682, 325584, 1310, 9180, 1312, 1309; -1188, -13090, -1, 52173, 27276, 281078, 29652, 295358, 1187, 8313, 1188, 1190],102,1]
%
% END OF LATTICE L103.1
%
\hbox{}\par\smallskip%
%
% LATTICE L103.2
%
\leavevmode%
\llap{\spacer}%
${L_{103.2}}=\pfillwitharg{2}({L_{103.1}})$%
{${}={}$}Nikulin $103$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{119}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{102}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{238}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43554\\-14637\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14637\\4919\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47123\\140301\\13923
\end{tabular}\endgroup%
\kern3pt%
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-15884\\-47292\\-4693
\end{tabular}\endgroup%
\kern3pt%
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572\\1703\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\3\\0
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\HardButStrongLineBreak\kern3pt%
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40\\119\\0
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\HardButStrongLineBreak\kern3pt%
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-1\\-3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1867\\-5559\\-561
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-487\\-1450\\-145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10032\\-29869\\-2975
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[43554, -14637, 0; -14637, 4919, 0; 0, 0, -1],[357, 1, 1],[1, 40, -1, -1867, -487, -10032, -529, -5276, -43, -377, -41, -440; 3, 119, -3, -5559, -1450, -29869, -1575, -15708, -128, -1122, -122, -1309; 0, 0, -1, -561, -145, -2975, -156, -1547, -12, -102, -11, -119],102,2]
%
% END OF LATTICE L103.2
%
\hbox{}\par\smallskip%
%
% LATTICE L103.3
%
\leavevmode%
\llap{\spacer}%
${L_{103.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{103.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22601670, 123879, -7555548; 123879, 669, -41415; -7555548, -41415, 2525755],[357, 3, 1],[-2343, -191984, -11030, -251116, -176368, -3453549, -56325, -1553389, -3637, 1777, 364, -37197; 2308, 189091, 10863, 247299, 173686, 3401020, 55468, 1529745, 3581, -1751, -358, 36652; -6971, -571200, -32817, -747133, -524739, -10275174, -167581, -4621722, -10821, 5287, 1083, -110670],102,3]
%
% END OF LATTICE L103.3
%
\hbox{}\par\smallskip%
%
% LATTICE L103.4
%
\leavevmode%
\llap{\spacer}%
${L_{103.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{103.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{17}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{714}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{34}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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34028526\\230979\\-14601300
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\kern3pt%
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230979\\1561\\-99113
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[34028526, 230979, -14601300; 230979, 1561, -99113; -14601300, -99113, 6265271],[357, 7, 1],[-9139, -35664, -14344, -979754, -229374, -641643, -219761, -288611, -4731, 6929, 474, -6907; 6924, 27013, 10863, 741897, 173686, 485860, 166404, 218535, 3581, -5253, -358, 5236; -21189, -82688, -33257, -2271591, -531811, -1487670, -509523, -669154, -10969, 16065, 1099, -16014],102,4]
%
% END OF LATTICE L103.4
%
\hbox{}\par\smallskip%
%
% LATTICE L103.5
%
\leavevmode%
\llap{\spacer}%
${L_{103.5}}=\pdualwitharg{3}({L_{103.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{1428}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{34}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{714}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[82502700, 7140, -117096; 7140, -6, -9; -117096, -9, 166],[1428, 3, 1],[-1, -11, 1, 77, 117, 1189, 41, 1201, 4, 8, 3, 1; -122, -1428, 118, 9248, 14060, 142919, 4930, 144466, 483, 969, 364, 119; -712, -7854, 711, 54791, 83256, 846090, 29176, 854658, 2847, 5695, 2136, 714],102,5]
%
% END OF LATTICE L103.5
%
\hbox{}\par\smallskip%
%
% LATTICE L103.6
%
\leavevmode%
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${L_{103.6}}=\pdualwitharg{2}({L_{103.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[62567977080, -194452188, 15598031148; -194452188, 604328, -48476416; 15598031148, -48476416, 3888547897],[1428, 4, 1],[887, -66393, -6729, -654725, -79228, -3153163, -158099, -1492626, -6372, -13426, 622, 35125; 0, 119, 11, 1173, 143, 5712, 288, 2737, 13, 51, 1, 0; -3558, 266322, 26992, 2626296, 317807, 12648272, 634182, 5987366, 25560, 53856, -2495, -140896],102,6]
%
% END OF LATTICE L103.6
%
\hbox{}\par\smallskip%
%
% LATTICE L103.7
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[65544486, 498729, -15438108; 498729, 3791, -117470; -15438108, -117470, 3636235],[357, 17, 1],[-5206, -8368, -8175, -32851, -130747, -150603, -125270, -67743, -2698, 232, 271, -1619; 6924, 11123, 10863, 43641, 173686, 200060, 166404, 89985, 3581, -309, -358, 2156; -21879, -35168, -34357, -138063, -549491, -632940, -526473, -284704, -11339, 975, 1139, -6804],102,7]
%
% END OF LATTICE L103.7
%
\hbox{}\par\smallskip%
%
% LATTICE L103.8
%
\leavevmode%
\llap{\spacer}%
${L_{103.8}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{103.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[341768595, -22225035, -110665359; -22225035, 1445262, 7196511; -110665359, 7196511, 35833666],[357, 21, 1],[14063, 150363, 57314, 1241706, 867101, 2418670, 274905, 1077751, 14928, -13592, -197, 37558; -160, -1717, -656, -14246, -9951, -27761, -3156, -12376, -173, 153, 3, -425; 43463, 464712, 177135, 3837631, 2679873, 7475172, 849625, 3330912, 46137, -42007, -609, 116076],102,8]
%
% END OF LATTICE L103.8
%
\hbox{}\par\smallskip%
%
% LATTICE L103.9
%
\leavevmode%
\llap{\spacer}%
${L_{103.9}}=\pdualwitharg{7}({L_{103.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16569084, 327012, -54264; 327012, 6454, -1071; -54264, -1071, 178],[1428, 7, 1],[133, 173, 1, 1, 3, 8, 11, 31, 5, 305, 141, 196; -6924, -9010, -53, -51, -152, -408, -564, -1598, -259, -15861, -7336, -10200; -1134, -1496, -14, 0, 0, -17, -42, -170, -35, -2499, -1176, -1649],102,9]
%
% END OF LATTICE L103.9
%
\hbox{}\par\smallskip%
%
% LATTICE L103.10
%
\leavevmode%
\llap{\spacer}%
${L_{103.10}}=\pdualwitharg{2.3}({L_{103.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[64245522936, -93015636, 16016136696; -93015636, 134664, -23188404; 16016136696, -23188404, 3992755027],[1428, 12, 1],[177, -241186, -22296, -726550, -263918, -10507165, -175709, -4980891, -21587, -19461, 620, 63011; 0, 119, 11, 391, 143, 5712, 96, 2737, 13, 17, 1, 0; -710, 967470, 89436, 2914412, 1058655, 42147420, 704822, 19979862, 86592, 78064, -2487, -252756],102,10]
%
% END OF LATTICE L103.10
%
\hbox{}\par\smallskip%
%
% LATTICE L103.11
%
\leavevmode%
\llap{\spacer}%
${L_{103.11}}=\pdualwitharg{3.17}(\pfillwitharg{2}({L_{103.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[144241566, 22501353, -49491981; 22501353, 3509922, -7720635; -49491981, -7720635, 16981625],[357, 51, 1],[-34559, -152466, -141325, -180349, -2141329, -2459656, -679025, -1096306, -37071, 1953, 577, -37888; -160, -707, -656, -838, -9951, -11431, -3156, -5096, -173, 9, 3, -175; -100793, -444675, -412182, -525998, -6245307, -7173726, -1980415, -3197439, -108120, 5696, 1683, -110502],102,11]
%
% END OF LATTICE L103.11
%
\hbox{}\par\smallskip%
%
% LATTICE L103.12
%
\leavevmode%
\llap{\spacer}%
${L_{103.12}}=\pdualwitharg{17}({L_{103.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-173637660, 88536, 165648; 88536, -34, -85; 165648, -85, -158],[1428, 17, 1],[-1, -11, -8, -44, -361, -211, -359, -199, -7, -1, 1, 1; -54, -546, -395, -2163, -17740, -10367, -17634, -9772, -342, -48, 52, 49; -1020, -11242, -8177, -44979, -369036, -215698, -366996, -203434, -7157, -1023, 1020, 1022],102,12]
%
% END OF LATTICE L103.12
%
\hbox{}\par\smallskip%
%
% LATTICE L103.13
%
\leavevmode%
\llap{\spacer}%
${L_{103.13}}=\pdualwitharg{3.7}({L_{103.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1986348, -124236, -17136; -124236, -7770, -1071; -17136, -1071, -146],[1428, 21, 1],[-3, -145, -42, -1310, -1897, -2693, -629, -2541, -37, -33, 1, 7; 50, 2448, 710, 22168, 32104, 45577, 10646, 43010, 627, 561, -16, -119; -14, -918, -273, -8687, -12600, -17901, -4186, -16932, -252, -238, 0, 51],102,13]
%
% END OF LATTICE L103.13
%
\hbox{}\par\smallskip%
%
% LATTICE L103.14
%
\leavevmode%
\llap{\spacer}%
${L_{103.14}}=\pdualwitharg{2.7}({L_{103.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6543096, 875364, -1633632; 875364, -92456, 218484; -1633632, 218484, -407873],[1428, 28, 1],[-122030, -153981, 705, 26699, -89, -13358, -12826, -37065, -11943, -599837, -67339, -368734; 1401, 1768, -8, -306, 1, 153, 147, 425, 137, 6885, 773, 4233; 489510, 617678, -2828, -107100, 357, 53584, 51450, 148682, 47908, 2406180, 270123, 1479136],102,14]
%
% END OF LATTICE L103.14
%
\hbox{}\par\smallskip%
%
% LATTICE L103.15
%
\leavevmode%
\llap{\spacer}%
${L_{103.15}}=\pdualwitharg{7.17}(\pfillwitharg{2}({L_{103.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[721557690, 76873881, 208334490; 76873881, 8189818, 22195999; 208334490, 22195999, 60151793],[357, 119, 1],[-283073, -59464, -385811, -1476953, -5845373, -959188, -5560735, -427520, -101173, 16001, 1565, -14780; 848739, 178291, 1156777, 4428345, 17526168, 2875931, 16672737, 1281832, 303346, -47976, -4692, 44315; 667233, 140163, 909398, 3481338, 13778177, 2260910, 13107255, 1007711, 238476, -37716, -3689, 34838],102,15]
%
% END OF LATTICE L103.15
%
\hbox{}\par\smallskip%
%
% LATTICE L103.16
%
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${L_{103.16}}=\pdualwitharg{3.17}({L_{103.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-61416852, 84252, 111384; 84252, -102, -153; 111384, -153, -202],[1428, 51, 1],[-1, -11, -7, -11, -267, -155, -87, -143, -4, 0, 3, 1; -10, -84, -50, -72, -1732, -1001, -558, -910, -21, 1, 28, 7; -544, -6006, -3825, -6017, -146064, -84798, -47600, -78246, -2193, -1, 1632, 546],102,16]
%
% END OF LATTICE L103.16
%
\hbox{}\par\smallskip%
%
% LATTICE L103.17
%
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${L_{103.17}}=\pdualwitharg{2.17}({L_{103.1}})$%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40316309880, -93024204, 10050733812; -93024204, 214744, -23190652; 10050733812, -23190652, 2505617465],[1428, 68, 1],[12943, 67203, 92915, 528415, 1087064, 2545261, 2170229, 1206314, 89016, 17378, 4522, 1773; 0, 7, 11, 69, 143, 336, 288, 161, 13, 3, 1, 0; -51918, -269570, -372708, -2119620, -4360517, -10209752, -8705394, -4838862, -357068, -69708, -18139, -7112],102,17]
%
% END OF LATTICE L103.17
%
\hbox{}\par\smallskip%
%
% LATTICE L103.18
%
\leavevmode%
\llap{\spacer}%
${L_{103.18}}=\pdualwitharg{2.3.7}({L_{103.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}17\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2413320, 672588, 599760; 672588, 213864, 167076; 599760, 167076, 149053],[1428, 84, 1],[1651, 48700, 26472, 795138, 286967, 1627117, 189601, 764121, 21341, 17683, -89, -1475; -19, -561, -305, -9163, -3307, -18751, -2185, -8806, -246, -204, 1, 17; -6622, -195330, -106176, -3189200, -1150989, -6526164, -760466, -3064794, -85596, -70924, 357, 5916],102,18]
%
% END OF LATTICE L103.18
%
\hbox{}\par\smallskip%
%
% LATTICE L103.19
%
\leavevmode%
\llap{\spacer}%
${L_{103.19}}=\pdualwitharg{3.7.17}(\pfillwitharg{2}({L_{103.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[270917121930, -484037287251, 161942326077; -484037287251, 864810956478, -289336176360; 161942326077, -289336176360, 96801991607],[357, 357, 1],[2308, 1589, 10863, 14547, 173686, 28580, 55468, 12855, 3581, -103, -358, 308; 1594900, 1098568, 7512167, 10061973, 120139303, 19769139, 38368616, 8892311, 2478712, -71094, -248755, 212619; 4763213, 3280905, 22435308, 30050384, 358799637, 59041128, 114589027, 26557155, 7402752, -212324, -742917, 634992],102,19]
%
% END OF LATTICE L103.19
%
\hbox{}\par\smallskip%
%
% LATTICE L103.20
%
\leavevmode%
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${L_{103.20}}=\pdualwitharg{7.17}({L_{103.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4071228, -1186668, 2856; -1186668, 345814, -833; 2856, -833, 2],[1428, 119, 1],[5, 1, 3, 7, 51, 4, 43, 3, -1, -1, -3, 0; 12, 2, 5, 3, 8, 0, -12, -2, -5, -3, -8, 0; -2142, -584, -2142, -8400, -66640, -5473, -63546, -4890, -595, 189, 952, -1],102,20]
%
% END OF LATTICE L103.20
%
\hbox{}\par\smallskip%
%
% LATTICE L103.21
%
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${L_{103.21}}=\pdualwitharg{2.3.17}({L_{103.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5751832632, -12362196, 1433900496; -12362196, 27336, -3081828; 1433900496, -3081828, 357463571],[1428, 204, 1],[2297, 57430, 83760, 167874, 1040554, 2441419, 696059, 1164693, 90829, 6715, 6980, 2827; 0, 7, 11, 23, 143, 336, 96, 161, 13, 1, 1, 0; -9214, -230370, -335988, -673396, -4173993, -9793308, -2792114, -4671954, -364344, -26936, -27999, -11340],102,21]
%
% END OF LATTICE L103.21
%
\hbox{}\par\smallskip%
%
% LATTICE L103.22
%
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${L_{103.22}}=\pdualwitharg{3.7.17}({L_{103.1}})$%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1428, 0, 0; 0, -52122, 1071; 0, 1071, -22],[1428, 357, 1],[-47, -11, -2, 0, 1, 0, -1, -1, -4, -6, -145, -12; -2, -2, -7, -1, 0, 1, 10, 4, 8, 2, 24, 1; -476, -186, -357, -49, 0, 48, 476, 186, 357, 49, 0, -48],102,22]
%
% END OF LATTICE L103.22
%
\hbox{}\par\smallskip%
%
% LATTICE L103.23
%
\leavevmode%
\llap{\spacer}%
${L_{103.23}}=\pdualwitharg{2.7.17}({L_{103.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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887059320\\-2217684\\-222809412
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-2217684\\4760\\556444
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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3749\\-11283\\15038
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40703\\-122501\\163268
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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163526\\-492155\\655936
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[887059320, -2217684, -222809412; -2217684, 4760, 556444; -222809412, 556444, 55964287],[1428, 476, 1],[-178, 3749, 40703, 237077, 488525, 163526, 977050, 77657, 40703, 7141, -89, -714; 537, -11283, -122501, -713517, -1470286, -492155, -2940573, -233720, -122502, -21492, 268, 2149; -714, 15038, 163268, 950964, 1959573, 655936, 3919146, 311498, 163268, 28644, -357, -2864],102,23]
%
% END OF LATTICE L103.23
%
\hbox{}\par\smallskip%
%
% LATTICE L103.24
%
\leavevmode%
\llap{\spacer}%
${L_{103.24}}=\pdualwitharg{2.3.7.17}({L_{103.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}17\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${476}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{2856}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{l}{2}{357}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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31416\\-1529388\\-381276
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-1529388\\74435928\\18556860
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-49\\355\\-1428
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-1\\-7\\28
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0\\-89\\357
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-2\\-15\\60
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31416, -1529388, -381276; -1529388, 74435928, 18556860; -381276, 18556860, 4626221],[1428, 1428, 1],[-568, -133, -49, -1, 0, -2, -16, -13, -97, -145, -876, -290; 4201, 983, 355, -7, -89, -15, 59, 82, 710, 1072, 6479, 2145; -16898, -3954, -1428, 28, 357, 60, -238, -330, -2856, -4312, -26061, -8628],102,24]
%
% END OF LATTICE L103.24
%

\medskip%
%------------------------------WEYL-GROUP-W104--------------------------------
%
\leavevmode\llap{}%
$W_{104}$%
\qquad\llap{24} lattices, $\chi=60$%
\hfill%
$22222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L104.1
%
\leavevmode%
\llap{\spacer}%
${L_{104.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{104}}%
\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{154}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{220}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-103180\\-30800\\1540
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-30800\\-9194\\461
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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125\\-418\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-87\\2
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-103180, -30800, 1540; -30800, -9194, 461; 1540, 461, 34],[1540, 1, 1],[-25, 125, 26, -23, -251, -251, -1249, -297, -631, -95, -1748, -1129, -554, -1281; 84, -418, -87, 77, 840, 840, 4180, 994, 2112, 318, 5852, 3780, 1855, 4290; -14, 0, 2, 0, -20, -21, -110, -28, -66, -11, -231, -160, -84, -220],103,1]
%
% END OF LATTICE L104.1
%
\hbox{}\par\smallskip%
%
% LATTICE L104.2
%
\leavevmode%
\llap{\spacer}%
${L_{104.2}}=\pfillwitharg{2}({L_{104.1}})$%
{${}={}$}Nikulin $104$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[5005, -770, -385; -770, 118, 55; -385, 55, -9],[385, 1, 1],[1, -16, 7, 1009, 511, 696, 1289, 216, 357, 86, 1378, 419, 389, 416; 7, -110, 48, 6930, 3510, 4781, 8855, 1484, 2453, 591, 9471, 2880, 2674, 2860; 0, 11, -5, -693, -350, -476, -880, -147, -242, -58, -924, -280, -259, -275],103,2]
%
% END OF LATTICE L104.2
%
\hbox{}\par\smallskip%
%
% LATTICE L104.3
%
\leavevmode%
\llap{\spacer}%
${L_{104.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{104.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10317230, 87780, -4118345; 87780, 745, -35040; -4118345, -35040, 1643926],[385, 5, 1],[43058, 42231, 3147, 7127, -66, -662, 1722, 6789, 36100, 18216, 541190, 42792, 249897, 76609; -40971, -40183, -2994, -6776, 63, 630, -1639, -6461, -34353, -17334, -514976, -40719, -237790, -72897; 106995, 104940, 7820, 17710, -164, -1645, 4279, 16870, 89705, 45265, 1344805, 106334, 620970, 190366],103,3]
%
% END OF LATTICE L104.3
%
\hbox{}\par\smallskip%
%
% LATTICE L104.4
%
\leavevmode%
\llap{\spacer}%
${L_{104.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{104.1}}))$%
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{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17191790, 133980, -2446675; 133980, 1043, -19068; -2446675, -19068, 348202],[385, 7, 1],[2277, 15633, 1165, 377, -122, -35, 3187, 359, 13363, 6743, 28619, 79202, 13215, 141793; -5853, -40183, -2994, -968, 315, 90, -8195, -923, -34353, -17334, -73568, -203595, -33970, -364485; 15679, 107646, 8022, 2596, -840, -241, 21945, 2472, 92015, 46431, 197065, 545370, 90996, 976360],103,4]
%
% END OF LATTICE L104.4
%
\hbox{}\par\smallskip%
%
% LATTICE L104.5
%
\leavevmode%
\llap{\spacer}%
${L_{104.5}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{104.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{tabular}\endgroup%
\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31255070, 226380, -14196875; 226380, 1639, -102828; -14196875, -102828, 6448594],[385, 11, 1],[52655, 4695, 3849, 793, -402, -809, 957, 8301, 4013, 22275, 60163, 261642, 305589, 42583; -40971, -3653, -2994, -616, 315, 630, -745, -6461, -3123, -17334, -46816, -203595, -237790, -33135; 115269, 10278, 8426, 1736, -880, -1771, 2095, 18172, 8785, 48763, 131705, 572770, 668976, 93220],103,5]
%
% END OF LATTICE L104.5
%
\hbox{}\par\smallskip%
%
% LATTICE L104.6
%
\leavevmode%
\llap{\spacer}%
${L_{104.6}}=\pdualwitharg{2}({L_{104.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3112583320\\-6894580\\778185100
\end{tabular}\endgroup%
\kern3pt%
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-6894580\\15272\\-1723732
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778185100\\-1723732\\194556093
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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27172144\\-31558065\\-108962700
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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3774\\-4375\\-15134
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2979\\-3454\\-11946
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-1\\0\\4
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\HardButStrongLineBreak\kern3pt%
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-461\\462\\1848
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\HardButStrongLineBreak\kern3pt%
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283\\-345\\-1135
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1550\\-1841\\-6216
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3112583320, -6894580, 778185100; -6894580, 15272, -1723732; 778185100, -1723732, 194556093],[1540, 4, 1],[3774, 2979, -1, -461, 283, 1550, 5294, 1777, 5744, 2380, 61522, 11612, 26205, 38321; -4375, -3454, 0, 462, -345, -1841, -6215, -2072, -6677, -2763, -71379, -13470, -30394, -44440; -15134, -11946, 4, 1848, -1135, -6216, -21230, -7126, -23034, -9544, -246708, -46565, -105084, -153670],103,6]
%
% END OF LATTICE L104.6
%
\hbox{}\par\smallskip%
%
% LATTICE L104.7
%
\leavevmode%
\llap{\spacer}%
${L_{104.7}}=\pdualwitharg{5}({L_{104.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{*}{2}{220}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{770}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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18382980\\9073680\\58520
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9073680\\4478690\\28885
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\kern3pt%
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58520\\28885\\186
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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-355741\\721644\\-152460
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\kern3pt%
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{$\left[\!\llap{\phantom{%
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41\\-83\\-10
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18382980, 9073680, 58520; 9073680, 4478690, 28885; 58520, 28885, 186],[1540, 5, 1],[-27, 207, 41, -38, -79, -394, -391, -463, -977, -146, -2657, -341, -831, -379; 56, -418, -83, 77, 160, 798, 792, 938, 1980, 296, 5390, 692, 1687, 770; -210, -220, -10, 0, 8, 35, 22, 0, -110, -35, -1155, -188, -560, -352],103,7]
%
% END OF LATTICE L104.7
%
\hbox{}\par\smallskip%
%
% LATTICE L104.8
%
\leavevmode%
\llap{\spacer}%
${L_{104.8}}=\pdualwitharg{7}({L_{104.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{308}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{1540}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5822740, 579040, -12320; 579040, -57582, 1225; -12320, 1225, -26],[1540, 7, 1],[1, -5, -4, -23, -137, -12, -273, -5, -39, -2, 1, 17, 2, 53; 10, -66, -45, -253, -1500, -131, -2970, -54, -418, -21, 11, 180, 21, 550; -4, -770, -231, -1045, -5880, -496, -10780, -178, -1232, -42, 44, 420, 41, 770],103,8]
%
% END OF LATTICE L104.8
%
\hbox{}\par\smallskip%
%
% LATTICE L104.9
%
\leavevmode%
\llap{\spacer}%
${L_{104.9}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{104.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16123030, 2060520, -6602365; 2060520, 263095, -843780; -6602365, -843780, 2703662],[385, 35, 1],[12069, 88362, 8462, 5152, 292, -679, -377, 976, 55133, 30673, 138343, 77978, 65904, 143872; 87, 649, 66, 44, 3, -6, -11, 5, 363, 210, 968, 549, 466, 1023; 29500, 215985, 20685, 12595, 714, -1660, -924, 2385, 134750, 74970, 338140, 190596, 161085, 351659],103,9]
%
% END OF LATTICE L104.9
%
\hbox{}\par\smallskip%
%
% LATTICE L104.10
%
\leavevmode%
\llap{\spacer}%
${L_{104.10}}=\pdualwitharg{11}({L_{104.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1566180, 15400, 3080; 15400, -22, -33; 3080, -33, -6],[1540, 11, 1],[45, 5, 4, 3, 11, 1, -1, -3, 1, 5, 18, 169, 104, 31; 434, 48, 38, 28, 100, 7, -10, -28, 10, 49, 175, 1640, 1008, 300; 20636, 2294, 1837, 1379, 5060, 462, -460, -1386, 456, 2288, 8246, 77440, 47663, 14210],103,10]
%
% END OF LATTICE L104.10
%
\hbox{}\par\smallskip%
%
% LATTICE L104.11
%
\leavevmode%
\llap{\spacer}%
${L_{104.11}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{104.1}}))$%
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{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240799790, -9988440, -95833045; -9988440, 413435, 3975180; -95833045, 3975180, 38139454],[385, 55, 1],[-563217, -53838, -57718, -22996, -2180, 33707, 425, -41828, -32597, -201587, -582049, -516430, -3058932, -86860; 609, 59, 66, 28, 3, -42, -1, 35, 33, 210, 616, 549, 3262, 93; -1415260, -135285, -145035, -57785, -5478, 84700, 1068, -105105, -81910, -506550, -1462580, -1297692, -7686525, -218263],103,11]
%
% END OF LATTICE L104.11
%
\hbox{}\par\smallskip%
%
% LATTICE L104.12
%
\leavevmode%
\llap{\spacer}%
${L_{104.12}}=\pdualwitharg{7.11}(\pfillwitharg{2}({L_{104.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[689161935, 19988045, 83478395; 19988045, 578809, 2421804; 83478395, 2421804, 10111306],[385, 77, 1],[49097, 32853, 35223, 2005, 6654, -2939, -1299, 3645, 19889, 123003, 50737, 1575606, 266649, 265009; -245398, -164206, -176049, -10021, -33255, 14689, 6490, -18220, -99412, -614805, -253597, -7875285, -1332779, -1324580; -346566, -231903, -248633, -14153, -46970, 20746, 9170, -25729, -140392, -868252, -358142, -11121880, -1882221, -1870645],103,12]
%
% END OF LATTICE L104.12
%
\hbox{}\par\smallskip%
%
% LATTICE L104.13
%
\leavevmode%
\llap{\spacer}%
${L_{104.13}}=\pdualwitharg{2.5}({L_{104.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L104.13
%
\hbox{}\par\smallskip%
%
% LATTICE L104.14
%
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\llap{\spacer}%
${L_{104.14}}=\pdualwitharg{2.7}({L_{104.1}})$%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[92008840, -768460, 22932140; -768460, 7112, -191520; 22932140, -191520, 5715571],[1540, 28, 1],[-2129, -48193, -20129, -75485, -100357, -32517, -335431, -4839, -28455, -775, -1163, -4163, -2323, -47809; -120, -2717, -1135, -4257, -5660, -1834, -18920, -273, -1606, -44, -66, -235, -131, -2695; 8538, 193270, 80724, 302720, 402465, 130404, 1345190, 19406, 114114, 3108, 4664, 16695, 9316, 191730],103,14]
%
% END OF LATTICE L104.14
%
\hbox{}\par\smallskip%
%
% LATTICE L104.15
%
\leavevmode%
\llap{\spacer}%
${L_{104.15}}=\pdualwitharg{5.7}({L_{104.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3278660\\-1684760\\7700
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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11439\\-24640\\-523600
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3278660, -1684760, 7700; -1684760, -865690, 3955; 7700, 3955, -18],[1540, 35, 1],[-1, 91, 31, 148, 169, 72, 317, 27, 195, 8, -5, -13, -7, -31; 2, -198, -67, -319, -364, -155, -682, -58, -418, -17, 11, 28, 15, 66; 10, -4620, -1470, -6820, -7728, -3275, -14322, -1200, -8470, -315, 275, 588, 300, 1232],103,15]
%
% END OF LATTICE L104.15
%
\hbox{}\par\smallskip%
%
% LATTICE L104.16
%
\leavevmode%
\llap{\spacer}%
${L_{104.16}}=\pdualwitharg{2.11}({L_{104.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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6971854120\\-4214980\\1738396660
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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5332\\13\\-21384
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4950\\7\\-19852
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8755\\0\\-35112
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0\\0\\0
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}}\!\right]$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6971854120, -4214980, 1738396660; -4214980, 2552, -1050984; 1738396660, -1050984, 433460439],[1540, 44, 1],[29145, 3238, 5332, 4950, 6377, 8755, 571, 192, 953, 7241, 24387, 56488, 137930, 20384; 112, 11, 13, 7, 5, 0, 0, 7, 6, 38, 112, 250, 595, 85; -116886, -12986, -21384, -19852, -25575, -35112, -2290, -770, -3822, -29040, -97804, -226545, -553168, -81750],103,16]
%
% END OF LATTICE L104.16
%
\hbox{}\par\smallskip%
%
% LATTICE L104.17
%
\leavevmode%
\llap{\spacer}%
${L_{104.17}}=\pdualwitharg{5.11}({L_{104.1}})$%
\nopagebreak\par%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-34392820, 127820, 43120; 127820, -110, -165; 43120, -165, -54],[1540, 55, 1],[-3, 13, 49, 158, 289, 874, 51, 353, 37, 26, 17, 11, 1, -1; -28, 130, 489, 1575, 2880, 8708, 508, 3514, 368, 258, 168, 108, 7, -10; -2310, 9980, 37620, 121310, 221892, 671055, 39158, 271040, 28410, 19965, 13055, 8448, 770, -768],103,17]
%
% END OF LATTICE L104.17
%
\hbox{}\par\smallskip%
%
% LATTICE L104.18
%
\leavevmode%
\llap{\spacer}%
${L_{104.18}}=\pdualwitharg{7.11}({L_{104.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-388879260, 260260, 344960; 260260, -154, -231; 344960, -231, -306],[1540, 77, 1],[-3, -11, -31, -12, -731, -61, -119, -21, -13, -5, 0, 11, 1, -1; -22, -74, -203, -77, -4660, -387, -750, -130, -78, -25, 1, 100, 7, -10; -3366, -12348, -34804, -13474, -820820, -68497, -133630, -23584, -14602, -5621, -1, 12320, 1122, -1120],103,18]
%
% END OF LATTICE L104.18
%
\hbox{}\par\smallskip%
%
% LATTICE L104.19
%
\leavevmode%
\llap{\spacer}%
${L_{104.19}}=\pdualwitharg{5.7.11}(\pfillwitharg{2}({L_{104.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[315223438915, -760572982015, 305385595515; -760572982015, 1835114999410, -736836175075; 305385595515, -736836175075, 295854782438],[385, 385, 1],[-5853, -3653, -2994, -88, 63, 90, -149, -923, -3123, -17334, -6688, -40719, -33970, -6627; -10464101, -6531214, -5354223, -157569, 111915, 160784, -266269, -1649698, -5582585, -30987174, -11956230, -72794709, -60729779, -11847530; -26055150, -16262435, -13331780, -392340, 278663, 400345, -662998, -4107675, -13900390, -77156695, -29770485, -181255613, -151214470, -29499827],103,19]
%
% END OF LATTICE L104.19
%
\hbox{}\par\smallskip%
%
% LATTICE L104.20
%
\leavevmode%
\llap{\spacer}%
${L_{104.20}}=\pdualwitharg{2.5.7}({L_{104.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1482068280, -6062980, 369635420; -6062980, 24920, -1512140; 369635420, -1512140, 92188967],[1540, 140, 1],[1713, 43199, 19409, 78389, 21404, 35387, 75147, 5897, 40511, 3421, 5321, 1426, 2663, 8563; -101, -2552, -1148, -4642, -1268, -2097, -4455, -350, -2409, -205, -319, -85, -158, -506; -6870, -173250, -77840, -314380, -85841, -141920, -301378, -23650, -162470, -13720, -21340, -5719, -10680, -34342],103,20]
%
% END OF LATTICE L104.20
%
\hbox{}\par\smallskip%
%
% LATTICE L104.21
%
\leavevmode%
\llap{\spacer}%
${L_{104.21}}=\pdualwitharg{2.5.11}({L_{104.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[86970141720, -162366820, 21685765640; -162366820, 303160, -40485720; 21685765640, -40485720, 5407286019],[1540, 220, 1],[960, 18913, 141365, 461911, 212138, 2574302, 75456, 528569, 56734, 85576, 67618, 14803, 73343, 191; 7, 6, 38, 112, 50, 595, 17, 112, 11, 13, 7, 1, 0, 0; -3850, -75850, -566940, -1852480, -850773, -10324160, -302614, -2119810, -227530, -343200, -271180, -59367, -294140, -766],103,21]
%
% END OF LATTICE L104.21
%
\hbox{}\par\smallskip%
%
% LATTICE L104.22
%
\leavevmode%
\llap{\spacer}%
${L_{104.22}}=\pdualwitharg{2.7.11}({L_{104.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L104.22
%
\hbox{}\par\smallskip%
%
% LATTICE L104.23
%
\leavevmode%
\llap{\spacer}%
${L_{104.23}}=\pdualwitharg{5.7.11}({L_{104.1}})$%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3469094860, 1650880, 2427040; 1650880, -770, -1155; 2427040, -1155, -1698],[1540, 385, 1],[23, 19, 17, 2, 11, 1, -1, -3, 1, 13, 8, 109, 49, 21; 126, 90, 59, 5, 20, -1, -2, 2, 38, 133, 59, 748, 321, 130; 32780, 27090, 24255, 2855, 15708, 1430, -1428, -4290, 1400, 18480, 11390, 155232, 69795, 29918],103,23]
%
% END OF LATTICE L104.23
%
\hbox{}\par\smallskip%
%
% LATTICE L104.24
%
\leavevmode%
\llap{\spacer}%
${L_{104.24}}=\pdualwitharg{2.5.7.11}({L_{104.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[89320, -52620260, -13120800; -52620260, 53898592440, 13439550740; -13120800, 13439550740, 3351136197],[1540, 1540, 1],[16, 11, 13, 1, 1, 0, 0, 1, 6, 38, 16, 50, 85, 17; 2834585, 1953097, 2326255, 183037, 191557, 19529, 2489, 175953, 1057441, 6705359, 2827501, 8840057, 15033161, 3008049; -11367950, -7832790, -9329320, -734060, -768229, -78320, -9982, -705650, -4240810, -26891480, -11339540, -35452571, -60289680, -12063618],103,24]
%
% END OF LATTICE L104.24
%

\medskip%
%------------------------------WEYL-GROUP-W105--------------------------------
%
\leavevmode\llap{}%
$W_{105}$%
\qquad\llap{32} lattices, $\chi=72$%
\hfill%
$2222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L105.1
%
\leavevmode%
\llap{\spacer}%
${L_{105.1}}$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-23541960, 3623880, 34320; 3623880, -557834, -5283; 34320, -5283, -50],[1560, 1, 1],[-13, -107, -172, -1317, -10685, -2302, -201073, -12427, -3299, -1089, -104, -63, -79, 2, 145, 7; -84, -689, -1107, -8475, -68757, -14813, -1293864, -79965, -21228, -7007, -669, -405, -507, 13, 936, 45; -48, -650, -1104, -8590, -69888, -15074, -1317576, -81455, -21660, -7189, -705, -455, -663, -1, 624, 50],104,1]
%
% END OF LATTICE L105.1
%
\hbox{}\par\smallskip%
%
% LATTICE L105.2
%
\leavevmode%
\llap{\spacer}%
${L_{105.2}}=\pfillwitharg{2}({L_{105.1}})$%
{${}={}$}Nikulin $105$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[15990, -2730, -390; -2730, 466, 65; -390, 65, -16],[390, 1, 1],[737, 566, 94, 138, 548, 76, 2429, 253, 1, -33, 29, 477, 4249, 949, 42343, 5282; 4350, 3341, 555, 815, 3237, 449, 14352, 1495, 6, -195, 171, 2815, 25077, 5601, 249912, 31175; -342, -260, -42, -60, -234, -32, -1014, -105, 0, 13, -15, -225, -1989, -443, -19734, -2460],104,2]
%
% END OF LATTICE L105.2
%
\hbox{}\par\smallskip%
%
% LATTICE L105.3
%
\leavevmode%
\llap{\spacer}%
${L_{105.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{105.1}}))$%
\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-62589946380, 323471850, -31295383470; 323471850, -1671736, 161738044; -31295383470, 161738044, -15647896877],[390, 2, 1],[-12202, 61521, 208355, 1849771, 15372793, 3343945, 73443035, 18201391, 1224808, 1690021, 195847, 171239, 391085, 24405, 121729, -12301; 12318, -62075, -210279, -1866905, -15515253, -3374939, -74123829, -18370120, -1236165, -1705704, -197670, -172840, -394758, -24636, -122889, 12415; 24531, -123682, -418878, -3718790, -30905550, -6722686, -147650295, -36592180, -2462361, -3397628, -393732, -344260, -786240, -49064, -244725, 24730],104,3]
%
% END OF LATTICE L105.3
%
\hbox{}\par\smallskip%
%
% LATTICE L105.4
%
\leavevmode%
\llap{\spacer}%
${L_{105.4}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{105.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-94044979470, 485620980, -31348737030; 485620980, -2507604, 161875779; -31348737030, 161875779, -10449715858],[390, 3, 1],[-5383, 40721, 45965, 1224211, 3391321, 2213075, 32403815, 12045956, 540397, 1118476, 43204, 113324, 86270, 16150, 53701, -8141; 8212, -62075, -70093, -1866905, -5171751, -3374939, -49415886, -18370120, -824110, -1705704, -65890, -172840, -131586, -24636, -81926, 12415; 16276, -123123, -138979, -3701505, -10253945, -6691419, -97975670, -36421965, -1633936, -3381807, -130631, -342645, -260845, -48831, -162370, 24615],104,4]
%
% END OF LATTICE L105.4
%
\hbox{}\par\smallskip%
%
% LATTICE L105.5
%
\leavevmode%
\llap{\spacer}%
${L_{105.5}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{105.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-156955167330, 809919240, -62782477680; 809919240, -4179340, 323969815; -62782477680, 323969815, -25113155372],[390, 5, 1],[-19217, 48482, 164138, 291430, 12109780, 2634156, 115707715, 2867583, 1929647, 1331273, 154267, 26975, 308015, 19219, 191705, -1938; 24636, -62075, -210279, -373381, -15515253, -3374939, -148247658, -3674024, -2472330, -1705704, -197670, -34568, -394758, -24636, -245778, 2483; 48360, -122005, -413055, -733387, -30474405, -6628885, -291179850, -7216307, -4855980, -3350165, -388215, -67883, -775125, -48365, -482430, 4877],104,5]
%
% END OF LATTICE L105.5
%
\hbox{}\par\smallskip%
%
% LATTICE L105.6
%
\leavevmode%
\llap{\spacer}%
${L_{105.6}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{105.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30416530960140, -142097841210, 15161324312550; -142097841210, 663842856, -70829624112; 15161324312550, -70829624112, 7557264015793],[390, 6, 1],[-30942, -14964951, -9140713, -218173133, -593974651, -384942231, -2806551379, -2082929548, -46262631, -185640532, -6290478, -13294732, -7841798, -837544, -1925383, -281237; -64, -30979, -18923, -451665, -1229657, -796915, -5810181, -4312125, -95774, -384319, -13023, -27525, -16237, -1735, -3991, -585; 62075, 30022278, 18337850, 437693010, 1191615802, 772260642, 5630427101, 4178716650, 92810833, 372426990, 12619786, 26671530, 15732002, 1680258, 3862651, 564210],104,6]
%
% END OF LATTICE L105.6
%
\hbox{}\par\smallskip%
%
% LATTICE L105.7
%
\leavevmode%
\llap{\spacer}%
${L_{105.7}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{105.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20890370747220, -152030275410, 10415339930190; -152030275410, 1106404760, -75797936630; 10415339930190, -75797936630, 5192789882671],[390, 10, 1],[-167125, -26952139, -49388575, -78588421, -3209348153, -693303471, -15164292664, -750295810, -249965312, -334350270, -33988904, -4789058, -42373528, -1508854, -10408514, -101501; -192, -30979, -56769, -90333, -3688971, -796915, -17430543, -862425, -287322, -384319, -39069, -5505, -48711, -1735, -11973, -117; 335205, 54058290, 99059370, 157625918, 6437035410, 1390568670, 30415238295, 1504879018, 501358995, 670611110, 68172030, 9605482, 84989190, 3026330, 20876505, 203582],104,7]
%
% END OF LATTICE L105.7
%
\hbox{}\par\smallskip%
%
% LATTICE L105.8
%
\leavevmode%
\llap{\spacer}%
${L_{105.8}}=\pdualwitharg{3}({L_{105.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-229059480, 88110360, 195000; 88110360, -33892662, -75009; 195000, -75009, -166],[1560, 3, 1],[-17, -66, -7, 38, 198, 153, 4895, 943, 101, 157, 14, 67, 75, 22, 201, 2; -44, -169, -17, 125, 585, 443, 14040, 2695, 284, 429, 37, 175, 195, 57, 520, 5; -88, -1170, -544, -11910, -31928, -20562, -597376, -110655, -9740, -9477, -275, -375, -13, 87, 1144, 90],104,8]
%
% END OF LATTICE L105.8
%
\hbox{}\par\smallskip%
%
% LATTICE L105.9
%
\leavevmode%
\llap{\spacer}%
${L_{105.9}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{105.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-408596168370, 2107112280, 94291012530; 2107112280, -10866284, -486254561; 94291012530, -486254561, -21759369598],[390, 13, 1],[10855, -2111, -92821, -823929, -526709, -1489417, -5032595, -8106964, -1091053, -57900, -87212, -76236, -13390, -10858, -8329, 5479; 24636, -4775, -210279, -1866905, -1193481, -3374939, -11403666, -18370120, -2472330, -131208, -197670, -172840, -30366, -24636, -18906, 12415; 46488, -9041, -397527, -3528655, -2255745, -6378749, -21553170, -34719815, -4672668, -247969, -373503, -326495, -57345, -46501, -35670, 23465],104,9]
%
% END OF LATTICE L105.9
%
\hbox{}\par\smallskip%
%
% LATTICE L105.10
%
\leavevmode%
\llap{\spacer}%
${L_{105.10}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{105.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{130}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{130}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20776739087130, -185691203490, 6901277359470; -185691203490, 1659607140, -61679866755; 6901277359470, -61679866755, 2292353433934],[390, 15, 1],[-90725, -21946749, -13405475, -63993451, -871109091, -564547311, -8232047491, -610955375, -135695503, -272256745, -9225563, -3899663, -11501391, -1228639, -5650341, -82651; -128, -30979, -18923, -90333, -1229657, -796915, -11620362, -862425, -191548, -384319, -13023, -5505, -16237, -1735, -7982, -117; 273130, 66071265, 40357535, 192653967, 2622496825, 1699584525, 24782795440, 1839297222, 408514880, 819636090, 27773800, 11740038, 34625240, 3698850, 17010500, 248823],104,10]
%
% END OF LATTICE L105.10
%
\hbox{}\par\smallskip%
%
% LATTICE L105.11
%
\leavevmode%
\llap{\spacer}%
${L_{105.11}}=\pdualwitharg{5}({L_{105.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{b}{2}{130}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{1560}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-148200, -45240, 6240; -45240, -13810, 1905; 6240, 1905, -262],[1560, 5, 1],[-61, -267, -388, -577, -23285, -5006, -436705, -5395, -7139, -2333, -212, -23, -115, 6, -95, -5; 192, 832, 1206, 1792, 72306, 15544, 1355952, 16751, 22164, 7241, 657, 71, 351, -19, 312, 16; -60, -325, -495, -747, -30225, -6505, -567840, -7017, -9300, -3055, -285, -33, -195, 5, 0, -3],104,11]
%
% END OF LATTICE L105.11
%
\hbox{}\par\smallskip%
%
% LATTICE L105.12
%
\leavevmode%
\llap{\spacer}%
${L_{105.12}}=\pdualwitharg{2.13}(\pfillwitharg{2}({L_{105.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[654992319540, -43407195390, 324345182070; -43407195390, 2876652376, -21494778292; 324345182070, -21494778292, 160612260623],[390, 26, 1],[-129116, -1602227, -38168699, -303676653, -190790461, -535804911, -901492679, -2899251968, -193180543, -19876624, -26267854, -18506112, -2519198, -1166364, -619063, -392917; -192, -2383, -56769, -451665, -283767, -796915, -1340811, -4312125, -287322, -29563, -39069, -27525, -3747, -1735, -921, -585; 260715, 3235266, 77071410, 613193230, 385249962, 1081913742, 1820321721, 5854258670, 390076089, 40135490, 53040858, 37368110, 5086842, 2355158, 1250031, 793390],104,12]
%
% END OF LATTICE L105.12
%
\hbox{}\par\smallskip%
%
% LATTICE L105.13
%
\leavevmode%
\llap{\spacer}%
${L_{105.13}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{105.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-53343792246243720, -1156568806770, -26589713810548200; -1156568806770, -25076040, -576502574700; -26589713810548200, -576502574700, -13253892360392563],[390, 30, 1],[5716534, -86532119, -97652967, -520153961, -7204646325, -4701529487, -34419831755, -5118157837, -574016404, -2376100651, -91780323, -48145909, -183251985, -34302763, -57027025, 3459007; 4106, -62075, -70093, -373381, -5171751, -3374939, -24707943, -3674024, -412055, -1705704, -65890, -34568, -131586, -24636, -40963, 2483; -11468405, 173599140, 195909580, 1043523276, 14453828480, 9432121680, 69052431175, 10267953786, 1151581115, 4766889270, 184127990, 96589434, 367636750, 68817570, 114406565, -6939396],104,13]
%
% END OF LATTICE L105.13
%
\hbox{}\par\smallskip%
%
% LATTICE L105.14
%
\leavevmode%
\llap{\spacer}%
${L_{105.14}}=\pdualwitharg{2}({L_{105.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{208}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{s}{2}{624}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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356069\\40786200\\255334560
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{$\left.\llap{\phantom{%
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0\\0\\0
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5\\559\\3536
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6835920, -57720, 18720; -57720, -400, 144; 18720, 144, -49],[1560, 8, 1],[-1, 5, 17, 151, 1255, 273, 2998, 1486, 100, 138, 16, 14, 32, 2, 5, -1; -117, 559, 1941, 17285, 143715, 31267, 343395, 170215, 11457, 15821, 1839, 1615, 3705, 233, 585, -115; -726, 3536, 12168, 108240, 899808, 195752, 2149797, 1065600, 71718, 99008, 11496, 10080, 23088, 1448, 3627, -720],104,14]
%
% END OF LATTICE L105.14
%
\hbox{}\par\smallskip%
%
% LATTICE L105.15
%
\leavevmode%
\llap{\spacer}%
${L_{105.15}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{105.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
137873579610\\-24391009500\\44777479110
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[137873579610, -24391009500, 44777479110; -24391009500, 4314978564, -7921517097; 44777479110, -7921517097, 14542471742],[390, 39, 1],[-32279, -600837, -4771103, -113879123, -23848887, -200927511, -225373921, -1087223113, -48295297, -7453759, -3283493, -6939817, -314901, -437389, -154767, -147347; -128, -2383, -18923, -451665, -94589, -796915, -893874, -4312125, -191548, -29563, -13023, -27525, -1249, -1735, -614, -585; 99320, 1848729, 14680315, 350397255, 73381181, 618238413, 693458126, 3345301440, 148600894, 22934640, 10103054, 21353280, 968926, 1345812, 476206, 453375],104,15]
%
% END OF LATTICE L105.15
%
\hbox{}\par\smallskip%
%
% LATTICE L105.16
%
\leavevmode%
\llap{\spacer}%
${L_{105.16}}=\pdualwitharg{13}({L_{105.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-946920, 4680, 1560; 4680, 26, -13; 1560, -13, -2],[1560, 13, 1],[-1, -1, -22, -171, -107, -300, -2017, -1621, -431, -11, -14, -9, -1, 0, 1, 1; -60, -53, -1155, -8955, -5601, -15701, -105552, -84825, -22548, -575, -729, -465, -51, 1, 48, 45; -468, -463, -10179, -79105, -49497, -138775, -933024, -749840, -199368, -5088, -6474, -4160, -462, 0, 456, 455],104,16]
%
% END OF LATTICE L105.16
%
\hbox{}\par\smallskip%
%
% LATTICE L105.17
%
\leavevmode%
\llap{\spacer}%
${L_{105.17}}=\pdualwitharg{3.5}({L_{105.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{b}{2}{390}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{130}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{520}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-903435000\\-456727440\\884520
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\kern3pt%
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-456727440\\-230896470\\447165
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
884520\\447165\\-866
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-903435000, -456727440, 884520; -456727440, -230896470, 447165; 884520, 447165, -866],[1560, 15, 1],[-12737, -12502, -381, -126, -220, 29, -145, -24, -103, -1408, -689, -3084, -41510, -26779, -778815, -28866; 27772, 27261, 831, 275, 481, -63, 312, 52, 224, 3068, 1502, 6724, 90506, 58388, 1698112, 62939; 1330720, 1306890, 39940, 13302, 23660, -2910, 13000, 2337, 10460, 146055, 71825, 321993, 4335175, 2797035, 81351920, 3015318],104,17]
%
% END OF LATTICE L105.17
%
\hbox{}\par\smallskip%
%
% LATTICE L105.18
%
\leavevmode%
\llap{\spacer}%
${L_{105.18}}=\pdualwitharg{5.13}(\pfillwitharg{2}({L_{105.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${390}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{390}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{s}{2}{130}\above{1pt}{1pt}{l}{2}{30}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25875467670, 13641362280, 10746283860; 13641362280, 7191630940, 5665364315; 10746283860, 5665364315, 4463015636],[390, 65, 1],[62075, 385144, 9175000, 14599568, 45862238, 128796858, 433401863, 139384515, 92873527, 4777945, 6314267, 889699, 605563, 280367, 297613, 18888; -384, -2383, -56769, -90333, -283767, -796915, -2681622, -862425, -574644, -29563, -39069, -5505, -3747, -1735, -1842, -117; -148980, -924345, -22019985, -35038939, -110069295, -309112245, -1040163750, -334522604, -222896310, -11467060, -15154230, -2135276, -1453350, -672880, -714270, -45331],104,18]
%
% END OF LATTICE L105.18
%
\hbox{}\par\smallskip%
%
% LATTICE L105.19
%
\leavevmode%
\llap{\spacer}%
${L_{105.19}}=\pdualwitharg{2.3.13}(\pfillwitharg{2}({L_{105.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${13}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{s}{2}{780}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{780}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2669820252302580, -417212363490, -1330761221741250; -417212363490, -65197704, -207957833148; -1330761221741250, -207957833148, -663312606068867],[390, 78, 1],[744094, -868351, -12726403, -338896577, -72214829, -612622603, -344998595, -3334531237, -74794708, -23815187, -11957135, -31356493, -1835785, -4465847, -570937, 2253607; 4106, -4775, -70093, -1866905, -397827, -3374939, -1900611, -18370120, -412055, -131208, -65890, -172840, -10122, -24636, -3151, 12415; -1492829, 1742118, 25532182, 679906890, 144880070, 1229066202, 692148985, 6689860320, 150055919, 47778912, 23988848, 62908560, 3683020, 8959548, 1145435, -4521270],104,19]
%
% END OF LATTICE L105.19
%
\hbox{}\par\smallskip%
%
% LATTICE L105.20
%
\leavevmode%
\llap{\spacer}%
${L_{105.20}}=\pdualwitharg{2.3}({L_{105.1}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-227760, -88920, 7800; -88920, -18672, 1704; 7800, 1704, -155],[1560, 24, 1],[70, 294, 12, 34, 28, 6, 6, -1, -1, 5, 11, 311, 869, 569, 2087, 3106; 3559, 14963, 613, 1745, 1443, 311, 312, -50, -52, 234, 556, 15790, 44148, 28914, 106067, 157865; 42598, 179088, 7336, 20880, 17264, 3720, 3731, -600, -622, 2808, 6656, 189000, 528424, 346080, 1269541, 1889520],104,20]
%
% END OF LATTICE L105.20
%
\hbox{}\par\smallskip%
%
% LATTICE L105.21
%
\leavevmode%
\llap{\spacer}%
${L_{105.21}}=\pdualwitharg{2.5.13}(\pfillwitharg{2}({L_{105.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-494369631319980, 231774913110, -246493390013130; 231774913110, -108662840, 115563296030; -246493390013130, 115563296030, -122901949211437],[390, 130, 1],[-1385969, 538941, 23699827, 42075031, 134485745, 380296349, 642492460, 413994313, 139290809, 14783819, 22268453, 3893249, 3419095, 2772651, 1063460, -279793; 12318, -4775, -210279, -373381, -1193481, -3374939, -5701833, -3674024, -1236165, -131208, -197670, -34568, -30366, -24636, -9453, 2483; 2779725, -1080910, -47532810, -84386458, -269727090, -762729370, -1288594725, -830314628, -279364215, -29650700, -44662020, -7808372, -6857400, -5560880, -2132895, 561158],104,21]
%
% END OF LATTICE L105.21
%
\hbox{}\par\smallskip%
%
% LATTICE L105.22
%
\leavevmode%
\llap{\spacer}%
${L_{105.22}}=\pdualwitharg{3.13}({L_{105.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1215240, -237120, 7800; -237120, 46254, -1521; 7800, -1521, 50],[1560, 39, 1],[-1, -1, -6, -131, -27, -226, -505, -1216, -107, -8, -3, -4, 0, 1, 1, 1; -12, -13, -81, -1795, -371, -3109, -6952, -16745, -1476, -111, -43, -65, -1, 9, 8, 5; -208, -240, -1534, -34320, -7106, -59592, -133312, -321165, -28340, -2139, -845, -1365, -31, 117, 88, 0],104,22]
%
% END OF LATTICE L105.22
%
\hbox{}\par\smallskip%
%
% LATTICE L105.23
%
\leavevmode%
\llap{\spacer}%
${L_{105.23}}=\pdualwitharg{2.5}({L_{105.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2542800, 32760, -1560; 32760, 80, -80; -1560, -80, 19],[1560, 40, 1],[1474, 1932, 178, 20, 110, -4, 5, 3, 11, 213, 317, 475, 19195, 4129, 45040, 4452; 149973, 196573, 18111, 2035, 11193, -407, 507, 305, 1119, 21671, 32253, 48329, 1953003, 420107, 4582617, 452971; 752190, 985920, 90840, 10208, 56160, -2040, 2535, 1528, 5610, 108680, 161760, 242392, 9795240, 2107040, 22984065, 2271872],104,23]
%
% END OF LATTICE L105.23
%
\hbox{}\par\smallskip%
%
% LATTICE L105.24
%
\leavevmode%
\llap{\spacer}%
${L_{105.24}}=\pdualwitharg{3.5.13}(\pfillwitharg{2}({L_{105.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-497551068406470, -284777055810, 165094144476300; -284777055810, -162994260, 94492861905; 165094144476300, 94492861905, -54780460280498],[390, 195, 1],[755261, -440531, -6457419, -34392161, -36642915, -310854509, -350115865, -338399248, -75904271, -12084304, -6067416, -3182344, -931590, -2266366, -579515, 228703; -1502310, 876287, 12844745, 68410941, 72888003, 618334079, 696430508, 673124472, 150984432, 24037400, 12068942, 6330120, 1853058, 4508096, 1152728, -454923; 2273570, -1326135, -19438835, -103531077, -110306545, -935768535, -1053957400, -1018686747, -228495410, -36377505, -18264805, -9579843, -2804375, -6822465, -1744520, 688467],104,24]
%
% END OF LATTICE L105.24
%
\hbox{}\par\smallskip%
%
% LATTICE L105.25
%
\leavevmode%
\llap{\spacer}%
${L_{105.25}}=\pdualwitharg{5.13}({L_{105.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2862600, -864240, 21840; -864240, -259610, 6565; 21840, 6565, -166],[1560, 65, 1],[-25, -11, -220, -333, -1037, -2902, -19489, -3131, -4151, -105, -128, -15, -7, 2, 1, -1; -528, -236, -4734, -7172, -22338, -62516, -419856, -67453, -89436, -2263, -2763, -325, -153, 41, 24, -20; -24180, -10785, -216255, -327587, -1020285, -2855385, -19176600, -3080857, -4084860, -103355, -126165, -14833, -6975, 1885, 1080, -923],104,25]
%
% END OF LATTICE L105.25
%
\hbox{}\par\smallskip%
%
% LATTICE L105.26
%
\leavevmode%
\llap{\spacer}%
${L_{105.26}}=\pdualwitharg{2.3.5.13}(\pfillwitharg{2}({L_{105.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1444639183667940, 733755306706110, 361271782080690; 733755306706110, 372686035523880, 183495706238520; 361271782080690, 183495706238520, 90345950742809],[390, 390, 1],[-64, -2383, -18923, -90333, -94589, -796915, -446937, -862425, -95774, -29563, -13023, -5505, -1249, -1735, -307, -117; 3612766, 134492375, 1067970737, 5098172693, 5338368223, 44975894171, 25224006019, 48673106590, 5405243045, 1668459562, 734981200, 310683118, 70487588, 97904254, 17321099, 6595709; -7337395, -273149070, -2169009310, -10354201338, -10842029630, -91344387810, -51229029025, -98853290328, -10977849935, -3388580040, -1492719800, -630985992, -143157700, -198839940, -35178515, -13395642],104,26]
%
% END OF LATTICE L105.26
%
\hbox{}\par\smallskip%
%
% LATTICE L105.27
%
\leavevmode%
\llap{\spacer}%
${L_{105.27}}=\pdualwitharg{2.13}({L_{105.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-321360, 276120, -4680; 276120, -216944, 3640; -4680, 3640, -61],[1560, 104, 1],[-5, -7, -131, -969, -601, -1679, -1408, -9044, -598, -60, -70, -36, -2, 4, 1, -1; -60, -86, -1620, -12010, -7452, -20822, -17463, -112175, -7419, -745, -873, -455, -27, 47, 12, -10; -3198, -4600, -86736, -643240, -399144, -1115296, -935391, -6008600, -397410, -39912, -46800, -24440, -1464, 2496, 639, -520],104,27]
%
% END OF LATTICE L105.27
%
\hbox{}\par\smallskip%
%
% LATTICE L105.28
%
\leavevmode%
\llap{\spacer}%
${L_{105.28}}=\pdualwitharg{2.3.5}({L_{105.1}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3120, 10920, -4680; 10920, -37200, 15960; -4680, 15960, -6847],[1560, 120, 1],[824, 3232, 98, 32, 54, -8, 6, 7, 7, 369, 179, 799, 10749, 6933, 25201, 7472; -5871, -23075, -707, -237, -429, 49, -26, -39, -45, -2561, -1265, -5679, -76479, -49349, -179426, -53205; -14290, -56160, -1720, -576, -1040, 120, -65, -96, -110, -6240, -3080, -13824, -186160, -120120, -436735, -129504],104,28]
%
% END OF LATTICE L105.28
%
\hbox{}\par\smallskip%
%
% LATTICE L105.29
%
\leavevmode%
\llap{\spacer}%
${L_{105.29}}=\pdualwitharg{3.5.13}({L_{105.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5904600, -1059240, 35880; -1059240, -189930, 6435; 35880, 6435, -218],[1560, 195, 1],[-1051, -80, -33, -12, -2, 1, 1, 0, -5, -8, -55, -252, -262, -2201, -4929, -2376; -5012, -383, -161, -61, -11, 1, 8, 4, -16, -36, -258, -1196, -1246, -10476, -23472, -11317; -321620, -24525, -10205, -3783, -655, 195, 400, 117, -1300, -2385, -16705, -76947, -80075, -672945, -1507360, -726687],104,29]
%
% END OF LATTICE L105.29
%
\hbox{}\par\smallskip%
%
% LATTICE L105.30
%
\leavevmode%
\llap{\spacer}%
${L_{105.30}}=\pdualwitharg{2.3.13}({L_{105.1}})$%
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\end{tabular}\endgroup%
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\kern3pt%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3120, -1560, 0; -1560, -3938064, 19344; 0, 19344, -95],[1560, 312, 1],[-3, -7, -33, -649, -131, -1087, -302, -5804, -126, -36, -10, 4, 2, 12, 1, -1; -6, -16, -82, -1680, -342, -2848, -793, -15255, -333, -97, -31, -15, 3, 23, 2, 0; -1222, -3264, -16744, -343200, -69872, -581880, -162023, -3116880, -68042, -19824, -6344, -3120, 608, 4680, 407, 0],104,30]
%
% END OF LATTICE L105.30
%
\hbox{}\par\smallskip%
%
% LATTICE L105.31
%
\leavevmode%
\llap{\spacer}%
${L_{105.31}}=\pdualwitharg{2.5.13}({L_{105.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${780}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{*}{2}{3120}\above{1pt}{1pt}{*}{2}{208}\above{1pt}{1pt}{s}{2}{240}\above{1pt}{1pt}{*}{2}{1040}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{208}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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132341\\-5274360\\-13060320
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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271\\-10083\\-24960
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397\\-15583\\-38584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1229\\-48705\\-120600
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-71760, -422760, 170040; -422760, -2488720, 1001000; 170040, 1001000, -402617],[1560, 520, 1],[13, 15, 271, 397, 1229, 3431, 2876, 3694, 1220, 122, 140, 14, 4, -6, 1, 5; -159, -469, -10083, -15583, -48705, -136501, -114690, -147461, -48981, -4991, -6297, -797, -435, 1, 30, 41; -390, -1160, -24960, -38584, -120600, -338000, -283995, -365144, -121290, -12360, -15600, -1976, -1080, 0, 75, 104],104,31]
%
% END OF LATTICE L105.31
%
\hbox{}\par\smallskip%
%
% LATTICE L105.32
%
\leavevmode%
\llap{\spacer}%
${L_{105.32}}=\pdualwitharg{2.3.5.13}({L_{105.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${260}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{*}{2}{1040}\above{1pt}{1pt}{*}{2}{624}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{*}{2}{3120}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{624}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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11\\961\\4810
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39\\3283\\16440
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237\\19727\\98800
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1045\\86649\\433992
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1079\\89405\\447800
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
260\\0\\-2080\\-12480\\-13520\\-115440\\-32630\\-126360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\240\\0\\-3120\\-3840\\-34320\\-9960\\-39000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2080\\0\\1040\\0\\-1040\\-12480\\-4160\\-17160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12480\\-3120\\0\\624\\0\\-3120\\-1560\\-7176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13520\\-3840\\-1040\\0\\80\\0\\-280\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-115440\\-34320\\-12480\\-3120\\0\\3120\\0\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32630\\-9960\\-4160\\-1560\\-280\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126360\\-39000\\-17160\\-7176\\-1560\\-1560\\0\\624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14300\\-4680\\-2600\\-1560\\-520\\-1560\\-130\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4680\\-1800\\-1560\\-1560\\-840\\-4680\\-960\\-3120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2600\\-1560\\-2600\\-4680\\-3640\\-26520\\-6760\\-24960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-1560\\-4680\\-14664\\-14040\\-113880\\-31200\\-119184
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-520\\-840\\-3640\\-14040\\-14120\\-117000\\-32480\\-124800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\-4680\\-26520\\-113880\\-117000\\-978120\\-273000\\-1051440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130\\-960\\-6760\\-31200\\-32480\\-273000\\-76445\\-294840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3120\\-24960\\-119184\\-124800\\-1051440\\-294840\\-1137864
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4308720, -3714360, 751920; -3714360, -3198000, 647400; 751920, 647400, -131059],[1560, 1560, 1],[11, 39, 237, 1045, 1079, 9039, 2526, 9734, 1072, 322, 124, 38, 4, -14, 1, 13; 961, 3283, 19727, 86649, 89405, 748747, 209205, 806115, 88739, 26617, 10173, 3051, 295, -1247, 95, 1185; 4810, 16440, 98800, 433992, 447800, 3750240, 1047845, 4037592, 444470, 133320, 50960, 15288, 1480, -6240, 475, 5928],104,32]
%
% END OF LATTICE L105.32
%

\medskip%
%------------------------------WEYL-GROUP-W106--------------------------------
%
\leavevmode\llap{}%
$W_{106}$%
\qquad\llap{24} lattices, $\chi=27$%
\hfill%
$22242222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L106.1
%
\leavevmode%
\llap{\spacer}%
${L_{106.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{106}}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{l}{2}{76}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{4}{2}\above{1pt}{1pt}{b}{2}{114}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{532}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165273780\\78204\\111720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78204\\-34\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
111720\\-55\\-74
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\2808\\3951
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\9120\\12844
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1407\\-1974
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-702\\-988
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\703\\987
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\23883\\33573
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\9130\\12838
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
275\\193116\\271586
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-21\\-18\\-9\\-57\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\76\\0\\-76\\-76\\-912\\-228\\-3724
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\42\\0\\-21\\-399\\-126\\-2394
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-76\\0\\4\\-2\\-114\\-48\\-1064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-76\\-21\\-2\\2\\0\\-8\\-266
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-912\\-399\\-114\\0\\114\\0\\-798
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-228\\-126\\-48\\-8\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3724\\-2394\\-1064\\-266\\-798\\0\\532
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-165273780, 78204, 111720; 78204, -34, -55; 111720, -55, -74],[1596, 1, 1],[4, 13, -2, -1, 1, 34, 13, 275; 2808, 9120, -1407, -702, 703, 23883, 9130, 193116; 3951, 12844, -1974, -988, 987, 33573, 12838, 271586],105,1]
%
% END OF LATTICE L106.1
%
\hbox{}\par\smallskip%
%
% LATTICE L106.2
%
\leavevmode%
\llap{\spacer}%
${L_{106.2}}=\pfillwitharg{2}({L_{106.1}})$%
{${}={}$}Nikulin $106$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{l}{2}{19}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{s}{2}{114}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{133}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23541\\-11172\\399
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11172\\5302\\-189
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
399\\-189\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-42\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\19\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
130\\273\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\170\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\235\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1114\\2337\\-228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
123\\258\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1839\\3857\\-399
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-21\\-9\\-9\\-57\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\19\\0\\-19\\-38\\-456\\-57\\-931
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\42\\0\\-21\\-399\\-63\\-1197
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-19\\0\\1\\-1\\-57\\-12\\-266
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-38\\-21\\-1\\2\\0\\-4\\-133
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-456\\-399\\-57\\0\\114\\0\\-399
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-57\\-63\\-12\\-4\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-931\\-1197\\-266\\-133\\-399\\0\\133
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[23541, -11172, 399; -11172, 5302, -189; 399, -189, 10],[399, 1, 1],[-20, 9, 130, 81, 112, 1114, 123, 1839; -42, 19, 273, 170, 235, 2337, 258, 3857; 3, 0, -21, -15, -22, -228, -26, -399],105,2]
%
% END OF LATTICE L106.2
%
\hbox{}\par\smallskip%
%
% LATTICE L106.3
%
\leavevmode%
\llap{\spacer}%
${L_{106.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{106.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{57}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{4}{6}\above{1pt}{1pt}{s}{2}{38}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{399}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-814530570\\-165390687\\-270711126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165390687\\-33578934\\-54967950
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270711126\\-54967950\\-89971471
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77471\\-1825\\234214
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-403266\\-9500\\1219173
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29123\\686\\-88046
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25809\\608\\-78027
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54976\\-1295\\166206
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-636382\\-14991\\1923940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-368174\\-8673\\1113081
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7830873\\-184471\\23674665
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-7\\-9\\-9\\-19\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\57\\0\\-57\\-114\\-456\\-171\\-2793
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\14\\0\\-21\\-133\\-63\\-1197
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-57\\0\\3\\-3\\-57\\-36\\-798
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-114\\-21\\-3\\6\\0\\-12\\-399
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-456\\-133\\-57\\0\\38\\0\\-399
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-171\\-63\\-36\\-12\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2793\\-1197\\-798\\-399\\-399\\0\\399
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-814530570, -165390687, -270711126; -165390687, -33578934, -54967950; -270711126, -54967950, -89971471],[399, 3, 1],[-77471, -403266, 29123, 25809, -54976, -636382, -368174, -7830873; -1825, -9500, 686, 608, -1295, -14991, -8673, -184471; 234214, 1219173, -88046, -78027, 166206, 1923940, 1113081, 23674665],105,3]
%
% END OF LATTICE L106.3
%
\hbox{}\par\smallskip%
%
% LATTICE L106.4
%
\leavevmode%
\llap{\spacer}%
${L_{106.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{106.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{l}{2}{133}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{4}{14}\above{1pt}{1pt}{s}{2}{798}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{19}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1145422866\\-299579175\\-163011450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-299579175\\-78350846\\-42634746
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-163011450\\-42634746\\-23199059
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95537\\-5475\\681366
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165766\\-9500\\1182237
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5131\\294\\-36594
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10609\\608\\-75663
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22600\\-1295\\161182
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-784802\\-44973\\5597172
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-151346\\-8673\\1079393
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-459863\\-26353\\3279723
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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42\\0\\-21\\-63\\-63\\-399\\-21\\0
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0\\133\\0\\-133\\-266\\-3192\\-399\\-931
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-21\\0\\6\\0\\-21\\-399\\-63\\-171
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-63\\-133\\0\\7\\-7\\-399\\-84\\-266
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-266\\-21\\-7\\14\\0\\-28\\-133
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-399\\-3192\\-399\\-399\\0\\798\\0\\-399
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-399\\-63\\-84\\-28\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-931\\-171\\-266\\-133\\-399\\0\\19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1145422866, -299579175, -163011450; -299579175, -78350846, -42634746; -163011450, -42634746, -23199059],[399, 7, 1],[-95537, -165766, 5131, 10609, -22600, -784802, -151346, -459863; -5475, -9500, 294, 608, -1295, -44973, -8673, -26353; 681366, 1182237, -36594, -75663, 161182, 5597172, 1079393, 3279723],105,4]
%
% END OF LATTICE L106.4
%
\hbox{}\par\smallskip%
%
% LATTICE L106.5
%
\leavevmode%
\llap{\spacer}%
${L_{106.5}}=\pdualwitharg{3}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{228}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{4}{6}\above{1pt}{1pt}{b}{2}{38}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{1596}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-273607068\\-3560676\\81396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3560676\\-46338\\1059
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81396\\1059\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\1310\\161
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-215\\16568\\2052
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-77\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-1310\\-162
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\77\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\6935\\836
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-123\\9478\\1152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2803\\215992\\26334
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2\\0\\-7\\-18\\-9\\-19\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\228\\0\\-228\\-228\\-912\\-684\\-11172
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\14\\0\\-21\\-133\\-126\\-2394
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-228\\0\\12\\-6\\-114\\-144\\-3192
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-228\\-21\\-6\\6\\0\\-24\\-798
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-912\\-133\\-114\\0\\38\\0\\-798
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-684\\-126\\-144\\-24\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11172\\-2394\\-3192\\-798\\-798\\0\\1596
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-273607068, -3560676, 81396; -3560676, -46338, 1059; 81396, 1059, -22],[1596, 3, 1],[-17, -215, 1, 17, -1, -90, -123, -2803; 1310, 16568, -77, -1310, 77, 6935, 9478, 215992; 161, 2052, -7, -162, 6, 836, 1152, 26334],105,5]
%
% END OF LATTICE L106.5
%
\hbox{}\par\smallskip%
%
% LATTICE L106.6
%
\leavevmode%
\llap{\spacer}%
${L_{106.6}}=\pdualwitharg{2}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{l}{2}{19}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{4}{8}\above{1pt}{1pt}{*}{2}{456}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{532}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
196524784680\\-351677004\\-49254832704
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-351677004\\629320\\88140496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49254832704\\88140496\\12344695091
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23134\\39\\-92304
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25424\\38\\-101441
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5179\\0\\-20664
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-599\\2\\-2390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7183\\17\\-28660
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-196687\\399\\-784776
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36901\\71\\-147234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-782072\\1463\\-3120446
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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24\\0\\-84\\-36\\-36\\-228\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\19\\0\\-38\\-76\\-912\\-114\\-1862
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\kern3pt%
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-84\\0\\168\\0\\-84\\-1596\\-252\\-4788
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\kern3pt%
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-36\\-38\\0\\4\\-4\\-228\\-48\\-1064
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-76\\-84\\-4\\8\\0\\-16\\-532
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-228\\-912\\-1596\\-228\\0\\456\\0\\-1596
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-114\\-252\\-48\\-16\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1862\\-4788\\-1064\\-532\\-1596\\0\\532
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[196524784680, -351677004, -49254832704; -351677004, 629320, 88140496; -49254832704, 88140496, 12344695091],[1596, 4, 1],[-23134, -25424, -5179, -599, -7183, -196687, -36901, -782072; 39, 38, 0, 2, 17, 399, 71, 1463; -92304, -101441, -20664, -2390, -28660, -784776, -147234, -3120446],105,6]
%
% END OF LATTICE L106.6
%
\hbox{}\par\smallskip%
%
% LATTICE L106.7
%
\leavevmode%
\llap{\spacer}%
${L_{106.7}}=\pdualwitharg{19}(\pfillwitharg{2}({L_{106.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${114}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{798}\above{1pt}{1pt}{l}{2}{19}\above{1pt}{1pt}{}{4}{38}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{19}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2318205162\\-702144639\\1098632934
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-702144639\\-212666582\\332757108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1098632934\\332757108\\-520658975
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
294319\\-5475\\617538
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26876\\-500\\56391
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110675\\2058\\-232218
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32681\\608\\-68571
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69636\\-1295\\146110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127258\\-2367\\267012
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
466272\\-8673\\978329
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
521959\\-9709\\1095171
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
114\\0\\-399\\-171\\-171\\-57\\-57\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\1\\0\\-19\\-38\\-24\\-57\\-49
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\kern3pt%
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-399\\0\\798\\0\\-399\\-399\\-1197\\-1197
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171\\-19\\0\\19\\-19\\-57\\-228\\-266
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171\\-38\\-399\\-19\\38\\0\\-76\\-133
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-24\\-399\\-57\\0\\6\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-57\\-1197\\-228\\-76\\0\\19\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-49\\-1197\\-266\\-133\\-21\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2318205162, -702144639, 1098632934; -702144639, -212666582, 332757108; 1098632934, 332757108, -520658975],[399, 19, 1],[294319, 26876, -110675, -32681, 69636, 127258, 466272, 521959; -5475, -500, 2058, 608, -1295, -2367, -8673, -9709; 617538, 56391, -232218, -68571, 146110, 267012, 978329, 1095171],105,7]
%
% END OF LATTICE L106.7
%
\hbox{}\par\smallskip%
%
% LATTICE L106.8
%
\leavevmode%
\llap{\spacer}%
${L_{106.8}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{106.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{l}{2}{399}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{4}{42}\above{1pt}{1pt}{s}{2}{266}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{57}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108577944891633\\32322515589\\36001097424852
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32322515589\\9622074\\10717149177
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36001097424852\\10717149177\\11936853447422
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109054\\8609\\328895
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-640725\\50464\\1932357
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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252\\-25\\-760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-8307\\665\\25053
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-127432\\10114\\384321
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1056762\\83676\\3187079
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-573719\\45390\\1730274
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1702275\\134615\\5133876
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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14\\0\\-7\\-63\\-63\\-133\\-21\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\399\\0\\-399\\-798\\-3192\\-1197\\-2793
\end{tabular}\endgroup%
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-7\\0\\2\\0\\-21\\-133\\-63\\-171
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-399\\0\\21\\-21\\-399\\-252\\-798
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-798\\-21\\-21\\42\\0\\-84\\-399
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-133\\-3192\\-133\\-399\\0\\266\\0\\-399
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-1197\\-63\\-252\\-84\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2793\\-171\\-798\\-399\\-399\\0\\57
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[108577944891633, 32322515589, 36001097424852; 32322515589, 9622074, 10717149177; 36001097424852, 10717149177, 11936853447422],[399, 21, 1],[-109054, -640725, 252, -8307, -127432, -1056762, -573719, -1702275; 8609, 50464, -25, 665, 10114, 83676, 45390, 134615; 328895, 1932357, -760, 25053, 384321, 3187079, 1730274, 5133876],105,8]
%
% END OF LATTICE L106.8
%
\hbox{}\par\smallskip%
%
% LATTICE L106.9
%
\leavevmode%
\llap{\spacer}%
${L_{106.9}}=\pdualwitharg{7}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{l}{2}{532}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{4}{14}\above{1pt}{1pt}{b}{2}{798}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{76}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-766828524\\10190460\\-212268
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10190460\\-135422\\2821
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-212268\\2821\\-58
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\-3972\\777
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-215\\-16112\\3192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\75\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\1274\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-225\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\-23085\\4389
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-10118\\1946
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-433\\-32452\\6270
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-21\\-126\\-63\\-399\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\532\\0\\-532\\-532\\-6384\\-1596\\-3724
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\6\\0\\-21\\-399\\-126\\-342
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-532\\0\\28\\-14\\-798\\-336\\-1064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-532\\-21\\-14\\14\\0\\-56\\-266
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-399\\-6384\\-399\\-798\\0\\798\\0\\-798
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-1596\\-126\\-336\\-56\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3724\\-342\\-1064\\-266\\-798\\0\\76
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-766828524, 10190460, -212268; 10190460, -135422, 2821; -212268, 2821, -58],[1596, 7, 1],[-53, -215, 1, 17, -3, -308, -135, -433; -3972, -16112, 75, 1274, -225, -23085, -10118, -32452; 777, 3192, -12, -252, 35, 4389, 1946, 6270],105,9]
%
% END OF LATTICE L106.9
%
\hbox{}\par\smallskip%
%
% LATTICE L106.10
%
\leavevmode%
\llap{\spacer}%
${L_{106.10}}=\pdualwitharg{2.3}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{57}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{4}{24}\above{1pt}{1pt}{*}{2}{152}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{1596}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2482900392\\-106352652\\-621820752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-106352652\\4556040\\26635092
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-621820752\\26635092\\155729585
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15184\\-263\\-60584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46043\\-798\\-183711
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1193\\-21\\-4760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2597\\-45\\-10362
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19185\\-332\\-76548
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-151467\\-2622\\-604352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-81325\\-1408\\-324486
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1682204\\-29127\\-6711978
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8\\0\\-28\\-36\\-36\\-76\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\57\\0\\-114\\-228\\-912\\-342\\-5586
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\kern3pt%
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-28\\0\\56\\0\\-84\\-532\\-252\\-4788
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\kern3pt%
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-36\\-114\\0\\12\\-12\\-228\\-144\\-3192
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\kern3pt%
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-36\\-228\\-84\\-12\\24\\0\\-48\\-1596
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76\\-912\\-532\\-228\\0\\152\\0\\-1596
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-342\\-252\\-144\\-48\\0\\12\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5586\\-4788\\-3192\\-1596\\-1596\\0\\1596
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2482900392, -106352652, -621820752; -106352652, 4556040, 26635092; -621820752, 26635092, 155729585],[1596, 12, 1],[-15184, -46043, -1193, -2597, -19185, -151467, -81325, -1682204; -263, -798, -21, -45, -332, -2622, -1408, -29127; -60584, -183711, -4760, -10362, -76548, -604352, -324486, -6711978],105,10]
%
% END OF LATTICE L106.10
%
\hbox{}\par\smallskip%
%
% LATTICE L106.11
%
\leavevmode%
\llap{\spacer}%
${L_{106.11}}=\pdualwitharg{3.19}(\pfillwitharg{2}({L_{106.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${38}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{266}\above{1pt}{1pt}{l}{2}{57}\above{1pt}{1pt}{}{4}{114}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{57}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3007636809933\\-8862878871\\1020083772666
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8862878871\\26117058\\-3005974287
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1020083772666\\-3005974287\\345976249466
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77284\\8609\\-227791
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23865\\2656\\-70341
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1444\\-175\\4256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5937\\665\\-17499
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90602\\10114\\-267045
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39488\\4404\\-116389
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
407119\\45390\\-1199964
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
444915\\49595\\-1311366
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\0\\-133\\-171\\-171\\-19\\-57\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\3\\0\\-57\\-114\\-24\\-171\\-147
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-133\\0\\266\\0\\-399\\-133\\-1197\\-1197
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171\\-57\\0\\57\\-57\\-57\\-684\\-798
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171\\-114\\-399\\-57\\114\\0\\-228\\-399
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-24\\-133\\-57\\0\\2\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-171\\-1197\\-684\\-228\\0\\57\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-147\\-1197\\-798\\-399\\-21\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3007636809933, -8862878871, 1020083772666; -8862878871, 26117058, -3005974287; 1020083772666, -3005974287, 345976249466],[399, 57, 1],[77284, 23865, -1444, 5937, 90602, 39488, 407119, 444915; 8609, 2656, -175, 665, 10114, 4404, 45390, 49595; -227791, -70341, 4256, -17499, -267045, -116389, -1199964, -1311366],105,11]
%
% END OF LATTICE L106.11
%
\hbox{}\par\smallskip%
%
% LATTICE L106.12
%
\leavevmode%
\llap{\spacer}%
${L_{106.12}}=\pdualwitharg{19}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${114}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{798}\above{1pt}{1pt}{b}{2}{76}\above{1pt}{1pt}{*}{4}{38}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{76}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10343676\\86184\\3926160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
86184\\-646\\-33079
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3926160\\-33079\\-1488398
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2044\\24600\\4845
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
351\\4224\\832
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1010\\-12159\\-2394
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-513\\-6174\\-1216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
505\\6079\\1197
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
910\\10953\\2157
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6621\\79690\\15694
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7377\\88788\\17486
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
114\\0\\-399\\-342\\-171\\-57\\-114\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-76\\-76\\-48\\-228\\-196
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-399\\0\\798\\0\\-399\\-399\\-2394\\-2394
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-342\\-76\\0\\76\\-38\\-114\\-912\\-1064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171\\-76\\-399\\-38\\38\\0\\-152\\-266
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-48\\-399\\-114\\0\\6\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114\\-228\\-2394\\-912\\-152\\0\\76\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-196\\-2394\\-1064\\-266\\-42\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10343676, 86184, 3926160; 86184, -646, -33079; 3926160, -33079, -1488398],[1596, 19, 1],[2044, 351, -1010, -513, 505, 910, 6621, 7377; 24600, 4224, -12159, -6174, 6079, 10953, 79690, 88788; 4845, 832, -2394, -1216, 1197, 2157, 15694, 17486],105,12]
%
% END OF LATTICE L106.12
%
\hbox{}\par\smallskip%
%
% LATTICE L106.13
%
\leavevmode%
\llap{\spacer}%
${L_{106.13}}=\pdualwitharg{3.7}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{l}{2}{1596}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{4}{42}\above{1pt}{1pt}{b}{2}{266}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{228}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-228228\\12768\\6384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12768\\-714\\-357
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6384\\-357\\-178
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\21\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\228\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\152\\-133
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\184\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\570\\-342
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-7\\-126\\-63\\-133\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1596\\0\\-1596\\-1596\\-6384\\-4788\\-11172
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\2\\0\\-21\\-133\\-126\\-342
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-1596\\0\\84\\-42\\-798\\-1008\\-3192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-1596\\-21\\-42\\42\\0\\-168\\-798
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-133\\-6384\\-133\\-798\\0\\266\\0\\-798
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-4788\\-126\\-1008\\-168\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11172\\-342\\-3192\\-798\\-798\\0\\228
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-228228, 12768, 6384; 12768, -714, -357; 6384, -357, -178],[1596, 21, 1],[1, 13, 0, -1, 0, 5, 7, 23; 21, 228, -1, -18, 10, 152, 184, 570; -7, 0, 2, 0, -21, -133, -126, -342],105,13]
%
% END OF LATTICE L106.13
%
\hbox{}\par\smallskip%
%
% LATTICE L106.14
%
\leavevmode%
\llap{\spacer}%
${L_{106.14}}=\pdualwitharg{2.7}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{l}{2}{133}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{4}{56}\above{1pt}{1pt}{*}{2}{3192}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{76}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2082553368\\145748316\\-521665368
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
145748316\\10204376\\-36508948
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-521665368\\-36508948\\130673605
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
118526\\-897\\472920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
118033\\-893\\470953
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
803\\-6\\3204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6607\\-50\\26362
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50849\\-385\\202888
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1197197\\-9063\\4776828
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
213617\\-1617\\852334
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
630046\\-4769\\2513890
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-84\\-252\\-252\\-1596\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\133\\0\\-266\\-532\\-6384\\-798\\-1862
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\24\\0\\-84\\-1596\\-252\\-684
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-266\\0\\28\\-28\\-1596\\-336\\-1064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-532\\-84\\-28\\56\\0\\-112\\-532
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1596\\-6384\\-1596\\-1596\\0\\3192\\0\\-1596
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-798\\-252\\-336\\-112\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1862\\-684\\-1064\\-532\\-1596\\0\\76
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2082553368, 145748316, -521665368; 145748316, 10204376, -36508948; -521665368, -36508948, 130673605],[1596, 28, 1],[118526, 118033, 803, 6607, 50849, 1197197, 213617, 630046; -897, -893, -6, -50, -385, -9063, -1617, -4769; 472920, 470953, 3204, 26362, 202888, 4776828, 852334, 2513890],105,14]
%
% END OF LATTICE L106.14
%
\hbox{}\par\smallskip%
%
% LATTICE L106.15
%
\leavevmode%
\llap{\spacer}%
${L_{106.15}}=\pdualwitharg{7.19}(\pfillwitharg{2}({L_{106.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${798}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{114}\above{1pt}{1pt}{l}{2}{133}\above{1pt}{1pt}{}{4}{266}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{133}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38372822419533\\48357337665\\-5502117698931
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48357337665\\60939802\\-6933755367
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5502117698931\\-6933755367\\788925527602
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540488\\1647291\\-3754989
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55633\\169555\\-386505
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1444\\-4407\\10032
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13841\\42188\\-96159
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-211214\\643756\\-1467389
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276164\\841704\\-1918623
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-949075\\2892615\\-6593608
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-148169\\451592\\-1029390
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
798\\0\\-399\\-1197\\-1197\\-399\\-399\\0
\end{tabular}\endgroup%
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0\\7\\0\\-133\\-266\\-168\\-399\\-49
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-399\\0\\114\\0\\-399\\-399\\-1197\\-171
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1197\\-133\\0\\133\\-133\\-399\\-1596\\-266
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1197\\-266\\-399\\-133\\266\\0\\-532\\-133
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-399\\-168\\-399\\-399\\0\\42\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-399\\-399\\-1197\\-1596\\-532\\0\\133\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-49\\-171\\-266\\-133\\-21\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[38372822419533, 48357337665, -5502117698931; 48357337665, 60939802, -6933755367; -5502117698931, -6933755367, 788925527602],[399, 133, 1],[-540488, -55633, 1444, -13841, -211214, -276164, -949075, -148169; 1647291, 169555, -4407, 42188, 643756, 841704, 2892615, 451592; -3754989, -386505, 10032, -96159, -1467389, -1918623, -6593608, -1029390],105,15]
%
% END OF LATTICE L106.15
%
\hbox{}\par\smallskip%
%
% LATTICE L106.16
%
\leavevmode%
\llap{\spacer}%
${L_{106.16}}=\pdualwitharg{3.19}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${38}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{266}\above{1pt}{1pt}{b}{2}{228}\above{1pt}{1pt}{*}{4}{114}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{228}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20748\\4788\\1596
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4788\\570\\171
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1596\\171\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\48\\-133
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\48\\-132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\49\\-133
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-42\\114
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-43\\114
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\158\\-456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\252\\-714
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\0\\-133\\-342\\-171\\-19\\-114\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-228\\-228\\-48\\-684\\-588
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-133\\0\\266\\0\\-399\\-133\\-2394\\-2394
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-342\\-228\\0\\228\\-114\\-114\\-2736\\-3192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171\\-228\\-399\\-114\\114\\0\\-456\\-798
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-48\\-133\\-114\\0\\2\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114\\-684\\-2394\\-2736\\-456\\0\\228\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-588\\-2394\\-3192\\-798\\-42\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20748, 4788, 1596; 4788, 570, 171; 1596, 171, 50],[1596, 57, 1],[-1, -1, -1, 1, 1, 0, -3, -5; 48, 48, 49, -42, -43, 1, 158, 252; -133, -132, -133, 114, 114, -4, -456, -714],105,16]
%
% END OF LATTICE L106.16
%
\hbox{}\par\smallskip%
%
% LATTICE L106.17
%
\leavevmode%
\llap{\spacer}%
${L_{106.17}}=\pdualwitharg{2.19}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${456}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{3192}\above{1pt}{1pt}{*}{2}{76}\above{1pt}{1pt}{*}{4}{152}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}{76}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7401657480\\2603128668\\-1843668876
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2603128668\\915508312\\-648409808
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1843668876\\-648409808\\459236993
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13730\\39\\55176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
832\\2\\3343
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4373\\0\\17556
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
217\\2\\874
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3553\\17\\14288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5627\\21\\22620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20619\\71\\82878
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23320\\77\\93730
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
456\\0\\-1596\\-684\\-684\\-228\\-228\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-38\\-76\\-48\\-114\\-98
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1596\\0\\3192\\0\\-1596\\-1596\\-4788\\-4788
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-684\\-38\\0\\76\\-76\\-228\\-912\\-1064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-684\\-76\\-1596\\-76\\152\\0\\-304\\-532
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-228\\-48\\-1596\\-228\\0\\24\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-228\\-114\\-4788\\-912\\-304\\0\\76\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-98\\-4788\\-1064\\-532\\-84\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7401657480, 2603128668, -1843668876; 2603128668, 915508312, -648409808; -1843668876, -648409808, 459236993],[1596, 76, 1],[13730, 832, 4373, 217, 3553, 5627, 20619, 23320; 39, 2, 0, 2, 17, 21, 71, 77; 55176, 3343, 17556, 874, 14288, 22620, 82878, 93730],105,17]
%
% END OF LATTICE L106.17
%
\hbox{}\par\smallskip%
%
% LATTICE L106.18
%
\leavevmode%
\llap{\spacer}%
${L_{106.18}}=\pdualwitharg{2.3.7}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{l}{2}{399}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{*}{4}{168}\above{1pt}{1pt}{*}{2}{1064}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{228}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1382136\\59052\\-346332
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59052\\6888\\-14784
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-346332\\-14784\\86783
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1067\\13\\-4256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3101\\38\\-12369
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-179\\2\\-714
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1432\\17\\-5712
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11070\\133\\-44156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5886\\71\\-23478
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17291\\209\\-68970
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-28\\-252\\-252\\-532\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\399\\0\\-798\\-1596\\-6384\\-2394\\-5586
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\8\\0\\-84\\-532\\-252\\-684
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-798\\0\\84\\-84\\-1596\\-1008\\-3192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-1596\\-84\\-84\\168\\0\\-336\\-1596
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-532\\-6384\\-532\\-1596\\0\\1064\\0\\-1596
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-2394\\-252\\-1008\\-336\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5586\\-684\\-3192\\-1596\\-1596\\0\\228
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1382136, 59052, -346332; 59052, 6888, -14784; -346332, -14784, 86783],[1596, 84, 1],[-1067, -3101, 1, -179, -1432, -11070, -5886, -17291; 13, 38, 0, 2, 17, 133, 71, 209; -4256, -12369, 4, -714, -5712, -44156, -23478, -68970],105,18]
%
% END OF LATTICE L106.18
%
\hbox{}\par\smallskip%
%
% LATTICE L106.19
%
\leavevmode%
\llap{\spacer}%
${L_{106.19}}=\pdualwitharg{3.7.19}(\pfillwitharg{2}({L_{106.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${266}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{38}\above{1pt}{1pt}{l}{2}{399}\above{1pt}{1pt}{}{4}{798}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{399}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18163856907105\\-23525271202545\\-7737986861619
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-23525271202545\\-30469210769634\\-10022003597298
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7737986861619\\-10022003597298\\-3296460707947
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1825\\36102429\\-109755457
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-500\\9890184\\-30067275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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98\\-1939403\\5896004
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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608\\-12026385\\36561567
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1295\\25625480\\-77904351
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-789\\15609994\\-47456143
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-8673\\171584632\\-521636640
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1387\\27439581\\-83419422
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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266\\0\\-133\\-1197\\-1197\\-133\\-399\\0
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-1197\\-399\\0\\399\\-399\\-399\\-4788\\-798
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1197\\-798\\-399\\-399\\798\\0\\-1596\\-399
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-133\\-168\\-133\\-399\\0\\14\\0\\-21
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-399\\-1197\\-1197\\-4788\\-1596\\0\\399\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-147\\-171\\-798\\-399\\-21\\0\\3
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18163856907105, -23525271202545, -7737986861619; -23525271202545, -30469210769634, -10022003597298; -7737986861619, -10022003597298, -3296460707947],[399, 399, 1],[-1825, -500, 98, 608, -1295, -789, -8673, -1387; 36102429, 9890184, -1939403, -12026385, 25625480, 15609994, 171584632, 27439581; -109755457, -30067275, 5896004, 36561567, -77904351, -47456143, -521636640, -83419422],105,19]
%
% END OF LATTICE L106.19
%
\hbox{}\par\smallskip%
%
% LATTICE L106.20
%
\leavevmode%
\llap{\spacer}%
${L_{106.20}}=\pdualwitharg{7.19}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${798}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{114}\above{1pt}{1pt}{b}{2}{532}\above{1pt}{1pt}{*}{4}{266}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{b}{2}{532}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45669540\\-14295372\\-197904
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14295372\\-4473322\\-61845
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-197904\\-61845\\-850
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
776\\-3048\\41097
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
167\\-656\\8848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\51\\-684
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-251\\986\\-13300
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\-149\\1995
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\HardButStrongLineBreak\kern3pt%
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233\\-915\\12327
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1955\\-7678\\103474
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
331\\-1300\\17522
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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798\\0\\-399\\-2394\\-1197\\-399\\-798\\0
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0\\28\\0\\-532\\-532\\-336\\-1596\\-196
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-399\\0\\114\\0\\-399\\-399\\-2394\\-342
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1197\\-532\\-399\\-266\\266\\0\\-1064\\-266
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\kern3pt%
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-399\\-336\\-399\\-798\\0\\42\\0\\-42
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-798\\-1596\\-2394\\-6384\\-1064\\0\\532\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-196\\-342\\-1064\\-266\\-42\\0\\4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-45669540, -14295372, -197904; -14295372, -4473322, -61845; -197904, -61845, -850],[1596, 133, 1],[776, 167, -13, -251, 38, 233, 1955, 331; -3048, -656, 51, 986, -149, -915, -7678, -1300; 41097, 8848, -684, -13300, 1995, 12327, 103474, 17522],105,20]
%
% END OF LATTICE L106.20
%
\hbox{}\par\smallskip%
%
% LATTICE L106.21
%
\leavevmode%
\llap{\spacer}%
${L_{106.21}}=\pdualwitharg{2.3.19}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${152}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{1064}\above{1pt}{1pt}{*}{2}{228}\above{1pt}{1pt}{*}{4}{456}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{228}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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72678648\\3959676\\-18216744
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3959676\\215688\\-992484
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18216744\\-992484\\4565987
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-838\\-3\\-3344
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-97\\0\\-387
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
533\\7\\2128
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\HardButStrongLineBreak\kern3pt%
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257\\3\\1026
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\HardButStrongLineBreak\kern3pt%
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-857\\-4\\-3420
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\HardButStrongLineBreak\kern3pt%
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-429\\-2\\-1712
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-4485\\-20\\-17898
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\HardButStrongLineBreak\kern3pt%
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-4894\\-21\\-19530
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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152\\0\\-532\\-684\\-684\\-76\\-228\\0
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0\\3\\0\\-114\\-228\\-48\\-342\\-294
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-532\\0\\1064\\0\\-1596\\-532\\-4788\\-4788
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-684\\-228\\-1596\\-228\\456\\0\\-912\\-1596
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-76\\-48\\-532\\-228\\0\\8\\0\\-84
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-228\\-342\\-4788\\-2736\\-912\\0\\228\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-294\\-4788\\-3192\\-1596\\-84\\0\\84
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[72678648, 3959676, -18216744; 3959676, 215688, -992484; -18216744, -992484, 4565987],[1596, 228, 1],[-838, -97, 533, 257, -857, -429, -4485, -4894; -3, 0, 7, 3, -4, -2, -20, -21; -3344, -387, 2128, 1026, -3420, -1712, -17898, -19530],105,21]
%
% END OF LATTICE L106.21
%
\hbox{}\par\smallskip%
%
% LATTICE L106.22
%
\leavevmode%
\llap{\spacer}%
${L_{106.22}}=\pdualwitharg{3.7.19}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${266}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{38}\above{1pt}{1pt}{b}{2}{1596}\above{1pt}{1pt}{*}{4}{798}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{b}{2}{1596}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-914554284\\911022336\\1549716
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
911022336\\-907503954\\-1543731
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1549716\\-1543731\\-2626
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\21\\-15295
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\HardButStrongLineBreak\kern3pt%
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-1\\12\\-7644
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\1178
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-18\\11172
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\10\\-12369
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\-7063
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\184\\-152418
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\30\\-24126
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
266\\0\\-133\\-2394\\-1197\\-133\\-798\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-798\\-4788\\-2394\\-19152\\-3192\\0\\1596\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-588\\-342\\-3192\\-798\\-42\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-914554284, 911022336, 1549716; 911022336, -907503954, -1543731; 1549716, -1543731, -2626],[1596, 399, 1],[-5, -1, 1, 1, -11, -4, -75, -11; 21, 12, -1, -18, 10, 8, 184, 30; -15295, -7644, 1178, 11172, -12369, -7063, -152418, -24126],105,22]
%
% END OF LATTICE L106.22
%
\hbox{}\par\smallskip%
%
% LATTICE L106.23
%
\leavevmode%
\llap{\spacer}%
${L_{106.23}}=\pdualwitharg{2.7.19}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3192}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{456}\above{1pt}{1pt}{*}{2}{532}\above{1pt}{1pt}{*}{4}{1064}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{*}{2}{532}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
136890436200\\-3978488052\\37307339496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3978488052\\115628072\\-1084274520
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37307339496\\-1084274520\\10167529775
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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135194\\-406839\\-539448
\end{tabular}\endgroup%
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6612\\-19897\\-26383
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-1657\\4989\\6612
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7533\\-22669\\-30058
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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63997\\-192592\\-255360
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76607\\-230538\\-305676
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255655\\-769354\\-1020110
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39312\\-118303\\-156862
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3192\\0\\-1596\\-4788\\-4788\\-1596\\-1596\\0
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0\\7\\0\\-266\\-532\\-336\\-798\\-98
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\kern3pt%
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\kern3pt%
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-1596\\-798\\-4788\\-6384\\-2128\\0\\532\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-98\\-684\\-1064\\-532\\-84\\0\\4
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[136890436200, -3978488052, 37307339496; -3978488052, 115628072, -1084274520; 37307339496, -1084274520, 10167529775],[1596, 532, 1],[135194, 6612, -1657, 7533, 63997, 76607, 255655, 39312; -406839, -19897, 4989, -22669, -192592, -230538, -769354, -118303; -539448, -26383, 6612, -30058, -255360, -305676, -1020110, -156862],105,23]
%
% END OF LATTICE L106.23
%
\hbox{}\par\smallskip%
%
% LATTICE L106.24
%
\leavevmode%
\llap{\spacer}%
${L_{106.24}}=\pdualwitharg{2.3.7.19}({L_{106.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1064}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{152}\above{1pt}{1pt}{*}{2}{1596}\above{1pt}{1pt}{*}{4}{3192}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{*}{2}{1596}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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130872\\90967212\\-22798860
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\kern3pt%
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90967212\\67619944056\\-16947414876
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22798860\\-16947414876\\4247487557
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3331\\332139\\1325212
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\HardButStrongLineBreak\kern3pt%
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-495\\49364\\196959
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-2781\\-11096
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-588\\58601\\233814
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-4703\\468808\\1870512
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-1877\\187126\\746620
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-2895\\288635\\1151634
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-294\\-684\\-3192\\-1596\\-84\\0\\12
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[130872, 90967212, -22798860; 90967212, 67619944056, -16947414876; -22798860, -16947414876, 4247487557],[1596, 1596, 1],[-3331, -495, 28, -588, -4703, -1877, -18811, -2895; 332139, 49364, -2781, 58601, 468808, 187126, 1875432, 288635; 1325212, 196959, -11096, 233814, 1870512, 746620, 7482846, 1151634],105,24]
%
% END OF LATTICE L106.24
%

\medskip%
%------------------------------WEYL-GROUP-W107--------------------------------
%
\leavevmode\llap{}%
$W_{107}$%
\qquad\llap{24} lattices, $\chi=60$%
\hfill%
$22222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L107.1
%
\leavevmode%
\llap{\spacer}%
${L_{107.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{107}}%
\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{858}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{286}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60078876\\27456\\20592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27456\\-10\\-13
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20592\\-13\\-2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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146431\\208995072\\148372224
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\kern3pt%
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-60\\-85636\\-60795
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\kern3pt%
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-60\\-85635\\-60796
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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7\\9984\\7098
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\HardButStrongLineBreak\kern3pt%
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2\\2854\\2027
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
511\\729300\\517803
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\44244\\31412
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
571\\814957\\578578
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89\\127026\\90180
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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0\\156\\0\\-6864\\-468\\-9438\\-1560
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\kern3pt%
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\kern3pt%
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\kern3pt%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-60078876, 27456, 20592; 27456, -10, -13; 20592, -13, -2],[1716, 1, 1],[-1, 7, 2, 511, 31, 571, 89, 89, 97, 2, 61, 1, 1, -1; -1430, 9984, 2854, 729300, 44244, 814957, 127026, 127028, 138450, 2855, 87087, 1428, 1430, -1428; -1012, 7098, 2027, 517803, 31412, 578578, 90180, 90178, 98280, 2026, 61776, 1012, 1001, -1014],106,1]
%
% END OF LATTICE L107.1
%
\hbox{}\par\smallskip%
%
% LATTICE L107.2
%
\leavevmode%
\llap{\spacer}%
${L_{107.2}}=\pfillwitharg{2}({L_{107.1}})$%
{${}={}$}Nikulin $107$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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21450\\3861\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[21450, 3861, 0; 3861, 695, 0; 0, 0, -1],[429, 1, 1],[-2, -7, 2, 1318, 43, 1679, 136, 145, 176, 11, 541, 9, 180, 7; 11, 39, -11, -7293, -238, -9295, -753, -803, -975, -61, -3003, -50, -1001, -39; 0, 0, -3, -1287, -41, -1573, -126, -132, -156, -9, -429, -7, -143, -6],106,2]
%
% END OF LATTICE L107.2
%
\hbox{}\par\smallskip%
%
% LATTICE L107.3
%
\leavevmode%
\llap{\spacer}%
${L_{107.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{107.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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3861\\-15\\-1290
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[39017550, 3861, -13036023; 3861, -15, -1290; -13036023, -1290, 4355422],[429, 3, 1],[2591, 2302, 1294, 49832, 3876, 123553, 2869, 6880, 1429, -433, -8791, -432, -430, 288; -11, -13, -10, -572, -49, -1716, -43, -121, -39, -2, 0, 1, 0, -1; 7755, 6890, 3873, 149149, 11601, 369798, 8587, 20592, 4277, -1296, -26312, -1293, -1287, 862],106,3]
%
% END OF LATTICE L107.3
%
\hbox{}\par\smallskip%
%
% LATTICE L107.4
%
\leavevmode%
\llap{\spacer}%
${L_{107.4}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{107.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[82356846, 10725, 14951079; 10725, -55, 1947; 14951079, 1947, 2714222],[429, 11, 1],[-356, -10436, -1955, -20525, -5853, -16959, -12994, -944, -6464, 655, 3625, 653, 59, -1306; -1, -39, -10, -156, -49, -156, -129, -11, -117, -2, 0, 1, 0, -3; 1961, 57486, 10769, 113061, 32241, 93418, 71577, 5200, 35607, -3608, -19968, -3597, -325, 7194],106,4]
%
% END OF LATTICE L107.4
%
\hbox{}\par\smallskip%
%
% LATTICE L107.5
%
\leavevmode%
\llap{\spacer}%
${L_{107.5}}=\pdualwitharg{3}({L_{107.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{286}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{858}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1903443828, -13218348, 61776; -13218348, -91794, 429; 61776, 429, -2],[1716, 3, 1],[57, 17, 0, -7, -1, 2, 1, 9, 9, 3, 150, 25, 427, 21; -8272, -2470, -1, 1001, 144, -286, -144, -1298, -1300, -434, -21736, -3624, -61919, -3046; -14454, -4966, -225, -1573, 0, 429, -2, -462, -910, -453, -30745, -5424, -97383, -4970],106,5]
%
% END OF LATTICE L107.5
%
\hbox{}\par\smallskip%
%
% LATTICE L107.6
%
\leavevmode%
\llap{\spacer}%
${L_{107.6}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{107.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${143}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[93792270, 12441, 14407107; 12441, -65, 1911; 14407107, 1911, 2213026],[429, 13, 1],[-3844, -788, -1919, -17047, -5745, -14085, -12754, -10192, -488, 643, 3011, 641, 49, -1282; -11, -3, -10, -132, -49, -132, -129, -121, -9, -2, 0, 1, 0, -3; 25025, 5130, 12493, 110979, 37401, 91696, 83031, 66352, 3177, -4186, -19602, -4173, -319, 8346],106,6]
%
% END OF LATTICE L107.6
%
\hbox{}\par\smallskip%
%
% LATTICE L107.7
%
\leavevmode%
\llap{\spacer}%
${L_{107.7}}=\pdualwitharg{2}({L_{107.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[65138285784, -214800300, 16246461660; -214800300, 708328, -53574404; 16246461660, -53574404, 4052110265],[1716, 4, 1],[2771, 33520, 14524, 3497174, 52662, 3859689, 299747, 298046, 321991, 12823, 413447, 2032, 50932, 1070; 0, 39, 21, 5577, 85, 6292, 492, 495, 546, 24, 858, 5, 143, 3; -11110, -134394, -58232, -14021436, -211141, -15474888, -1201794, -1194974, -1290978, -51412, -1657656, -8147, -204204, -4290],106,7]
%
% END OF LATTICE L107.7
%
\hbox{}\par\smallskip%
%
% LATTICE L107.8
%
\leavevmode%
\llap{\spacer}%
${L_{107.8}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{107.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[144058912998, 2383524, -48237038421; 2383524, 33, -798105; -48237038421, -798105, 16151807807],[429, 33, 1],[9319, 217961, 221659, 1380173, 1341048, 4382591, 1231143, 325943, 1234841, 113603, 72629, 0, -24185, -1849; 2, 39, 38, 234, 227, 741, 208, 55, 208, 19, 13, 1, 0, 1; 27831, 650936, 661980, 4121858, 4005012, 13088517, 3676783, 973422, 3687827, 339273, 216905, 0, -72228, -5522],106,8]
%
% END OF LATTICE L107.8
%
\hbox{}\par\smallskip%
%
% LATTICE L107.9
%
\leavevmode%
\llap{\spacer}%
${L_{107.9}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{107.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L107.9
%
\hbox{}\par\smallskip%
%
% LATTICE L107.10
%
\leavevmode%
\llap{\spacer}%
${L_{107.10}}=\pdualwitharg{11}({L_{107.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-110484660, 125268, 125268; 125268, -110, -143; 125268, -143, -142],[1716, 11, 1],[39, 487, 12, 41, 11, 6, -1, -1, 7, 7, 206, 141, 241, 419; 992, 12402, 307, 1053, 284, 156, -24, -26, 156, 176, 5226, 3580, 6123, 10650; 33394, 416988, 10274, 35100, 9416, 5135, -858, -856, 6006, 5995, 176397, 120736, 206362, 358776],106,10]
%
% END OF LATTICE L107.10
%
\hbox{}\par\smallskip%
%
% LATTICE L107.11
%
\leavevmode%
\llap{\spacer}%
${L_{107.11}}=\pdualwitharg{2.3}({L_{107.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[152686248, 475332, 38081472; 475332, 65640, 118404; 38081472, 118404, 9497899],[1716, 12, 1],[5864800, 1995731, 169895, 1155457, -107, -15408, 15620, 264257, 418746, 387292, 25403790, 1114481, 79693149, 2027399; -54439, -18525, -1577, -10725, 1, 143, -145, -2453, -3887, -3595, -235807, -10345, -739739, -18819; -23514018, -8001578, -681168, -4632628, 429, 61776, -62626, -1059498, -1678898, -1552788, -101852608, -4468341, -319517484, -8128546],106,11]
%
% END OF LATTICE L107.11
%
\hbox{}\par\smallskip%
%
% LATTICE L107.12
%
\leavevmode%
\llap{\spacer}%
${L_{107.12}}=\pdualwitharg{13}({L_{107.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[89398452, 22308, -58344; 22308, -26, -13; -58344, -13, 38],[1716, 13, 1],[-295, -23, -4, -5, 1, 1, -1, -13, -5, -10, -149, -113, -155, -307; -23078, -1800, -314, -396, 76, 77, -78, -1012, -390, -781, -11649, -8836, -12122, -24012; -461032, -35946, -6253, -7821, 1560, 1562, -1560, -20306, -7812, -15626, -232848, -176592, -242231, -479778],106,12]
%
% END OF LATTICE L107.12
%
\hbox{}\par\smallskip%
%
% LATTICE L107.13
%
\leavevmode%
\llap{\spacer}%
${L_{107.13}}=\pdualwitharg{3.11}({L_{107.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-842556, -70356, -25740; -70356, -5874, -2145; -25740, -2145, -766],[1716, 33, 1],[-27, -529, -255, -1550, -3001, -4889, -2741, -723, -2721, -120, -75, 1, 14, -9; 350, 6864, 3310, 20124, 38964, 63479, 35590, 9388, 35334, 1559, 975, -12, -182, 116; -72, -1430, -693, -4225, -8184, -13338, -7480, -1974, -7436, -330, -208, 0, 39, -22],106,13]
%
% END OF LATTICE L107.13
%
\hbox{}\par\smallskip%
%
% LATTICE L107.14
%
\leavevmode%
\llap{\spacer}%
${L_{107.14}}=\pdualwitharg{11.13}(\pfillwitharg{2}({L_{107.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[135642557193, -4814667, -24547342248; -4814667, 143, 871299; -24547342248, 871299, 4442352185],[429, 143, 1],[10174, 54913, 241993, 347719, 1464068, 368048, 4032242, 355843, 311104, 124024, 18298, 0, -2031, -6055; -30520, -164730, -725941, -1043103, -4391977, -1104087, -12096102, -1067474, -933264, -372053, -54891, 1, 6093, 18168; 56225, 303468, 1337336, 1921614, 8090940, 2033959, 22283547, 1966510, 1719267, 685399, 101121, 0, -11224, -33462],106,14]
%
% END OF LATTICE L107.14
%
\hbox{}\par\smallskip%
%
% LATTICE L107.15
%
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${L_{107.15}}=\pdualwitharg{3.13}({L_{107.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7102524, 41184, -8580; 41184, -78, -39; -8580, -39, 10],[1716, 39, 1],[-13, -1, -4, -13, -27, -34, -21, -53, -1, 1, 2, 3, 1, -1; -440, -34, -137, -451, -940, -1188, -736, -1870, -36, 32, 66, 100, 33, -34; -12870, -992, -3978, -12980, -26988, -34023, -21034, -53196, -1010, 975, 1969, 2964, 990, -988],106,15]
%
% END OF LATTICE L107.15
%
\hbox{}\par\smallskip%
%
% LATTICE L107.16
%
\leavevmode%
\llap{\spacer}%
${L_{107.16}}=\pdualwitharg{2.11}({L_{107.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[136724863704, -106733484, 34101165384; -106733484, 83336, -26620880; 34101165384, -26620880, 8505325579],[1716, 44, 1],[178294, 2135483, 88189, 262439, 14379, 28546, 214, 213, 185108, 90318, 2058152, 342635, 2292045, 1963363; 45, 546, 24, 78, 5, 13, 3, 0, 39, 21, 507, 85, 572, 492; -714850, -8561982, -353584, -1052220, -57651, -114452, -858, -854, -742170, -362120, -8251932, -1373757, -9189700, -7871886],106,16]
%
% END OF LATTICE L107.16
%
\hbox{}\par\smallskip%
%
% LATTICE L107.17
%
\leavevmode%
\llap{\spacer}%
${L_{107.17}}=\pdualwitharg{2.13}({L_{107.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[92649794952, -139001148, 23108318376; -139001148, 208520, -34669076; 23108318376, -34669076, 5763578629],[1716, 52, 1],[-1378365, -113744, -56340, -119938, -4802, 4269, 15155, 2782, -11309, -68713, -1255645, -245304, -1379852, -1390738; 484, 39, 17, 33, 1, 0, 0, 11, 6, 28, 462, 89, 495, 495; 5526378, 456042, 225888, 480876, 19253, -17116, -60762, -11154, 45342, 275496, 5034348, 983515, 5532340, 5575986],106,17]
%
% END OF LATTICE L107.17
%
\hbox{}\par\smallskip%
%
% LATTICE L107.18
%
\leavevmode%
\llap{\spacer}%
${L_{107.18}}=\pdualwitharg{3.11.13}(\pfillwitharg{2}({L_{107.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L107.18
%
\hbox{}\par\smallskip%
%
% LATTICE L107.19
%
\leavevmode%
\llap{\spacer}%
${L_{107.19}}=\pdualwitharg{2.3.11}({L_{107.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1980264, -1597596, 497640; -1597596, 1319208, -401544; 497640, -401544, 125057],[1716, 132, 1],[3091, 56010, 52492, 314682, 151985, 988647, 276791, 72848, 273059, 23447, 14387, -107, -1440, 1652; -28, -507, -475, -2847, -1375, -8944, -2504, -659, -2470, -212, -130, 1, 13, -15; -12390, -224510, -210408, -1261364, -609213, -3962868, -1109482, -292002, -1094522, -93984, -57668, 429, 5772, -6622],106,19]
%
% END OF LATTICE L107.19
%
\hbox{}\par\smallskip%
%
% LATTICE L107.20
%
\leavevmode%
\llap{\spacer}%
${L_{107.20}}=\pdualwitharg{11.13}({L_{107.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-63308388, -1595880, 73788; -1595880, -39754, 1859; 73788, 1859, -86],[1716, 143, 1],[35, 37, 16, 5, 21, 1, -1, -3, -5, 0, 11, 107, 15, 353; 48, 54, 27, 9, 40, 2, 0, -6, -12, -6, 12, 128, 19, 462; 31018, 32868, 14300, 4482, 18876, 901, -858, -2704, -4554, -143, 9675, 94380, 13256, 312312],106,20]
%
% END OF LATTICE L107.20
%
\hbox{}\par\smallskip%
%
% LATTICE L107.21
%
\leavevmode%
\llap{\spacer}%
${L_{107.21}}=\pdualwitharg{2.3.13}({L_{107.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24551474376, -76051404, 6123503100; -76051404, 235560, -18968352; 6123503100, -18968352, 1527292807],[1716, 156, 1],[3638, -1063, -19571, -114461, -66699, -373178, -124890, -368717, -10000, -13618, -7232, 321, 8099, 3197; 11, 2, 28, 154, 89, 495, 165, 484, 13, 17, 11, 1, 0, 0; -14586, 4262, 78468, 458920, 267423, 1496220, 500734, 1478334, 40094, 54600, 28996, -1287, -32472, -12818],106,21]
%
% END OF LATTICE L107.21
%
\hbox{}\par\smallskip%
%
% LATTICE L107.22
%
\leavevmode%
\llap{\spacer}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10266828, -153008856, -1062204; -153008856, -2272331490, -15774759; -1062204, -15774759, -109510],[1716, 429, 1],[57, 19, 21, 2, 23, 1, -1, -3, -1, 6, 7, 193, 26, 199; -17014, -5714, -6467, -625, -7304, -323, 282, 970, 366, -1555, -2049, -56872, -7699, -59110; 2450292, 822910, 931359, 90011, 1051908, 46518, -40612, -139698, -52712, 223938, 295088, 8190468, 1108779, 8512790],106,22]
%
% END OF LATTICE L107.22
%
\hbox{}\par\smallskip%
%
% LATTICE L107.23
%
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\llap{\spacer}%
${L_{107.23}}=\pdualwitharg{2.11.13}({L_{107.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[89812457592, 66352572, -22350894852; 66352572, 49192, -16512496; -22350894852, -16512496, 5562285263],[1716, 572, 1],[-451762, -412571, -212143, -46433, -30352, -4260, -214, -2769, -41960, -242388, -408098, -878710, -450271, -4999627; 1354931, 1237389, 636263, 139263, 91033, 12777, 645, 8305, 125847, 726973, 1223973, 2635439, 1350459, 14994951; -1811290, -1654158, -850564, -186168, -121693, -17080, -858, -11102, -168234, -971828, -1636224, -3523091, -1805312, -20045454],106,23]
%
% END OF LATTICE L107.23
%
\hbox{}\par\smallskip%
%
% LATTICE L107.24
%
\leavevmode%
\llap{\spacer}%
${L_{107.24}}=\pdualwitharg{2.3.11.13}({L_{107.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3957003336, 569832331068, 142125984000; 569832331068, 82059290319864, 20466998355804; 142125984000, 20466998355804, 5104821404909],[1716, 1716, 1],[-45820, -13943, -21463, -1559, -3025, -412, 48, -275, -1422, -24632, -13806, -89165, -45683, -169063; 447831, 136327, 210297, 15343, 30088, 4223, 71, 2745, 13865, 240279, 134849, 871064, 446353, 1652041; -1794234, -546194, -842556, -61472, -120549, -16920, -286, -10998, -55550, -962676, -540272, -3489915, -1788312, -6618898],106,24]
%
% END OF LATTICE L107.24
%

\medskip%
%------------------------------WEYL-GROUP-W108--------------------------------
%
\leavevmode\llap{}%
$W_{108}$%
\qquad\llap{24} lattices, $\chi=28$%
\hfill%
$22222262$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L108.1
%
\leavevmode%
\llap{\spacer}%
${L_{108.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{108}}%
\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{116}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{870}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{580}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{b}{2}$}%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-964563780, 5310480, -210540; 5310480, -29014, 1117; -210540, 1117, -38],[1740, 1, 1],[1, -335, -52, -2486, -79, -1593, -14, 13; 230, -77024, -11956, -571590, -18164, -366270, -3219, 2989; 1220, -408030, -63337, -3028035, -96226, -1940390, -17055, 15834],107,1]
%
% END OF LATTICE L108.1
%
\hbox{}\par\smallskip%
%
% LATTICE L108.2
%
\leavevmode%
\llap{\spacer}%
${L_{108.2}}=\pfillwitharg{2}({L_{108.1}})$%
{${}={}$}Nikulin $108$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-29\\-9\\-435\\-7\\-145\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-18705, 7395, 3915; 7395, -2923, -1566; 3915, -1566, -271],[435, 1, 1],[2, 350, -37, -15749, -494, -19191, -1406, -455; 5, 870, -92, -39150, -1228, -47705, -3495, -1131; 0, 29, -3, -1305, -41, -1595, -117, -38],107,2]
%
% END OF LATTICE L108.2
%
\hbox{}\par\smallskip%
%
% LATTICE L108.3
%
\leavevmode%
\llap{\spacer}%
${L_{108.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{108.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{87}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{290}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{435}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8256201690\\26534565\\-2752419435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26534565\\-85278\\8845986
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2752419435\\8845986\\-917590561
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90288\\198710\\272745
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-769507\\1693629\\2324553
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-166887\\367312\\504138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2116703\\4658850\\6394210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78617\\173039\\237489
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1273391\\2802850\\3846705
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2545\\-5599\\-7688
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2018\\4437\\6096
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\-435\\-30\\-870\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\87\\0\\-870\\-87\\-3480\\-87\\-87
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\6\\0\\-6\\-435\\-15\\-27
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-435\\-870\\0\\290\\0\\-3045\\-145\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-87\\-6\\0\\3\\0\\-3\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-870\\-3480\\-435\\-3045\\0\\435\\0\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-87\\-15\\-145\\-3\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-87\\-27\\-435\\-21\\-435\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8256201690, 26534565, -2752419435; 26534565, -85278, 8845986; -2752419435, 8845986, -917590561],[435, 3, 1],[-90288, -769507, -166887, -2116703, -78617, -1273391, 2545, -2018; 198710, 1693629, 367312, 4658850, 173039, 2802850, -5599, 4437; 272745, 2324553, 504138, 6394210, 237489, 3846705, -7688, 6096],107,3]
%
% END OF LATTICE L108.3
%
\hbox{}\par\smallskip%
%
% LATTICE L108.4
%
\leavevmode%
\llap{\spacer}%
${L_{108.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{108.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{145}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{174}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{29}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13965338310\\44552265\\5585780625
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44552265\\-142130\\-17819775
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5585780625\\-17819775\\-2234170379
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21945\\39742\\54549
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
935142\\1693629\\2324495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
202807\\367312\\504120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1543361\\2795310\\3836352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
95536\\173039\\237475
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
309482\\560570\\769283
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9281\\-16797\\-23070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2454\\4437\\6100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-5\\-87\\-10\\-58\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\145\\0\\-870\\-145\\-1160\\-435\\-145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\10\\0\\-10\\-145\\-75\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87\\-870\\0\\174\\0\\-609\\-435\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-145\\-10\\0\\5\\0\\-15\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58\\-1160\\-145\\-609\\0\\29\\0\\-145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-435\\-75\\-435\\-15\\0\\30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-145\\-45\\-435\\-35\\-145\\-15\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13965338310, 44552265, 5585780625; 44552265, -142130, -17819775; 5585780625, -17819775, -2234170379],[435, 5, 1],[21945, 935142, 202807, 1543361, 95536, 309482, -9281, 2454; 39742, 1693629, 367312, 2795310, 173039, 560570, -16797, 4437; 54549, 2324495, 504120, 3836352, 237475, 769283, -23070, 6100],107,4]
%
% END OF LATTICE L108.4
%
\hbox{}\par\smallskip%
%
% LATTICE L108.5
%
\leavevmode%
\llap{\spacer}%
${L_{108.5}}=\pdualwitharg{3}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{348}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{290}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{1740}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5542180140\\-885660\\6918240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-885660\\-114\\1089
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6918240\\1089\\-8626
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\15140\\25170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-251\\-131022\\-217848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-27145\\-45132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-978\\-510545\\-848830
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-107\\-55858\\-92868
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2517\\-1313990\\-2184570
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-7309\\-12151
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\6786\\11283
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-30\\-870\\-120\\-3480\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\348\\0\\-1740\\-348\\-13920\\-174\\-174
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\6\\0\\-12\\-870\\-15\\-27
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-870\\-1740\\0\\290\\0\\-6090\\-145\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-348\\-12\\0\\12\\0\\-6\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3480\\-13920\\-870\\-6090\\0\\1740\\0\\-870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-174\\-15\\-145\\-6\\0\\2\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-174\\-27\\-435\\-42\\-870\\-3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5542180140, -885660, 6918240; -885660, -114, 1089; 6918240, 1089, -8626],[1740, 3, 1],[29, -251, -52, -978, -107, -2517, -14, 13; 15140, -131022, -27145, -510545, -55858, -1313990, -7309, 6786; 25170, -217848, -45132, -848830, -92868, -2184570, -12151, 11283],107,5]
%
% END OF LATTICE L108.5
%
\hbox{}\par\smallskip%
%
% LATTICE L108.6
%
\leavevmode%
\llap{\spacer}%
${L_{108.6}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{108.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{435}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{58}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{87}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13384235730\\1623288195\\4610131740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1623288195\\196878315\\559133340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4610131740\\559133340\\1587936367
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-398736\\3698\\1156317
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17644766\\163647\\51169050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3887919\\36059\\11274795
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10002581\\92771\\29007047
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1894270\\17569\\5493300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6281555\\58261\\18216234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33116\\-307\\-96035
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\0\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-15\\-87\\-30\\-174\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\435\\0\\-870\\-435\\-3480\\-435\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\30\\0\\-30\\-435\\-75\\-135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87\\-870\\0\\58\\0\\-609\\-145\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-435\\-30\\0\\15\\0\\-15\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-174\\-3480\\-435\\-609\\0\\87\\0\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-435\\-75\\-145\\-15\\0\\10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-435\\-135\\-435\\-105\\-435\\-15\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13384235730, 1623288195, 4610131740; 1623288195, 196878315, 559133340; 4610131740, 559133340, 1587936367],[435, 15, 1],[-398736, -17644766, -3887919, -10002581, -1894270, -6281555, 33116, -31; 3698, 163647, 36059, 92771, 17569, 58261, -307, 0; 1156317, 51169050, 11274795, 29007047, 5493300, 18216234, -96035, 90],107,6]
%
% END OF LATTICE L108.6
%
\hbox{}\par\smallskip%
%
% LATTICE L108.7
%
\leavevmode%
\llap{\spacer}%
${L_{108.7}}=\pdualwitharg{2}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{116}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{3480}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{580}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93060519524520\\-40517217180\\-23318789159100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40517217180\\17640616\\10152666776
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23318789159100\\10152666776\\5843143049543
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1521\\0\\-6070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
136802\\-2987\\545954
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47558\\-1005\\189796
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2703632\\-56115\\10789740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50388\\-1033\\201090
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1291503\\-26245\\5154170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54527\\-1089\\217608
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1309\\-28\\5224
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-1740\\-40\\-1160\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\116\\0\\-3480\\-116\\-4640\\-348\\-116
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\8\\0\\-8\\-580\\-60\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1740\\-3480\\0\\3480\\0\\-12180\\-1740\\-1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-116\\-8\\0\\4\\0\\-12\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1160\\-4640\\-580\\-12180\\0\\580\\0\\-580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-348\\-60\\-1740\\-12\\0\\24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-116\\-36\\-1740\\-28\\-580\\-12\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[93060519524520, -40517217180, -23318789159100; -40517217180, 17640616, 10152666776; -23318789159100, 10152666776, 5843143049543],[1740, 4, 1],[-1521, 136802, 47558, 2703632, 50388, 1291503, 54527, 1309; 0, -2987, -1005, -56115, -1033, -26245, -1089, -28; -6070, 545954, 189796, 10789740, 201090, 5154170, 217608, 5224],107,7]
%
% END OF LATTICE L108.7
%
\hbox{}\par\smallskip%
%
% LATTICE L108.8
%
\leavevmode%
\llap{\spacer}%
${L_{108.8}}=\pdualwitharg{5}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{580}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{174}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{116}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5596673460\\28495980\\-1097940
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28495980\\-145070\\5585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1097940\\5585\\-214
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\230\\872
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-335\\-77024\\-291450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-12416\\-46985
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-532\\-122322\\-462927
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87\\-20004\\-75710
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-365\\-83926\\-317666
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-4599\\-17415
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\2989\\11310
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-10\\-174\\-40\\-232\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\580\\0\\-1740\\-580\\-4640\\-870\\-290
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\0\\-20\\-290\\-75\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-174\\-1740\\0\\174\\0\\-1218\\-435\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-580\\-20\\0\\20\\0\\-30\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-232\\-4640\\-290\\-1218\\0\\116\\0\\-290
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-870\\-75\\-435\\-30\\0\\30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-290\\-45\\-435\\-70\\-290\\-15\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5596673460, 28495980, -1097940; 28495980, -145070, 5585; -1097940, 5585, -214],[1740, 5, 1],[1, -335, -54, -532, -87, -365, -20, 13; 230, -77024, -12416, -122322, -20004, -83926, -4599, 2989; 872, -291450, -46985, -462927, -75710, -317666, -17415, 11310],107,8]
%
% END OF LATTICE L108.8
%
\hbox{}\par\smallskip%
%
% LATTICE L108.9
%
\leavevmode%
\llap{\spacer}%
${L_{108.9}}=\pdualwitharg{29}(\pfillwitharg{2}({L_{108.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${145}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{58}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{29}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{174}\above{1pt}{1pt}{}{6}{58}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90821391045\\10886745\\6254108970
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10886745\\-1247\\-749679
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6254108970\\-749679\\-430668134
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
415603\\0\\6035335
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
122146\\-1\\1773789
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
768235\\-9\\11156213
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008001\\-15\\14638065
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
361912\\-7\\5255641
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
202144\\-5\\2935510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35131\\-3\\-510168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9280\\2\\134763
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
145\\0\\-145\\-435\\-290\\-290\\-435\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-30\\-29\\-40\\-87\\-29
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145\\0\\58\\0\\-58\\-145\\-435\\-261
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-435\\-30\\0\\30\\0\\-105\\-435\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-290\\-29\\-58\\0\\29\\0\\-87\\-203
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-290\\-40\\-145\\-105\\0\\5\\0\\-145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-435\\-87\\-435\\-435\\-87\\0\\174\\-87
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-29\\-261\\-435\\-203\\-145\\-87\\58
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-90821391045, 10886745, 6254108970; 10886745, -1247, -749679; 6254108970, -749679, -430668134],[435, 29, 1],[415603, 122146, 768235, 1008001, 361912, 202144, -35131, 9280; 0, -1, -9, -15, -7, -5, -3, 2; 6035335, 1773789, 11156213, 14638065, 5255641, 2935510, -510168, 134763],107,9]
%
% END OF LATTICE L108.9
%
\hbox{}\par\smallskip%
%
% LATTICE L108.10
%
\leavevmode%
\llap{\spacer}%
${L_{108.10}}=\pdualwitharg{2.3}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{348}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{1160}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{1740}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21921376299240\\-91351306740\\-5493000898320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-91351306740\\380681448\\22890570516
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5493000898320\\22890570516\\1376421738173
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1962\\5\\-7830
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
425323\\116\\1697370
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
149667\\34\\597288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2942743\\580\\11743840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
169421\\30\\676122
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4518101\\725\\18030750
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68989\\9\\275320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17792\\1\\71004
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-1740\\-120\\-3480\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\348\\0\\-3480\\-348\\-13920\\-348\\-348
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\24\\0\\-24\\-1740\\-60\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1740\\-3480\\0\\1160\\0\\-12180\\-580\\-1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-348\\-24\\0\\12\\0\\-12\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3480\\-13920\\-1740\\-12180\\0\\1740\\0\\-1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-348\\-60\\-580\\-12\\0\\8\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-348\\-108\\-1740\\-84\\-1740\\-12\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[21921376299240, -91351306740, -5493000898320; -91351306740, 380681448, 22890570516; -5493000898320, 22890570516, 1376421738173],[1740, 12, 1],[-1962, 425323, 149667, 2942743, 169421, 4518101, 68989, 17792; 5, 116, 34, 580, 30, 725, 9, 1; -7830, 1697370, 597288, 11743840, 676122, 18030750, 275320, 71004],107,10]
%
% END OF LATTICE L108.10
%
\hbox{}\par\smallskip%
%
% LATTICE L108.11
%
\leavevmode%
\llap{\spacer}%
${L_{108.11}}=\pdualwitharg{3.5}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{1740}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{58}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{348}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53169180\\-2121060\\57420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2121060\\-82830\\2295
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57420\\2295\\-62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\852
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-251\\522\\-213150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\83\\-33975
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\261\\-107039
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59\\122\\-50130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-225\\464\\-191226
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\-1705
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-27\\11040
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-30\\-174\\-120\\-696\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1740\\0\\-1740\\-1740\\-13920\\-870\\-870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\0\\-60\\-870\\-75\\-135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-174\\-1740\\0\\58\\0\\-1218\\-145\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-1740\\-60\\0\\60\\0\\-30\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-696\\-13920\\-870\\-1218\\0\\348\\0\\-870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-870\\-75\\-145\\-30\\0\\10\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-870\\-135\\-435\\-210\\-870\\-15\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-53169180, -2121060, 57420; -2121060, -82830, 2295; 57420, 2295, -62],[1740, 15, 1],[1, -251, -40, -126, -59, -225, -2, 13; -2, 522, 83, 261, 122, 464, 4, -27; 852, -213150, -33975, -107039, -50130, -191226, -1705, 11040],107,11]
%
% END OF LATTICE L108.11
%
\hbox{}\par\smallskip%
%
% LATTICE L108.12
%
\leavevmode%
\llap{\spacer}%
${L_{108.12}}=\pdualwitharg{2.5}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{580}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{696}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{116}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{}{6}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9607828828440\\-29319586380\\-2407504451520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29319586380\\89472680\\7346824760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2407504451520\\7346824760\\603266126779
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-217\\0\\-866
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
734586\\-2987\\2931610
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
248274\\-1005\\990820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2779844\\-11223\\11093892
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
256340\\-1033\\1023010
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1304379\\-5249\\5205558
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
271387\\-1089\\1083060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6981\\-28\\27860
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-348\\-40\\-232\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\580\\0\\-3480\\-580\\-4640\\-1740\\-580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\40\\0\\-40\\-580\\-300\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-348\\-3480\\0\\696\\0\\-2436\\-1740\\-1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-580\\-40\\0\\20\\0\\-60\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-232\\-4640\\-580\\-2436\\0\\116\\0\\-580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-1740\\-300\\-1740\\-60\\0\\120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-580\\-180\\-1740\\-140\\-580\\-60\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9607828828440, -29319586380, -2407504451520; -29319586380, 89472680, 7346824760; -2407504451520, 7346824760, 603266126779],[1740, 20, 1],[-217, 734586, 248274, 2779844, 256340, 1304379, 271387, 6981; 0, -2987, -1005, -11223, -1033, -5249, -1089, -28; -866, 2931610, 990820, 11093892, 1023010, 5205558, 1083060, 27860],107,12]
%
% END OF LATTICE L108.12
%
\hbox{}\par\smallskip%
%
% LATTICE L108.13
%
\leavevmode%
\llap{\spacer}%
${L_{108.13}}=\pdualwitharg{3.29}(\pfillwitharg{2}({L_{108.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${435}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{174}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{87}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{58}\above{1pt}{1pt}{}{6}{174}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16577898529035\\-46511505\\5477933171700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46511505\\174\\-15369072
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5477933171700\\-15369072\\1810105893763
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46661476\\-870\\141211875
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14240709\\-266\\43096734
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90998711\\-1701\\275389887
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40365079\\-755\\122157055
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44337235\\-830\\134178012
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25349623\\-475\\76715700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
774747\\14\\-2344621
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
435\\0\\-435\\-435\\-870\\-870\\-435\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-30\\-87\\-120\\-87\\-87
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-435\\0\\174\\0\\-174\\-435\\-435\\-783
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-435\\-30\\0\\10\\0\\-105\\-145\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-870\\-87\\-174\\0\\87\\0\\-87\\-609
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-870\\-120\\-435\\-105\\0\\15\\0\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-435\\-87\\-435\\-145\\-87\\0\\58\\-87
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-87\\-783\\-435\\-609\\-435\\-87\\174
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16577898529035, -46511505, 5477933171700; -46511505, 174, -15369072; 5477933171700, -15369072, 1810105893763],[435, 87, 1],[-46661476, -14240709, -90998711, -40365079, -44337235, -25349623, 774747, 0; -870, -266, -1701, -755, -830, -475, 14, 1; 141211875, 43096734, 275389887, 122157055, 134178012, 76715700, -2344621, 0],107,13]
%
% END OF LATTICE L108.13
%
\hbox{}\par\smallskip%
%
% LATTICE L108.14
%
\leavevmode%
\llap{\spacer}%
${L_{108.14}}=\pdualwitharg{29}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${580}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{58}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{116}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{174}\above{1pt}{1pt}{}{6}{58}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-2039180820\\41422440\\-31720200
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41422440\\-841406\\644061
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-31720200\\644061\\-489374
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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161\\8370\\580
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-1675\\-87076\\-6030
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-7548\\-392388\\-27173
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-12454\\-647430\\-44835
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-11487\\-597160\\-41354
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-7997\\-415730\\-28790
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\HardButStrongLineBreak\kern3pt%
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-2054\\-106779\\-7395
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1885\\97993\\6786
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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580\\0\\-290\\-870\\-1160\\-1160\\-870\\0
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0\\4\\0\\-60\\-116\\-160\\-174\\-58
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-1160\\-160\\-290\\-210\\0\\20\\0\\-290
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-870\\-174\\-435\\-435\\-174\\0\\174\\-87
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-58\\-261\\-435\\-406\\-290\\-87\\58
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2039180820, 41422440, -31720200; 41422440, -841406, 644061; -31720200, 644061, -489374],[1740, 29, 1],[161, -1675, -7548, -12454, -11487, -7997, -2054, 1885; 8370, -87076, -392388, -647430, -597160, -415730, -106779, 97993; 580, -6030, -27173, -44835, -41354, -28790, -7395, 6786],107,14]
%
% END OF LATTICE L108.14
%
\hbox{}\par\smallskip%
%
% LATTICE L108.15
%
\leavevmode%
\llap{\spacer}%
${L_{108.15}}=\pdualwitharg{5.29}(\pfillwitharg{2}({L_{108.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${29}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{290}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{145}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{870}\above{1pt}{1pt}{}{6}{290}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6793508233545\\38438775\\2741174750250
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38438775\\290\\15509925
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2741174750250\\15509925\\1106061662666
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-9417823\\28253295\\23339983
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\HardButStrongLineBreak\kern3pt%
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-14371221\\43113397\\35615880
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-91832688\\275496363\\227586925
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-24441008\\73322571\\60571611
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-44743573\\134229889\\110887010
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-5116389\\15349072\\12679834
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2345542\\-7036584\\-5812905
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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29\\0\\-145\\-87\\-290\\-58\\-435\\0
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-87\\-30\\0\\6\\0\\-21\\-435\\-435
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-435\\-435\\-2175\\-435\\-435\\0\\870\\-435
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-145\\-1305\\-435\\-1015\\-145\\-435\\290
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6793508233545, 38438775, 2741174750250; 38438775, 290, 15509925; 2741174750250, 15509925, 1106061662666],[435, 145, 1],[-9417823, -14371221, -91832688, -24441008, -44743573, -5116389, 2345542, 0; 28253295, 43113397, 275496363, 73322571, 134229889, 15349072, -7036584, 1; 23339983, 35615880, 227586925, 60571611, 110887010, 12679834, -5812905, 0],107,15]
%
% END OF LATTICE L108.15
%
\hbox{}\par\smallskip%
%
% LATTICE L108.16
%
\leavevmode%
\llap{\spacer}%
${L_{108.16}}=\pdualwitharg{2.3.5}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}29\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{1740}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{s}{2}{232}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{348}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{}{6}{120}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2726016716760\\230910180\\-683078269320
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230910180\\19560\\-57860880
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-683078269320\\-57860880\\171163998793
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-651\\-2\\-2598
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\HardButStrongLineBreak\kern3pt%
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-100063\\-1015\\-399330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31257\\-341\\-124740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-114611\\-1305\\-457388
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-31475\\-369\\-125610
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\HardButStrongLineBreak\kern3pt%
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-162586\\-1943\\-648846
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-11722\\-145\\-46780
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\HardButStrongLineBreak\kern3pt%
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-3473\\-38\\-13860
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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12\\0\\-60\\-348\\-120\\-696\\-60\\0
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-60\\-1740\\-300\\-580\\-60\\0\\40\\-60
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0\\-1740\\-540\\-1740\\-420\\-1740\\-60\\120
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2726016716760, 230910180, -683078269320; 230910180, 19560, -57860880; -683078269320, -57860880, 171163998793],[1740, 60, 1],[-651, -100063, -31257, -114611, -31475, -162586, -11722, -3473; -2, -1015, -341, -1305, -369, -1943, -145, -38; -2598, -399330, -124740, -457388, -125610, -648846, -46780, -13860],107,16]
%
% END OF LATTICE L108.16
%
\hbox{}\par\smallskip%
%
% LATTICE L108.17
%
\leavevmode%
\llap{\spacer}%
${L_{108.17}}=\pdualwitharg{3.29}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1740}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{174}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{348}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{58}\above{1pt}{1pt}{}{6}{174}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-261468060\\-260448420\\8666940
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\kern3pt%
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-260448420\\-259432086\\8633097
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8666940\\8633097\\-287282
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-899\\1800\\26970
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\HardButStrongLineBreak\kern3pt%
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269\\-538\\-8052
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\HardButStrongLineBreak\kern3pt%
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1615\\-3231\\-48372
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\HardButStrongLineBreak\kern3pt%
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1047\\-2095\\-31370
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\HardButStrongLineBreak\kern3pt%
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3321\\-6646\\-99528
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\HardButStrongLineBreak\kern3pt%
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2693\\-5390\\-80730
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434\\-869\\-13021
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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1740\\0\\-870\\-870\\-3480\\-3480\\-870\\0
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-870\\0\\174\\0\\-348\\-870\\-435\\-783
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-870\\-60\\0\\10\\0\\-210\\-145\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3480\\-348\\-348\\0\\348\\0\\-174\\-1218
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3480\\-480\\-870\\-210\\0\\60\\0\\-870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-870\\-174\\-435\\-145\\-174\\0\\58\\-87
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-174\\-783\\-435\\-1218\\-870\\-87\\174
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-261468060, -260448420, 8666940; -260448420, -259432086, 8633097; 8666940, 8633097, -287282],[1740, 87, 1],[-899, 269, 1615, 1047, 3321, 2693, 434, -404; 1800, -538, -3231, -2095, -6646, -5390, -869, 808; 26970, -8052, -48372, -31370, -99528, -80730, -13021, 12093],107,17]
%
% END OF LATTICE L108.17
%
\hbox{}\par\smallskip%
%
% LATTICE L108.18
%
\leavevmode%
\llap{\spacer}%
${L_{108.18}}=\pdualwitharg{3.5.29}(\pfillwitharg{2}({L_{108.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${87}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{870}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{435}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{290}\above{1pt}{1pt}{}{6}{870}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-721793460268470\\1318068594638325\\437985459418500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1318068594638325\\-2406927903618810\\-799806192137490
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
437985459418500\\-799806192137490\\-265770297489883
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-252892619\\761052249
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-371626407\\1118368395
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-2337337001\\7033956090
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-204454599\\615283406
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-1101108391\\3313663395
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-123003602\\370165677
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\35628578\\-107220250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-28234485\\84968550
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
87\\0\\-435\\-87\\-870\\-174\\-435\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-30\\-435\\-120\\-435\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-435\\0\\870\\0\\-870\\-435\\-2175\\-3915
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87\\-30\\0\\2\\0\\-21\\-145\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-870\\-435\\-870\\0\\435\\0\\-435\\-3045
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-174\\-120\\-435\\-21\\0\\3\\0\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-435\\-435\\-2175\\-145\\-435\\0\\290\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-435\\-3915\\-435\\-3045\\-435\\-435\\870
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-721793460268470, 1318068594638325, 437985459418500; 1318068594638325, -2406927903618810, -799806192137490; 437985459418500, -799806192137490, -265770297489883],[435, 435, 1],[0, -1, -9, -1, -7, -1, -1, 2; -252892619, -371626407, -2337337001, -204454599, -1101108391, -123003602, 35628578, -28234485; 761052249, 1118368395, 7033956090, 615283406, 3313663395, 370165677, -107220250, 84968550],107,18]
%
% END OF LATTICE L108.18
%
\hbox{}\par\smallskip%
%
% LATTICE L108.19
%
\leavevmode%
\llap{\spacer}%
${L_{108.19}}=\pdualwitharg{2.29}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${580}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{232}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}{116}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{696}\above{1pt}{1pt}{}{6}{232}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7325577480\\63471140580\\-1793503260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63471140580\\549934228888\\-15539484468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1793503260\\-15539484468\\439099027
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\0\\290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39134\\-103\\156198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
381770\\-1005\\1523776
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
734972\\-1935\\2933520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
392336\\-1033\\1565942
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
343703\\-905\\1371830
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
413539\\-1089\\1650564
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10637\\-28\\42456
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
580\\0\\-580\\-1740\\-1160\\-1160\\-1740\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-120\\-116\\-160\\-348\\-116
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-580\\0\\232\\0\\-232\\-580\\-1740\\-1044
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1740\\-120\\0\\120\\0\\-420\\-1740\\-1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1160\\-116\\-232\\0\\116\\0\\-348\\-812
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1160\\-160\\-580\\-420\\0\\20\\0\\-580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1740\\-348\\-1740\\-1740\\-348\\0\\696\\-348
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-116\\-1044\\-1740\\-812\\-580\\-348\\232
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7325577480, 63471140580, -1793503260; 63471140580, 549934228888, -15539484468; -1793503260, -15539484468, 439099027],[1740, 116, 1],[71, 39134, 381770, 734972, 392336, 343703, 413539, 10637; 0, -103, -1005, -1935, -1033, -905, -1089, -28; 290, 156198, 1523776, 2933520, 1565942, 1371830, 1650564, 42456],107,19]
%
% END OF LATTICE L108.19
%
\hbox{}\par\smallskip%
%
% LATTICE L108.20
%
\leavevmode%
\llap{\spacer}%
${L_{108.20}}=\pdualwitharg{5.29}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${116}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{290}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{580}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{870}\above{1pt}{1pt}{}{6}{290}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39830340\\-17925480\\-3279900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17925480\\-8066930\\-1475955
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3279900\\-1475955\\-270026
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\98\\-232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
385\\-1516\\3610
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1779\\-7004\\16675
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
599\\-2358\\5613
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2817\\-11088\\26390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
403\\-1586\\3774
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
605\\-2379\\5655
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-433\\1705\\-4060
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
116\\0\\-290\\-174\\-1160\\-232\\-870\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-60\\-580\\-160\\-870\\-290
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-290\\0\\290\\0\\-580\\-290\\-2175\\-1305
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-174\\-60\\0\\6\\0\\-42\\-435\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1160\\-580\\-580\\0\\580\\0\\-870\\-2030
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-232\\-160\\-290\\-42\\0\\4\\0\\-290
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-870\\-870\\-2175\\-435\\-870\\0\\870\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-290\\-1305\\-435\\-2030\\-290\\-435\\290
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-39830340, -17925480, -3279900; -17925480, -8066930, -1475955; -3279900, -1475955, -270026],[1740, 145, 1],[-25, 385, 1779, 599, 2817, 403, 605, -433; 98, -1516, -7004, -2358, -11088, -1586, -2379, 1705; -232, 3610, 16675, 5613, 26390, 3774, 5655, -4060],107,20]
%
% END OF LATTICE L108.20
%
\hbox{}\par\smallskip%
%
% LATTICE L108.21
%
\leavevmode%
\llap{\spacer}%
${L_{108.21}}=\pdualwitharg{2.3.29}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1740}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{696}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}{348}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{232}\above{1pt}{1pt}{}{6}{696}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1125887312760\\-13267727940\\-282042669120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13267727940\\156350136\\3323658912
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-282042669120\\3323658912\\70653666937
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
218\\5\\870
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15327\\806\\61146
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
153613\\8086\\612828
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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102897\\5420\\410500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
170405\\8980\\679818
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
155639\\8205\\620910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68069\\3591\\271556
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17010\\899\\67860
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1740\\0\\-1740\\-1740\\-3480\\-3480\\-1740\\0
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-3480\\-348\\-696\\0\\348\\0\\-348\\-2436
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3480\\-480\\-1740\\-420\\0\\60\\0\\-1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1740\\-348\\-1740\\-580\\-348\\0\\232\\-348
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-348\\-3132\\-1740\\-2436\\-1740\\-348\\696
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1125887312760, -13267727940, -282042669120; -13267727940, 156350136, 3323658912; -282042669120, 3323658912, 70653666937],[1740, 348, 1],[218, 15327, 153613, 102897, 170405, 155639, 68069, 17010; 5, 806, 8086, 5420, 8980, 8205, 3591, 899; 870, 61146, 612828, 410500, 679818, 620910, 271556, 67860],107,21]
%
% END OF LATTICE L108.21
%
\hbox{}\par\smallskip%
%
% LATTICE L108.22
%
\leavevmode%
\llap{\spacer}%
${L_{108.22}}=\pdualwitharg{3.5.29}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${348}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{870}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{1740}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{290}\above{1pt}{1pt}{}{6}{870}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13072620\\18677160\\445440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18677160\\-26683770\\-636405
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
445440\\-636405\\-15178
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-2\\348
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\18\\-2310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\83\\-10875
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\9\\-1199
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-389\\122\\-16530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\16\\-2226
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\4\\-725
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-27\\3480
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
348\\0\\-870\\-174\\-3480\\-696\\-870\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-60\\-1740\\-480\\-870\\-870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-870\\0\\870\\0\\-1740\\-870\\-2175\\-3915
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\kern3pt%
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-174\\-60\\0\\2\\0\\-42\\-145\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3480\\-1740\\-1740\\0\\1740\\0\\-870\\-6090
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-696\\-480\\-870\\-42\\0\\12\\0\\-870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-870\\-870\\-2175\\-145\\-870\\0\\290\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-870\\-3915\\-435\\-6090\\-870\\-435\\870
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13072620, 18677160, 445440; 18677160, -26683770, -636405; 445440, -636405, -15178],[1740, 435, 1],[9, -53, -252, -28, -389, -53, -19, 80; -2, 18, 83, 9, 122, 16, 4, -27; 348, -2310, -10875, -1199, -16530, -2226, -725, 3480],107,22]
%
% END OF LATTICE L108.22
%
\hbox{}\par\smallskip%
%
% LATTICE L108.23
%
\leavevmode%
\llap{\spacer}%
${L_{108.23}}=\pdualwitharg{2.5.29}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${116}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{1160}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{580}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{3480}\above{1pt}{1pt}{}{6}{1160}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1007212920\\-919188060\\944967900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-919188060\\838873720\\-862396200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
944967900\\-862396200\\886579591
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
247\\-743\\-986
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15344\\-46175\\-61270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
148446\\-446723\\-592760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56876\\-171159\\-227112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
151278\\-455247\\-604070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26435\\-79552\\-105558
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
158177\\-476010\\-631620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4067\\-12239\\-16240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
116\\0\\-580\\-348\\-1160\\-232\\-1740\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-120\\-580\\-160\\-1740\\-580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-580\\0\\1160\\0\\-1160\\-580\\-8700\\-5220
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-348\\-120\\0\\24\\0\\-84\\-1740\\-1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1160\\-580\\-1160\\0\\580\\0\\-1740\\-4060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-232\\-160\\-580\\-84\\0\\4\\0\\-580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1740\\-1740\\-8700\\-1740\\-1740\\0\\3480\\-1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-580\\-5220\\-1740\\-4060\\-580\\-1740\\1160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1007212920, -919188060, 944967900; -919188060, 838873720, -862396200; 944967900, -862396200, 886579591],[1740, 580, 1],[247, 15344, 148446, 56876, 151278, 26435, 158177, 4067; -743, -46175, -446723, -171159, -455247, -79552, -476010, -12239; -986, -61270, -592760, -227112, -604070, -105558, -631620, -16240],107,23]
%
% END OF LATTICE L108.23
%
\hbox{}\par\smallskip%
%
% LATTICE L108.24
%
\leavevmode%
\llap{\spacer}%
${L_{108.24}}=\pdualwitharg{2.3.5.29}({L_{108.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${348}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{3480}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{1740}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{1160}\above{1pt}{1pt}{}{6}{3480}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
567240\\66654180\\-16705740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66654180\\7845287640\\-1966288740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16705740\\-1966288740\\492817037
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\567\\2262
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\7723\\30810
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1133\\74583\\297540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-149\\9793\\39068
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1219\\80035\\319290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-221\\14500\\57846
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-477\\31258\\124700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\8287\\33060
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
348\\0\\-1740\\-348\\-3480\\-696\\-1740\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-120\\-1740\\-480\\-1740\\-1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1740\\0\\3480\\0\\-3480\\-1740\\-8700\\-15660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-348\\-120\\0\\8\\0\\-84\\-580\\-1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3480\\-1740\\-3480\\0\\1740\\0\\-1740\\-12180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-696\\-480\\-1740\\-84\\0\\12\\0\\-1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1740\\-1740\\-8700\\-580\\-1740\\0\\1160\\-1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1740\\-15660\\-1740\\-12180\\-1740\\-1740\\3480
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[567240, 66654180, -16705740; 66654180, 7845287640, -1966288740; -16705740, -1966288740, 492817037],[1740, 1740, 1],[-8, -117, -1133, -149, -1219, -221, -477, -126; 567, 7723, 74583, 9793, 80035, 14500, 31258, 8287; 2262, 30810, 297540, 39068, 319290, 57846, 124700, 33060],107,24]
%
% END OF LATTICE L108.24
%

\medskip%
%------------------------------WEYL-GROUP-W109--------------------------------
%
\leavevmode\llap{}%
$W_{109}$%
\qquad\llap{24} lattices, $\chi=90$%
\hfill%
$222422222222422222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L109.1
%
\leavevmode%
\llap{\spacer}%
${L_{109.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{109}}%
\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-6396119940, 2136697380, 6986100; 2136697380, -713788318, -2333787; 6986100, -2333787, -7630],[1740, 1, 1],[1236, 5367, 4129, -1, -1987, -6871, -563347, -59522, -1997591, -33114, -89409, -73681, -8693, -3809, -1823, -24947, -1336, -19417; 3700, 16066, 12360, -3, -5948, -20568, -1686350, -178176, -5979684, -99125, -267641, -220560, -26022, -11402, -5457, -74675, -3999, -58116; -25, -29, 0, 2, -2, -21, -2175, -235, -8004, -140, -406, -360, -47, -28, -24, -870, -80, -2436],108,1]
%
% END OF LATTICE L109.1
%
\hbox{}\par\smallskip%
%
% LATTICE L109.2
%
\leavevmode%
\llap{\spacer}%
${L_{109.2}}=\pfillwitharg{2}({L_{109.1}})$%
{${}={}$}Nikulin $109$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-435, 0, 0; 0, 1, 0; 0, 0, 1],[435, 1, 1],[-8, -24, -11, -3, -1, -2, -84, -8, -125, -3, -5, -1, 0, 0, -1, -119, -13, -223; -25, -87, -45, -14, -6, -15, -725, -71, -1131, -30, -58, -15, -1, 1, 0, -290, -34, -609; -165, -493, -225, -61, -20, -39, -1595, -151, -2349, -55, -87, -15, 1, 0, -21, -2465, -269, -4611],108,2]
%
% END OF LATTICE L109.2
%
\hbox{}\par\smallskip%
%
% LATTICE L109.3
%
\leavevmode%
\llap{\spacer}%
${L_{109.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{109.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-32223930, 64815, -10727535; 64815, -129, 21576; -10727535, 21576, -3571258],[435, 3, 1],[-1703, -1699, 48, 142, -143, -665, -195969, -21077, -119290, -12388, -35450, -5172, -3985, -1138, -616, -64074, -5824, -29306; 4870, 4872, -135, -406, 405, 1892, 557960, 60014, 339677, 35280, 100978, 14735, 11356, 3245, 1758, 182990, 16636, 83723; 5145, 5133, -145, -429, 432, 2009, 592035, 63675, 360383, 37425, 107097, 15625, 12039, 3438, 1861, 193575, 17595, 88537],108,3]
%
% END OF LATTICE L109.3
%
\hbox{}\par\smallskip%
%
% LATTICE L109.4
%
\leavevmode%
\llap{\spacer}%
${L_{109.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{109.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\end{tabular}\endgroup%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-45952530, 99615, 18393975; 99615, -215, -39875; 18393975, -39875, -7362778],[435, 5, 1],[410, 2042, -35, -171, 173, 2408, 47290, 25430, 431771, 2989, 42763, 3743, 4806, 1372, 2227, 15437, 7015, 105889; 974, 4872, -81, -406, 405, 5676, 111592, 60014, 1019031, 7056, 100978, 8841, 11356, 3245, 5274, 36598, 16636, 251169; 1019, 5075, -87, -425, 430, 5985, 117537, 63205, 1073145, 7429, 106285, 9303, 11945, 3410, 5535, 38367, 17435, 263175],108,4]
%
% END OF LATTICE L109.4
%
\hbox{}\par\smallskip%
%
% LATTICE L109.5
%
\leavevmode%
\llap{\spacer}%
${L_{109.5}}=\pdualwitharg{3}({L_{109.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{b}{2}{174}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{*}{4}{12}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{870}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{116}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10399980\\-577680\\40020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-577680\\-32082\\2223
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40020\\2223\\-154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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59971\\-81780\\14393280
\end{tabular}\endgroup%
\kern3pt%
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3322\\-4531\\797280
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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108\\-140\\26025
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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307\\-406\\73863
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\HardButStrongLineBreak\kern3pt%
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89\\-120\\21380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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34\\-47\\8151
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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19\\-28\\4530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-8\\1183
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\HardButStrongLineBreak\kern3pt%
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509\\-870\\119625
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46\\-80\\10791
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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461\\-812\\107996
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10399980, -577680, 40020; -577680, -32082, 2223; 40020, 2223, -154],[1740, 3, 1],[108, 307, 89, 34, 19, 5, 509, 46, 461, 13, 12, -1, -1, 3, 6, 1724, 185, 2091; -140, -406, -120, -47, -28, -8, -870, -80, -812, -25, -29, 0, 2, -2, -7, -2175, -235, -2668; 26025, 73863, 21380, 8151, 4530, 1183, 119625, 10791, 107996, 3015, 2697, -260, -231, 750, 1457, 416295, 44649, 504484],108,5]
%
% END OF LATTICE L109.5
%
\hbox{}\par\smallskip%
%
% LATTICE L109.6
%
\leavevmode%
\llap{\spacer}%
${L_{109.6}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{109.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{870}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{4}{15}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{174}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{145}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53895630\\8553840\\18563625
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8553840\\1357665\\2946255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18563625\\2946255\\6393991
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63931514\\-391935\\-185431365
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10121450\\-62051\\-29356950
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22020393\\-134997\\-63869464
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3582\\23\\10389
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22048\\145\\63945
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\HardButStrongLineBreak\kern3pt%
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-140\\1\\406
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-31\\0\\90
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-2384\\14\\6915
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-7790\\47\\22595
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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-232414\\1413\\674115
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1311178\\7975\\3803060
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[53895630, 8553840, 18563625; 8553840, 1357665, 2946255; 18563625, 2946255, 6393991],[435, 15, 1],[-3582, -22048, -140, -31, -2384, -7790, -434356, -232414, -1311178, -26921, -380037, -10952, -41564, -11481, -6075, -128563, -58751, -297162; 23, 145, 1, 0, 14, 47, 2639, 1413, 7975, 164, 2320, 67, 255, 71, 38, 812, 372, 1885; 10389, 63945, 406, 90, 6915, 22595, 1259847, 674115, 3803060, 78084, 1102290, 31766, 120555, 33300, 17620, 372882, 170400, 861880],108,6]
%
% END OF LATTICE L109.6
%
\hbox{}\par\smallskip%
%
% LATTICE L109.7
%
\leavevmode%
\llap{\spacer}%
${L_{109.7}}=\pdualwitharg{2}({L_{109.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2483996970840, 4273410420, -620999245320; 4273410420, 7351880, -1068352608; -620999245320, -1068352608, 155249811983],[1740, 4, 1],[351174, 447108, 23984, 1, 41456, 421066, 39494152, 4230182, 35817959, 2456499, 6954939, 1507474, 765728, 213787, 344663, 12320177, 1129921, 8594323; -470245, -598705, -32115, 0, -55511, -563829, -52884835, -5664447, -47962230, -3289390, -9313060, -2018595, -1025355, -286274, -461526, -16497520, -1513038, -11508360; 1401460, 1784312, 95715, 4, 165442, 1680384, 157612680, 16881748, 142941783, 9803360, 27755668, 6016005, 3055856, 853178, 1375476, 49167180, 4509272, 34298097],108,7]
%
% END OF LATTICE L109.7
%
\hbox{}\par\smallskip%
%
% LATTICE L109.8
%
\leavevmode%
\llap{\spacer}%
${L_{109.8}}=\pdualwitharg{5}({L_{109.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5761140, -579420, 40020; -579420, -58270, 4025; 40020, 4025, -278],[1740, 5, 1],[3, 12, -1, -1, 5, 26, 484, 259, 8767, 30, 423, 73, 46, 25, 19, 125, 56, 1673; -5, -29, 0, 2, -2, -21, -435, -235, -8004, -28, -406, -72, -47, -28, -24, -174, -80, -2436; 359, 1305, -144, -115, 690, 3435, 63307, 33845, 1144920, 3909, 54955, 9456, 5935, 3190, 2385, 15457, 6895, 205320],108,8]
%
% END OF LATTICE L109.8
%
\hbox{}\par\smallskip%
%
% LATTICE L109.9
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-219604965, -353220, 15127125; -353220, -406, 24331; 15127125, 24331, -1042007],[435, 29, 1],[10088, 348, -839, -841, 839, 11758, 39856, 124320, 72791, 73083, 7213, 91571, 23524, 6722, 10925, 13071, 34461, 17941; -25, -1, 0, 2, -1, -21, -75, -235, -138, -140, -14, -180, -47, -14, -24, -30, -80, -42; 146450, 5052, -12180, -12209, 12180, 170694, 578600, 1804786, 1056726, 1060965, 104713, 1329360, 341504, 97585, 158601, 189755, 500279, 260454],108,9]
%
% END OF LATTICE L109.9
%
\hbox{}\par\smallskip%
%
% LATTICE L109.10
%
\leavevmode%
\llap{\spacer}%
${L_{109.10}}=\pdualwitharg{2.3}({L_{109.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8693280120, 394980, -2178401700; 394980, 24, -98976; -2178401700, -98976, 545873813],[1740, 12, 1],[-197906, -557228, -40032, -60531, -16528, -8586, -892088, -81138, -204455, -24041, -27905, -322, 0, -3649, -11591, -3203843, -342579, -965811; -14795, -41673, -2995, -4531, -1239, -645, -67135, -6109, -15399, -1815, -2117, -25, 1, -271, -865, -239395, -25601, -72181; -789780, -2223720, -159755, -241560, -65958, -34264, -3560040, -323796, -815915, -95940, -111360, -1285, 0, -14562, -46256, -12785520, -1367124, -3854245],108,10]
%
% END OF LATTICE L109.10
%
\hbox{}\par\smallskip%
%
% LATTICE L109.11
%
\leavevmode%
\llap{\spacer}%
${L_{109.11}}=\pdualwitharg{3.5}({L_{109.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1472549820, -16371660, 1129260; -16371660, -182010, 12555; 1129260, 12555, -866],[1740, 15, 1],[5, 12, -1, -1, 15, 22, 1180, 629, 7079, 72, 1003, 57, 106, 55, 13, 241, 106, 1041; -5, -29, 0, 2, -2, -7, -435, -235, -2668, -28, -406, -24, -47, -28, -8, -174, -80, -812; 6447, 15225, -1304, -1275, 19530, 28585, 1532331, 816765, 9191840, 93477, 1301955, 73976, 137535, 71310, 16835, 311721, 137055, 1345600],108,11]
%
% END OF LATTICE L109.11
%
\hbox{}\par\smallskip%
%
% LATTICE L109.12
%
\leavevmode%
\llap{\spacer}%
${L_{109.12}}=\pdualwitharg{2.5}({L_{109.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}29\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2116957080, 193140, -530494680; 193140, 40, -48400; -530494680, -48400, 132938267],[1740, 20, 1],[-4663, -25435, -106, 0, -4027, -37303, -681629, -364129, -3078035, -41966, -589220, -25328, -63585, -17168, -26282, -178888, -80966, -609895; -387, -2117, -9, 1, -331, -3075, -56231, -30041, -253953, -3463, -48633, -2091, -5251, -1419, -2175, -14819, -6709, -50547; -18608, -101500, -423, 0, -16070, -148860, -2720084, -1453080, -12283095, -167468, -2351320, -101073, -253740, -68510, -104880, -713864, -323100, -2433825],108,12]
%
% END OF LATTICE L109.12
%
\hbox{}\par\smallskip%
%
% LATTICE L109.13
%
\leavevmode%
\llap{\spacer}%
${L_{109.13}}=\pdualwitharg{3.29}(\pfillwitharg{2}({L_{109.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[112048693305, 4213410, 37024886235; 4213410, 174, 1392261; 37024886235, 1392261, 12234343483],[435, 87, 1],[-366967, -15965, -16003, 0, -44588, -149767, -1450079, -4503273, -876441, -2613472, -254984, -1067793, -812847, -226361, -121182, -446564, -1186656, -207355; 60, 2, 0, 1, 14, 40, 370, 1144, 222, 655, 63, 260, 194, 51, 25, 85, 221, 38; 1110555, 48315, 48430, 0, 134937, 453241, 4388385, 13628289, 2652380, 7909170, 771660, 3231470, 2459925, 685038, 366734, 1351440, 3591186, 627520],108,13]
%
% END OF LATTICE L109.13
%
\hbox{}\par\smallskip%
%
% LATTICE L109.14
%
\leavevmode%
\llap{\spacer}%
${L_{109.14}}=\pdualwitharg{29}({L_{109.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${290}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{1740}\above{1pt}{1pt}{r}{2}{58}\above{1pt}{1pt}{*}{4}{116}\above{1pt}{1pt}{b}{2}{174}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{58}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4383060\\984840\\1131000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
984840\\-221270\\-254127
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1131000\\-254127\\-291842
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-481069\\14940\\-1877460
\end{tabular}\endgroup%
\kern3pt%
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124131\\-3855\\484444
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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144\\-25\\580
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\HardButStrongLineBreak\kern3pt%
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-9\\-1\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-449\\0\\-1740
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\HardButStrongLineBreak\kern3pt%
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-22\\2\\-87
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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239\\-2\\928
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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983\\-21\\3828
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2953\\-75\\11510
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\HardButStrongLineBreak\kern3pt%
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9106\\-235\\35496
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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10583\\-276\\41256
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\-132\\-8700\\-2436\\-4524\\-10092\\-24300\\-73080\\-83532
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4383060, 984840, 1131000; 984840, -221270, -254127; 1131000, -254127, -291842],[1740, 29, 1],[144, -9, -449, -22, 239, 983, 2953, 9106, 10583, 5169, 492, 12041, 1471, 727, 466, 428, 1037, 1009; -25, -1, 0, 2, -2, -21, -75, -235, -276, -140, -14, -360, -47, -28, -24, -30, -80, -84; 580, -34, -1740, -87, 928, 3828, 11510, 35496, 41256, 20155, 1919, 46980, 5742, 2842, 1827, 1685, 4089, 3984],108,14]
%
% END OF LATTICE L109.14
%
\hbox{}\par\smallskip%
%
% LATTICE L109.15
%
\leavevmode%
\llap{\spacer}%
${L_{109.15}}=\pdualwitharg{5.29}(\pfillwitharg{2}({L_{109.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${58}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{87}\above{1pt}{1pt}{r}{2}{290}\above{1pt}{1pt}{}{4}{145}\above{1pt}{1pt}{r}{2}{870}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{290}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45886581825\\-3480870\\18519751335
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3480870\\290\\-1404905
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18519751335\\-1404905\\7474542178
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11802289469\\-35409674265\\-29249311695
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-820230\\2460884\\2032755
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4763294338\\-14291015431\\-11804750354
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[45886581825, -3480870, 18519751335; -3480870, 290, -1404905; 18519751335, -1404905, 7474542178],[435, 145, 1],[-74060, -16110, -9689, 0, -44993, -453382, -292650, -4544176, -2653205, -527442, -257300, -646495, -820230, -228417, -366848, -90124, -1197434, -627715; 222192, 48332, 29067, 1, 134993, 1360266, 878024, 13633672, 7960281, 1582457, 771963, 1939641, 2460884, 685302, 1100619, 270389, 3592523, 1883259; 183541, 39925, 24012, 0, 111505, 1123605, 725267, 11261715, 6575370, 1307146, 637660, 1602192, 2032755, 566080, 909150, 223352, 2967570, 1555650],108,15]
%
% END OF LATTICE L109.15
%
\hbox{}\par\smallskip%
%
% LATTICE L109.16
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[359075263560, 2211540, -89975817600; 2211540, 120, -554160; -89975817600, -554160, 22545824161],[1740, 60, 1],[-137789, -708505, -326, 0, -131921, -399023, -21735887, -11604267, -32684135, -1334858, -18703660, -267428, -2008155, -537464, -270362, -5441144, -2452818, -6140195; -411, -2117, -1, 1, -391, -1185, -64583, -34481, -97121, -3967, -55593, -795, -5971, -1599, -805, -16211, -7309, -18299; -549888, -2827500, -1301, 0, -526470, -1592420, -86743524, -46310280, -130435765, -5327148, -74642520, -1067251, -8014140, -2144910, -1078960, -21714504, -9788700, -24504275],108,16]
%
% END OF LATTICE L109.16
%
\hbox{}\par\smallskip%
%
% LATTICE L109.17
%
\leavevmode%
\llap{\spacer}%
${L_{109.17}}=\pdualwitharg{3.29}({L_{109.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-28489020, 33637680, -4924200; 33637680, -39716718, 5814123; -4924200, 5814123, -851126],[1740, 87, 1],[-3413, -94, 401, 258, -1055, -1898, -18406, -57165, -22251, -33173, -3232, -27009, -10236, -5597, -1428, -4876, -12681, -4357; -25, -1, 0, 2, -2, -7, -75, -235, -92, -140, -14, -120, -47, -28, -8, -30, -80, -28; 19575, 537, -2320, -1479, 6090, 10933, 105975, 329121, 128104, 190965, 18603, 155440, 58899, 32190, 8207, 28005, 72819, 25016],108,17]
%
% END OF LATTICE L109.17
%
\hbox{}\par\smallskip%
%
% LATTICE L109.18
%
\leavevmode%
\llap{\spacer}%
${L_{109.18}}=\pdualwitharg{3.5.29}(\pfillwitharg{2}({L_{109.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1744976432625, 3186263880120, 1058774375235; 3186263880120, -5818002644505, -1933283728785; 1058774375235, -1933283728785, -642417373171],[435, 435, 1],[-5, -1, 0, 2, -1, -7, -15, -235, -46, -28, -14, -12, -47, -14, -8, -6, -80, -14; -6134806, -1058124, 170094, 2557193, -2551266, -11917714, -24238312, -378024094, -73779344, -44445141, -21932761, -18562802, -71529848, -20439619, -11073171, -7948927, -104784491, -18184196; 18462009, 3184305, -511879, -7695585, 7677750, 35865025, 72942483, 1137621135, 222030665, 133752669, 66004185, 55862671, 215261055, 61510740, 33323465, 23921403, 315337155, 54723295],108,18]
%
% END OF LATTICE L109.18
%
\hbox{}\par\smallskip%
%
% LATTICE L109.19
%
\leavevmode%
\llap{\spacer}%
${L_{109.19}}=\pdualwitharg{2.29}({L_{109.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1160}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{435}\above{1pt}{1pt}{r}{2}{232}\above{1pt}{1pt}{*}{4}{116}\above{1pt}{1pt}{*}{2}{696}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}{232}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
998760\\-1740\\-231420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231420\\348\\53635
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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151856\\158625\\654240
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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-65\\-68\\-280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-101\\-105\\-435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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0\\1\\0
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-1777\\-1854\\-7656
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-5747\\-6000\\-24760
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[998760, -1740, -231420; -1740, 232, 348; -231420, 348, 53635],[1740, 116, 1],[-1481, -65, -101, 0, -175, -1777, -5747, -17851, -5212, -10366, -1012, -6361, -3231, -902, -1454, -1792, -4766, -1250; -1550, -68, -105, 1, -182, -1854, -6000, -18638, -5442, -10825, -1057, -6645, -3376, -943, -1521, -1875, -4987, -1308; -6380, -280, -435, 0, -754, -7656, -24760, -76908, -22455, -44660, -4360, -27405, -13920, -3886, -6264, -7720, -20532, -5385],108,19]
%
% END OF LATTICE L109.19
%
\hbox{}\par\smallskip%
%
% LATTICE L109.20
%
\leavevmode%
\llap{\spacer}%
${L_{109.20}}=\pdualwitharg{5.29}({L_{109.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${58}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{348}\above{1pt}{1pt}{r}{2}{290}\above{1pt}{1pt}{*}{4}{580}\above{1pt}{1pt}{b}{2}{870}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{290}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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66626340\\-78115850\\-32137075
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27410220\\-32137075\\-13221262
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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34552340\\-35671\\71720480
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14213246\\-14673\\29502511
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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29468\\-28\\61161
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70919\\-72\\147204
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44357\\-47\\92075
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17387\\-24\\36105
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3831\\-6\\7957
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49354\\-80\\102515
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50519\\-84\\104940
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-56826660, 66626340, 27410220; 66626340, -78115850, -32137075; 27410220, -32137075, -13221262],[1740, 145, 1],[29468, 14249, 70919, 44357, 23607, 17387, 3831, 49354, 50519, 2498, 281, -1175, -767, 5453, 26203, 16511, 255776, 298201; -28, -14, -72, -47, -28, -24, -6, -80, -84, -5, -1, 0, 2, -2, -21, -15, -235, -276; 61161, 29575, 147204, 92075, 49010, 36105, 7957, 102515, 104940, 5191, 585, -2436, -1595, 11310, 54375, 34267, 530845, 618900],108,20]
%
% END OF LATTICE L109.20
%
\hbox{}\par\smallskip%
%
% LATTICE L109.21
%
\leavevmode%
\llap{\spacer}%
${L_{109.21}}=\pdualwitharg{2.3.29}({L_{109.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[45153664680, 4369140, -11303723820; 4369140, 696, -1093764; -11303723820, -1093764, 2829763057],[1740, 348, 1],[367639, 14735, 4973, 0, 55625, 176821, 1685693, 5227269, 508176, 3019954, 293228, 610953, 923889, 252338, 131142, 470048, 1239954, 107750; -11795, -473, -160, 1, -1781, -5665, -54015, -167501, -16284, -96775, -9397, -19580, -29611, -8089, -4205, -15075, -39769, -3456; 1468560, 58860, 19865, 0, 222198, 706324, 6733620, 20880696, 2029945, 12063420, 1171320, 2440495, 3690540, 1007982, 523856, 1877640, 4953084, 430415],108,21]
%
% END OF LATTICE L109.21
%
\hbox{}\par\smallskip%
%
% LATTICE L109.22
%
\leavevmode%
\llap{\spacer}%
${L_{109.22}}=\pdualwitharg{3.5.29}({L_{109.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${174}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{116}\above{1pt}{1pt}{r}{2}{870}\above{1pt}{1pt}{*}{4}{1740}\above{1pt}{1pt}{b}{2}{290}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{870}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L109.22
%
\hbox{}\par\smallskip%
%
% LATTICE L109.23
%
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\kern3pt%
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\kern3pt%
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\kern3pt%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16354388760, 2566500, 4076541600; 2566500, 1160, 639160; 4076541600, 639160, 1016130823],[1740, 580, 1],[674694, 326660, 407216, 1022325, 276048, 422642, 99208, 1302206, 338255, 75011, 14115, 1714, 0, 64727, 599683, 377877, 5854069, 1706395; -2047621, -991377, -1235856, -3102646, -837777, -1282677, -301087, -3952075, -1026573, -227652, -42838, -5202, 1, -196438, -1819968, -1146814, -17766440, -5178717; -2705468, -1309880, -1632903, -4099440, -1106930, -1694760, -397816, -5221740, -1356375, -300788, -56600, -6873, 0, -259550, -2404680, -1515256, -23474340, -6842505],108,23]
%
% END OF LATTICE L109.23
%
\hbox{}\par\smallskip%
%
% LATTICE L109.24
%
\leavevmode%
\llap{\spacer}%
${L_{109.24}}=\pdualwitharg{2.3.5.29}({L_{109.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}29\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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3480\\1717380\\-429780
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1717380\\54111442200\\-13541713560
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-429780\\-13541713560\\3388895189
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3480, 1717380, -429780; 1717380, 54111442200, -13541713560; -429780, -13541713560, 3388895189],[1740, 1740, 1],[-15971, -7717, -3200, -24031, -6433, -3237, -2247, -29377, -2536, -1651, -293, -4, 1, -1577, -4773, -8967, -138833, -13484; 2933074, 1417180, 587641, 4412805, 1181146, 594238, 412448, 5392122, 465470, 302983, 53755, 733, 0, 289789, 876697, 1646861, 25497513, 2476400; 11720292, 5662920, 2348159, 17633160, 4719750, 2374520, 1648104, 21546420, 1859975, 1210692, 214800, 2929, 0, 1157970, 3503200, 6580704, 101885700, 9895465],108,24]
%
% END OF LATTICE L109.24
%

\medskip%
%------------------------------WEYL-GROUP-W110--------------------------------
%
\leavevmode\llap{}%
$W_{110}$%
\qquad\llap{24} lattices, $\chi=72$%
\hfill%
$2222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L110.1
%
\leavevmode%
\llap{\spacer}%
${L_{110.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{110}}%
\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{260}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{364}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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1334060\\-267540\\-1820
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-267540\\53654\\365
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-1820\\365\\2
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4864\\24240\\2279
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\-1
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73\\364\\0
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1\\5\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\GramMatrix{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[1334060, -267540, -1820; -267540, 53654, 365; -1820, 365, 2],[1820, 1, 1],[-7844, -15507, -365, -2739, -61, 0, 73, 1, -60, -1517, -243, -13069, -2371, -4864, -442697, -24929; -39091, -77280, -1819, -13650, -304, 0, 364, 5, -299, -7560, -1211, -65130, -11816, -24240, -2206204, -124235; -3679, -7280, -172, -1300, -30, -1, 0, 0, -26, -700, -113, -6110, -1110, -2279, -207480, -11685],109,1]
%
% END OF LATTICE L110.1
%
\hbox{}\par\smallskip%
%
% LATTICE L110.2
%
\leavevmode%
\llap{\spacer}%
${L_{110.2}}=\pfillwitharg{2}({L_{110.1}})$%
{${}={}$}Nikulin $110$%
\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-910, 455, 3185; 455, -223, -1365; 3185, -1365, 354],[455, 1, 1],[38253, 38617, 1968, 8429, 302, 193, 1921, -112, -280, 2277, 953, 29544, 5540, 23175, 1061323, 119873; 88790, 89635, 4568, 19565, 701, 448, 4459, -260, -650, 5285, 2212, 68575, 12859, 53792, 2463461, 278240; -1768, -1785, -91, -390, -14, -9, -91, 5, 13, -105, -44, -1365, -256, -1071, -49049, -5540],109,2]
%
% END OF LATTICE L110.2
%
\hbox{}\par\smallskip%
%
% LATTICE L110.3
%
\leavevmode%
\llap{\spacer}%
${L_{110.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{110.1}}))$%
\nopagebreak\par%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5175170, -447265, -1030575; -447265, -38490, -89065; -1030575, -89065, -205227],[455, 5, 1],[-22882, -54931, -87499, -469148, -425246, -1743971, -79352209, -1787263, -2810923, -555427, -130486, -97613, -10724, 91, 5669, -214; -1833, -4403, -7014, -37609, -34090, -139807, -6361355, -143278, -225342, -44527, -10461, -7826, -860, 7, 455, -17; 115700, 277753, 442430, 2372201, 2150215, 8818220, 401236745, 9037122, 14213160, 2808463, 659790, 493571, 54225, -460, -28665, 1082],109,3]
%
% END OF LATTICE L110.3
%
\hbox{}\par\smallskip%
%
% LATTICE L110.4
%
\leavevmode%
\llap{\spacer}%
${L_{110.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{110.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${182}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{455}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-5712070\\-555555\\2451995
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-555555\\-53886\\238483
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2451995\\238483\\-1052557
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5712070, -555555, 2451995; -555555, -53886, 238483; 2451995, 238483, -1052557],[455, 7, 1],[49554, 84959, 189459, 5079106, 920760, 3776111, 24545181, 19349227, 6086303, 859019, 282530, 1056751, 23218, -199, -1753, 2322; -1833, -3145, -7014, -188045, -34090, -139807, -908765, -716390, -225342, -31805, -10461, -39130, -860, 7, 65, -85; 115024, 197205, 439768, 11789505, 2137247, 8765022, 56973709, 44912980, 14127386, 1993935, 655802, 2452905, 53893, -462, -4069, 5390],109,4]
%
% END OF LATTICE L110.4
%
\hbox{}\par\smallskip%
%
% LATTICE L110.5
%
\leavevmode%
\llap{\spacer}%
${L_{110.5}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{110.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{455}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-7737730\\-880425\\-1782235
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\kern3pt%
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-880425\\-100074\\-202787
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7737730, -880425, -1782235; -880425, -100074, -202787; -1782235, -202787, -410503],[455, 13, 1],[-1986, -309697, -98655, -203438, -479436, -1966195, -6881799, -10074983, -243775, -3130957, -147106, -42323, -12086, 107, 491, -1218; -141, -22015, -7014, -14465, -34090, -139807, -489335, -716390, -17334, -222635, -10461, -3010, -860, 7, 35, -85; 8692, 1355445, 431782, 890385, 2098343, 8605428, 30119509, 44095090, 1066928, 13703235, 643838, 185235, 52897, -468, -2149, 5330],109,5]
%
% END OF LATTICE L110.5
%
\hbox{}\par\smallskip%
%
% LATTICE L110.6
%
\leavevmode%
\llap{\spacer}%
${L_{110.6}}=\pdualwitharg{2}({L_{110.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${104}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{260}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{91}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[180394760, 507780, -45095960; 507780, 1416, -126940; -45095960, -126940, 11273307],[1820, 4, 1],[184731, 92471, 9141, 37229, 1140, 454, 114, -907, -443, 6761, 5035, 147059, 27212, 112954, 2579918, 582123; -147589, -73885, -7306, -29770, -913, -365, -91, 730, 364, -5390, -4019, -117455, -21737, -90235, -2061059, -465055; 737308, 369075, 36484, 148590, 4550, 1812, 455, -3620, -1768, 26985, 20096, 586950, 108610, 450828, 10297105, 2323400],109,6]
%
% END OF LATTICE L110.6
%
\hbox{}\par\smallskip%
%
% LATTICE L110.7
%
\leavevmode%
\llap{\spacer}%
${L_{110.7}}=\pdualwitharg{5}({L_{110.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-482339746980, -5244841420, 13924820; -5244841420, -57031090, 151415; 13924820, 151415, -402],[1820, 5, 1],[503, 193, 20, 23, -1, -2, 61, 2, 17, 31, 20, 185, 161, 322, 29059, 326; -45968, -17640, -1829, -2106, 90, 183, -5460, -181, -1547, -2828, -1826, -16900, -14710, -29423, -2655380, -29790; 109265, 41104, 3875, 3458, -740, -350, 56420, 1103, 6175, 8624, 5005, 42718, 36260, 71390, 6408220, 71713],109,7]
%
% END OF LATTICE L110.7
%
\hbox{}\par\smallskip%
%
% LATTICE L110.8
%
\leavevmode%
\llap{\spacer}%
${L_{110.8}}=\pdualwitharg{7}({L_{110.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[43513016820, 415393160, 2993900; 415393160, 3965514, 28581; 2993900, 28581, 206],[1820, 7, 1],[8561, 2421, 401, 3039, 71, 3, 1, -1, 58, 231, 262, 14199, 2581, 5301, 68951, 27184; -896129, -253420, -41975, -318110, -7432, -314, -104, 105, -6071, -24180, -27425, -1486290, -270168, -554886, -7217496, -2845505; -89908, -25420, -4207, -31850, -742, -35, -104, -35, -637, -2440, -2758, -149240, -27118, -55685, -724256, -285530],109,8]
%
% END OF LATTICE L110.8
%
\hbox{}\par\smallskip%
%
% LATTICE L110.9
%
\leavevmode%
\llap{\spacer}%
${L_{110.9}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{110.1}}))$%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14558943490, -15589665, 2939512940; -15589665, 16695, -3147620; 2939512940, -3147620, 593500231],[455, 35, 1],[159307, 52731, 585433, 3130722, 2835879, 11625600, 75558033, 11911937, 18729043, 528449, 867517, 646948, 70021, -2000, -4633, 2253; 9880, 3264, 36229, 193713, 175463, 719289, 4674826, 736997, 1158755, 32694, 53666, 40014, 4327, -129, -286, 142; -788970, -261151, -2899365, -15504944, -14044730, -57575945, -374202205, -58994033, -92755845, -2617151, -4296390, -3204019, -346780, 9905, 22945, -11158],109,9]
%
% END OF LATTICE L110.9
%
\hbox{}\par\smallskip%
%
% LATTICE L110.10
%
\leavevmode%
\llap{\spacer}%
${L_{110.10}}=\pdualwitharg{13}({L_{110.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16380, -3640, -1820; -3640, 806, 377; -1820, 377, -58],[1820, 13, 1],[-4, -117, 5, 87, 245, 552, 3979, 2951, 76, 2133, 67, 57, 43, 24, 53, 1; -19, -560, 23, 410, 1156, 2606, 18788, 13935, 359, 10080, 317, 270, 204, 114, 252, 5; 2, 0, -13, -90, -234, -507, -3612, -2665, -67, -1820, -52, -40, -26, -13, -28, 0],109,10]
%
% END OF LATTICE L110.10
%
\hbox{}\par\smallskip%
%
% LATTICE L110.11
%
\leavevmode%
\llap{\spacer}%
${L_{110.11}}=\pdualwitharg{5.13}(\pfillwitharg{2}({L_{110.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25784031910, 28270515, 5130354320; 28270515, 31005, 5625100; 5130354320, 5625100, 1020807589],[455, 65, 1],[-36431, -1096509, -1738937, -715290, -8422911, -34529160, -120838323, -35379457, -4278959, -10986583, -2576453, -147788, -207869, 6040, 7403, -6741; 760, 22848, 36229, 14901, 175463, 719289, 2517214, 736997, 89135, 228858, 53666, 3078, 4327, -129, -154, 142; 183090, 5510687, 8739315, 3594808, 42330730, 173531995, 607292945, 177805303, 21504615, 55214887, 12948390, 742733, 1044680, -30355, -37205, 33878],109,11]
%
% END OF LATTICE L110.11
%
\hbox{}\par\smallskip%
%
% LATTICE L110.12
%
\leavevmode%
\llap{\spacer}%
${L_{110.12}}=\pdualwitharg{2.5}({L_{110.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${520}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{455}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\GramMatrix{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31525031720, 85454460, -7898767240; 85454460, 370840, -21411380; -7898767240, -21411380, 1979078863],[1820, 20, 1],[-131994054, -12696199, -5337321, -3272107, -43543, -43315, -15787066, -1032265, -5075151, -2065100, -5206464, -24141734, -21044027, -84306965, -1903181804, -85426772; -1145833, -110215, -46333, -28405, -378, -376, -137046, -8961, -44057, -17927, -45197, -209573, -182682, -731864, -16521414, -741585; -526818240, -50673413, -21302460, -13059722, -173790, -172880, -63009765, -4120004, -20256080, -8242283, -20780180, -96355142, -83991490, -336488240, -7596030715, -340957644],109,12]
%
% END OF LATTICE L110.12
%
\hbox{}\par\smallskip%
%
% LATTICE L110.13
%
\leavevmode%
\llap{\spacer}%
${L_{110.13}}=\pdualwitharg{7.13}(\pfillwitharg{2}({L_{110.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{l}{2}{65}\above{1pt}{1pt}{r}{2}{182}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{91}\above{1pt}{1pt}{r}{2}{182}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{910}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26769248415, -34083140, 11568349520; -34083140, 43407, -14729078; 11568349520, -14729078, 4999270431],[455, 91, 1],[-95455, -2052161, -4556287, -9370858, -22069343, -90471794, -45230755, -463498523, -11211541, -20561831, -6750709, -1936138, -544649, 15826, 2771, -88313; 478035, 10277125, 22817664, 46928795, 110522178, 453078259, 226513377, 2321177600, 56146840, 102972625, 33807211, 9696080, 2727572, -79259, -13877, 442275; 222292, 4778995, 10610509, 21822500, 51394252, 210687295, 105331673, 1079377845, 26109013, 47883615, 15720796, 4508805, 1268358, -36855, -6453, 205660],109,13]
%
% END OF LATTICE L110.13
%
\hbox{}\par\smallskip%
%
% LATTICE L110.14
%
\leavevmode%
\llap{\spacer}%
${L_{110.14}}=\pdualwitharg{2.7}({L_{110.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${728}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{1820}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-727100920\\57331820\\182096460
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57331820\\-3757768\\-14359324
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
182096460\\-14359324\\-45604563
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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94648\\515\\377762
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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203432\\1107\\811944
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\HardButStrongLineBreak\kern3pt%
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47779\\260\\190697
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\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-727100920, 57331820, 182096460; 57331820, -3757768, -14359324; 182096460, -14359324, -45604563],[1820, 28, 1],[-23969353, -1617956, -873001, -1970603, 94648, 203432, 47779, -381321, -1005753, -303206, -1048161, -23004503, -3936972, -15572728, -49997991, -78386321; -130442, -8805, -4751, -10725, 515, 1107, 260, -2075, -5473, -1650, -5704, -125190, -21425, -84747, -272090, -426580; -95667208, -6457635, -3484348, -7865130, 377762, 811944, 190697, -1521940, -4014192, -1210165, -4183452, -91816270, -15713362, -62154344, -199553497, -312857860],109,14]
%
% END OF LATTICE L110.14
%
\hbox{}\par\smallskip%
%
% LATTICE L110.15
%
\leavevmode%
\llap{\spacer}%
${L_{110.15}}=\pdualwitharg{5.7}({L_{110.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${910}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{364}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{260}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103740\\507780\\-67340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
507780\\-2235590\\293615
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67340\\293615\\-38526
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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510119\\1311960\\9107280
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\kern3pt%
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\kern3pt%
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222033\\571039\\3964001
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22454\\57746\\400855
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1333\\3428\\23796
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1304\\3353\\23275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2417\\6214\\43134
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-103740, 507780, -67340; 507780, -2235590, 293615; -67340, 293615, -38526],[1820, 35, 1],[22454, 1333, 1304, 2417, 463, -47, -3439, -370, -1199, -25, 331, 6085, 6077, 13127, 173479, 13777; 57746, 3428, 3353, 6214, 1190, -121, -8840, -951, -3081, -64, 852, 15652, 15630, 33761, 446160, 35432; 400855, 23796, 23275, 43134, 8260, -840, -61360, -6601, -21385, -444, 5915, 108654, 108500, 234360, 3097120, 245959],109,15]
%
% END OF LATTICE L110.15
%
\hbox{}\par\smallskip%
%
% LATTICE L110.16
%
\leavevmode%
\llap{\spacer}%
${L_{110.16}}=\pdualwitharg{2.13}({L_{110.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[133926520, 4038580, -33489820; 4038580, 121784, -1009892; -33489820, -1009892, 8374503],[1820, 52, 1],[1, 8093, 5797, 12809, 30698, 127156, 223256, 654669, 15957, 103613, 10161, 3149, 1218, 508, 156, -391; 0, -490, -351, -775, -1857, -7691, -13503, -39595, -965, -6265, -614, -190, -73, -29, -7, 30; 4, 32305, 23140, 51130, 122538, 507572, 891177, 2613260, 63696, 413595, 40560, 12570, 4862, 2028, 623, -1560],109,16]
%
% END OF LATTICE L110.16
%
\hbox{}\par\smallskip%
%
% LATTICE L110.17
%
\leavevmode%
\llap{\spacer}%
${L_{110.17}}=\pdualwitharg{5.13}({L_{110.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-225641780, -31213000, 129220; -31213000, -4317690, 17875; 129220, 17875, -74],[1820, 65, 1],[-1, -11, -4, -1, -5, -2, 5, 2, 1, 11, 4, 1, 5, 2, -5, -2; -8, 0, 19, 6, 34, 15, -28, -13, -7, -84, -34, -12, -94, -135, -812, -110; -3695, -19292, -2405, -298, -520, 130, 1960, 351, 55, -1092, -1235, -1158, -14040, -29250, -205800, -30199],109,17]
%
% END OF LATTICE L110.17
%
\hbox{}\par\smallskip%
%
% LATTICE L110.18
%
\leavevmode%
\llap{\spacer}%
${L_{110.18}}=\pdualwitharg{7.13}({L_{110.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[134255940, 53773720, 171080; 53773720, 21538062, 68523; 171080, 68523, 218],[1820, 91, 1],[3, 37, 27, 71, 143, 263, 253, 1273, 28, 87, 4, -9, -19, -15, -5, -2; -7, -80, -55, -130, -248, -436, -412, -2055, -43, -120, 5, 30, 48, 36, 12, 5; -154, -3900, -3913, -14910, -34398, -69615, -69312, -354445, -8491, -30680, -4732, -2380, -182, 455, 152, 0],109,18]
%
% END OF LATTICE L110.18
%
\hbox{}\par\smallskip%
%
% LATTICE L110.19
%
\leavevmode%
\llap{\spacer}%
${L_{110.19}}=\pdualwitharg{5.7.13}(\pfillwitharg{2}({L_{110.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{910}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{455}\above{1pt}{1pt}{r}{2}{910}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{182}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1444318886465\\2113715364670\\421266926080
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1444318886465, 2113715364670, 421266926080; 2113715364670, -3093356095830, -616510924575; 421266926080, -616510924575, -122871634673],[455, 455, 1],[-141, -629, -7014, -2893, -34090, -139807, -69905, -143278, -17334, -6361, -10461, -602, -860, 7, 5, -17; -6435292, -28672041, -319675117, -131841414, -1553532818, -6371128803, -3185618347, -6529262345, -789912473, -289867065, -476672648, -27428139, -39162712, 346683, 227287, -789328; 32288725, 143860392, 1603952350, 661507056, 7794765615, 31966788945, 15983665140, 32760215306, 3963342460, 1454392082, 2391678835, 137619181, 196496755, -1739465, -1140400, 3960411],109,19]
%
% END OF LATTICE L110.19
%
\hbox{}\par\smallskip%
%
% LATTICE L110.20
%
\leavevmode%
\llap{\spacer}%
${L_{110.20}}=\pdualwitharg{2.5.7}({L_{110.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[202810669880, -2897077820, -50846304600; -2897077820, 41383720, 726321260; -50846304600, 726321260, 12747587161],[1820, 140, 1],[-473124, -6568, -20211, -13615, -893, -595, -6683, -2851, -13661, -921, -17234, -83742, -73997, -298965, -966887, -304198; 67223, 934, 2885, 1963, 142, 118, 1027, 425, 1937, 130, 2435, 11869, 10498, 42442, 137293, 43199; -1890980, -26251, -80780, -54418, -3570, -2380, -26715, -11396, -54600, -3681, -68880, -334698, -295750, -1194900, -3864445, -1215816],109,20]
%
% END OF LATTICE L110.20
%
\hbox{}\par\smallskip%
%
% LATTICE L110.21
%
\leavevmode%
\llap{\spacer}%
${L_{110.21}}=\pdualwitharg{2.5.13}({L_{110.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[757618680, 2775500, -189827820; 2775500, 45240, -695500; -189827820, -695500, 47562979],[1820, 260, 1],[-530486, -678235, -313469, -17971, -28435, -28207, -60850, -39581, -10919, -79732, -227936, -90310, -1054375, -4303033, -7518820, -4399940; -4517, -5775, -2669, -153, -242, -240, -518, -337, -93, -679, -1941, -769, -8978, -36640, -64022, -37465; -2117280, -2706977, -1251120, -71726, -113490, -112580, -242865, -157976, -43580, -318227, -909740, -360446, -4208230, -17174300, -30009175, -17561076],109,21]
%
% END OF LATTICE L110.21
%
\hbox{}\par\smallskip%
%
% LATTICE L110.22
%
\leavevmode%
\llap{\spacer}%
${L_{110.22}}=\pdualwitharg{2.7.13}({L_{110.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1222315640, -12734540, -306191340; -12734540, 114296, 3190096; -306191340, 3190096, 76701249],[1820, 364, 1],[1375, -103234, -523847, -1160209, -2781706, -11524178, -2890669, -59336799, -1446423, -1341814, -921201, -284839, -108390, -36206, 573, 89831; 24, -1805, -9159, -20285, -48635, -201487, -50540, -1037435, -25289, -23460, -16106, -4980, -1895, -633, 10, 1570; 5488, -412035, -2090816, -4630710, -11102546, -45996132, -11537447, -236829320, -5773068, -5355545, -3676764, -1136870, -432614, -144508, 2287, 358540],109,22]
%
% END OF LATTICE L110.22
%
\hbox{}\par\smallskip%
%
% LATTICE L110.23
%
\leavevmode%
\llap{\spacer}%
${L_{110.23}}=\pdualwitharg{5.7.13}({L_{110.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{l}{2}{52}\above{1pt}{1pt}{r}{2}{910}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{1820}\above{1pt}{1pt}{b}{2}{910}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{182}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20185620, 3654560, -52780; 3654560, 661570, -9555; -52780, -9555, 138],[1820, 455, 1],[0, -1, -6, -5, -59, -121, -121, -124, -15, -11, -9, -1, -1, 1, 1, 1; -1, -20, -118, -100, -1186, -2441, -2444, -2506, -304, -224, -187, -22, -34, 1, 4, 5; -70, -1768, -10465, -8834, -104650, -215215, -215420, -220857, -26775, -19708, -16380, -1904, -2730, 455, 660, 728],109,23]
%
% END OF LATTICE L110.23
%
\hbox{}\par\smallskip%
%
% LATTICE L110.24
%
\leavevmode%
\llap{\spacer}%
${L_{110.24}}=\pdualwitharg{2.5.7.13}({L_{110.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[112840, -14012180, 3510780; -14012180, 1594942440, -399615580; 3510780, -399615580, 100124373],[1820, 1820, 1],[16, 4, 1, -23, -341, -1571, -407, -1695, -221, -45, -206, -18, -69, -69, -3, 14; 9927, 2368, -2280, -15578, -227321, -1040159, -268952, -1119140, -145362, -29455, -133155, -11509, -43549, -43321, -1918, 8573; 39620, 9451, -9100, -62174, -907270, -4151420, -1073425, -4466644, -580160, -117559, -531440, -45934, -173810, -172900, -7655, 34216],109,24]
%
% END OF LATTICE L110.24
%

\medskip%
%------------------------------WEYL-GROUP-W111--------------------------------
%
\leavevmode\llap{}%
$W_{111}$%
\qquad\llap{24} lattices, $\chi=60$%
\hfill%
$22222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L111.1
%
\leavevmode%
\llap{\spacer}%
${L_{111.1}}$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{111}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-526380, -262260, 0; -262260, -130666, -3; 0, -3, 14],[1860, 1, 1],[-275, -1081, -127, 170, 299, 146, 6951, 287, 1143, 142, -139, -279, -139, -6641; 552, 2170, 255, -341, -600, -293, -13950, -576, -2294, -285, 279, 560, 279, 13330; 114, 434, 45, -93, -140, -66, -3100, -126, -496, -60, 62, 120, 59, 2790],110,1]
%
% END OF LATTICE L111.1
%
\hbox{}\par\smallskip%
%
% LATTICE L111.2
%
\leavevmode%
\llap{\spacer}%
${L_{111.2}}=\pfillwitharg{2}({L_{111.1}})$%
{${}={}$}Nikulin $111$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[35805, 14415, 5115; 14415, 5803, 2046; 5115, 2046, 331],[465, 1, 1],[37, 370, -179, -4045, -2371, -2299, -58764, -2665, -13948, -6391, -9533, -2009, -767, -9126; -93, -930, 450, 10168, 5960, 5779, 147715, 6699, 35061, 16065, 23963, 5050, 1928, 22940; 3, 31, -15, -341, -200, -194, -4960, -225, -1178, -540, -806, -170, -65, -775],110,2]
%
% END OF LATTICE L111.2
%
\hbox{}\par\smallskip%
%
% LATTICE L111.3
%
\leavevmode%
\llap{\spacer}%
${L_{111.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{111.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{93}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{186}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{465}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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3255\\-3\\-1089
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-322710, 3255, 106950; 3255, -3, -1089; 106950, -1089, -35441],[465, 3, 1],[-73, -872, -72, 218, 216, 215, 5139, 71, 841, 67, -249, -226, -222, -5294; -75, -899, -75, 217, 220, 220, 5270, 73, 868, 70, -248, -230, -227, -5425; -218, -2604, -215, 651, 645, 642, 15345, 212, 2511, 200, -744, -675, -663, -15810],110,3]
%
% END OF LATTICE L111.3
%
\hbox{}\par\smallskip%
%
% LATTICE L111.4
%
\leavevmode%
\llap{\spacer}%
${L_{111.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{111.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-787710, 4185, -315270; 4185, -5, 1670; -315270, 1670, -126181],[465, 5, 1],[311, 1236, 61, -309, -61, -303, -1446, -299, -1174, -55, 371, 65, 317, 1508; -225, -899, -45, 217, 44, 220, 1054, 219, 868, 42, -248, -46, -227, -1085; -780, -3100, -153, 775, 153, 760, 3627, 750, 2945, 138, -930, -163, -795, -3782],110,4]
%
% END OF LATTICE L111.4
%
\hbox{}\par\smallskip%
%
% LATTICE L111.5
%
\leavevmode%
\llap{\spacer}%
${L_{111.5}}=\pdualwitharg{3}({L_{111.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24252540, 12129060, -33480; 12129060, 6065922, -16743; -33480, -16743, 46],[1860, 3, 1],[-183, -3189, -272, -1349, -623, -131, -3217, 3, 399, 47, -46, -277, -184, -10733; 370, 6448, 550, 2728, 1260, 265, 6510, -6, -806, -95, 93, 560, 372, 21700; 1478, 25854, 2215, 11067, 5160, 1104, 27900, -2, -2976, -370, 372, 2220, 1479, 86490],110,5]
%
% END OF LATTICE L111.5
%
\hbox{}\par\smallskip%
%
% LATTICE L111.6
%
\leavevmode%
\llap{\spacer}%
${L_{111.6}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{111.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L111.6
%
\hbox{}\par\smallskip%
%
% LATTICE L111.7
%
\leavevmode%
\llap{\spacer}%
${L_{111.7}}=\pdualwitharg{2}({L_{111.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16694470920, -16265700, 4172746320; -16265700, 15848, -4065576; 4172746320, -4065576, 1042968773],[1860, 4, 1],[1654, 11121, 6631, 13735, 2012, 2365, 43231, 1178, 3511, 449, 897, 348, 939, 29929; -3237, -21731, -12945, -26815, -3930, -4624, -84630, -2313, -6944, -930, -1860, -695, -1851, -58745; -6630, -44578, -26580, -55056, -8065, -9480, -173290, -4722, -14074, -1800, -3596, -1395, -3764, -119970],110,7]
%
% END OF LATTICE L111.7
%
\hbox{}\par\smallskip%
%
% LATTICE L111.8
%
\leavevmode%
\llap{\spacer}%
${L_{111.8}}=\pdualwitharg{5}({L_{111.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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112226820\\56119920\\-91140
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56119920\\28063210\\-45575
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[112226820, 56119920, -91140; 56119920, 28063210, -45575; -91140, -45575, 74],[1860, 5, 1],[-23, 151, 16, -15, -31, -108, -1291, -337, -2011, -106, -915, -89, -102, -569; 48, -310, -33, 31, 64, 223, 2666, 696, 4154, 219, 1891, 184, 211, 1178; 1230, -4960, -618, 620, 1236, 4325, 51894, 13590, 81530, 4323, 37665, 3704, 4320, 24676],110,8]
%
% END OF LATTICE L111.8
%
\hbox{}\par\smallskip%
%
% LATTICE L111.9
%
\leavevmode%
\llap{\spacer}%
${L_{111.9}}=\pdualwitharg{31}(\pfillwitharg{2}({L_{111.1}}))$%
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{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4518870, -16275, -292020; -16275, 217, -1054; -292020, -1054, -18871],[465, 31, 1],[481, 62, 481, -15, -471, -471, -364, -469, -60, -451, 17, 491, 485, 374; -63, -8, -60, 2, 60, 59, 45, 57, 7, 45, -3, -70, -66, -50; -7440, -959, -7440, 232, 7285, 7285, 5630, 7254, 928, 6975, -263, -7595, -7502, -5785],110,9]
%
% END OF LATTICE L111.9
%
\hbox{}\par\smallskip%
%
% LATTICE L111.10
%
\leavevmode%
\llap{\spacer}%
${L_{111.10}}=\pdualwitharg{2.3}({L_{111.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28366201560, -31127100, 7081432920; -31127100, 34152, -7770672; 7081432920, -7770672, 1767832471],[1860, 12, 1],[-6687, -153350, -33208, -217242, -32633, -38745, -702719, -6227, -46817, 923, 17075, 348, -6454, -298116; 4501, 103199, 22345, 146165, 21955, 26066, 472750, 4189, 31496, -620, -11470, -230, 4349, 200725; 26806, 614730, 133120, 870852, 130815, 155316, 2816970, 24962, 187674, -3700, -68448, -1395, 25872, 1195050],110,10]
%
% END OF LATTICE L111.10
%
\hbox{}\par\smallskip%
%
% LATTICE L111.11
%
\leavevmode%
\llap{\spacer}%
${L_{111.11}}=\pdualwitharg{3.5}({L_{111.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{1860}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{930}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{372}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-53940, -26040, 1860; -26040, -12570, 915; 1860, 915, 218],[1860, 15, 1],[-61, 151, 16, -15, -95, -348, -4267, -379, -6971, -126, -3395, -345, -422, -2553; 126, -310, -33, 31, 196, 718, 8804, 782, 14384, 260, 7006, 712, 871, 5270; -10, 0, 2, 0, -12, -45, -558, -50, -930, -17, -465, -48, -60, -372],110,11]
%
% END OF LATTICE L111.11
%
\hbox{}\par\smallskip%
%
% LATTICE L111.12
%
\leavevmode%
\llap{\spacer}%
${L_{111.12}}=\pdualwitharg{2.5}({L_{111.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{620}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}{1240}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{124}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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68337028680\\-35512980\\17050960380
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\kern3pt%
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-35512980\\18440\\-8860940
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[68337028680, -35512980, 17050960380; -35512980, 18440, -8860940; 17050960380, -8860940, 4254432121],[1860, 20, 1],[-54340, -140359, 925, 47637, 347, -16185, -157001, -53876, -418757, -55033, -606753, -18383, -110067, -402115; 21867, 56482, -372, -19158, -139, 6519, 63209, 21687, 168547, 22149, 244187, 7398, 44294, 161820; 217830, 562650, -3708, -190960, -1391, 64880, 629362, 215970, 1678650, 220608, 2432260, 73691, 441220, 1611938],110,12]
%
% END OF LATTICE L111.12
%
\hbox{}\par\smallskip%
%
% LATTICE L111.13
%
\leavevmode%
\llap{\spacer}%
${L_{111.13}}=\pdualwitharg{3.31}(\pfillwitharg{2}({L_{111.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${31}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{310}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{465}\above{1pt}{1pt}{r}{2}{186}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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3844\\5\\11470
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18233\\5\\54405
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24747\\-2\\73842
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8877284310, -12254145, -2975084415; -12254145, 16926, 4106787; -2975084415, 4106787, 997053487],[465, 93, 1],[10358, 2977, 3844, 744, 18233, 24747, 25492, 14576, 9490, 58491, 11723, 106437, 62522, 36843; -2, 0, 5, 1, 5, -2, -5, -4, -3, -20, -4, -35, -19, -10; 30907, 8883, 11470, 2220, 54405, 73842, 76065, 43493, 28317, 174530, 34980, 317595, 186558, 109935],110,13]
%
% END OF LATTICE L111.13
%
\hbox{}\par\smallskip%
%
% LATTICE L111.14
%
\leavevmode%
\llap{\spacer}%
${L_{111.14}}=\pdualwitharg{31}({L_{111.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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382\\-8029\\-1116
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-247380, -13020, 9300; -13020, -682, 465; 9300, 465, -166],[1860, 31, 1],[61, 33, 473, 46, 1059, 382, 521, 575, 73, 322, 7, 1, -11, 9; -1284, -694, -9945, -967, -22260, -8029, -10950, -12084, -1534, -6765, -147, -20, 231, -190; -186, -98, -1395, -135, -3100, -1116, -1520, -1674, -212, -930, -20, 0, 31, -30],110,14]
%
% END OF LATTICE L111.14
%
\hbox{}\par\smallskip%
%
% LATTICE L111.15
%
\leavevmode%
\llap{\spacer}%
${L_{111.15}}=\pdualwitharg{5.31}(\pfillwitharg{2}({L_{111.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6624262590, 13665885, -2622606975; 13665885, 28210, -5410430; -2622606975, -5410430, 1038314441],[465, 155, 1],[40939, 3922, 3038, 980, 4804, 32603, 6717, 57611, 12503, 46237, 15445, 28046, 82372, 9708; -122823, -11766, -9111, -2939, -14411, -97811, -20152, -172845, -37512, -138723, -46339, -84145, -247135, -29126; 102765, 9845, 7626, 2460, 12059, 81840, 16861, 144615, 31385, 116064, 38770, 70401, 206770, 24369],110,15]
%
% END OF LATTICE L111.15
%
\hbox{}\par\smallskip%
%
% LATTICE L111.16
%
\leavevmode%
\llap{\spacer}%
${L_{111.16}}=\pdualwitharg{2.3.5}({L_{111.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1665455160, -28593780, 416392620; -28593780, 490920, -7148940; 416392620, -7148940, 104105363],[1860, 60, 1],[1335, 10443, -1, -929, 110, 2440, 17774, 1779, 37824, 1558, 49196, 1447, 8459, 30493; -151, -1178, 0, 62, -15, -301, -2139, -211, -4433, -181, -5673, -166, -966, -3472; -5350, -41850, 4, 3720, -441, -9780, -71238, -7130, -151590, -6244, -197160, -5799, -33900, -122202],110,16]
%
% END OF LATTICE L111.16
%
\hbox{}\par\smallskip%
%
% LATTICE L111.17
%
\leavevmode%
\llap{\spacer}%
${L_{111.17}}=\pdualwitharg{3.31}({L_{111.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1860, 0, 0; 0, -4278, 93; 0, 93, -2],[1860, 93, 1],[-1, -1, -4, -1, -21, -7, -9, -3, -1, -1, 0, 1, 0, -1; -2, -4, -20, -6, -140, -51, -70, -26, -10, -15, -1, 0, 2, 0; -124, -222, -1085, -321, -7440, -2697, -3690, -1364, -522, -775, -51, 0, 93, -30],110,17]
%
% END OF LATTICE L111.17
%
\hbox{}\par\smallskip%
%
% LATTICE L111.18
%
\leavevmode%
\llap{\spacer}%
${L_{111.18}}=\pdualwitharg{3.5.31}(\pfillwitharg{2}({L_{111.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-162854411565, 26590653930, -8570485215; 26590653930, -4341686865, 1399377510; -8570485215, 1399377510, -451036078],[465, 465, 1],[-21, -8, -4, 2, 12, 59, 9, 19, 7, 3, -3, -14, -66, -10; 92694, 35863, 18570, -8666, -54511, -272711, -42170, -90608, -34820, -17527, 9709, 56597, 280012, 43213; 287990, 111420, 57691, -26925, -169353, -847230, -131007, -281480, -108165, -54436, 30180, 175863, 870015, 134262],110,18]
%
% END OF LATTICE L111.18
%
\hbox{}\par\smallskip%
%
% LATTICE L111.19
%
\leavevmode%
\llap{\spacer}%
${L_{111.19}}=\pdualwitharg{2.31}({L_{111.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[71427720, -14707020, 17841120; -14707020, 3028328, -3673500; 17841120, -3673500, 4456331],[1860, 124, 1],[12202, 4643, 118771, 10581, 58032, 80541, 53441, 56462, 6801, 52889, 863, -812, -433, 3779; -81, -31, -795, -71, -390, -542, -360, -381, -46, -360, -6, 5, 3, -25; -48918, -18614, -476160, -42420, -232655, -322896, -214250, -226362, -27266, -212040, -3460, 3255, 1736, -15150],110,19]
%
% END OF LATTICE L111.19
%
\hbox{}\par\smallskip%
%
% LATTICE L111.20
%
\leavevmode%
\llap{\spacer}%
${L_{111.20}}=\pdualwitharg{5.31}({L_{111.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-90606180, -30481680, -252960; -30481680, -10254490, -85095; -252960, -85095, -706],[1860, 155, 1],[-235, -127, -364, -177, -815, -1470, -401, -2213, -281, -248, -27, -1, 42, -7; 876, 474, 1359, 661, 3044, 5491, 1498, 8268, 1050, 927, 101, 4, -157, 26; -21390, -11630, -33387, -16255, -74896, -135160, -36884, -203670, -25880, -22878, -2500, -124, 3875, -626],110,20]
%
% END OF LATTICE L111.20
%
\hbox{}\par\smallskip%
%
% LATTICE L111.21
%
\leavevmode%
\llap{\spacer}%
${L_{111.21}}=\pdualwitharg{2.3.31}({L_{111.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[39792840, -172980, 9926820; -172980, 744, -43152; 9926820, -43152, 2476369],[1860, 372, 1],[201, 196, 1546, 380, 1971, 2597, 1653, 541, 175, 309, -9, -116, 0, 202; -16, -19, -165, -45, -250, -351, -235, -84, -31, -85, -5, 0, 1, -15; -806, -786, -6200, -1524, -7905, -10416, -6630, -2170, -702, -1240, 36, 465, 0, -810],110,21]
%
% END OF LATTICE L111.21
%
\hbox{}\par\smallskip%
%
% LATTICE L111.22
%
\leavevmode%
\llap{\spacer}%
${L_{111.22}}=\pdualwitharg{3.5.31}({L_{111.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[200933940, -201279900, -269700; -201279900, 201604470, 270165; -269700, 270165, 362],[1860, 465, 1],[-1375, -527, -157, -52, -1, 95, -5, -121, -221, -217, -322, -1495, -2713, -743; -26, -10, -3, -1, 0, 2, 0, -2, -4, -4, -6, -28, -51, -14; -1005020, -385170, -114731, -37995, -744, 69285, -3726, -88660, -161670, -158689, -235425, -1092936, -1983225, -543114],110,22]
%
% END OF LATTICE L111.22
%
\hbox{}\par\smallskip%
%
% LATTICE L111.23
%
\leavevmode%
\llap{\spacer}%
${L_{111.23}}=\pdualwitharg{2.5.31}({L_{111.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31843203720, -469826700, -8294482140; -469826700, 6932840, 122380560; -8294482140, 122380560, 2160537911],[1860, 620, 1],[165884, 74843, 409165, 193787, 220413, 1577483, 214193, 1174180, 148501, 260867, 29557, 2931, -14075, 9151; -496221, -223884, -1223970, -579692, -659341, -4718867, -640735, -3512433, -444225, -780357, -88417, -8768, 42104, -27374; 664950, 300010, 1640148, 776800, 883531, 6323380, 858598, 4706730, 595270, 1045692, 118480, 11749, -56420, 36682],110,23]
%
% END OF LATTICE L111.23
%
\hbox{}\par\smallskip%
%
% LATTICE L111.24
%
\leavevmode%
\llap{\spacer}%
${L_{111.24}}=\pdualwitharg{2.3.5.31}({L_{111.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3720, -128163300, -31971540; -128163300, 4369329247560, 1089970252320; -31971540, 1089970252320, 271903325117],[1860, 1860, 1],[-3314, -1065, -365, 67, 149, 1, -253, -1258, -1221, -1921, -2353, -2435, -16003, -2033; -372665, -119299, -40151, 8307, 17191, 0, -28844, -143297, -138682, -217830, -266288, -275172, -1805867, -229137; 1493890, 478230, 160952, -33300, -68913, 0, 115626, 574430, 555930, 873208, 1067460, 1103073, 7239120, 918534],110,24]
%
% END OF LATTICE L111.24
%

\medskip%
%------------------------------WEYL-GROUP-W112--------------------------------
%
\leavevmode\llap{}%
$W_{112}$%
\qquad\llap{24} lattices, $\chi=66$%
\hfill%
$22222242222224\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L112.1
%
\leavevmode%
\llap{\spacer}%
${L_{112.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{112}}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-648168612, 162288, 233772; 162288, -38, -61; 233772, -61, -82],[1932, 1, 1],[1, 703, 234, 1210, 751, 1037, 271, 8, 125, 18, 32, 5, -2, -1; 1575, 1106852, 368424, 1905090, 1182412, 1632701, 426674, 12595, 196788, 28336, 50370, 7868, -3151, -1574; 1679, 1180636, 392987, 2032119, 1261260, 1741583, 455130, 13436, 209944, 30233, 53751, 8400, -3358, -1680],111,1]
%
% END OF LATTICE L112.1
%
\hbox{}\par\smallskip%
%
% LATTICE L112.2
%
\leavevmode%
\llap{\spacer}%
${L_{112.2}}=\pfillwitharg{2}({L_{112.1}})$%
{${}={}$}Nikulin $112$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-483, 0, -3381; 0, 10, 483; -3381, 483, -338],[483, 1, 1],[456, 3137, 766, 950, 1, -160, 19, 128, 23727, 15586, 79642, 24527, 67320, 8741; 3146, 21643, 5285, 6555, 7, -1104, 131, 883, 163691, 107527, 549447, 169211, 464439, 60304; -67, -460, -112, -138, 0, 23, -3, -19, -3496, -2296, -11730, -3612, -9913, -1287],111,2]
%
% END OF LATTICE L112.2
%
\hbox{}\par\smallskip%
%
% LATTICE L112.3
%
\leavevmode%
\llap{\spacer}%
${L_{112.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{112.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-59355870, 804195, 19733931; 804195, -9906, -267498; 19733931, -267498, -6560885],[483, 3, 1],[-7069, -2539102, -1690873, -2915343, -2714694, -7498237, -979923, -58036, -455728, -132067, -79487, -19716, 11162, 3348; -2753, -988885, -658532, -1135418, -1057273, -2920287, -381644, -22603, -177491, -51436, -30958, -7679, 4347, 1304; -21150, -7596831, -5058984, -8722520, -8122191, -22434246, -2931867, -173640, -1363509, -395136, -237820, -58989, 33396, 10017],111,3]
%
% END OF LATTICE L112.3
%
\hbox{}\par\smallskip%
%
% LATTICE L112.4
%
\leavevmode%
\llap{\spacer}%
${L_{112.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{112.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-209565006, 2241603, 29878863; 2241603, -23114, -319662; 29878863, -319662, -4259993],[483, 7, 1],[-5321, -1911202, -181819, -6583201, -291910, -5643977, -737595, -43684, -343028, -14201, -179489, -2120, 8402, 2520; -2753, -988885, -94076, -3406254, -151039, -2920287, -381644, -22603, -177491, -7348, -92874, -1097, 4347, 1304; -37114, -13330639, -1268188, -45917844, -2036073, -39366754, -5144727, -304696, -2392621, -99052, -1251936, -14787, 58604, 17577],111,4]
%
% END OF LATTICE L112.4
%
\hbox{}\par\smallskip%
%
% LATTICE L112.5
%
\leavevmode%
\llap{\spacer}%
${L_{112.5}}=\pdualwitharg{3}({L_{112.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-88147500, 104328, 20181672; 104328, -114, -24729; 20181672, -24729, -4545734],[1932, 3, 1],[392, 275875, 91829, 158283, 294723, 406964, 106353, 3140, 49069, 7067, 4189, 1965, -784, -393; 104293, 73397140, 24431302, 42111528, 78411676, 108273627, 28295438, 835403, 13054892, 1880186, 1114488, 522788, -208587, -104558; 1173, 825516, 274785, 473639, 881916, 1217781, 318246, 9396, 146832, 21147, 12535, 5880, -2346, -1176],111,5]
%
% END OF LATTICE L112.5
%
\hbox{}\par\smallskip%
%
% LATTICE L112.6
%
\leavevmode%
\llap{\spacer}%
${L_{112.6}}=\pdualwitharg{2}({L_{112.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[820553008296, -1596085092, -205563982764; -1596085092, 3104600, 399849376; -205563982764, 399849376, 51497649247],[1932, 4, 1],[-15469, -2132631, -2828034, -14573212, -2256353, -12441163, -1622811, -92799, -357533, -203470, -367772, -17275, -9657, -1209; 19, 2323, 3073, 15801, 2443, 13455, 1753, 98, 368, 203, 345, 14, 0, 2; -61748, -8512875, -11288732, -58172244, -9006739, -49661692, -6477814, -370428, -1427173, -812196, -1468044, -68957, -38548, -4826],111,6]
%
% END OF LATTICE L112.6
%
\hbox{}\par\smallskip%
%
% LATTICE L112.7
%
\leavevmode%
\llap{\spacer}%
${L_{112.7}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{112.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1351290307017, -7647964002, -455339277792; -7647964002, 43285557, 2577106035; -455339277792, 2577106035, 153433985890],[483, 21, 1],[27455, 6674471, 630587, 7564595, 1002393, 19321151, 2517072, 140480, 1050539, 41153, 160415, 5217, -8626, 1953; -179, -46414, -4391, -52739, -6994, -134895, -17585, -994, -7544, -301, -1219, -44, 0, -10; 81480, 19808313, 1871439, 22450001, 2974875, 57340794, 7470099, 416913, 3117765, 122133, 476077, 15483, -25599, 5796],111,7]
%
% END OF LATTICE L112.7
%
\hbox{}\par\smallskip%
%
% LATTICE L112.8
%
\leavevmode%
\llap{\spacer}%
${L_{112.8}}=\pdualwitharg{23}(\pfillwitharg{2}({L_{112.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1265535831, 1762467, 604827573; 1762467, -1909, -842352; 604827573, -842352, -289060477],[483, 23, 1],[-34653, -541195, -8289202, -1864168, -13308313, -1598209, -4803904, -284513, -97137, -647446, -50828, -96659, 2379, 16414; -3991, -62324, -954583, -214677, -1532580, -184049, -553215, -32764, -11186, -74557, -5853, -11130, 274, 1890; -72496, -1132210, -17341471, -3899943, -27841730, -3343542, -10050034, -595217, -203216, -1354493, -106335, -202216, 4977, 34339],111,8]
%
% END OF LATTICE L112.8
%
\hbox{}\par\smallskip%
%
% LATTICE L112.9
%
\leavevmode%
\llap{\spacer}%
${L_{112.9}}=\pdualwitharg{7}({L_{112.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14306460, 61824, 1027824; 61824, -266, -4501; 1027824, -4501, -70822],[1932, 7, 1],[384, 6013, 124, 1550, 35, -94, -49, 47, 33639, 1600, 57928, 5137, 49659, 12979; 66979, 1048800, 21628, 270342, 6104, -16399, -8546, 8199, 5867576, 279084, 10104222, 896032, 8661869, 2263886; 1316, 20608, 425, 5313, 120, -322, -168, 161, 115276, 5483, 198513, 17604, 170177, 44478],111,9]
%
% END OF LATTICE L112.9
%
\hbox{}\par\smallskip%
%
% LATTICE L112.10
%
\leavevmode%
\llap{\spacer}%
${L_{112.10}}=\pdualwitharg{2.3}({L_{112.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15253267512, 25014869460, -3789871092; 25014869460, 41023583544, -6215267028; -3789871092, -6215267028, 941642365],[1932, 12, 1],[-1835, -167860, -220448, -375358, -173358, -951473, -123519, -6419, -21982, -10664, -4326, 24, 2263, -291; 19, 2323, 3073, 5267, 2443, 13455, 1753, 98, 368, 203, 115, 14, 0, 2; -7260, -660261, -866964, -1475956, -681597, -3740628, -485562, -25188, -86043, -41580, -16652, 189, 9108, -1158],111,10]
%
% END OF LATTICE L112.10
%
\hbox{}\par\smallskip%
%
% LATTICE L112.11
%
\leavevmode%
\llap{\spacer}%
${L_{112.11}}=\pdualwitharg{3.23}(\pfillwitharg{2}({L_{112.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[910247823429, 16491952890, -299073574884; 16491952890, 298802706, -5418642243; -299073574884, -5418642243, 98264451605],[483, 69, 1],[83587, 906781, 13800589, 1028797, 21955763, 2629245, 7880224, 442124, 144641, 919319, 22625, 124099, -690, 5441; 20, 277, 4235, 317, 6783, 814, 2445, 143, 49, 329, 9, 63, 1, 0; 254403, 2759856, 42003129, 3131221, 66824016, 8002305, 23984055, 1345638, 440226, 2798019, 68861, 377706, -2100, 16560],111,11]
%
% END OF LATTICE L112.11
%
\hbox{}\par\smallskip%
%
% LATTICE L112.12
%
\leavevmode%
\llap{\spacer}%
${L_{112.12}}=\pdualwitharg{3.7}({L_{112.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2226094836, 70548912, -3294060; 70548912, -2235786, 104391; -3294060, 104391, -4874],[1932, 21, 1],[-20, -14359, -683, -8243, -2193, -21200, -5541, -164, -2569, -53, -221, -15, 40, 21; -1139, -818432, -38930, -469844, -125000, -1208397, -315838, -9349, -146464, -3022, -12604, -856, 2277, 1198; -10878, -7824600, -372195, -4492061, -1195104, -11553360, -3019716, -89397, -1400700, -28905, -120589, -8196, 21735, 11466],111,12]
%
% END OF LATTICE L112.12
%
\hbox{}\par\smallskip%
%
% LATTICE L112.13
%
\leavevmode%
\llap{\spacer}%
${L_{112.13}}=\pdualwitharg{23}({L_{112.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3348156, 54096, 27048; 54096, -874, -437; 27048, -437, -218],[1932, 23, 1],[0, 13, 101, 23, 331, 20, 121, 4, 3, 11, 1, 5, 0, -1; 11, 996, 7700, 1746, 25060, 1511, 9122, 291, 212, 756, 66, 308, -1, -62; -23, -404, -3059, -681, -9660, -577, -3450, -92, -56, -161, -9, 0, 2, 0],111,13]
%
% END OF LATTICE L112.13
%
\hbox{}\par\smallskip%
%
% LATTICE L112.14
%
\leavevmode%
\llap{\spacer}%
${L_{112.14}}=\pdualwitharg{2.7}({L_{112.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[32090520, 141760500, -7923132; 141760500, 626232824, -35000588; -7923132, -35000588, 1956217],[1932, 28, 1],[-28535, -107005, -8418, -99788, -573, 159, -593, -5573, -677479, -128014, -4607116, -101751, -3922579, -511027; 98, 368, 29, 345, 2, 0, 2, 19, 2323, 439, 15801, 349, 13455, 1753; -113820, -426811, -33576, -397992, -2285, 644, -2366, -22232, -2702385, -510632, -18377184, -405871, -15646624, -2038414],111,14]
%
% END OF LATTICE L112.14
%
\hbox{}\par\smallskip%
%
% LATTICE L112.15
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L112.15
%
\hbox{}\par\smallskip%
%
% LATTICE L112.16
%
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${L_{112.16}}=\pdualwitharg{3.23}({L_{112.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-41297209044, 4412104536, -63792708; 4412104536, -471379710, 6815475; -63792708, 6815475, -98542],[1932, 69, 1],[23, 443, 3362, 250, 10653, 637, 3813, 104, 65, 194, 4, 15, -2, -3; 333, 6652, 50526, 3760, 160300, 9589, 57422, 1579, 996, 3010, 64, 308, -29, -62; 8142, 173292, 1318107, 98213, 4190508, 250836, 1503096, 41883, 26808, 82593, 1837, 11592, -711, -2346],111,16]
%
% END OF LATTICE L112.16
%
\hbox{}\par\smallskip%
%
% LATTICE L112.17
%
\leavevmode%
\llap{\spacer}%
${L_{112.17}}=\pdualwitharg{2.3.7}({L_{112.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}23\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1752265873848, -8023756356, -438980139276; -8023756356, 36741432, 2010122856; -438980139276, 2010122856, 109973928931],[1932, 84, 1],[-133393, -18239198, -3454684, -41533150, -2755692, -106353355, -13871577, -792109, -3047348, -247324, -1039942, -20814, -79375, -10953; 796, 109066, 20659, 248377, 16480, 636042, 82960, 4739, 18239, 1481, 6233, 125, 483, 65; -532476, -72806937, -13790352, -165791360, -11000127, -424539612, -55372338, -3161928, -12164355, -987264, -4151224, -83085, -316848, -43722],111,17]
%
% END OF LATTICE L112.17
%
\hbox{}\par\smallskip%
%
% LATTICE L112.18
%
\leavevmode%
\llap{\spacer}%
${L_{112.18}}=\pdualwitharg{2.23}({L_{112.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2013144, 152628, -504252; 152628, 80776, -38180; -504252, -38180, 126305],[1932, 92, 1],[-6569, -34826, -1059546, -236860, -842256, -201681, -604337, -33767, -5510, -69862, -5156, -4800, 1, -703; 19, 101, 3073, 687, 2443, 585, 1753, 98, 16, 203, 15, 14, 0, 2; -26220, -139007, -4229148, -945420, -3361841, -805004, -2412194, -134780, -21993, -278852, -20580, -19159, 4, -2806],111,18]
%
% END OF LATTICE L112.18
%
\hbox{}\par\smallskip%
%
% LATTICE L112.19
%
\leavevmode%
\llap{\spacer}%
${L_{112.19}}=\pdualwitharg{3.7.23}(\pfillwitharg{2}({L_{112.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14031978670542, 46973983815108, -15726199275087; 46973983815108, -157251889097499, 52645620905073; -15726199275087, 52645620905073, -17624980001113],[483, 483, 1],[-3991, -62324, -136369, -71559, -218940, -184049, -553215, -32764, -11186, -10651, -1951, -1590, 274, 1890; 38568287, 602343107, 1317967988, 691598402, 2115997543, 1778786271, 5346684050, 316659409, 108112285, 102942796, 18856990, 15368633, -2647791, -18268610; 115206609, 1799247828, 3936877524, 2065860650, 6320656680, 5313379197, 15970979913, 945887397, 322940184, 307498500, 56327362, 45907356, -7909167, -54569823],111,19]
%
% END OF LATTICE L112.19
%
\hbox{}\par\smallskip%
%
% LATTICE L112.20
%
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${L_{112.20}}=\pdualwitharg{7.23}({L_{112.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-207592571172, 40339588128, -83445012; 40339588128, -7838827570, 16215115; -83445012, 16215115, -33542],[1932, 161, 1],[23, 495, 538, 842, 1711, 717, 4297, 120, 77, 34, 16, 5, -2, -7; 149, 3420, 3722, 5832, 11860, 4973, 29822, 843, 548, 246, 120, 44, -13, -62; 14812, 421876, 460897, 724647, 1476876, 620354, 3726828, 108997, 73360, 34339, 18207, 8832, -1309, -12558],111,20]
%
% END OF LATTICE L112.20
%
\hbox{}\par\smallskip%
%
% LATTICE L112.21
%
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${L_{112.21}}=\pdualwitharg{2.3.23}({L_{112.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[93421176072, -515888436, -23404145604; -515888436, 2908488, 129241968; -23404145604, 129241968, 5863274843],[1932, 276, 1],[-1317293, -6943545, -211224116, -15737874, -167876053, -40196177, -120440731, -6721845, -1095593, -13871748, -340402, -945845, 487, -146339; 8092, 42653, 1297513, 96675, 1031233, 246918, 739846, 41291, 6730, 85211, 2091, 5810, -3, 899; -5258352, -27717147, -843161508, -62822228, -670125309, -160454544, -480773646, -26832168, -4373373, -55373052, -1358812, -3775611, 1944, -584154],111,21]
%
% END OF LATTICE L112.21
%
\hbox{}\par\smallskip%
%
% LATTICE L112.22
%
\leavevmode%
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${L_{112.22}}=\pdualwitharg{3.7.23}({L_{112.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8796396, 16454844, 179676; 16454844, -30621234, -334719; 179676, -334719, -3658],[1932, 483, 1],[-133, -5177, -5680, -2988, -18313, -7707, -46391, -1406, -979, -476, -90, -155, 12, 221; 277, 10804, 11854, 6236, 38220, 16085, 96822, 2935, 2044, 994, 188, 324, -25, -462; -31878, -1242864, -1363647, -717367, -4396680, -1850352, -11137980, -337617, -235116, -114333, -21623, -37260, 2877, 53130],111,22]
%
% END OF LATTICE L112.22
%
\hbox{}\par\smallskip%
%
% LATTICE L112.23
%
\leavevmode%
\llap{\spacer}%
${L_{112.23}}=\pdualwitharg{2.7.23}({L_{112.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[321240259032, 718713660, 79936787868; 718713660, 1809640, 178691968; 79936787868, 178691968, 19891425023],[1932, 644, 1],[4522481, 23788109, 103372308, 161736502, 82152303, 137690613, 412556279, 23014769, 3749141, 6776692, 3486998, 459447, -4363, 502471; -13545689, -71249902, -309619685, -484431525, -246061742, -412409523, -1235684369, -68933602, -11229389, -20297479, -10444221, -1376131, 13068, -1504996; -18052608, -94956155, -412636284, -645611484, -327931355, -549626336, -1646821022, -91869176, -14965629, -27050852, -13919220, -1833997, 17416, -2005738],111,23]
%
% END OF LATTICE L112.23
%
\hbox{}\par\smallskip%
%
% LATTICE L112.24
%
\leavevmode%
\llap{\spacer}%
${L_{112.24}}=\pdualwitharg{2.3.7.23}({L_{112.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}23\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[106248408, 12016284588, -3010396032; 12016284588, 1358995442280, -340464180672; -3010396032, -340464180672, 85295251709],[1932, 1932, 1],[-836, -4319, -18761, -9781, -14901, -24970, -74802, -4157, -674, -1211, -205, -78, 5, -93; 45989, 243283, 1057328, 551494, 840437, 1408689, 4221047, 235753, 38459, 69640, 11990, 4789, 21, 5083; 183540, 970935, 4219764, 2200996, 3354159, 5622036, 16846074, 940884, 153489, 277932, 47852, 19113, 84, 20286],111,24]
%
% END OF LATTICE L112.24
%

\medskip%
%------------------------------WEYL-GROUP-W113--------------------------------
%
\leavevmode\llap{}%
$W_{113}$%
\qquad\llap{32} lattices, $\chi=80$%
\hfill%
$2226222222262222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L113.1
%
\leavevmode%
\llap{\spacer}%
${L_{113.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-4758603960, 3057480, -704520; 3057480, -1946, 429; -704520, 429, -74],[2280, 1, 1],[-109, -268, -9401, -183, -512, -499, -1199, -277, -107, -92, -1543, -9, 8, 19, -169, -107; -206868, -508630, -17841912, -347311, -971712, -947040, -2275554, -525712, -203073, -174605, -2928432, -17081, 15183, 36060, -320739, -203072; -161541, -397185, -13932624, -271213, -758805, -739540, -1776975, -410528, -158580, -136350, -2286840, -13339, 11856, 28160, -250458, -158576],112,1]
%
% END OF LATTICE L113.1
%
\hbox{}\par\smallskip%
%
% LATTICE L113.2
%
\leavevmode%
\llap{\spacer}%
${L_{113.2}}=\pfillwitharg{2}({L_{113.1}})$%
{${}={}$}Nikulin $113$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[78090, 10260, -570; 10260, 1348, -75; -570, -75, 4],[570, 1, 1],[2, 2, -1, -1, -8, -7, -56, -9, -11, -17, -227, -7, -13, -3, -1, 1; -15, -15, 0, 7, 57, 50, 399, 64, 78, 120, 1596, 49, 90, 20, 0, -8; 3, 5, -114, -10, -69, -60, -513, -86, -111, -185, -2622, -86, -183, -60, -171, -10],112,2]
%
% END OF LATTICE L113.2
%
\hbox{}\par\smallskip%
%
% LATTICE L113.3
%
\leavevmode%
\llap{\spacer}%
${L_{113.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{113.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17278980, 34770, -8630940; 34770, -56, 17366; -8630940, 17366, -4311199],[570, 2, 1],[-8606, -7284, -30136, -663, 662, 332, -15196, -2314, -9259, -22491, -196154, -15217, -42347, -10257, -97949, -5626; 2217, 1875, 7752, 170, -171, -85, 3933, 598, 2391, 5805, 50616, 3926, 10923, 2645, 25251, 1450; 17238, 14590, 60363, 1328, -1326, -665, 30438, 4635, 18546, 45050, 392901, 30480, 84822, 20545, 196194, 11269],112,3]
%
% END OF LATTICE L113.3
%
\hbox{}\par\smallskip%
%
% LATTICE L113.4
%
\leavevmode%
\llap{\spacer}%
${L_{113.4}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{113.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21261570, 46740, -7079400; 46740, -84, 15561; -7079400, 15561, -2357206],[570, 3, 1],[-2457, -6239, -17209, -568, 189, 569, -4337, -3963, -2643, -19261, -111991, -13032, -12089, -17569, -27963, -9637; 739, 1875, 5168, 170, -57, -170, 1311, 1196, 797, 5805, 33744, 3926, 3641, 5290, 8417, 2900; 7384, 18750, 51718, 1707, -568, -1710, 13034, 11910, 7943, 57885, 336566, 39165, 36331, 52800, 84037, 28962],112,4]
%
% END OF LATTICE L113.4
%
\hbox{}\par\smallskip%
%
% LATTICE L113.5
%
\leavevmode%
\llap{\spacer}%
${L_{113.5}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{113.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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494\\170\\2465
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-493\\-171\\-2460
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-29746590, 70680, 5956500; 70680, -140, -14155; 5956500, -14155, -1192738],[570, 5, 1],[6409, 1085, 44893, 494, -493, -99, 11309, 3445, 6893, 3349, 292091, 11330, 31531, 3055, 72937, 8379; 2217, 375, 15504, 170, -171, -34, 3933, 1196, 2391, 1161, 101232, 3926, 10923, 1058, 25251, 2900; 31980, 5414, 224010, 2465, -2460, -494, 56430, 17190, 34395, 16711, 1457490, 56535, 157335, 15244, 363945, 41810],112,5]
%
% END OF LATTICE L113.5
%
\hbox{}\par\smallskip%
%
% LATTICE L113.6
%
\leavevmode%
\llap{\spacer}%
${L_{113.6}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{113.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[105052857060, -66395310, 52415768730; -66395310, 41964, -33127716; 52415768730, -33127716, 26152671031],[570, 6, 1],[59225, 147349, 207280, 15160, 1421, 2133, 161577, 62782, 77233, 534941, 1521188, 348728, 316981, 225297, 702657, 119162; 56, 130, 171, 9, 0, 5, 190, 71, 85, 575, 1615, 367, 329, 230, 703, 117; -118700, -295320, -415435, -30384, -2848, -4275, -323836, -125829, -154792, -1072140, -3048797, -698928, -635300, -451545, -1408280, -238827],112,6]
%
% END OF LATTICE L113.6
%
\hbox{}\par\smallskip%
%
% LATTICE L113.7
%
\leavevmode%
\llap{\spacer}%
${L_{113.7}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{113.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17008858140, -34490130, 8454628170; -34490130, 69940, -17144080; 8454628170, -17144080, 4202559449],[570, 10, 1],[97675, 16137, 339290, 8152, 2207, 255, 270299, 34912, 128603, 59289, 2526898, 192984, 525775, 24887, 1162795, 65656; 168, 26, 513, 9, 0, 1, 570, 71, 255, 115, 4845, 367, 987, 46, 2109, 117; -196500, -32464, -682575, -16400, -4440, -513, -543780, -70235, -258720, -119276, -5083545, -388240, -1057740, -50067, -2339280, -132085],112,7]
%
% END OF LATTICE L113.7
%
\hbox{}\par\smallskip%
%
% LATTICE L113.8
%
\leavevmode%
\llap{\spacer}%
${L_{113.8}}=\pdualwitharg{3}({L_{113.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[148015320, -351120, -6840; -351120, -7134, 1359; -6840, 1359, -226],[2280, 3, 1],[-135, -948, -10835, -623, -568, -1619, -1261, -853, -105, -252, -1325, -17, 8, 19, -259, -427; -51013, -358225, -4094272, -235416, -214633, -611780, -476501, -322328, -39677, -95225, -500688, -6424, 3023, 7180, -97869, -161352; -302671, -2125425, -24292184, -1396773, -1273463, -3629820, -2827181, -1912440, -235412, -564990, -2970688, -38115, 17936, 42600, -580678, -957336],112,8]
%
% END OF LATTICE L113.8
%
\hbox{}\par\smallskip%
%
% LATTICE L113.9
%
\leavevmode%
\llap{\spacer}%
${L_{113.9}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{113.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5022602310, -22954470, 1652104530; -22954470, 104910, -7550505; 1652104530, -7550505, 543433306],[570, 15, 1],[14350, 7112, 99685, 3592, 324, 225, 39718, 30779, 18896, 26134, 742547, 85064, 77250, 21939, 170840, 57877; 56, 26, 342, 9, 0, 2, 190, 142, 85, 115, 3230, 367, 329, 92, 703, 234; -43625, -21621, -303050, -10920, -985, -684, -120745, -93570, -57445, -79449, -2257390, -258600, -234845, -66696, -519365, -175950],112,9]
%
% END OF LATTICE L113.9
%
\hbox{}\par\smallskip%
%
% LATTICE L113.10
%
\leavevmode%
\llap{\spacer}%
${L_{113.10}}=\pdualwitharg{19}(\pfillwitharg{2}({L_{113.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-91184610, 238260, -33587820; 238260, -532, 87761; -33587820, 87761, -12372062],[570, 19, 1],[-37115, -31419, -13685, -2862, 2855, 2869, -3445, -19943, -39907, -96951, -89011, -65602, -182573, -88449, -22229, -48521; 2217, 1875, 816, 170, -171, -170, 207, 1196, 2391, 5805, 5328, 3926, 10923, 5290, 1329, 2900; 100776, 85310, 37158, 7771, -7752, -7790, 9354, 54150, 108357, 263245, 241686, 178125, 495729, 240160, 60357, 131746],112,10]
%
% END OF LATTICE L113.10
%
\hbox{}\par\smallskip%
%
% LATTICE L113.11
%
\leavevmode%
\llap{\spacer}%
${L_{113.11}}=\pdualwitharg{5}({L_{113.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-52966680, 759240, -52440; 759240, -9730, 625; -52440, 625, -38],[2280, 5, 1],[-88, -41, -7007, -134, -365, -69, -800, -179, -65, -10, -745, -2, 8, 1, -169, -93; -16533, -7703, -1316472, -25176, -68577, -12964, -150309, -33632, -12213, -1879, -139992, -376, 1503, 188, -31749, -17472; -150495, -70119, -11983680, -229175, -624255, -118012, -1368285, -306160, -111180, -17106, -1274520, -3425, 13680, 1712, -288990, -159040],112,11]
%
% END OF LATTICE L113.11
%
\hbox{}\par\smallskip%
%
% LATTICE L113.12
%
\leavevmode%
\llap{\spacer}%
${L_{113.12}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{113.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-705808404060, -26662890, -352157307660; -26662890, -840, -13303230; -352157307660, -13303230, -175705997017],[570, 30, 1],[2004314, 1017952, 7019814, 463477, -154178, -46442, 3536664, 1616040, 2155663, 3142025, 45673228, 10629795, 9860791, 1433101, 22809857, 3930596; 739, 375, 2584, 170, -57, -17, 1311, 598, 797, 1161, 16872, 3926, 3641, 529, 8417, 1450; -4017130, -2040222, -14069405, -928920, 309010, 93081, -7088330, -3238935, -4320470, -6297378, -91540195, -21304680, -19763410, -2872281, -45716470, -7877865],112,12]
%
% END OF LATTICE L113.12
%
\hbox{}\par\smallskip%
%
% LATTICE L113.13
%
\leavevmode%
\llap{\spacer}%
${L_{113.13}}=\pdualwitharg{2}({L_{113.1}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36136846320, -375878520, -201499560; -375878520, -3909712, -2095904; -201499560, -2095904, -1123559],[2280, 8, 1],[2372, 5946, 26306, 4121, 11632, 2862, 27814, 811, 2551, 2259, 4873, 255, -145, -127, 3365, 283; -225303, -564785, -2498709, -391440, -1104891, -271855, -2642007, -77036, -242319, -214585, -462897, -24224, 13773, 12065, -319599, -26880; -5112, -12800, -56601, -8864, -25008, -6150, -59736, -1741, -5472, -4840, -10431, -544, 312, 270, -7296, -611],112,13]
%
% END OF LATTICE L113.13
%
\hbox{}\par\smallskip%
%
% LATTICE L113.14
%
\leavevmode%
\llap{\spacer}%
${L_{113.14}}=\pdualwitharg{2.19}(\pfillwitharg{2}({L_{113.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[373540531620, -315077190, 186392896170; -315077190, 265772, -157220288; 186392896170, -157220288, 93008144503],[570, 38, 1],[975575, 801697, 176590, 79032, 20251, 14079, 144733, 353928, 1300615, 2991953, 1340870, 1944248, 5290859, 1250471, 613997, 657712; 168, 130, 27, 9, 0, 5, 30, 71, 255, 575, 255, 367, 987, 230, 111, 117; -1955100, -1606640, -353895, -158384, -40584, -28215, -290052, -709289, -2606496, -5996020, -2687169, -3896368, -10603140, -2506005, -1230480, -1318087],112,14]
%
% END OF LATTICE L113.14
%
\hbox{}\par\smallskip%
%
% LATTICE L113.15
%
\leavevmode%
\llap{\spacer}%
${L_{113.15}}=\pdualwitharg{3.19}(\pfillwitharg{2}({L_{113.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[365854140330, -381898290, 121646445690; -381898290, 398658, -126981123; 121646445690, -126981123, 40447424582],[570, 57, 1],[262650, 647512, 95085, 63832, 5452, 22743, 38966, 571721, 350160, 2416538, 721995, 1570328, 1424438, 2019957, 165304, 1062439; 56, 130, 18, 9, 0, 10, 10, 142, 85, 575, 170, 367, 329, 460, 37, 234; -789925, -1947405, -285970, -191976, -16397, -68400, -117191, -1719462, -1053113, -7267785, -2171414, -4722792, -4284025, -6075060, -497155, -3195306],112,15]
%
% END OF LATTICE L113.15
%
\hbox{}\par\smallskip%
%
% LATTICE L113.16
%
\leavevmode%
\llap{\spacer}%
${L_{113.16}}=\pdualwitharg{3.5}({L_{113.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-20276040, 855000, -54720; 855000, -35670, 2235; -54720, 2235, -134],[2280, 15, 1],[-91, -42, -1059, -10, 8, 1, -259, -413, -128, -177, -10037, -574, -519, -293, -1128, -755; -3265, -1507, -38000, -359, 287, 36, -9291, -14816, -4592, -6350, -360088, -20593, -18620, -10512, -40470, -27088; -17300, -7986, -201400, -1905, 1520, 192, -49210, -78480, -24325, -33639, -1907600, -109095, -98645, -55692, -214415, -143520],112,16]
%
% END OF LATTICE L113.16
%
\hbox{}\par\smallskip%
%
% LATTICE L113.17
%
\leavevmode%
\llap{\spacer}%
${L_{113.17}}=\pdualwitharg{19}({L_{113.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-289635240, 3276360, 9961320; 3276360, -36974, -110371; 9961320, -110371, -282046],[2280, 19, 1],[22393, 19258, 17003, 1886, -1672, -3981, 1859, 22373, 22796, 56057, 103501, 38282, 107113, 104401, 13204, 57963; 2239527, 1925995, 1700472, 188619, -167217, -398140, 185919, 2237528, 2279832, 5606270, 10351152, 3828589, 10712388, 10441160, 1320534, 5796888; -85500, -73530, -64920, -7201, 6384, 15200, -7098, -85424, -87039, -214035, -395184, -146167, -408975, -398620, -50415, -221312],112,17]
%
% END OF LATTICE L113.17
%
\hbox{}\par\smallskip%
%
% LATTICE L113.18
%
\leavevmode%
\llap{\spacer}%
${L_{113.18}}=\pdualwitharg{5.19}(\pfillwitharg{2}({L_{113.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[400027048230, -515540490, -80252395890; -515540490, 664430, 103426405; -80252395890, 103426405, 16100028922],[570, 95, 1],[-641800, -105482, -232345, -51992, -13322, -3705, -95216, -465679, -855638, -393664, -1764239, -1279064, -3480700, -329059, -403930, -865377; 168, 26, 54, 9, 0, 2, 30, 142, 255, 115, 510, 367, 987, 92, 111, 234; -3199125, -525787, -1158150, -259160, -66405, -18468, -474615, -2321230, -4265025, -1962263, -8794050, -6375640, -17349945, -1640232, -2013435, -4313570],112,18]
%
% END OF LATTICE L113.18
%
\hbox{}\par\smallskip%
%
% LATTICE L113.19
%
\leavevmode%
\llap{\spacer}%
${L_{113.19}}=\pdualwitharg{2.3}({L_{113.1}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{l}{2}{19}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{}{6}{16}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{2}{304}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15234960, 1990440, -26044440; 1990440, -259824, 3400584; -26044440, 3400584, -44503757],[2280, 24, 1],[8020, 56826, 81545, 37625, 34464, 24692, 77418, 6584, 6557, 16059, 10776, 1247, -451, -397, 14903, 3134; -152681, -1081835, -1552433, -716297, -656121, -470085, -1473887, -125347, -124834, -305740, -205162, -23743, 8586, 7560, -283708, -59663; -16360, -115920, -166345, -76752, -70304, -50370, -157928, -13431, -13376, -32760, -21983, -2544, 920, 810, -30400, -6393],112,19]
%
% END OF LATTICE L113.19
%
\hbox{}\par\smallskip%
%
% LATTICE L113.20
%
\leavevmode%
\llap{\spacer}%
${L_{113.20}}=\pdualwitharg{2.3.19}(\pfillwitharg{2}({L_{113.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-60655865248740, -475989330, -30264020426460; -475989330, -3192, -237493122; -30264020426460, -237493122, -15100121457623],[570, 114, 1],[33481526, 85029448, 6172634, 7745447, -2575502, -3882194, 3107744, 26985948, 36000205, 262379207, 40148520, 177539169, 164699541, 119685079, 20052833, 65656360; 739, 1875, 136, 170, -57, -85, 69, 598, 797, 5805, 888, 3926, 3641, 2645, 443, 1450; -67104466, -170418030, -12371339, -15523608, 5161882, 7780785, -6228614, -54085875, -72152462, -525866610, -80466613, -355828200, -330094714, -239875665, -40190362, -131590029],112,20]
%
% END OF LATTICE L113.20
%
\hbox{}\par\smallskip%
%
% LATTICE L113.21
%
\leavevmode%
\llap{\spacer}%
${L_{113.21}}=\pdualwitharg{2.5}({L_{113.1}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1544277055440, 84716972760, 32479240680; 84716972760, -4647459760, -1781767680; 32479240680, -1781767680, -683103499],[2280, 40, 1],[92483, 17831, 206159, 13481, -1613, -1343, 38629, 5879, 58798, 32596, 751954, 120991, 354602, 18152, 919484, 27739; 1757151, 338785, 3916983, 256138, -30645, -25517, 733875, 111696, 1117125, 619307, 14286765, 2298774, 6737289, 344881, 17469873, 527032; -186000, -35864, -414675, -27120, 3240, 2702, -77520, -11815, -118200, -65536, -1511925, -243280, -713040, -36502, -1849080, -55785],112,21]
%
% END OF LATTICE L113.21
%
\hbox{}\par\smallskip%
%
% LATTICE L113.22
%
\leavevmode%
\llap{\spacer}%
${L_{113.22}}=\pdualwitharg{2.5.19}(\pfillwitharg{2}({L_{113.1}}))$%
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{$1\above{1pt}{1pt}{-}{3}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8577400450620, -231079710, -4263568388940; -231079710, -5320, -114862790; -4263568388940, -114862790, -2119291912721],[570, 190, 1],[48578894, 8224708, 8955976, 3745999, -3736838, -375516, 4509076, 13051450, 52233283, 25379351, 58252142, 85864825, 238965251, 11576869, 29094983, 31753962; 2217, 375, 408, 170, -171, -17, 207, 598, 2391, 1161, 2664, 3926, 10923, 529, 1329, 1450; -97730490, -16546378, -18017535, -7536160, 7517730, 755459, -9071310, -26256765, -105082350, -51057902, -117191025, -172741920, -480747690, -23290219, -58532970, -63882275],112,22]
%
% END OF LATTICE L113.22
%
\hbox{}\par\smallskip%
%
% LATTICE L113.23
%
\leavevmode%
\llap{\spacer}%
${L_{113.23}}=\pdualwitharg{3.19}({L_{113.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3513480, 305520, -1525320; 305520, -135546, 381159; -1525320, 381159, -966214],[2280, 57, 1],[-33348, -234183, -140873, -153902, -140317, -399959, -16396, -210733, -25941, -62262, -17231, -4202, 1976, 4699, -3367, -105475; -666673, -4681645, -2816248, -3076716, -2805133, -7995740, -327779, -4212848, -518597, -1244705, -344472, -84004, 39503, 93940, -67311, -2108592; -210349, -1477155, -888584, -970767, -885077, -2522820, -103421, -1329240, -163628, -392730, -108688, -26505, 12464, 29640, -21238, -665304],112,23]
%
% END OF LATTICE L113.23
%
\hbox{}\par\smallskip%
%
% LATTICE L113.24
%
\leavevmode%
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${L_{113.24}}=\pdualwitharg{3.5.19}(\pfillwitharg{2}({L_{113.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2527752362190, 153640080, 831546988050; 153640080, -7980, -50541615; 831546988050, -50541615, -273551476874],[570, 285, 1],[-7125183, -3619013, -2627189, -1648304, 548091, 330467, -661357, -11485725, -7661181, -11167351, -17087963, -37782000, -35049607, -10188043, -4267431, -27944579; 14251105, 7238401, 5254650, 3296778, -1096239, -660968, 1322783, 22972646, 15323159, 22335863, 34177702, 75567926, 70102855, 20377144, 8535305, 55892058; -21661900, -11002482, -7987150, -5011155, 1666300, 1004682, -2010650, -34918770, -23291435, -33950853, -51950630, -114864405, -106557415, -30973572, -12973795, -84956790],112,24]
%
% END OF LATTICE L113.24
%
\hbox{}\par\smallskip%
%
% LATTICE L113.25
%
\leavevmode%
\llap{\spacer}%
${L_{113.25}}=\pdualwitharg{5.19}({L_{113.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-59492040, 3317400, 4959000; 3317400, -184870, -275595; 4959000, -275595, -405862],[2280, 95, 1],[18497, 8632, 77707, 28251, 77020, 14575, 8905, 37901, 13795, 2132, 8405, 453, -1672, -223, 1859, 19507; 407121, 189991, 1710336, 621806, 1695213, 320796, 195999, 834200, 303627, 46925, 184992, 9970, -36801, -4908, 40917, 429352; -50445, -23541, -211920, -77045, -210045, -39748, -24285, -103360, -37620, -5814, -22920, -1235, 4560, 608, -5070, -53200],112,25]
%
% END OF LATTICE L113.25
%
\hbox{}\par\smallskip%
%
% LATTICE L113.26
%
\leavevmode%
\llap{\spacer}%
${L_{113.26}}=\pdualwitharg{2.3.5}({L_{113.1}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8295191760, -500444040, 38565148920; -500444040, -30191280, 2326594440; 38565148920, 2326594440, -179291545273],[2280, 120, 1],[113636, 173442, 1287703, 607921, 575904, 85664, 1399818, 123394, 131221, 71415, 263058, 45031, -4571, -4045, 150943, 39844; -19432119, -29659153, -220201507, -103956535, -98481543, -14648839, -239373761, -21100809, -22439262, -12212228, -44983906, -7700481, 781654, 691712, -25811652, -6813445; -227720, -347568, -2580485, -1218240, -1154080, -171666, -2805160, -247275, -262960, -143112, -527155, -90240, 9160, 8106, -302480, -79845],112,26]
%
% END OF LATTICE L113.26
%
\hbox{}\par\smallskip%
%
% LATTICE L113.27
%
\leavevmode%
\llap{\spacer}%
${L_{113.27}}=\pdualwitharg{2.3.5.19}(\pfillwitharg{2}({L_{113.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L113.27
%
\hbox{}\par\smallskip%
%
% LATTICE L113.28
%
\leavevmode%
\llap{\spacer}%
${L_{113.28}}=\pdualwitharg{2.19}({L_{113.1}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-107310480, -26058120, 337440; -26058120, -6325936, 81928; 337440, 81928, -1061],[2280, 152, 1],[337, 299, 34, 34, -19, -17, 23, 37, 311, 781, 182, 542, 1531, 377, 193, 107; 1659, 1475, 168, 169, -93, -85, 111, 180, 1518, 3820, 891, 2655, 7506, 1850, 948, 526; 235296, 209000, 23787, 23864, -13224, -11970, 15888, 25669, 216144, 543400, 126693, 377416, 1066584, 262770, 134592, 74651],112,28]
%
% END OF LATTICE L113.28
%
\hbox{}\par\smallskip%
%
% LATTICE L113.29
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4119960, 4404960, 487920; 4404960, -4655190, -510435; 487920, -510435, -55466],[2280, 285, 1],[-9445, -4352, -5765, -1018, 834, 93, -1423, -43057, -13334, -18427, -54979, -59726, -53985, -30465, -6170, -78431; -19083, -8793, -11648, -2057, 1685, 188, -2875, -86992, -26940, -37230, -111080, -120671, -109072, -61552, -12466, -158464; 92530, 42636, 56480, 9975, -8170, -912, 13940, 421800, 130625, 180519, 538600, 585105, 528865, 298452, 60445, 768360],112,29]
%
% END OF LATTICE L113.29
%
\hbox{}\par\smallskip%
%
% LATTICE L113.30
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1578120240, -178521720, -58760160; -178521720, -20190768, -6645744; -58760160, -6645744, -2187431],[2280, 456, 1],[277, 679, 24, 53, -19, -17, 33, 132, 338, 2396, 181, 1587, 1454, 1042, 172, 278; -194366, -476540, -16847, -37241, 13326, 11970, -23132, -92554, -237031, -1680445, -126952, -1113151, -1019919, -730965, -120667, -195044; 583072, 1429560, 50539, 111720, -39976, -35910, 69392, 277647, 711056, 5041080, 380837, 3339288, 3059608, 2192790, 361984, 585105],112,30]
%
% END OF LATTICE L113.30
%
\hbox{}\par\smallskip%
%
% LATTICE L113.31
%
\leavevmode%
\llap{\spacer}%
${L_{113.31}}=\pdualwitharg{2.5.19}({L_{113.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1011959760, 966093000, -2414520; 966093000, -922304080, 2305080; -2414520, 2305080, -5761],[2280, 760, 1],[-37, -17, -19, -55, -149, -7, -17, -9, -26, -4, -2, -1, 2, 0, -4, -5; 645, 367, 450, 1384, 4089, 211, 567, 327, 1101, 215, 132, 168, -15, -17, 15, 61; 273600, 153976, 188025, 576840, 1698600, 87362, 234000, 134615, 451440, 87704, 53655, 67640, -6840, -6802, 7680, 26505],112,31]
%
% END OF LATTICE L113.31
%
\hbox{}\par\smallskip%
%
% LATTICE L113.32
%
\leavevmode%
\llap{\spacer}%
${L_{113.32}}=\pdualwitharg{2.3.5.19}({L_{113.1}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2540043120, -2651054040, 848889600; -2651054040, -2766912240, 885989760; 848889600, 885989760, -283701307],[2280, 2280, 1],[184921, 283159, 110802, 994785, 943617, 140555, 121061, 203006, 216338, 118100, 22951, 75239, -7346, -6802, 12692, 64500; 474, 724, 283, 2539, 2406, 358, 308, 516, 549, 299, 58, 189, -19, -17, 33, 166; 554800, 849528, 332425, 2984520, 2831000, 421686, 363200, 609045, 649040, 354312, 68855, 225720, -22040, -20406, 38080, 193515],112,32]
%
% END OF LATTICE L113.32
%

\medskip%
%------------------------------WEYL-GROUP-W114--------------------------------
%
\leavevmode\llap{}%
$W_{114}$%
\qquad\llap{24} lattices, $\chi=80$%
\hfill%
$2262222222622222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L114.1
%
\leavevmode%
\llap{\spacer}%
${L_{114.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}41\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{114}}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1509227220, 341940, 496920; 341940, -74, -117; 496920, -117, -158],[2460, 1, 1],[37, 109, 2, -2, -4, 3, 387, 19, 53, 629, 61, 218, 742, 105, 3303, 62; 76210, 224516, 4120, -4119, -8241, 6178, 797040, 39132, 109160, 1295518, 125639, 449007, 1528275, 216266, 6803130, 127701; 59930, 176546, 3239, -3240, -6478, 4860, 626890, 30777, 85850, 1018850, 98807, 353112, 1201874, 170076, 5350090, 100425],113,1]
%
% END OF LATTICE L114.1
%
\hbox{}\par\smallskip%
%
% LATTICE L114.2
%
\leavevmode%
\llap{\spacer}%
${L_{114.2}}=\pfillwitharg{2}({L_{114.1}})$%
{${}={}$}Nikulin $114$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}41\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-41205, -16605, 7995; -16605, -6691, 3198; 7995, 3198, -547],[615, 1, 1],[-2, -686, 51, 2860, 15762, 1821, 85588, 5359, 4422, 36930, 6137, 18356, 56726, 3483, 95632, 2597; 5, 1722, -128, -7179, -39565, -4571, -214840, -13452, -11100, -92701, -15405, -46077, -142393, -8743, -240055, -6519; 0, 41, -3, -171, -943, -109, -5125, -321, -265, -2214, -368, -1101, -3403, -209, -5740, -156],113,2]
%
% END OF LATTICE L114.2
%
\hbox{}\par\smallskip%
%
% LATTICE L114.3
%
\leavevmode%
\llap{\spacer}%
${L_{114.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{114.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-230484165, 232470, -76716330; 232470, -222, 77373; -76716330, 77373, -25534921],[615, 3, 1],[-14758, -43474, -1595, 532, 3190, -1197, -154388, -5053, -21142, -250906, -48665, -57972, -591950, -41883, -1317512, -16487; 15350, 45223, 1660, -553, -3321, 1244, 160515, 5254, 21985, 260924, 50609, 60289, 615615, 43558, 1370220, 17147; 44385, 130749, 4797, -1600, -9594, 3600, 464325, 15197, 63585, 754605, 146361, 174352, 1780302, 125964, 3962445, 49585],113,3]
%
% END OF LATTICE L114.3
%
\hbox{}\par\smallskip%
%
% LATTICE L114.4
%
\leavevmode%
\llap{\spacer}%
${L_{114.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{114.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-297837735, 339480, -119036325; 339480, -370, 135675; -119036325, 135675, -47575054],[615, 5, 1],[-4697, -69181, -2538, 2540, 5076, -1905, -49139, -24124, -6729, -399285, -77444, -276764, -942008, -66651, -419327, -78710; 3070, 45223, 1660, -1659, -3321, 1244, 32103, 15762, 4397, 260924, 50609, 180867, 615615, 43558, 274044, 51441; 11761, 173225, 6355, -6360, -12710, 4770, 123041, 60405, 16849, 999785, 193915, 693000, 2358730, 166890, 1049969, 197085],113,4]
%
% END OF LATTICE L114.4
%
\hbox{}\par\smallskip%
%
% LATTICE L114.5
%
\leavevmode%
\llap{\spacer}%
${L_{114.5}}=\pdualwitharg{3}({L_{114.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}41\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1784289660, -1168500, 924960; -1168500, -714, 597; 924960, 597, -478],[2460, 3, 1],[153, 1777, 171, 202, 2054, 289, 9043, 56, 97, 273, 4, -2, -4, 11, 1207, 19; 64190, 745544, 71744, 84751, 861779, 121254, 3794140, 23496, 40700, 114554, 1679, -839, -1681, 4614, 506350, 7971; 376230, 4369698, 420495, 496726, 5050872, 710664, 22237170, 137707, 238530, 671334, 9837, -4918, -9840, 27048, 2967990, 46721],113,5]
%
% END OF LATTICE L114.5
%
\hbox{}\par\smallskip%
%
% LATTICE L114.6
%
\leavevmode%
\llap{\spacer}%
${L_{114.6}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{114.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[999751451955, -2894727510, 332510935005; -2894727510, 8381535, -962767845; 332510935005, -962767845, 110591009077],[615, 15, 1],[126847, 2010680, 100312, 4199, 47659, 52623, 1227026, 197160, 166426, 10013513, 1952599, 2341985, 24000727, 1707015, 10797167, 682659; 43, 697, 37, 2, 0, 14, 369, 61, 53, 3239, 635, 766, 7872, 562, 3567, 227; -381387, -6045450, -301605, -12625, -143295, -158220, -3689262, -592795, -500388, -30107325, -5870820, -7041575, -72162255, -5132430, -32463513, -2052530],113,6]
%
% END OF LATTICE L114.6
%
\hbox{}\par\smallskip%
%
% LATTICE L114.7
%
\leavevmode%
\llap{\spacer}%
${L_{114.7}}=\pdualwitharg{2}({L_{114.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}41\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3206795774280, -6646307460, -803005746000; -6646307460, 13774936, 1664285304; -803005746000, 1664285304, 201078669643],[2460, 4, 1],[-218107, -691256, -34458, -4309, -16591, -18145, -2112782, -203626, -286378, -3445521, -671821, -2417219, -8256963, -587237, -18571103, -704447; 215, 697, 37, 6, 0, 14, 1845, 183, 265, 3239, 635, 2298, 7872, 562, 17835, 681; -871010, -2760530, -137608, -17208, -66256, -72462, -8437390, -813180, -1143650, -13759682, -2682916, -9653160, -32974168, -2345130, -74163670, -2813208],113,7]
%
% END OF LATTICE L114.7
%
\hbox{}\par\smallskip%
%
% LATTICE L114.8
%
\leavevmode%
\llap{\spacer}%
${L_{114.8}}=\pdualwitharg{5}({L_{114.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-34890180, 118080, 5311140; 118080, -370, -18405; 5311140, -18405, -802234],[2460, 5, 1],[1535, 91075, 8832, 31562, 107424, 15201, 95633, 8975, 1071, 15771, 289, -290, -578, 435, 11213, 2752; 110728, 6569758, 637103, 2276751, 7749123, 1096538, 6898578, 647421, 77258, 1137668, 20848, -20919, -41697, 31378, 808848, 198516; 7622, 452230, 43855, 156720, 533410, 75480, 474862, 44565, 5318, 78310, 1435, -1440, -2870, 2160, 55678, 13665],113,8]
%
% END OF LATTICE L114.8
%
\hbox{}\par\smallskip%
%
% LATTICE L114.9
%
\leavevmode%
\llap{\spacer}%
${L_{114.9}}=\pdualwitharg{41}(\pfillwitharg{2}({L_{114.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1610584755, -3324690, 432053490; -3324690, -6478, 891873; 432053490, 891873, -115902139],[615, 41, 1],[174918, 12569, 18916, -18905, -923, 14176, 44613, 179614, 250527, 72520, 576707, 2061041, 171101, 496358, 380832, 586187; 7210, 518, 779, -780, -38, 585, 1840, 7407, 10330, 2990, 23777, 84972, 7054, 20463, 15700, 24165; 652105, 46858, 70520, -70479, -3441, 52849, 166320, 669612, 933980, 270359, 2149999, 7683687, 637875, 1850453, 1419765, 2185341],113,9]
%
% END OF LATTICE L114.9
%
\hbox{}\par\smallskip%
%
% LATTICE L114.10
%
\leavevmode%
\llap{\spacer}%
${L_{114.10}}=\pdualwitharg{2.3}({L_{114.1}})$%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18678019481160, -15569465460, -4677109932000; -15569465460, 12978264, 3898705728; -4677109932000, 3898705728, 1171181844953],[2460, 12, 1],[-578779, -6875630, -1334618, -1592817, -16282905, -1154197, -36388164, -457378, -417116, -1290709, -59635, -615, -49773, -40877, -4443531, -140159; -475, -5535, -1067, -1264, -12874, -908, -28495, -355, -315, -943, -39, 0, -82, -40, -3895, -119; -2311350, -27457782, -5329788, -6360904, -65025672, -4609278, -145315890, -1826536, -1665750, -5154438, -238152, -2456, -198768, -163242, -17745210, -559724],113,10]
%
% END OF LATTICE L114.10
%
\hbox{}\par\smallskip%
%
% LATTICE L114.11
%
\leavevmode%
\llap{\spacer}%
${L_{114.11}}=\pdualwitharg{3.5}({L_{114.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}41\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{2460}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{s}{2}{1230}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{492}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\HardButStrongLineBreak\kern3pt%
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2019254\\145426467\\5043000
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-337858860, -1138980, 168126240; -1138980, -3570, 559005; 168126240, 559005, -83439158],[2460, 15, 1],[30081, 1746901, 168105, 198582, 2019254, 284113, 1778027, 55054, 19073, 268413, 3934, -1966, -3940, 10811, 237287, 18677; 2166430, 125811616, 12106904, 14301855, 145426467, 20461790, 128053332, 3964984, 1373636, 19331090, 283327, -141591, -283761, 778606, 17089374, 1345115; 75126, 4362810, 419835, 495950, 5043000, 709560, 4440546, 137495, 47634, 670350, 9825, -4910, -9840, 27000, 592614, 46645],113,11]
%
% END OF LATTICE L114.11
%
\hbox{}\par\smallskip%
%
% LATTICE L114.12
%
\leavevmode%
\llap{\spacer}%
${L_{114.12}}=\pdualwitharg{2.5}({L_{114.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}41\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{820}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{}{6}{120}\above{1pt}{1pt}{s}{2}{1640}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{164}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4806116760\\-2919387780\\-1211577060
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4806116760, -2919387780, -1211577060; -2919387780, 1773328760, 735950340; -1211577060, 735950340, 305427239],[2460, 20, 1],[-7648, -462623, -90377, -325843, -1114163, -79347, -502485, -95527, -5957, -95182, -4858, -677, -1447, -2283, -55320, -26918; 53, 3239, 635, 2298, 7872, 562, 3567, 681, 43, 697, 37, 6, 0, 14, 369, 183; -30466, -1842950, -360040, -1298100, -4438660, -316110, -2001866, -380580, -23734, -379250, -19360, -2700, -5740, -9090, -220334, -107220],113,12]
%
% END OF LATTICE L114.12
%
\hbox{}\par\smallskip%
%
% LATTICE L114.13
%
\leavevmode%
\llap{\spacer}%
${L_{114.13}}=\pdualwitharg{3.41}(\pfillwitharg{2}({L_{114.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}41\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[118268307465, -1339796565, 38438059035; -1339796565, 15177831, -435443616; 38438059035, -435443616, 12492648403],[615, 123, 1],[-374771, -29150, -60643, -2745, -507, -29342, -85806, -113837, -483904, -142585, -1141502, -1371135, -342963, -1001068, -772869, -401368; 265, 20, 38, 1, 1, 27, 70, 90, 370, 107, 851, 1015, 253, 735, 565, 291; 1153125, 89691, 186591, 8446, 1560, 90282, 264015, 350263, 1488915, 438717, 3512265, 4218818, 1055256, 3080166, 2378025, 1234961],113,13]
%
% END OF LATTICE L114.13
%
\hbox{}\par\smallskip%
%
% LATTICE L114.14
%
\leavevmode%
\llap{\spacer}%
${L_{114.14}}=\pdualwitharg{41}({L_{114.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}41\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1148820, 59040, 29520; 59040, -3034, -1517; 29520, -1517, -758],[2460, 41, 1],[33, 9, 35, 122, 10, 57, 43, 32, 17, 1, 0, -2, 0, 3, 7, 13; 840, 238, 939, 3327, 275, 1586, 1210, 921, 530, 36, 20, -39, -1, 58, 160, 312; -410, -130, -533, -1968, -166, -984, -770, -615, -410, -34, -41, 0, 2, 0, -50, -123],113,14]
%
% END OF LATTICE L114.14
%
\hbox{}\par\smallskip%
%
% LATTICE L114.15
%
\leavevmode%
\llap{\spacer}%
${L_{114.15}}=\pdualwitharg{5.41}(\pfillwitharg{2}({L_{114.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}41\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[202762696830, -2264764560, -84821516475; -2264764560, 25296385, 947416725; -84821516475, 947416725, 35483300314],[615, 205, 1],[-163182, -63461, -132016, -17923, -1105, -63892, -37365, -743554, -210711, -310432, -2485235, -8955509, -746681, -2179466, -336528, -2621483; 489599, 190403, 396086, 53772, 3316, 191703, 112109, 2230932, 632207, 931403, 7456556, 26869572, 2240296, 6539133, 1009697, 7865322; -403153, -156785, -326155, -44280, -2730, -157850, -92313, -1837005, -520577, -766945, -6139955, -22125240, -1844730, -5384530, -831417, -6476565],113,15]
%
% END OF LATTICE L114.15
%
\hbox{}\par\smallskip%
%
% LATTICE L114.16
%
\leavevmode%
\llap{\spacer}%
${L_{114.16}}=\pdualwitharg{2.3.5}({L_{114.1}})$%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1352234974440, 1580860749660, -337145076240; 1580860749660, 1848140860920, -394147044000; -337145076240, -394147044000, 84058469557],[2460, 60, 1],[108805, 6417998, 1242718, 1479223, 15101935, 1068591, 6726944, 421454, 76144, 1164911, 51925, 369, 63883, 41103, 855965, 133393; -95, -5535, -1067, -1264, -12874, -908, -5699, -355, -63, -943, -39, 0, -82, -40, -779, -119; 435954, 25715610, 4979340, 5927000, 60511080, 4281690, 26953974, 1688720, 305106, 4667850, 208080, 1480, 255840, 164670, 3429486, 534460],113,16]
%
% END OF LATTICE L114.16
%
\hbox{}\par\smallskip%
%
% LATTICE L114.17
%
\leavevmode%
\llap{\spacer}%
${L_{114.17}}=\pdualwitharg{3.41}({L_{114.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}41\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6872947260, -159639240, 4093440; -159639240, -3707958, 95079; 4093440, 95079, -2438],[2460, 123, 1],[133, 37, 145, 170, 42, 241, 183, 46, 77, 5, 2, -2, 0, 11, 27, 17; 2790, 784, 3084, 3631, 899, 5174, 3940, 996, 1700, 114, 59, -39, -1, 214, 550, 351; 332100, 92694, 363711, 427015, 105573, 606390, 460890, 116071, 195570, 12840, 5658, -4879, -39, 26814, 66780, 42230],113,17]
%
% END OF LATTICE L114.17
%
\hbox{}\par\smallskip%
%
% LATTICE L114.18
%
\leavevmode%
\llap{\spacer}%
${L_{114.18}}=\pdualwitharg{3.5.41}(\pfillwitharg{2}({L_{114.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}41\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6976729378995, 1231071561855, 419698342335; 1231071561855, -217227458085, -74057436540; 419698342335, -74057436540, -25247747017],[615, 615, 1],[1442, 518, 779, -260, -38, 585, 368, 2469, 2066, 2990, 23777, 28324, 7054, 20463, 3140, 8055; -2293463, -823984, -1239956, 413177, 60501, -929491, -584994, -3925292, -3284962, -4754441, -37809019, -45040577, -11217351, -32541059, -4993431, -12809983; 6751224, 2425545, 3650025, -1216265, -178095, 2736135, 1722039, 11554825, 9669891, 13995570, 111297780, 132585185, 33020325, 95790555, 14699076, 37708520],113,18]
%
% END OF LATTICE L114.18
%
\hbox{}\par\smallskip%
%
% LATTICE L114.19
%
\leavevmode%
\llap{\spacer}%
${L_{114.19}}=\pdualwitharg{2.41}({L_{114.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}41\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3389880, 218940, -848700; 218940, 41656, -54776; -848700, -54776, 212483],[2460, 164, 1],[-47338, -14121, -113529, -410947, -34339, -100529, -77823, -121867, -38507, -3048, -6654, -1109, 1, -2485, -8022, -32654; 265, 79, 635, 2298, 192, 562, 435, 681, 215, 17, 37, 6, 0, 14, 45, 183; -189010, -56382, -453296, -1640820, -137108, -401390, -310730, -486588, -153750, -12170, -26568, -4428, 4, -9922, -32030, -130380],113,19]
%
% END OF LATTICE L114.19
%
\hbox{}\par\smallskip%
%
% LATTICE L114.20
%
\leavevmode%
\llap{\spacer}%
${L_{114.20}}=\pdualwitharg{5.41}({L_{114.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}41\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-169368537540, 50690760, -171922020; 50690760, -15170, 51455; -171922020, 51455, -174514],[2460, 205, 1],[115, 175, 708, 2578, 216, 1269, 197, 775, 99, 39, 41, -10, -2, 15, 17, 188; 168, 238, 939, 3327, 275, 1586, 242, 921, 106, 36, 20, -39, -1, 58, 32, 312; -113242, -172330, -697205, -2538720, -212710, -1249680, -194002, -763215, -97498, -38410, -40385, 9840, 1970, -14760, -16738, -185115],113,20]
%
% END OF LATTICE L114.20
%
\hbox{}\par\smallskip%
%
% LATTICE L114.21
%
\leavevmode%
\llap{\spacer}%
${L_{114.21}}=\pdualwitharg{2.3.41}({L_{114.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}41\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[434430405960, -696910620, -108784489020; -696910620, 1122744, 174511416; -108784489020, 174511416, 27240416161],[2460, 492, 1],[-7999099, -2358894, -18888974, -22691105, -5675849, -16566953, -12789404, -6640670, -6193596, -479645, -980799, -17207, -1229, -464281, -1403291, -1874947; -6040, -1781, -14261, -17131, -4285, -12507, -9655, -5013, -4675, -362, -740, -13, -1, -351, -1060, -1416; -31944330, -9420222, -75432948, -90616724, -22666452, -66159978, -51074370, -26519456, -24734070, -1915458, -3916812, -68716, -4908, -1854102, -5604030, -7487584],113,21]
%
% END OF LATTICE L114.21
%
\hbox{}\par\smallskip%
%
% LATTICE L114.22
%
\leavevmode%
\llap{\spacer}%
${L_{114.22}}=\pdualwitharg{3.5.41}({L_{114.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}41\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-502403340, 598916520, -100975620; 598916520, -713969490, 120373335; -100975620, 120373335, -20294602],[2460, 615, 1],[-26627, -37335, -146752, -172634, -42724, -245741, -37405, -47225, -16059, -5351, -2649, 1890, 38, -10375, -5281, -16804; 558, 784, 3084, 3631, 899, 5174, 788, 996, 340, 114, 59, -39, -1, 214, 110, 351; 135792, 190410, 748455, 880475, 217905, 1253370, 190782, 240875, 81918, 27300, 13530, -9635, -195, 52890, 26928, 85690],113,22]
%
% END OF LATTICE L114.22
%
\hbox{}\par\smallskip%
%
% LATTICE L114.23
%
\leavevmode%
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${L_{114.23}}=\pdualwitharg{2.5.41}({L_{114.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[701508886440, -244954500, 175845247680; -244954500, 208280, -61494260; 175845247680, -61494260, 44078699959],[2460, 820, 1],[9099020, 13571991, 109117113, 394984615, 33005479, 96626255, 14960517, 117139639, 7403101, 2930130, 6397734, 1066289, -1237, 2386471, 1541604, 31379450; -27297007, -40715894, -327350704, -1184951547, -99016245, -289878203, -44881464, -351418236, -22209260, -8790373, -19193165, -3198861, 3711, -7159399, -4624803, -94138167; -36337234, -54200190, -435762760, -1577383980, -131808460, -385880110, -59745314, -467800980, -29564526, -11701570, -25549560, -4258260, 4940, -9530450, -6156446, -125314860],113,23]
%
% END OF LATTICE L114.23
%
\hbox{}\par\smallskip%
%
% LATTICE L114.24
%
\leavevmode%
\llap{\spacer}%
${L_{114.24}}=\pdualwitharg{2.3.5.41}({L_{114.1}})$%
\nopagebreak\par%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5613720, -2470432860, 618170940; -2470432860, 1087605364920, -272149079220; 618170940, -272149079220, 68099260733],[2460, 2460, 1],[-7518, -11087, -88785, -106663, -26681, -77881, -12025, -31221, -5825, -2256, -4616, -81, -5, -2177, -1318, -8808; 1751683, 2583384, 20688166, 24854443, 6217219, 18148081, 2802134, 7275442, 1357482, 525781, 1075989, 18877, 1111, 506909, 307031, 2052053; 7000422, 10324230, 82678140, 99328240, 24846480, 72526950, 11198442, 29075560, 5425038, 2101230, 4300080, 75440, 4440, 2025810, 1227018, 8200820],113,24]
%
% END OF LATTICE L114.24
%

\medskip%
%------------------------------WEYL-GROUP-W115--------------------------------
%
\leavevmode\llap{}%
$W_{115}$%
\qquad\llap{24} lattices, $\chi=84$%
\hfill%
$222222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L115.1
%
\leavevmode%
\llap{\spacer}%
${L_{115.1}}$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}43\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{115}}%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[1684740, -330240, -2580; -330240, 64706, 513; -2580, 513, 2],[2580, 1, 1],[-2113, -4059, -261, -12073, -237, -71, -1301, 0, 347, 23, -26, -24, -4267, -238, -309, -19124, -475, -41197; -10476, -20124, -1294, -59856, -1175, -352, -6450, 0, 1720, 114, -129, -119, -21156, -1180, -1532, -94815, -2355, -204250; -38514, -74003, -4761, -220332, -4330, -1300, -23865, -1, 6450, 432, -430, -432, -77400, -4325, -5624, -348300, -8654, -750780],114,1]
%
% END OF LATTICE L115.1
%
\hbox{}\par\smallskip%
%
% LATTICE L115.2
%
\leavevmode%
\llap{\spacer}%
${L_{115.2}}=\pfillwitharg{2}({L_{115.1}})$%
{${}={}$}Nikulin $115$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}43\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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101910\\-34185\\-2580
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[101910, -34185, -2580; -34185, 11467, 860; -2580, 860, -181],[645, 1, 1],[16510, 65191, 4433, 108349, 4794, 931, 47737, 431, 6408, 1, -648, 70, 20737, 2711, 2071, 274978, 7074, 316307; 49299, 194661, 13237, 323532, 14315, 2780, 142545, 1287, 19135, 3, -1935, 209, 61920, 8095, 6184, 821085, 21123, 944495; -1122, -4429, -301, -7353, -325, -63, -3225, -29, -430, 0, 43, -5, -1419, -185, -141, -18705, -481, -21500],114,2]
%
% END OF LATTICE L115.2
%
\hbox{}\par\smallskip%
%
% LATTICE L115.3
%
\leavevmode%
\llap{\spacer}%
${L_{115.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{115.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}43\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{258}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{43}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{430}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{645}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\GramMatrix{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12896130, 19350, 4305375; 19350, -21, -6465; 4305375, -6465, -1437347],[645, 3, 1],[-153, -10495, -2733, -61128, -18878, -11368, -453423, -33185, -1417281, -24014, -272918, -17040, -127814, -14077, -1814, -18932, 230, 216; -286, -19737, -5149, -115283, -35615, -21455, -855915, -62649, -2675890, -45342, -515355, -32183, -241445, -26605, -3433, -35905, 429, 430; -457, -31347, -8163, -182578, -56385, -33954, -1354285, -99117, -4233135, -71725, -815151, -50895, -381754, -42045, -5418, -56545, 687, 645],114,3]
%
% END OF LATTICE L115.3
%
\hbox{}\par\smallskip%
%
% LATTICE L115.4
%
\leavevmode%
\llap{\spacer}%
${L_{115.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{115.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}43\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{430}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{645}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{258}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{43}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3166305\\-5175\\-634577
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\GramMatrix{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15798630, 25800, 3166305; 25800, -35, -5175; 3166305, -5175, -634577],[645, 5, 1],[-287, -6539, -1701, -114068, -2348, -7068, -169129, -20629, -176197, -44780, -169632, -10590, -238274, -1749, -1126, -7042, 144, 26; -858, -19737, -5149, -345849, -7123, -21455, -513549, -62649, -535178, -136026, -515355, -32183, -724335, -5321, -3433, -21543, 429, 86; -1425, -32465, -8445, -566310, -11657, -35090, -839661, -102415, -874749, -222315, -842155, -52575, -1182930, -8683, -5590, -34959, 715, 129],114,4]
%
% END OF LATTICE L115.4
%
\hbox{}\par\smallskip%
%
% LATTICE L115.5
%
\leavevmode%
\llap{\spacer}%
${L_{115.5}}=\pdualwitharg{3}({L_{115.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}43\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{258}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{172}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{430}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{2580}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3439140\\-690306\\-1557
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7740\\-1557\\-2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17133780, 3439140, 7740; 3439140, -690306, -1557; 7740, -1557, -2],[2580, 3, 1],[23, -26, -70, -4719, -813, -1091, -22858, -1717, -149987, -2575, -14938, -974, -15233, -927, -301, -2082, -23, 347; 114, -129, -347, -23392, -4030, -5408, -113305, -8511, -743470, -12764, -74046, -4828, -75508, -4595, -1492, -10320, -114, 1720; 260, -258, -780, -52804, -9105, -12228, -256280, -19254, -1682160, -28882, -167571, -10929, -170968, -10410, -3384, -23435, -261, 3870],114,5]
%
% END OF LATTICE L115.5
%
\hbox{}\par\smallskip%
%
% LATTICE L115.6
%
\leavevmode%
\llap{\spacer}%
${L_{115.6}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{115.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}43\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{1290}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{215}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{86}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{129}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[759202410, -26615925, -256041135; -26615925, 933105, 8976225; -256041135, 8976225, 86349914],[645, 15, 1],[3856, 530701, 159099, 3822859, 241738, 746527, 6028733, 2220881, 19081350, 1622175, 18538208, 1171290, 8889747, 201725, 140399, 327570, -3566, -12441; -13, -1806, -542, -13029, -824, -2545, -20554, -7572, -65059, -5531, -63210, -3994, -30315, -688, -479, -1118, 12, 43; 11435, 1573800, 471810, 11336735, 716877, 2213835, 17878282, 6586050, 56585979, 4810580, 54975285, 3473475, 26362655, 598218, 416355, 971413, -10575, -36894],114,6]
%
% END OF LATTICE L115.6
%
\hbox{}\par\smallskip%
%
% LATTICE L115.7
%
\leavevmode%
\llap{\spacer}%
${L_{115.7}}=\pdualwitharg{2}({L_{115.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}43\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{344}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{129}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{5160}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{860}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18084608040, -9600180, 4512205860; -9600180, 5096, -2395296; 4512205860, -2395296, 1125819353],[2580, 4, 1],[111572, 419286, 25724, 284422, 19939, 1126, 56672, -642, 6118, 1607, 23141, 5143, 158005, 31506, 9163, 2155463, 52087, 2206017; 89691, 337077, 20683, 228717, 16040, 907, 45795, -513, 4945, 1290, 18576, 4130, 126936, 25315, 7364, 1732470, 41868, 1773320; -446982, -1679752, -103056, -1139457, -79880, -4511, -227040, 2572, -24510, -6438, -92708, -20604, -633003, -126220, -36709, -8635260, -208672, -8837790],114,7]
%
% END OF LATTICE L115.7
%
\hbox{}\par\smallskip%
%
% LATTICE L115.8
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7118220, 5160, 7740; 5160, 10, -15; 7740, -15, -2],[2580, 5, 1],[-89, -171, -11, -509, -2, -3, -11, 0, 3, 1, -1, -1, -179, -2, -13, -161, -20, -347; -38514, -74003, -4761, -220332, -866, -1300, -4773, -1, 1290, 432, -430, -432, -77400, -865, -5624, -69660, -8654, -150156; -56400, -108360, -6970, -322500, -1267, -1900, -6966, 0, 1892, 630, -645, -635, -113520, -1268, -8240, -102039, -12675, -219902],114,8]
%
% END OF LATTICE L115.8
%
\hbox{}\par\smallskip%
%
% LATTICE L115.9
%
\leavevmode%
\llap{\spacer}%
${L_{115.9}}=\pdualwitharg{43}(\pfillwitharg{2}({L_{115.1}}))$%
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{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}43\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-67824330, -287025, -31549530; -287025, -43, -133515; -31549530, -133515, -14675749],[645, 43, 1],[17762, 9502, 106592, 166502, 737283, 444150, 1236188, 1296926, 1288252, 2815937, 248107, 666235, 348716, 550762, 71068, 51857, -8881, -207; -27, -13, -141, -216, -950, -568, -1575, -1649, -1635, -3570, -314, -840, -438, -685, -86, -60, 14, 0; -38184, -20427, -229147, -357939, -1584980, -954815, -2657505, -2788077, -2769430, -6053583, -533370, -1432244, -749655, -1184005, -152779, -111480, 19092, 445],114,9]
%
% END OF LATTICE L115.9
%
\hbox{}\par\smallskip%
%
% LATTICE L115.10
%
\leavevmode%
\llap{\spacer}%
${L_{115.10}}=\pdualwitharg{2.3}({L_{115.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}43\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11893970280, -37660260, 2966349840; -37660260, 119256, -9392448; 2966349840, -9392448, 739806067],[2580, 12, 1],[321, 41731, 13485, 171636, 110468, 34844, 2843463, 210677, 9096141, 155120, 1781168, 113694, 435997, 101497, 7549, 194702, 1288, 18354; 86, 11180, 3614, 46010, 29615, 9342, 762390, 56488, 2438960, 41593, 477601, 30487, 116917, 27220, 2025, 52245, 347, 4945; -1286, -167184, -54024, -687613, -442560, -139593, -11391560, -844020, -36441210, -621446, -7135764, -455484, -1746703, -406620, -30243, -780020, -5160, -73530],114,10]
%
% END OF LATTICE L115.10
%
\hbox{}\par\smallskip%
%
% LATTICE L115.11
%
\leavevmode%
\llap{\spacer}%
${L_{115.11}}=\pdualwitharg{3.5}({L_{115.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}43\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17804580, 48728460, -9739500; 48728460, -59628810, 11897445; -9739500, 11897445, -2373826],[2580, 15, 1],[69929, 405706, 26458, 413863, 5039, 8187, 11334, 629, -1889, -629, 644, 1892, 127937, 4411, 29613, 124116, 46621, 814589; 18181518, 105483429, 6879071, 107604232, 1310138, 2128616, 2946833, 163539, -491146, -163540, 167442, 491920, 33263596, 1146859, 7699372, 32270124, 12121446, 211792888; 90837550, 527010795, 34368855, 537606640, 6545643, 10634880, 14722813, 817065, -2453838, -817070, 836565, 2457705, 166189840, 5729877, 38467200, 161226307, 60560535, 1058148558],114,11]
%
% END OF LATTICE L115.11
%
\hbox{}\par\smallskip%
%
% LATTICE L115.12
%
\leavevmode%
\llap{\spacer}%
${L_{115.12}}=\pdualwitharg{2.5}({L_{115.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}43\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12457958280, -66411780, 3109630140; -66411780, 354040, -16577040; 3109630140, -16577040, 776194573],[2580, 20, 1],[217169, 832043, 53237, 614533, 9616, 3683, 60793, 1273, 6719, 1610, 23184, 6430, 240694, 10273, 16206, 794434, 98172, 848508; 162, 602, 36, 387, 5, 1, 0, -2, 0, 3, 43, 9, 258, 10, 14, 645, 77, 645; -870030, -3333360, -213280, -2461965, -38524, -14755, -243552, -5100, -26918, -6450, -92880, -25760, -964275, -41156, -64925, -3182688, -393300, -3399322],114,12]
%
% END OF LATTICE L115.12
%
\hbox{}\par\smallskip%
%
% LATTICE L115.13
%
\leavevmode%
\llap{\spacer}%
${L_{115.13}}=\pdualwitharg{3.43}(\pfillwitharg{2}({L_{115.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}43\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${43}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{258}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{1290}\above{1pt}{1pt}{l}{2}{129}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{258}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2280083825535\\-147495375\\-761503627455
\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2280083825535, -147495375, -761503627455; -147495375, 9546, 49260585; -761503627455, 49260585, 254327392762],[645, 129, 1],[-199017, -642462, -8290082, -4634327, -63013912, -38924425, -36553594, -115808808, -115701473, -84592723, -22482558, -61084978, -10782353, -52611188, -7325615, -1988106, 183708, 76373; -1, -2, -24, -13, -175, -107, -100, -316, -315, -230, -61, -165, -29, -140, -19, -5, 1, 0; -595894, -1923651, -24822051, -13876039, -188675400, -116546985, -109448275, -346753161, -346431780, -253286383, -67316970, -182899812, -32284375, -157527705, -21934257, -5952760, 550056, 228675],114,13]
%
% END OF LATTICE L115.13
%
\hbox{}\par\smallskip%
%
% LATTICE L115.14
%
\leavevmode%
\llap{\spacer}%
${L_{115.14}}=\pdualwitharg{43}({L_{115.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}43\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${516}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{86}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{430}\above{1pt}{1pt}{l}{2}{172}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{86}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2580, 0, 2580; 0, 86, 473; 2580, 473, 22],[2580, 43, 1],[-1, 2, 2, -61, -187, -305, -482, -539, -1131, -2551, -118, -350, -403, -393, -159, -98, -41, -29; 6, -11, -11, 336, 1030, 1680, 2655, 2969, 6230, 14052, 650, 1928, 2220, 2165, 876, 540, 226, 160; 0, 2, 0, -72, -215, -344, -540, -602, -1260, -2838, -131, -387, -444, -430, -172, -105, -43, -30],114,14]
%
% END OF LATTICE L115.14
%
\hbox{}\par\smallskip%
%
% LATTICE L115.15
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2638234767345, 204825555, 528753621120; 204825555, 15910, 41051025; 528753621120, 41051025, 105972521974],[645, 215, 1],[497549, 535392, 6908492, 11585969, 10502456, 32437445, 18277036, 96508602, 19283831, 211484573, 18735710, 50904814, 26956235, 8768646, 6104759, 994066, -153092, -12729; -1492650, -1606178, -20725500, -34757946, -31507403, -97312442, -54831168, -289526122, -57851556, -634454409, -56207191, -152714607, -80868792, -26305966, -18314296, -2982201, 459277, 38187; -2481960, -2670735, -34462135, -57795135, -52390168, -161810075, -91172673, -481420905, -96194942, -1054963935, -93460710, -253932200, -134467755, -43741277, -30452815, -4958772, 763680, 63497],114,15]
%
% END OF LATTICE L115.15
%
\hbox{}\par\smallskip%
%
% LATTICE L115.16
%
\leavevmode%
\llap{\spacer}%
${L_{115.16}}=\pdualwitharg{2.3.5}({L_{115.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}43\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[56103147753258120, -261952703713740, 14003974364861520; -261952703713740, 1223090356440, -65386330259880; 14003974364861520, -65386330259880, 3495548928452231],[2580, 60, 1],[781931, 9131717, 603699, 2395894, 120560, 53110, 332725, 35419, 101427, 312, 41860, 36318, 661965, 92779, 160229, 2725134, 1031276, 9072150; 173517, 2034803, 135643, 542531, 27753, 12611, 81313, 9445, 27047, -3, 43, 6323, 135751, 19569, 34711, 596969, 227165, 2007799; -3129350, -36545700, -2416020, -9588355, -482472, -212535, -1331452, -141720, -405834, -1250, -167700, -145380, -2649445, -371328, -641265, -10906348, -4127280, -36307566],114,16]
%
% END OF LATTICE L115.16
%
\hbox{}\par\smallskip%
%
% LATTICE L115.17
%
\leavevmode%
\llap{\spacer}%
${L_{115.17}}=\pdualwitharg{3.43}({L_{115.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}43\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-202651260, -111569520, 237360; -111569520, -61424382, 130677; 237360, 130677, -278],[2580, 129, 1],[-1187, -165, -491, -189, -556, -227, -47, -60, -43, -1, 3, 5, -27, -254, -421, -223, -750, -1577; 2792, 388, 1154, 444, 1305, 532, 110, 140, 100, 2, -7, -11, 64, 600, 992, 525, 1765, 3710; 298850, 41493, 123195, 47324, 138675, 56244, 11575, 14577, 10290, 86, -729, -903, 7028, 65145, 106812, 56365, 189243, 397350],114,17]
%
% END OF LATTICE L115.17
%
\hbox{}\par\smallskip%
%
% LATTICE L115.18
%
\leavevmode%
\llap{\spacer}%
${L_{115.18}}=\pdualwitharg{3.5.43}(\pfillwitharg{2}({L_{115.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7526280084630, 20039924735685, -6699376485450; 20039924735685, -53359505354310, 17838161577990; -6699376485450, 17838161577990, -5963323804627],[645, 645, 1],[-9, -13, -141, -72, -190, -568, -105, -1649, -327, -1190, -314, -840, -146, -137, -86, -4, 14, 0; -14618982, -23461500, -263186518, -137036500, -364084291, -1096647428, -203484082, -3202228176, -636162128, -2317600721, -612598419, -1644993239, -287003446, -271975628, -175472410, -8535891, 21928581, 102209; -43729925, -70180650, -787272810, -409918835, -1089089466, -3280413885, -608684242, -9578861460, -1902958989, -6932665325, -1832472600, -4920686940, -858516665, -813563784, -524892615, -25533508, 65595210, 305739],114,18]
%
% END OF LATTICE L115.18
%
\hbox{}\par\smallskip%
%
% LATTICE L115.19
%
\leavevmode%
\llap{\spacer}%
${L_{115.19}}=\pdualwitharg{2.43}({L_{115.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}43\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[29943480, -3661020, 7489740; -3661020, 447544, -915728; 7489740, -915728, 1873403],[2580, 172, 1],[-451, -1, 1331, 1780, 18032, 6236, 36977, 40099, 40993, 91124, 8244, 23398, 6463, 23183, 2007, 4238, 1116, 222; 6, 0, -22, -30, -305, -106, -630, -684, -700, -1557, -141, -401, -111, -400, -35, -75, -21, -5; 1806, 4, -5332, -7131, -72240, -24983, -148140, -160648, -164230, -365070, -33028, -93740, -25893, -92880, -8041, -16980, -4472, -890],114,19]
%
% END OF LATTICE L115.19
%
\hbox{}\par\smallskip%
%
% LATTICE L115.20
%
\leavevmode%
\llap{\spacer}%
${L_{115.20}}=\pdualwitharg{5.43}({L_{115.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}43\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[669899580, -111427620, 647580; -111427620, 18534290, -107715; 647580, -107715, 626],[2580, 215, 1],[1, -1, -1, 31, 19, 155, 49, 274, 115, 1297, 60, 178, 205, 40, 81, 10, 21, 3; 6, -7, -11, 192, 120, 992, 315, 1765, 742, 8376, 388, 1154, 1332, 261, 532, 66, 140, 20; 0, -170, -860, 960, 989, 10320, 3504, 20210, 8692, 99330, 4685, 14405, 17100, 3526, 7740, 1011, 2365, 338],114,20]
%
% END OF LATTICE L115.20
%
\hbox{}\par\smallskip%
%
% LATTICE L115.21
%
\leavevmode%
\llap{\spacer}%
${L_{115.21}}=\pdualwitharg{2.3.43}({L_{115.1}})$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[151815915240, -287228820, 37896090060; -287228820, 543864, -71697684; 37896090060, -71697684, 9459572401],[2580, 516, 1],[2486616, 680062, 1960708, 183476, 2043397, 190848, 148064, 164090, 63778, 9295, 641, 81401, 43425, 1360118, 489855, 986441, 3239425, 3339737; -42763, -11695, -33717, -3155, -35135, -3281, -2545, -2819, -1095, -159, -11, -1401, -747, -23395, -8425, -16965, -55711, -57435; -9961982, -2724492, -7855068, -735049, -8186340, -764583, -593180, -657384, -255510, -37238, -2568, -326112, -173971, -5448960, -1962477, -3951920, -12977916, -13379790],114,21]
%
% END OF LATTICE L115.21
%
\hbox{}\par\smallskip%
%
% LATTICE L115.22
%
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${L_{115.22}}=\pdualwitharg{3.5.43}({L_{115.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[297259860, 84048660, -216720; 84048660, 23763090, -61275; -216720, -61275, 158],[2580, 645, 1],[351, 49, 147, 57, 34, 71, 3, 20, 3, 1, -1, -3, 7, 14, 121, 13, 220, 93; 900, 126, 380, 148, 89, 188, 8, 54, 8, 2, -3, -11, 16, 34, 304, 33, 561, 238; 830330, 116055, 348945, 135560, 81141, 170280, 7217, 48375, 7218, 2150, -2535, -8385, 15800, 32379, 283800, 30623, 519225, 219822],114,22]
%
% END OF LATTICE L115.22
%
\hbox{}\par\smallskip%
%
% LATTICE L115.23
%
\leavevmode%
\llap{\spacer}%
${L_{115.23}}=\pdualwitharg{2.5.43}({L_{115.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}43\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4401361320, 66060900, -1049120880; 66060900, 906440, -15810240; -1049120880, -15810240, 250023599],[2580, 860, 1],[167119, 1293, -279283, -347002, -693034, -1173858, -1378309, -7435239, -1513101, -16772114, -1510838, -4249352, -1162543, -816511, -335687, -131236, -112486, 3556; -501873, -3883, 838711, 1042077, 2081241, 3525197, 4139181, 22328665, 4543973, 50368107, 4537177, 12761171, 3491217, 2452053, 1008097, 394113, 337805, -10679; 669510, 5180, -1118860, -1390155, -2776424, -4702695, -5521764, -29786960, -6061766, -67192230, -6052700, -17023700, -4657365, -3271096, -1344825, -525756, -450640, 14246],114,23]
%
% END OF LATTICE L115.23
%
\hbox{}\par\smallskip%
%
% LATTICE L115.24
%
\leavevmode%
\llap{\spacer}%
${L_{115.24}}=\pdualwitharg{2.3.5.43}({L_{115.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}43\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[500520, -426992580, -106582380; -426992580, 363283933080, 90679951440; -106582380, 90679951440, 22634784653],[2580, 2580, 1],[7279, 1959, 5461, 493, 1015, 391, 45, 3, -25, -129, -9, 389, 157, 935, 1559, 605, 9747, 1975; -4092494, -1101220, -3068646, -276910, -569551, -218798, -25042, 1288, 14448, 73523, 5137, -219603, -88451, -526532, -877285, -340319, -5481703, -1110531; 16395470, 4411740, 12293700, 1109365, 2281752, 876555, 100324, -5160, -57882, -294550, -20580, 879780, 354355, 2109408, 3514605, 1363396, 21960960, 4449042],114,24]
%
% END OF LATTICE L115.24
%

\medskip%
%------------------------------WEYL-GROUP-W116--------------------------------
%
\leavevmode\llap{}%
$W_{116}$%
\qquad\llap{24} lattices, $\chi=90$%
\hfill%
$224222222224222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L116.1
%
\leavevmode%
\llap{\spacer}%
${L_{116.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{116}}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{434}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{*}{2}{868}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{124}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{62}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2637852\\627564\\-5208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
627564\\149302\\-1239
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5208\\-1239\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1702273\\7166208\\1374912
\end{tabular}\endgroup%
\kern3pt%
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-404690\\1703659\\326865
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1716\\-7224\\-1387
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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796\\-3351\\-645
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6134\\-25823\\-4991
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\-341\\-67
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-80\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
309\\-1302\\-434
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\21\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59\\248\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-5\\21\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\-341\\-62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6\\0\\-6\\-66\\-10416\\-516\\-4836\\-903\\-651
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\kern3pt%
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0\\434\\0\\-434\\-77252\\-3906\\-37324\\-7161\\-5425
\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-217\\-9\\-26\\-3038\\-141\\-1240\\-210\\-124
\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-1953\\-141\\-750\\-105462\\-5109\\-46872\\-8484\\-5766
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-2604\\-210\\-1218\\-174468\\-8484\\-78120\\-14217\\-9765
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1302\\-124\\-806\\-118048\\-5766\\-53320\\-9765\\-6789
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[2637852, 627564, -5208; 627564, 149302, -1239; -5208, -1239, 10],[2604, 1, 1],[796, 6134, 81, 19, 309, -5, -59, -5, 81, 62, 2732, 205, 1125, 159279, 7727, 70987, 12875, 8785; -3351, -25823, -341, -80, -1302, 21, 248, 21, -341, -261, -11501, -863, -4736, -670530, -32529, -298840, -54201, -36983; -645, -4991, -67, -18, -434, -3, 0, 0, -62, -48, -2170, -164, -906, -128464, -6234, -57288, -10395, -7099],115,1]
%
% END OF LATTICE L116.1
%
\hbox{}\par\smallskip%
%
% LATTICE L116.2
%
\leavevmode%
\llap{\spacer}%
${L_{116.2}}=\pfillwitharg{2}({L_{116.1}})$%
{${}={}$}Nikulin $116$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{434}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{}{2}{217}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{31}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{62}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54033\\-5208\\-651
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5208\\502\\63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-651\\63\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4081211\\42990738\\-4140360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-393300\\-4142951\\399000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60858\\-641067\\61739
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[54033, -5208, -651; -5208, 502, 63; -651, 63, 10],[651, 1, 1],[1142, 8734, 112, 11, 185, -4, 3, 26, 162, 100, 4100, 302, 817, 115321, 11182, 51333, 18604, 12672; 12030, 92008, 1180, 116, 1953, -42, 31, 273, 1705, 1053, 43183, 3181, 8606, 1214766, 117789, 540733, 195972, 133486; -1161, -8897, -115, -12, -217, 3, 0, -21, -155, -99, -4123, -305, -828, -116963, -11343, -52080, -18879, -12865],115,2]
%
% END OF LATTICE L116.2
%
\hbox{}\par\smallskip%
%
% LATTICE L116.3
%
\leavevmode%
\llap{\spacer}%
${L_{116.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{116.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{1302}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{4}{3}\above{1pt}{1pt}{}{2}{651}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{93}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{186}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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569714187\\-2454921\\189860895
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\kern3pt%
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-2454921\\10578\\-818118
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\kern3pt%
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189860895\\-818118\\63272357
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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592\\-587\\-1784
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77338\\-76601\\-233058
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5780\\-5723\\-17418
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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15789\\-15628\\-47580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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72206\\-71463\\-217592
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\HardButStrongLineBreak\kern3pt%
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120249\\-119007\\-362369
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\HardButStrongLineBreak\kern3pt%
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245995\\-243443\\-741303
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[569714187, -2454921, 189860895; -2454921, 10578, -818118; 189860895, -818118, 63272357],[651, 3, 1],[592, 77338, 5780, 15789, 2233088, 72206, 994812, 120249, 245995, 7415, 170879, 2227, 225, 865, -143, -1944, -144, 2222; -587, -76601, -5723, -15628, -2210145, -71463, -984560, -119007, -243443, -7337, -169043, -2201, -220, -651, 147, 1984, 147, -2201; -1784, -233058, -17418, -47580, -6729387, -217592, -2997855, -362369, -741303, -22345, -514941, -6711, -678, -2604, 431, 5859, 434, -6696],115,3]
%
% END OF LATTICE L116.3
%
\hbox{}\par\smallskip%
%
% LATTICE L116.4
%
\leavevmode%
\llap{\spacer}%
${L_{116.4}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{116.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1254313599, -5564097, -537605565; -5564097, 24682, 2384802; -537605565, 2384802, 230420641],[651, 7, 1],[-2384, -14850, -7772, -21239, -429168, -291422, -1338372, -69335, -330977, -29935, -32859, -3001, -307, -213, 551, 2520, 80, -2986; -1761, -10943, -5723, -15628, -315735, -214389, -984560, -51003, -243443, -22011, -24149, -2201, -220, -93, 441, 1984, 63, -2201; -5544, -34534, -18074, -49392, -998045, -677712, -3112431, -161241, -769699, -69615, -76415, -6979, -714, -496, 1281, 5859, 186, -6944],115,4]
%
% END OF LATTICE L116.4
%
\hbox{}\par\smallskip%
%
% LATTICE L116.5
%
\leavevmode%
\llap{\spacer}%
${L_{116.5}}=\pdualwitharg{3}({L_{116.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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{$\left.\llap{\phantom{%
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3247188, 1395744, -2604; 1395744, 599934, -1119; -2604, -1119, 2],[2604, 3, 1],[32, 4352, 329, 1817, 257613, 4167, 114877, 6948, 14234, 431, 10001, 134, 35, 741, 0, -53, -3, 119; -75, -10199, -771, -4258, -603694, -9765, -269204, -16282, -33356, -1010, -23436, -314, -82, -1736, 0, 124, 7, -279; -290, -39060, -2946, -16242, -2301936, -37232, -1026348, -62069, -127131, -3847, -89187, -1191, -306, -6510, -1, 372, 14, -1116],115,5]
%
% END OF LATTICE L116.5
%
\hbox{}\par\smallskip%
%
% LATTICE L116.6
%
\leavevmode%
\llap{\spacer}%
${L_{116.6}}=\pdualwitharg{2}({L_{116.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[502941768, -762972, -125747160; -762972, 1144, 190764; -125747160, 190764, 31439719],[2604, 4, 1],[30515, 247327, 3893, 972, 33361, 1295, 163, -2587, -2577, 655, 74763, 6497, 19808, 2865449, 279235, 644017, 470137, 324667; 29097, 235879, 3715, 929, 31899, 1239, 155, -2478, -2480, 618, 71176, 6190, 18881, 2731596, 266196, 613955, 448203, 309535; 121872, 987784, 15548, 3882, 133238, 5172, 651, -10332, -10292, 2616, 298592, 25948, 79110, 11444146, 1115220, 2572101, 1877652, 1296668],115,6]
%
% END OF LATTICE L116.6
%
\hbox{}\par\smallskip%
%
% LATTICE L116.7
%
\leavevmode%
\llap{\spacer}%
${L_{116.7}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{116.1}}))$%
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{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[540841035, 19768266, 183143226; 19768266, 721686, 6694065; 183143226, 6694065, 62017190],[651, 21, 1],[230, 150034, 98492, 314133, 6546587, 1491060, 20666499, 360052, 5245012, 166510, 588386, 68248, 19347, 99193, 9600, 9261, -2752, -76072; -2, -992, -650, -2071, -43152, -9828, -136214, -2373, -34565, -1097, -3875, -449, -127, -651, -63, -62, 18, 496; -679, -442959, -290787, -927444, -19328097, -4402195, -61015626, -1063015, -15485337, -491603, -1737147, -201495, -57120, -292857, -28343, -27342, 8125, 224595],115,7]
%
% END OF LATTICE L116.7
%
\hbox{}\par\smallskip%
%
% LATTICE L116.8
%
\leavevmode%
\llap{\spacer}%
${L_{116.8}}=\pdualwitharg{7}({L_{116.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3221148, -1083264, 2604; -1083264, -364294, 875; 2604, 875, -2],[2604, 7, 1],[106, 686, 363, 2005, 40611, 13795, 126769, 3286, 15708, 1427, 1577, 148, 39, 123, 2, -41, -1, 133; -321, -2077, -1099, -6070, -122946, -41763, -383780, -9948, -47554, -4320, -4774, -448, -118, -372, -6, 124, 3, -403; -2499, -15965, -8421, -46396, -939238, -319011, -2931236, -75969, -363041, -32949, -36363, -3395, -868, -2666, -21, 868, 9, -3255],115,8]
%
% END OF LATTICE L116.8
%
\hbox{}\par\smallskip%
%
% LATTICE L116.9
%
\leavevmode%
\llap{\spacer}%
${L_{116.9}}=\pdualwitharg{31}(\pfillwitharg{2}({L_{116.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6372142287, 1414623, -615807591; 1414623, 310, -136710; -615807591, -136710, 59512009],[651, 31, 1],[-5779, -8109, -18781, -51286, -233967, -703555, -104226, -1171627, -25771, -72233, -17895, -7223, -722, -69, 1447, 210, 1447, -233; -48, -70, -164, -453, -2072, -6234, -924, -10395, -229, -645, -161, -67, -9, -7, -3, 0, 0, -2; -59799, -83909, -194339, -530689, -2421006, -7280133, -1078493, -12123573, -266669, -747441, -185171, -74741, -7471, -714, 14973, 2173, 14973, -2411],115,9]
%
% END OF LATTICE L116.9
%
\hbox{}\par\smallskip%
%
% LATTICE L116.10
%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1473306744, -1403556, -368692548; -1403556, 1320, 351240; -368692548, 351240, 92264693],[2604, 12, 1],[-3895, -428367, -30503, -80147, -11238931, -362765, -2494733, -601576, -1224558, -36340, -818918, -9732, -323, 13366, 650, 163, -1949, -22727; -2244, -246946, -17588, -46221, -6481790, -209218, -1438803, -346955, -706273, -20961, -472409, -5617, -189, 7595, 373, 93, -1120, -13082; -15556, -1710828, -121824, -320094, -44886450, -1448824, -9963555, -2402596, -4890684, -145136, -3270624, -38868, -1290, 53382, 2596, 651, -7784, -90768],115,10]
%
% END OF LATTICE L116.10
%
\hbox{}\par\smallskip%
%
% LATTICE L116.11
%
\leavevmode%
\llap{\spacer}%
${L_{116.11}}=\pdualwitharg{3.7}({L_{116.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-643188, 5208, -2604; 5208, -42, 21; -2604, 21, -10],[2604, 21, 1],[-13, -43, -4, -1, -3, 0, 1, 0, -4, -1, -19, -10, -55, -1113, -126, -3473, -30, -430; -1655, -5487, -515, -136, -434, -3, 124, 1, -465, -123, -2387, -1263, -6976, -141298, -15999, -441068, -3811, -54653; -301, -1023, -105, -42, -186, -7, 0, 2, 0, -14, -372, -210, -1218, -24924, -2828, -78120, -677, -9765],115,11]
%
% END OF LATTICE L116.11
%
\hbox{}\par\smallskip%
%
% LATTICE L116.12
%
\leavevmode%
\llap{\spacer}%
${L_{116.12}}=\pdualwitharg{3.31}(\pfillwitharg{2}({L_{116.1}}))$%
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{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[91039845603, -6921432, 30149670558; -6921432, -186, -2292171; 30149670558, -2292171, 9984668018],[651, 93, 1],[3470, 394046, 1143560, 3643777, 17144061, 17291892, 7731047, 29226402, 1961810, 1930172, 1539596, 790082, 223507, 258717, 110886, 3521, -221772, -28168; 1, -7, -23, -78, -371, -375, -168, -637, -43, -43, -35, -19, -6, -7, -3, 0, 7, 1; -10478, -1189860, -3453090, -11002737, -51768150, -52214540, -23344644, -88251947, -5923875, -5828341, -4648959, -2385729, -674901, -781221, -334831, -10632, 669662, 85056],115,12]
%
% END OF LATTICE L116.12
%
\hbox{}\par\smallskip%
%
% LATTICE L116.13
%
\leavevmode%
\llap{\spacer}%
${L_{116.13}}=\pdualwitharg{2.7}({L_{116.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{s}{2}{248}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{*}{4}{28}\above{1pt}{1pt}{b}{2}{124}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{l}{2}{217}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{1736}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2113088712, -2762844, -528953124; -2762844, 3640, 691600; -528953124, 691600, 132408737],[2604, 28, 1],[-55273, -334881, -174267, -475005, -9594049, -6514159, -14957105, -1549534, -7395232, -668450, -733472, -66974, -7477, -21128, 652, 163, -1301, -83239; -10605, -64263, -33443, -91160, -1841245, -1250169, -2870507, -297381, -1419273, -128289, -140771, -12855, -1436, -4061, 123, 31, -249, -15965; -220752, -1337464, -695996, -1897098, -38317178, -26016564, -59736411, -6188604, -29535436, -2669688, -2929376, -267484, -29862, -84382, 2604, 651, -5196, -332444],115,13]
%
% END OF LATTICE L116.13
%
\hbox{}\par\smallskip%
%
% LATTICE L116.14
%
\leavevmode%
\llap{\spacer}%
${L_{116.14}}=\pdualwitharg{31}({L_{116.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${186}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{62}\above{1pt}{1pt}{*}{4}{124}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{186}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{1302}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
466116\\-23436\\-10416
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\kern3pt%
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-10416\\527\\202
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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49631\\935676\\122388
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
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\end{tabular}\endgroup%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-465\\-217\\-279\\-806\\-3038\\-4371\\-1240\\-6510\\-124
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-124\\-806\\-3808\\-5766\\-1720\\-9765\\-219
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[466116, -23436, -10416; -23436, 1178, 527; -10416, 527, 202],[2604, 31, 1],[2, -16, -49, -325, -1541, -2335, -697, -3961, -89, -266, -72, -39, -27, -43, -41, -7, 1, 1; 39, -301, -923, -6126, -29050, -44019, -13140, -74676, -1678, -5016, -1358, -736, -510, -812, -774, -132, 21, 19; 0, -42, -124, -806, -3808, -5766, -1720, -9765, -219, -651, -175, -93, -62, -98, -93, -16, 0, 2],115,14]
%
% END OF LATTICE L116.14
%
\hbox{}\par\smallskip%
%
% LATTICE L116.15
%
\leavevmode%
\llap{\spacer}%
${L_{116.15}}=\pdualwitharg{7.31}(\pfillwitharg{2}({L_{116.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1302}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{434}\above{1pt}{1pt}{}{4}{217}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{1302}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{186}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8117979765\\3152142\\3513581757
\end{tabular}\endgroup%
\kern3pt%
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3152142\\-434\\1366449
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3513581757\\1366449\\1520727370
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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43119722225\\-129360737772\\-99510213621
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38039694\\-114120469\\-87786699
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18635168536\\-55906184592\\-43005601757
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6206\\-18615\\-14322
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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33558\\-100675\\-77444
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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681720\\-2045183\\-1573250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2172195\\-6516663\\-5012917
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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1169510\\-3508573\\-2698955
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8117979765, 3152142, 3513581757; 3152142, -434, 1366449; 3513581757, 1366449, 1520727370],[651, 217, 1],[6206, 33558, 681720, 2172195, 1460033, 30925076, 4608773, 7466990, 1169510, 3451948, 131116, 470998, 133241, 22033, 198310, 2099, -56660, -16792; -18615, -100675, -2045183, -6516663, -4380152, -92776353, -13826487, -22401243, -3508573, -10355973, -393353, -1413013, -399729, -66100, -594939, -6297, 169983, 50377; -14322, -77444, -1573250, -5012917, -3369414, -71367828, -10635968, -17232063, -2698955, -7966287, -302585, -1086953, -307489, -50847, -457653, -4844, 130758, 38752],115,15]
%
% END OF LATTICE L116.15
%
\hbox{}\par\smallskip%
%
% LATTICE L116.16
%
\leavevmode%
\llap{\spacer}%
${L_{116.16}}=\pdualwitharg{2.3.7}({L_{116.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}31\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{s}{2}{744}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{*}{4}{84}\above{1pt}{1pt}{b}{2}{372}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{651}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{5208}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3400824\\-44268\\-851508
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44268\\11256\\11088
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3400824, -44268, -851508; -44268, 11256, 11088; -851508, 11088, 213203],[2604, 84, 1],[133811, 443815, 41727, 5626, 19513, 659, 163, 1, 38467, 9941, 192801, 102025, 281878, 5710031, 1293109, 8912461, 308039, 4417845; -215, -713, -67, -9, -31, -1, 0, 0, -62, -16, -310, -164, -453, -9176, -2078, -14322, -495, -7099; 534436, 1772580, 166656, 22470, 77934, 2632, 651, 4, 153636, 39704, 770040, 407484, 1125810, 22805646, 5164628, 35596029, 1230296, 17644704],115,16]
%
% END OF LATTICE L116.16
%
\hbox{}\par\smallskip%
%
% LATTICE L116.17
%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3152316468, -393878436, 908796; -393878436, -49214670, 113553; 908796, 113553, -262],[2604, 93, 1],[0, -20, -59, -383, -1809, -913, -817, -1546, -104, -103, -83, -44, -29, -45, -14, -7, 1, 1; -1, 175, 519, 3380, 15974, 8063, 7216, 13657, 919, 911, 735, 391, 260, 406, 127, 64, -7, -9; -434, 6468, 20274, 136338, 648060, 327484, 293400, 556171, 37539, 37541, 30639, 16833, 12090, 19866, 6479, 3456, 434, -432],115,17]
%
% END OF LATTICE L116.17
%
\hbox{}\par\smallskip%
%
% LATTICE L116.18
%
\leavevmode%
\llap{\spacer}%
${L_{116.18}}=\pdualwitharg{2.31}({L_{116.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3192504, 1486884, -796824; 1486884, 682744, -371132; -796824, -371132, 198881],[2604, 124, 1],[4285, 8021, 19779, 56651, 261063, 786655, 58384, 1316494, 29124, 83090, 21104, 9346, 1599, 1062, -280, -134, -5869, 1; -30, -56, -138, -395, -1820, -5484, -407, -9177, -203, -579, -147, -65, -11, -7, 3, 1, 42, 0; 17112, 32032, 78988, 226238, 1042566, 3141540, 233159, 5257476, 116308, 331824, 84280, 37324, 6386, 4242, -1116, -535, -23436, 4],115,18]
%
% END OF LATTICE L116.18
%
\hbox{}\par\smallskip%
%
% LATTICE L116.19
%
\leavevmode%
\llap{\spacer}%
${L_{116.19}}=\pdualwitharg{3.7.31}(\pfillwitharg{2}({L_{116.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${434}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{1302}\above{1pt}{1pt}{}{4}{651}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{434}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{62}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22745381538999, -40241699464050, -13387023162009; -40241699464050, 71196623937579, 23684657119431; -13387023162009, 23684657119431, 7879067178197],[651, 651, 1],[-16, -10, -164, -453, -296, -2078, -924, -495, -229, -215, -23, -67, -9, -1, -1, 0, 0, -2; 3215479, 1933757, 31351403, 85613602, 55795839, 391491479, 173989022, 93135713, 43020773, 40194237, 4267631, 12058313, 1205830, 16653, -804035, -350158, -114893, 388943; -9665831, -5812935, -94243317, -257357226, -167724078, -1176835919, -523016571, -279968936, -129321822, -120825166, -12828636, -36247680, -3624768, -50061, 2416946, 1052583, 345371, -1169175],115,19]
%
% END OF LATTICE L116.19
%
\hbox{}\par\smallskip%
%
% LATTICE L116.20
%
\leavevmode%
\llap{\spacer}%
${L_{116.20}}=\pdualwitharg{7.31}({L_{116.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1302}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{434}\above{1pt}{1pt}{*}{4}{868}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{1302}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{186}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-302064\\75299\\-818
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-111542340, 27805512, -302064; 27805512, -6931414, 75299; -302064, 75299, -818],[2604, 217, 1],[4, 4, 79, 499, 335, 3547, 1057, 856, 134, 395, 15, 54, 33, 7, 44, 7, -1, -1; 9, 17, 343, 2196, 1478, 15657, 4668, 3783, 593, 1755, 67, 245, 156, 34, 219, 36, -3, -5; -651, 87, 2387, 17794, 12290, 130851, 39200, 31992, 5082, 15624, 626, 2604, 2170, 544, 3906, 728, 93, -91],115,20]
%
% END OF LATTICE L116.20
%
\hbox{}\par\smallskip%
%
% LATTICE L116.21
%
\leavevmode%
\llap{\spacer}%
${L_{116.21}}=\pdualwitharg{2.3.31}({L_{116.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${248}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{*}{2}{744}\above{1pt}{1pt}{*}{4}{372}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{248}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{1736}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[54430729752, -156680076, -13628398560; -156680076, 465000, 39229632; -13628398560, 39229632, 3412286555],[2604, 372, 1],[-134809, -1076517, -2827729, -8562964, -39929129, -40201677, -8971786, -67666201, -4520979, -4388775, -3447067, -1695855, -484204, -783127, -510579, -75382, -187559, -2605; -724, -5782, -15188, -45993, -214466, -215930, -48189, -363447, -24283, -23573, -18515, -9109, -2601, -4207, -2743, -405, -1008, -14; -538408, -4299456, -11293548, -34199262, -159471270, -160559788, -35832039, -270249196, -18056148, -17528144, -13767096, -6773004, -1933842, -3127698, -2039180, -301065, -749084, -10404],115,21]
%
% END OF LATTICE L116.21
%
\hbox{}\par\smallskip%
%
% LATTICE L116.22
%
\leavevmode%
\llap{\spacer}%
${L_{116.22}}=\pdualwitharg{3.7.31}({L_{116.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${434}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{1302}\above{1pt}{1pt}{*}{4}{2604}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{434}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{62}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2604\\1372308\\-2604
\end{tabular}\endgroup%
\kern3pt%
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1372308\\-716017974\\1358637
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2604\\1358637\\-2578
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3025\\784\\416640
\end{tabular}\endgroup%
\kern3pt%
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1437534\\-372695\\-198060240
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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190\\-42\\-22351
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\HardButStrongLineBreak\kern3pt%
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22\\-4\\-2133
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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83\\-6\\-3255
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57\\10\\5208
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\HardButStrongLineBreak\kern3pt%
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13\\4\\2094
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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29\\12\\6293
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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15\\8\\4200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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0\\1\\527
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2\\-1\\-525
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-126\\-2604\\-16926\\-11424\\-40362\\-36120\\-9765\\-4599
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2604, 1372308, -2604; 1372308, -716017974, 1358637; -2604, 1358637, -2578],[2604, 651, 1],[190, 22, 83, 57, 13, 29, 15, 0, -2, -1, 5, 106, 699, 473, 1672, 1497, 405, 191; -42, -4, -6, 10, 4, 12, 8, 1, -1, -7, -3, -43, -206, -130, -453, -400, -106, -48; -22351, -2133, -3255, 5208, 2094, 6293, 4200, 527, -525, -3689, -1587, -22785, -109368, -69054, -240653, -212520, -56327, -25515],115,22]
%
% END OF LATTICE L116.22
%
\hbox{}\par\smallskip%
%
% LATTICE L116.23
%
\leavevmode%
\llap{\spacer}%
${L_{116.23}}=\pdualwitharg{2.7.31}({L_{116.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}31\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1105403208, 17709804, 263488344; 17709804, 341992, 4177684; 263488344, 4177684, 62838911],[2604, 868, 1],[-183209, -61949, -1118379, -3334382, -2214025, -46781537, -3477888, -11231261, -1747759, -5062045, -187825, -630039, -162454, -33941, -425749, -18948, -8569, -659; 548796, 185566, 3350064, 9988021, 6632032, 140132406, 10417888, 33642837, 5235349, 15163173, 562623, 1887259, 486625, 101669, 1275315, 56758, 25668, 1974; 731724, 247420, 4466728, 13317290, 8842662, 186842208, 13890443, 44856876, 6980428, 20217456, 750160, 2516332, 648830, 135558, 1700412, 75677, 34224, 2632],115,23]
%
% END OF LATTICE L116.23
%
\hbox{}\par\smallskip%
%
% LATTICE L116.24
%
\leavevmode%
\llap{\spacer}%
${L_{116.24}}=\pdualwitharg{2.3.7.31}({L_{116.1}})$%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\kern3pt%
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0\\-504\\-10416\\-33852\\-22848\\-161448\\-36120\\-39060\\-18396
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[67704, -35276388, 8832768; -35276388, 18381974520, -4602616872; 8832768, -4602616872, 1152437789],[2604, 2604, 1],[-51309, -5699, -18977, -4744, -957, -3865, -493, -25, -17, -1933, -1907, -34265, -101740, -67497, -475313, -105991, -114065, -53225; 80968, 8994, 29956, 7489, 1508, 6078, 772, 31, 21, 3039, 3007, 54051, 160525, 106501, 749985, 167242, 179984, 83986; 323764, 35964, 119784, 29946, 6030, 24304, 3087, 124, 84, 12152, 12024, 216132, 641886, 425862, 2998940, 668745, 719696, 335832],115,24]
%
% END OF LATTICE L116.24
%

\medskip%
%------------------------------WEYL-GROUP-W117--------------------------------
%
\leavevmode\llap{}%
$W_{117}$%
\qquad\llap{24} lattices, $\chi=104$%
\hfill%
$62222222226222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L117.1
%
\leavevmode%
\llap{\spacer}%
${L_{117.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}53\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{117}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{b}{2}{1060}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{1590}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{636}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{106}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4060860\\810900\\-3180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
810900\\161926\\-635
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3180\\-635\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23937769\\119905080\\8000880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4777812\\23932219\\1596920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16605\\83175\\5549
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
221\\-1107\\-73
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1217\\-6096\\-405
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
74701\\-374180\\-24910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2103\\-10534\\-702
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
525817\\-2633835\\-175695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16605\\-83175\\-5549
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1408231\\-7053876\\-470640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48709\\-243985\\-16280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
117014\\-586127\\-39114
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39301\\-196860\\-13140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[4060860, 810900, -3180; 810900, 161926, -635; -3180, -635, 2],[3180, 1, 1],[221, 1217, 74701, 2103, 525817, 16605, 1408231, 48709, 117014, 39301, 3100, 2104, 42641, 111, 2222, 0, -127, -1, 328, 551; -1107, -6096, -374180, -10534, -2633835, -83175, -7053876, -243985, -586127, -196860, -15528, -10539, -213590, -556, -11130, 0, 636, 5, -1643, -2760; -73, -405, -24910, -702, -175695, -5549, -470640, -16280, -39114, -13140, -1037, -705, -14310, -38, -795, -1, 0, 0, -106, -180],116,1]
%
% END OF LATTICE L117.1
%
\hbox{}\par\smallskip%
%
% LATTICE L117.2
%
\leavevmode%
\llap{\spacer}%
${L_{117.2}}=\pfillwitharg{2}({L_{117.1}})$%
{${}={}$}Nikulin $117$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}53\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{l}{2}{265}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{1590}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{159}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{106}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-141510\\-46905\\18285
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46905\\-15547\\6095
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18285\\6095\\4834
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
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-45516586\\137067979\\-652125
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18635888\\-56119840\\266999
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75449\\227206\\-1081
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54019\\162672\\-774
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-573316\\1726475\\-8215
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2372\\7143\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-166319\\500850\\-2385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2021\\6086\\-29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33211\\100011\\-477
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
352\\-1060\\5
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-141510, -46905, 18285; -46905, -15547, 6095; 18285, 6095, 4834],[795, 1, 1],[-75449, -54019, -573316, -2372, -166319, -2021, -33211, 352, 3696, -1051, -3073, -24503, -813906, -23802, -12318679, -390589, -16613453, -1152008, -2778140, -470081; 227206, 162672, 1726475, 7143, 500850, 6086, 100011, -1060, -11130, 3165, 9254, 73788, 2450985, 71677, 37096290, 1176214, 50029509, 3469140, 8366050, 1415595; -1081, -774, -8215, -34, -2385, -29, -477, 5, 53, -15, -44, -351, -11660, -341, -176490, -5596, -238023, -16505, -39803, -6735],116,2]
%
% END OF LATTICE L117.2
%
\hbox{}\par\smallskip%
%
% LATTICE L117.3
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24828840570, 95290290, -8276691735; 95290290, -365709, 31765008; -8276691735, 31765008, -2759034433],[795, 3, 1],[-35474, -70507, -6606218, -187607, -31508804, -2987770, -42259765, -8774909, -21098027, -1183001, -562027, -128660, -3949633, -11560, -158749, 265, 36361, 11561, -13675, -12001; 35637, 70834, 6636925, 188480, 31655575, 3001689, 42456657, 8815795, 21196343, 1188515, 564648, 129261, 3968110, 11615, 159530, -264, -36517, -11615, 13727, 12055; 106827, 212326, 19894080, 564963, 94886165, 8997423, 127261798, 26424915, 63534969, 3562510, 1692498, 387449, 11893995, 34812, 478060, -798, -109498, -34815, 41181, 36140],116,3]
%
% END OF LATTICE L117.3
%
\hbox{}\par\smallskip%
%
% LATTICE L117.4
%
\leavevmode%
\llap{\spacer}%
${L_{117.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{117.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}53\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-41818706070, 159653490, 8363327655; 159653490, -609515, -31929120; 8363327655, -31929120, -1672582823],[795, 5, 1],[21434, 127801, 798290, 113351, 11422420, 1805186, 76598983, 1060345, 12747243, 428855, 339571, 233204, 477261, 6984, 57539, -161, -65923, -1397, 8267, 4351; 35637, 212502, 1327385, 188480, 18993345, 3001689, 127369971, 1763159, 21196343, 713109, 564648, 387783, 793622, 11615, 95718, -264, -109551, -2323, 13727, 7233; 106495, 634980, 3966308, 563185, 56752347, 8969075, 380582400, 5268329, 63334735, 2130768, 1687160, 1158675, 2371273, 34700, 285882, -800, -327540, -6941, 41075, 21618],116,4]
%
% END OF LATTICE L117.4
%
\hbox{}\par\smallskip%
%
% LATTICE L117.5
%
\leavevmode%
\llap{\spacer}%
${L_{117.5}}=\pdualwitharg{3}({L_{117.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}53\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10406395423380, 24489405780, 107862420; 24489405780, 57631002, 253833; 107862420, 253833, 1118],[3180, 3, 1],[6719, 1523, 92761, 247, 1729, 7, 1, -1, 680, 387, 472, 874, 161321, 4547, 379394, 35948, 1016327, 105469, 253402, 28377; -2854622, -647059, -39410270, -104940, -734580, -2974, -424, 425, -288903, -164420, -200533, -371326, -68538540, -1931830, -161188635, -15272801, -431794816, -44809365, -107659907, -12056200; -116583, -26422, -1609080, -4278, -29945, -123, -212, -15, -11925, -6740, -8202, -15173, -2799990, -78912, -6583660, -623802, -17636068, -1830165, -4397145, -492400],116,5]
%
% END OF LATTICE L117.5
%
\hbox{}\par\smallskip%
%
% LATTICE L117.6
%
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${L_{117.6}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{117.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[900496351335, -14350058460, 297306520470; -14350058460, 228678555, -4737793710; 297306520470, -4737793710, 98158273471],[795, 15, 1],[-546565, -1138209, -21642194, -3095506, -104751357, -49707953, -703556187, -29232824, -351730012, -3951067, -9422285, -2184741, -13581076, -227444, -918453, -136297, -909213, -9136, -502688, -36473; -3159, -6576, -125027, -17882, -605101, -287139, -4064093, -168863, -2031755, -22823, -54426, -12619, -78440, -1313, -5300, -786, -5247, -53, -2915, -211; 1655310, 3447145, 65544888, 9374955, 317246764, 150543990, 2130768805, 88533639, 1065238785, 11966081, 28536045, 6616640, 41131233, 688830, 2781599, 412785, 2753615, 27669, 1522425, 110461],116,6]
%
% END OF LATTICE L117.6
%
\hbox{}\par\smallskip%
%
% LATTICE L117.7
%
\leavevmode%
\llap{\spacer}%
${L_{117.7}}=\pdualwitharg{2}({L_{117.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}53\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1024793160, -1605900, -256199880; -1605900, 2504, 401480; -256199880, 401480, 64050367],[3180, 4, 1],[3999, 27154, 873076, 25147, 12834659, 406291, 8630003, 1195703, 2879069, 242843, 77418, 54263, 565291, 1992, 110644, 794, 199, -1587, -2335, 1408; 3173, 21579, 694035, 19993, 10205415, 323065, 6862281, 950790, 2289388, 193110, 61567, 43161, 449705, 1587, 88245, 635, 159, -1270, -1908, 1110; 15976, 108480, 3487930, 100462, 51274320, 1623128, 34476765, 4776820, 11501848, 970155, 309284, 216780, 2258330, 7958, 442020, 3172, 795, -6340, -9328, 5625],116,7]
%
% END OF LATTICE L117.7
%
\hbox{}\par\smallskip%
%
% LATTICE L117.8
%
\leavevmode%
\llap{\spacer}%
${L_{117.8}}=\pdualwitharg{5}({L_{117.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}53\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-571328340, -12720, 340260; -12720, 10, 5; 340260, 5, -202],[3180, 5, 1],[-1, 1, 83, 17, 1171, 191, 16601, 117, 1448, 103, 46, 44, 235, 13, 260, 34, 2479, 15, 142, 5; -425, 426, 35192, 7206, 496239, 80939, 7034796, 49579, 613581, 43644, 19490, 18639, 99534, 5504, 110028, 14386, 1048764, 6345, 60049, 2112; -1695, 1695, 140662, 28810, 1984479, 323685, 28133460, 198278, 2453900, 174552, 77955, 74565, 398242, 22030, 440589, 57615, 4200780, 25418, 240620, 8472],116,8]
%
% END OF LATTICE L117.8
%
\hbox{}\par\smallskip%
%
% LATTICE L117.9
%
\leavevmode%
\llap{\spacer}%
${L_{117.9}}=\pdualwitharg{2.3}({L_{117.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-40712668680, 2745996780, 10188815400; 2745996780, -182773896, -687219312; 10188815400, -687219312, -2549868691],[3180, 12, 1],[-145522700, -31454427, -916686687, -1052752, 12262114, 2026700, 1285394, -3980961, -42434917, -3893197, -14272069, -21810496, -1906354232, -52212171, -8481384871, -801000545, -5647367232, -2339675651, -5603646921, -155901572; -456021, -98568, -2872600, -3299, 38425, 6351, 4028, -12475, -132977, -12200, -44724, -68347, -5973895, -163616, -26577910, -2510076, -17697018, -7331785, -17560013, -488545; -581359560, -125659652, -3662140470, -4205718, 48986840, 8096616, 5135117, -15903840, -169526436, -15553225, -57016560, -87132388, -7615837650, -208586322, -33882921200, -3199977216, -22561091357, -9346945920, -22386429804, -622822895],116,9]
%
% END OF LATTICE L117.9
%
\hbox{}\par\smallskip%
%
% LATTICE L117.10
%
\leavevmode%
\llap{\spacer}%
${L_{117.10}}=\pdualwitharg{53}(\pfillwitharg{2}({L_{117.1}}))$%
\nopagebreak\par%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-502896302970, 23333250, -66454276575; 23333250, -1007, 3083328; -66454276575, 3083328, -8781474131],[795, 53, 1],[-263769, -1572844, -926859, -1395044, -13262281, -22217149, -17787455, -65250541, -2960109, -26390557, -4179270, -2870195, -554154, -85969, -66836, 1954, 15299, 85969, -1917, -267677; -166, -999, -590, -889, -8460, -14174, -11349, -41635, -1889, -16845, -2669, -1836, -355, -56, -45, -1, 9, 55, -1, -165; 1996086, 11902581, 7014055, 10557070, 100363020, 168129462, 134607516, 493786955, 22400783, 199711950, 31626849, 21720354, 4193590, 650575, 505785, -14787, -115776, -650575, 14507, 2025660],116,10]
%
% END OF LATTICE L117.10
%
\hbox{}\par\smallskip%
%
% LATTICE L117.11
%
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${L_{117.11}}=\pdualwitharg{3.5}({L_{117.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-233707740, -3498000, 219420; -3498000, -52230, 3285; 219420, 3285, -206],[3180, 15, 1],[29, 64, 2457, 353, 5993, 2845, 80559, 1674, 20149, 453, 541, 126, 1571, 27, 49, 5, 1, -2, -9, 3; -191, -420, -16112, -2314, -39273, -18643, -527880, -10969, -132023, -2968, -3544, -825, -10282, -176, -318, -32, 0, 13, 53, -20; 27840, 61465, 2359878, 339060, 5756542, 2732760, 77381060, 1607967, 19354275, 435136, 519675, 121040, 1509228, 25950, 47117, 4815, 1060, -1923, -8745, 2876],116,11]
%
% END OF LATTICE L117.11
%
\hbox{}\par\smallskip%
%
% LATTICE L117.12
%
\leavevmode%
\llap{\spacer}%
${L_{117.12}}=\pdualwitharg{2.5}({L_{117.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}53\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[929734202280, -1467992940, -232726611420; -1467992940, 2317880, 367460960; -232726611420, 367460960, 58255010443],[3180, 20, 1],[-1587, -1592, -779366, -168735, -24675599, -4044839, -88223143, -2494077, -31005623, -556025, -1010808, -1010803, -2797763, -168730, -7518562, -1017136, -19090727, -475615, -4765667, -51412; -1, 3, -53, -13, -2067, -341, -7473, -212, -2650, -48, -89, -93, -265, -17, -795, -109, -2067, -52, -530, -6; -6340, -6360, -3113538, -674090, -98578092, -16158980, -352448145, -9963744, -123866300, -2221299, -4038140, -4038120, -11176958, -674070, -30036372, -4063420, -76266735, -1900064, -19038660, -205389],116,12]
%
% END OF LATTICE L117.12
%
\hbox{}\par\smallskip%
%
% LATTICE L117.13
%
\leavevmode%
\llap{\spacer}%
${L_{117.13}}=\pdualwitharg{3.53}(\pfillwitharg{2}({L_{117.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}53\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${318}\above{1pt}{1pt}{}{6}{106}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{159}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{318}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{1590}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{265}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[370736608462815, 4399122960, 123244921071840; 4399122960, 52311, 1462411722; 123244921071840, 1462411722, 40970625029407],[795, 159, 1],[21684123, 45139089, 80963384, 122745806, 391843327, 1970981823, 526354321, 5795553526, 263139394, 783308989, 373591281, 86619379, 50795084, 9012662, 3432077, 5395197, 679551, 1819066, 377542, 7241819; 40, 79, 140, 211, 670, 3368, 899, 9895, 449, 1335, 635, 146, 85, 14, 5, 7, 1, 5, 1, 15; -65228637, -135784198, -243548295, -369235206, -1178715235, -5928967377, -1583341643, -17433772035, -791557215, -2356294405, -1123810728, -260562257, -152798160, -27111249, -10324130, -16229448, -2044177, -5471985, -1135695, -21784325],116,13]
%
% END OF LATTICE L117.13
%
\hbox{}\par\smallskip%
%
% LATTICE L117.14
%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3180, 0, -3180; 0, -106, -583; -3180, -583, -26],[3180, 53, 1],[-46, -430, -553, -867, -4276, -7194, -11557, -21251, -968, -17399, -1403, -1019, -413, -99, -71, -51, -35, -1, 2, 11; 253, 2364, 3040, 4766, 23505, 39545, 63528, 116815, 5321, 95640, 7712, 5601, 2270, 544, 390, 280, 192, 5, -11, -60; -53, -477, -610, -954, -4695, -7897, -12684, -23320, -1062, -19080, -1537, -1113, -450, -106, -75, -53, -36, 0, 2, 0],116,14]
%
% END OF LATTICE L117.14
%
\hbox{}\par\smallskip%
%
% LATTICE L117.15
%
\leavevmode%
\llap{\spacer}%
${L_{117.15}}=\pdualwitharg{2.3.5}({L_{117.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}53\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{}{6}{40}\above{1pt}{1pt}{*}{2}{636}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{424}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{l}{2}{265}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{6360}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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197939100\\25320\\-49547280
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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506708315613\\2080833669\\2024277429671
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-187947\\-778\\-750840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-370342\\-1525\\-1479500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-6946486\\-28567\\-27750906
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1547825892360, 197939100, -387445243860; 197939100, 25320, -49547280; -387445243860, -49547280, 96983658001],[3180, 60, 1],[-187947, -370342, -6946486, -986917, -33188793, -15737535, -111310048, -9245851, -111167497, -623514, -2964318, -680413, -4191517, -63838, -225216, -23790, -69982, -793, -212533, -6939; -778, -1525, -28567, -4056, -136316, -64634, -457125, -37969, -456489, -2560, -12167, -2790, -17172, -259, -901, -91, -265, -3, -901, -29; -750840, -1479500, -27750906, -3942690, -132587768, -62870760, -444678745, -36936768, -444109260, -2490911, -11842320, -2718220, -16744926, -255030, -899728, -95040, -279575, -3168, -849060, -27721],116,15]
%
% END OF LATTICE L117.15
%
\hbox{}\par\smallskip%
%
% LATTICE L117.16
%
\leavevmode%
\llap{\spacer}%
${L_{117.16}}=\pdualwitharg{5.53}(\pfillwitharg{2}({L_{117.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}53\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${530}\above{1pt}{1pt}{}{6}{1590}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{265}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{530}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{106}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{159}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L117.16
%
\hbox{}\par\smallskip%
%
% LATTICE L117.17
%
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${L_{117.17}}=\pdualwitharg{3.53}({L_{117.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[393261060, 157473600, 368880; 157473600, 63057174, 147711; 368880, 147711, 346],[3180, 159, 1],[-1966, -477, -581, -141, -34, -74, -17, -2, 3, 9, -62, -199, -771, -1211, -1994, -10066, -5391, -29742, -1355, -8121; 4638, 1125, 1370, 332, 80, 174, 40, 5, -7, -20, 147, 470, 1820, 2858, 4705, 23751, 12720, 70175, 3197, 19160; 115911, 28249, 34530, 8586, 2095, 4611, 1048, 0, -210, -1060, 3339, 11501, 44970, 70914, 117155, 591639, 316952, 1749000, 79710, 478060],116,17]
%
% END OF LATTICE L117.17
%
\hbox{}\par\smallskip%
%
% LATTICE L117.18
%
\leavevmode%
\llap{\spacer}%
${L_{117.18}}=\pdualwitharg{3.5.53}(\pfillwitharg{2}({L_{117.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}53\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21198097595234910, 56395015288759485, 18754065600479775; 56395015288759485, -150032225067883590, -49892959097572185; 18754065600479775, -49892959097572185, -16591817967011413],[795, 795, 1],[-166, -333, -118, -889, -564, -14174, -3783, -8327, -1889, -1123, -2669, -612, -71, -56, -3, -1, 3, 11, -1, -11; -1404168075, -2790998556, -986822069, -7426481044, -4706760757, -118272420087, -31563685991, -69471914299, -15758063831, -9365963185, -22248235410, -5093135939, -590004830, -457653451, -23719940, 10402662, 27148011, 91530743, -10205159, -94998209; 4222448520, 8392761470, 2967454863, 22332037335, 14153615464, 355654863360, 94914591415, 208907741712, 47385789810, 28164219133, 66902267820, 15315477310, 1774192893, 1376201445, 71327804, -31281660, -81636295, -275240448, 30687750, 285667403],116,18]
%
% END OF LATTICE L117.18
%
\hbox{}\par\smallskip%
%
% LATTICE L117.19
%
\leavevmode%
\llap{\spacer}%
${L_{117.19}}=\pdualwitharg{2.53}({L_{117.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}53\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[45518520, -5561820, -11374860; -5561820, 679672, 1389872; -11374860, 1389872, 2842523],[3180, 212, 1],[5731, 39002, 23726, 36269, 349859, 587095, 235328, 1728279, 78533, 351214, 112070, 78799, 15541, 2998, 3544, 1910, 274, -531, 1, 2189; 77, 519, 315, 481, 4635, 7777, 3117, 22890, 1040, 4650, 1483, 1041, 205, 39, 45, 23, 3, -10, 0, 30; 22896, 155820, 94790, 144902, 1397760, 2345568, 940185, 6904840, 313756, 1403175, 447744, 314820, 62090, 11978, 14160, 7632, 1095, -2120, 4, 8745],116,19]
%
% END OF LATTICE L117.19
%
\hbox{}\par\smallskip%
%
% LATTICE L117.20
%
\leavevmode%
\llap{\spacer}%
${L_{117.20}}=\pdualwitharg{5.53}({L_{117.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}53\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-445158660, 129426000, -143100; 129426000, -37629470, 41605; -143100, 41605, -46],[3180, 265, 1],[-5, -2, 1, 13, 17, 147, 241, 90, 21, 79, 35, 32, 3, 7, 1, 3, -1, -2, -1, -7; -21, -6, 4, 50, 63, 543, 888, 331, 77, 288, 126, 111, 10, 20, 0, -18, -48, -23, -7, -36; -3445, 795, 506, 4770, 4077, 33655, 53160, 19292, 4290, 14628, 5035, 795, -294, -3710, -3123, -25705, -40440, -14628, -3230, -10812],116,20]
%
% END OF LATTICE L117.20
%
\hbox{}\par\smallskip%
%
% LATTICE L117.21
%
\leavevmode%
\llap{\spacer}%
${L_{117.21}}=\pdualwitharg{2.3.53}({L_{117.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}53\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20209542360, -129937980, -5058391020; -129937980, 825528, 32523132; -5058391020, 32523132, 1266100897],[3180, 636, 1],[-7898512, -1817631, -1057421, -172174, -46338, -1592, 5793, 212531, 799, -71308, -471389, -972336, -1732546, -2617793, -8323563, -41847913, -5585727, -122971079, -5580733, -8298593; -79213, -18229, -10605, -1727, -465, -17, 58, 2130, 8, -715, -4727, -9751, -17375, -26253, -83475, -419683, -56018, -1233250, -55968, -83225; -31554504, -7261424, -4224390, -687834, -185120, -6360, 23143, 849060, 3192, -284875, -1883196, -3884476, -6921510, -10458066, -33252580, -167182140, -22314943, -491268660, -22294992, -33152825],116,21]
%
% END OF LATTICE L117.21
%
\hbox{}\par\smallskip%
%
% LATTICE L117.22
%
\leavevmode%
\llap{\spacer}%
${L_{117.22}}=\pdualwitharg{3.5.53}({L_{117.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}53\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-32719020, 36471420, -76320; 36471420, -40651530, 85065; -76320, 85065, -178],[3180, 795, 1],[554, 135, 33, 41, 2, 22, 5, 0, -1, -1, 16, 55, 43, 339, 112, 2828, 1515, 1672, 381, 457; 1564, 383, 94, 120, 6, 68, 16, 1, -3, -4, 41, 152, 120, 950, 315, 7957, 4264, 4707, 1073, 1288; 509595, 125080, 30756, 39750, 2009, 23055, 5500, 477, -1005, -1484, 12720, 49025, 38886, 308460, 102454, 2588520, 1387340, 1531647, 349215, 419336],116,22]
%
% END OF LATTICE L117.22
%
\hbox{}\par\smallskip%
%
% LATTICE L117.23
%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15107931960, 34544340, 3755824860; 34544340, 103880, 8569040; 3755824860, 8569040, 933710311],[3180, 1060, 1],[-207225, 796, -17136, -206827, -682249, -6006341, -2497262, -3769903, -895053, -870146, -1657800, -1865821, -107199, -414848, -436766, -3318784, -1232788, -1700837, -348297, -248869; 620768, -2385, 51333, 619576, 2043762, 17992744, 7480860, 11293215, 2681243, 2606631, 4966147, 5589300, 321128, 1242729, 1308387, 9941831, 3692970, 5095069, 1043367, 745518; 827860, -3180, 68458, 826270, 2725572, 23995220, 9976515, 15060692, 3575720, 3476217, 6622880, 7453920, 428258, 1657310, 1744872, 13258480, 4924965, 6794812, 1391440, 994227],116,23]
%
% END OF LATTICE L117.23
%
\hbox{}\par\smallskip%
%
% LATTICE L117.24
%
\leavevmode%
\llap{\spacer}%
${L_{117.24}}=\pdualwitharg{2.3.5.53}({L_{117.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}53\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${6360}\above{1pt}{1pt}{}{6}{2120}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{3180}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{6360}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{1272}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{53}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\kern3pt%
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-254345940\\85587607560\\-21423920760
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63666780\\-21423920760\\5362743437
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\kern3pt%
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\kern3pt%
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95590373\\-17324681179\\-69212463409
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-31607\\5727950\\22883280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7502\\1359287\\5430380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-898\\162681\\649914
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-957\\173128\\691650
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-89\\16068\\64192
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\HardButStrongLineBreak\kern3pt%
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-955\\171934\\686880
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[756840, -254345940, 63666780; -254345940, 85587607560, -21423920760; 63666780, -21423920760, 5362743437],[3180, 3180, 1],[-31607, -7502, -898, -957, -89, -955, -66, -95, -9, -32, -1438, -3513, -1305, -10058, -6520, -164270, -21964, -96837, -22021, -6577; 5727950, 1359287, 162681, 173128, 16068, 171934, 11851, 16875, 1607, 5824, 261085, 637058, 236588, 1823217, 1181739, 29773241, 3980839, 17550961, 3991083, 1191983; 22883280, 5430380, 649914, 691650, 64192, 686880, 47345, 67416, 6420, 23267, 1043040, 2545060, 945174, 7283790, 4721072, 118944720, 15903535, 70116456, 15944460, 4761997],116,24]
%
% END OF LATTICE L117.24
%

\medskip%
%------------------------------WEYL-GROUP-W118--------------------------------
%
\leavevmode\llap{}%
$W_{118}$%
\qquad\llap{48} lattices, $\chi=36$%
\hfill%
$2222222222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L118.1
%
\leavevmode%
\llap{\spacer}%
${L_{118.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N_{118}}%
\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{1540}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{924}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8560860\\-1714020\\-4620
\end{tabular}\endgroup%
\kern3pt%
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-1714020\\343174\\925
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\925\\2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-829\\-4140\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-663\\-3311\\-187
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-497\\-2482\\-138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22357\\-111650\\-6160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-331\\-1653\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-826\\-4125\\-220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-164\\-819\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
185\\924\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-4620\\-120\\-660\\-420\\-240\\-4620\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\0\\-770\\-33\\-275\\-231\\-165\\-3696\\-11
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-770\\0\\1540\\0\\-3080\\-4620\\-4620\\-124740\\-770
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-33\\-6\\0\\6\\0\\-42\\-60\\-1848\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-275\\-110\\-3080\\0\\110\\0\\-110\\-4620\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-231\\-126\\-4620\\-42\\0\\42\\0\\-924\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-165\\-110\\-4620\\-60\\-110\\0\\10\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-3696\\-2772\\-124740\\-1848\\-4620\\-924\\0\\924\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-14\\-770\\-15\\-55\\-21\\-5\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[8560860, -1714020, -4620; -1714020, 343174, 925; -4620, 925, 2],[4620, 1, 1],[-829, -663, -497, -22357, -331, -826, -164, 1, 185, 0; -4140, -3311, -2482, -111650, -1653, -4125, -819, 5, 924, 0; -240, -187, -138, -6160, -90, -220, -42, 0, 0, -1],117,1]
%
% END OF LATTICE L118.1
%
\hbox{}\par\smallskip%
%
% LATTICE L118.2
%
\leavevmode%
\llap{\spacer}%
${L_{118.2}}=\pfillwitharg{2}({L_{118.1}})$%
{${}={}$}Nikulin $118$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{385}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{231}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54285\\-25410\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25410\\11894\\539
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\539\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\15\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-385\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\216\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11521\\24640\\-770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
728\\1557\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5066\\10835\\-330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3898\\8337\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2644\\5655\\-170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28837\\61677\\-1848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\308\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-15\\-1155\\-60\\-330\\-210\\-120\\-1155\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\0\\-385\\-33\\-275\\-231\\-165\\-1848\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\1\\0\\-3\\-55\\-63\\-55\\-693\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-385\\0\\385\\0\\-1540\\-2310\\-2310\\-31185\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-33\\-3\\0\\6\\0\\-42\\-60\\-924\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-275\\-55\\-1540\\0\\110\\0\\-110\\-2310\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-231\\-63\\-2310\\-42\\0\\42\\0\\-462\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-165\\-55\\-2310\\-60\\-110\\0\\10\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-1848\\-693\\-31185\\-924\\-2310\\-462\\0\\231\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-7\\-385\\-15\\-55\\-21\\-5\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[54285, -25410, -1155; -25410, 11894, 539; -1155, 539, -38],[1155, 1, 1],[7, -180, 101, 11521, 728, 5066, 3898, 2644, 28837, 144; 15, -385, 216, 24640, 1557, 10835, 8337, 5655, 61677, 308; 0, 11, -7, -770, -48, -330, -252, -170, -1848, -9],117,2]
%
% END OF LATTICE L118.2
%
\hbox{}\par\smallskip%
%
% LATTICE L118.3
%
\leavevmode%
\llap{\spacer}%
${L_{118.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{1155}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{330}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{77}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-118750170\\21357105\\-39548355
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21357105\\-3840738\\7112733
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39548355\\7112733\\-13171117
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
237419\\-2235\\-714095
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1147375\\-10802\\-3451008
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432816\\-4075\\-1301799
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19545671\\-184030\\-58788345
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
194246\\-1829\\-584242
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1477666\\-13915\\-4444440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
103296\\-973\\-310688
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10044\\-95\\-30210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24551\\231\\73843
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1151\\11\\3462
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-15\\-1155\\-20\\-330\\-70\\-120\\-385\\0
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\kern3pt%
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0\\66\\0\\-1155\\-33\\-825\\-231\\-495\\-1848\\-33
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\kern3pt%
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-15\\0\\3\\0\\-3\\-165\\-63\\-165\\-693\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-1155\\0\\1155\\0\\-4620\\-2310\\-6930\\-31185\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-33\\-3\\0\\2\\0\\-14\\-60\\-308\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-825\\-165\\-4620\\0\\330\\0\\-330\\-2310\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-231\\-63\\-2310\\-14\\0\\14\\0\\-154\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-495\\-165\\-6930\\-60\\-330\\0\\30\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-1848\\-693\\-31185\\-308\\-2310\\-154\\0\\77\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-33\\-21\\-1155\\-15\\-165\\-21\\-15\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-118750170, 21357105, -39548355; 21357105, -3840738, 7112733; -39548355, 7112733, -13171117],[1155, 3, 1],[237419, 1147375, 432816, 19545671, 194246, 1477666, 103296, 10044, -24551, -1151; -2235, -10802, -4075, -184030, -1829, -13915, -973, -95, 231, 11; -714095, -3451008, -1301799, -58788345, -584242, -4444440, -310688, -30210, 73843, 3462],117,3]
%
% END OF LATTICE L118.3
%
\hbox{}\par\smallskip%
%
% LATTICE L118.4
%
\leavevmode%
\llap{\spacer}%
${L_{118.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{77}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{1155}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-147639030\\30742635\\-29497545
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30742635\\-6401230\\6142225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29497545\\6142225\\-5893463
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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85841\\-1341\\-431043
\end{tabular}\endgroup%
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691381\\-10802\\-3471710
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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260798\\-4075\\-1309575
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2355459\\-36806\\-11827739
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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351122\\-5487\\-1763130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
178062\\-2783\\-894124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
186688\\-2919\\-937440
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1208\\-19\\-6066
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44393\\693\\222915
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-689\\11\\3460
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-15\\-231\\-60\\-66\\-210\\-24\\-1155\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\110\\0\\-385\\-165\\-275\\-1155\\-165\\-9240\\-55
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\5\\0\\-15\\-55\\-315\\-55\\-3465\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-385\\0\\77\\0\\-308\\-2310\\-462\\-31185\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-165\\-15\\0\\30\\0\\-210\\-60\\-4620\\-75
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\kern3pt%
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-55\\-35\\-385\\-75\\-55\\-105\\-5\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-147639030, 30742635, -29497545; 30742635, -6401230, 6142225; -29497545, 6142225, -5893463],[1155, 5, 1],[85841, 691381, 260798, 2355459, 351122, 178062, 186688, 1208, -44393, -689; -1341, -10802, -4075, -36806, -5487, -2783, -2919, -19, 693, 11; -431043, -3471710, -1309575, -11827739, -1763130, -894124, -937440, -6066, 222915, 3460],117,4]
%
% END OF LATTICE L118.4
%
\hbox{}\par\smallskip%
%
% LATTICE L118.5
%
\leavevmode%
\llap{\spacer}%
${L_{118.5}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{55}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{770}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-179678730\\40128165\\-51308565
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40128165\\-8961722\\11458888
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51308565\\11458888\\-14651533
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
617632\\-6705\\-2168145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
994871\\-10802\\-3492412
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
375269\\-4075\\-1317351
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2420918\\-26290\\-8498435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
505220\\-5487\\-1773534
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1280986\\-13915\\-4496800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38368\\-417\\-134688
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8674\\-95\\-30450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9128\\99\\32043
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-985\\11\\3458
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\0\\-105\\-1155\\-420\\-2310\\-210\\-840\\-1155\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\154\\0\\-385\\-231\\-1925\\-231\\-1155\\-1848\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\0\\7\\0\\-21\\-385\\-63\\-385\\-693\\-49
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-385\\0\\55\\0\\-1540\\-330\\-2310\\-4455\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-231\\-21\\0\\42\\0\\-42\\-420\\-924\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-1925\\-385\\-1540\\0\\770\\0\\-770\\-2310\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-231\\-63\\-330\\-42\\0\\6\\0\\-66\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-1155\\-385\\-2310\\-420\\-770\\0\\70\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-1848\\-693\\-4455\\-924\\-2310\\-66\\0\\33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-77\\-49\\-385\\-105\\-385\\-21\\-35\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-179678730, 40128165, -51308565; 40128165, -8961722, 11458888; -51308565, 11458888, -14651533],[1155, 7, 1],[617632, 994871, 375269, 2420918, 505220, 1280986, 38368, 8674, -9128, -985; -6705, -10802, -4075, -26290, -5487, -13915, -417, -95, 99, 11; -2168145, -3492412, -1317351, -8498435, -1773534, -4496800, -134688, -30450, 32043, 3458],117,5]
%
% END OF LATTICE L118.5
%
\hbox{}\par\smallskip%
%
% LATTICE L118.6
%
\leavevmode%
\llap{\spacer}%
${L_{118.6}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${165}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{462}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-246336090\\58899225\\-44762025
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58899225\\-14082706\\10702648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44762025\\10702648\\-8133761
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
397072\\-6705\\-2194005
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58141\\-982\\-321256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
241229\\-4075\\-1332903
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
990286\\-16730\\-5471795
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
324740\\-5487\\-1794342
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
74846\\-1265\\-413560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
172576\\-2919\\-953568
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5554\\-95\\-30690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3736\\63\\20643
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-625\\11\\3454
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
165\\0\\-165\\-1155\\-660\\-330\\-2310\\-1320\\-1155\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-35\\-33\\-25\\-231\\-165\\-168\\-11
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\0\\11\\0\\-33\\-55\\-693\\-605\\-693\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-35\\0\\35\\0\\-140\\-2310\\-2310\\-2835\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-33\\-33\\0\\66\\0\\-462\\-660\\-924\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-25\\-55\\-140\\0\\10\\0\\-110\\-210\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-231\\-693\\-2310\\-462\\0\\462\\0\\-462\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\-165\\-605\\-2310\\-660\\-110\\0\\110\\0\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-168\\-693\\-2835\\-924\\-210\\-462\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-77\\-385\\-165\\-55\\-231\\-55\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-246336090, 58899225, -44762025; 58899225, -14082706, 10702648; -44762025, 10702648, -8133761],[1155, 11, 1],[397072, 58141, 241229, 990286, 324740, 74846, 172576, 5554, -3736, -625; -6705, -982, -4075, -16730, -5487, -1265, -2919, -95, 63, 11; -2194005, -321256, -1332903, -5471795, -1794342, -413560, -953568, -30690, 20643, 3454],117,6]
%
% END OF LATTICE L118.6
%
\hbox{}\par\smallskip%
%
% LATTICE L118.7
%
\leavevmode%
\llap{\spacer}%
${L_{118.7}}=\pdualwitharg{3}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{4620}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{330}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{308}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216627180\\-86661960\\69300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-86661960\\-34669218\\27723
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69300\\27723\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-399\\1000\\3280
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-553\\1386\\4587
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-233\\584\\1962
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6451\\16170\\55440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
417\\-1045\\-3300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\-203\\-658
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1229\\3080\\9856
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81\\203\\657
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-60\\-4620\\-40\\-660\\-140\\-240\\-1540\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\66\\0\\-2310\\-33\\-825\\-231\\-495\\-3696\\-33
\end{tabular}\endgroup%
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-60\\0\\12\\0\\-6\\-330\\-126\\-330\\-2772\\-42
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-2310\\0\\4620\\0\\-9240\\-4620\\-13860\\-124740\\-2310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-33\\-6\\0\\2\\0\\-14\\-60\\-616\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-825\\-330\\-9240\\0\\330\\0\\-330\\-4620\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-231\\-126\\-4620\\-14\\0\\14\\0\\-308\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-495\\-330\\-13860\\-60\\-330\\0\\30\\0\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1540\\-3696\\-2772\\-124740\\-616\\-4620\\-308\\0\\308\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-33\\-42\\-2310\\-15\\-165\\-21\\-15\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-216627180, -86661960, 69300; -86661960, -34669218, 27723; 69300, 27723, -22],[4620, 3, 1],[-399, -553, -233, -6451, 2, 417, 81, -2, -1229, -81; 1000, 1386, 584, 16170, -5, -1045, -203, 5, 3080, 203; 3280, 4587, 1962, 55440, -2, -3300, -658, 0, 9856, 657],117,7]
%
% END OF LATTICE L118.7
%
\hbox{}\par\smallskip%
%
% LATTICE L118.8
%
\leavevmode%
\llap{\spacer}%
${L_{118.8}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{231}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{385}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7509698480130\\-188961465\\2499162654450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-188961465\\4755\\-62884740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2499162654450\\-62884740\\831699700051
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-433205\\-34014\\1301729
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-10543235\\-827871\\31681155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3997208\\-313879\\12011130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36200492\\-2842686\\108778131
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1807207\\-141918\\5430440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2774439\\-217888\\8336856
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-993667\\-78050\\2985850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26435\\-2080\\79434
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
202053\\15862\\-607145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23976\\1891\\-72045
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-15\\-231\\-20\\-66\\-70\\-24\\-385\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\330\\0\\-1155\\-165\\-825\\-1155\\-495\\-9240\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\15\\0\\-15\\-165\\-315\\-165\\-3465\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-1155\\0\\231\\0\\-924\\-2310\\-1386\\-31185\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-165\\-15\\0\\10\\0\\-70\\-60\\-1540\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-825\\-165\\-924\\0\\66\\0\\-66\\-2310\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-1155\\-315\\-2310\\-70\\0\\70\\0\\-770\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-495\\-165\\-1386\\-60\\-66\\0\\6\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-9240\\-3465\\-31185\\-1540\\-2310\\-770\\0\\385\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-105\\-1155\\-75\\-165\\-105\\-15\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7509698480130, -188961465, 2499162654450; -188961465, 4755, -62884740; 2499162654450, -62884740, 831699700051],[1155, 15, 1],[-433205, -10543235, -3997208, -36200492, -1807207, -2774439, -993667, -26435, 202053, 23976; -34014, -827871, -313879, -2842686, -141918, -217888, -78050, -2080, 15862, 1891; 1301729, 31681155, 12011130, 108778131, 5430440, 8336856, 2985850, 79434, -607145, -72045],117,8]
%
% END OF LATTICE L118.8
%
\hbox{}\par\smallskip%
%
% LATTICE L118.9
%
\leavevmode%
\llap{\spacer}%
${L_{118.9}}=\pdualwitharg{2}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{1540}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}{440}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{231}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3074877960\\3358740\\-768712560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3358740\\3656\\-839680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-768712560\\-839680\\192176407
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12427\\-9945\\49665
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33535\\-26829\\134024
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10989\\-8789\\43918
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
456427\\-364980\\1824130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11575\\-9252\\46260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19707\\-15730\\78760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1135\\924\\-4536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2307\\1850\\-9220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
289\\-231\\1155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1154\\-925\\4612
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-30\\-2310\\-120\\-660\\-420\\-240\\-1155\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\88\\0\\-1540\\-132\\-1100\\-924\\-660\\-3696\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\4\\0\\-12\\-220\\-252\\-220\\-1386\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-1540\\0\\1540\\0\\-6160\\-9240\\-9240\\-62370\\-1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-132\\-12\\0\\24\\0\\-168\\-240\\-1848\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-1100\\-220\\-6160\\0\\440\\0\\-440\\-4620\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-924\\-252\\-9240\\-168\\0\\168\\0\\-924\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-660\\-220\\-9240\\-240\\-440\\0\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-3696\\-1386\\-62370\\-1848\\-4620\\-924\\0\\231\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-44\\-28\\-1540\\-60\\-220\\-84\\-20\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3074877960, 3358740, -768712560; 3358740, 3656, -839680; -768712560, -839680, 192176407],[4620, 4, 1],[12427, 33535, 10989, 456427, 11575, 19707, -1135, -2307, 289, 1154; -9945, -26829, -8789, -364980, -9252, -15730, 924, 1850, -231, -925; 49665, 134024, 43918, 1824130, 46260, 78760, -4536, -9220, 1155, 4612],117,9]
%
% END OF LATTICE L118.9
%
\hbox{}\par\smallskip%
%
% LATTICE L118.10
%
\leavevmode%
\llap{\spacer}%
${L_{118.10}}=\pdualwitharg{5}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{308}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{4620}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25728780\\8482320\\23100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8482320\\2796470\\7615
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23100\\7615\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103\\312\\156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\209\\110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\106\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-661\\2002\\1078
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\315\\165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-207\\627\\319
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1033\\3129\\1575
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-172\\521\\261
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21353\\64680\\32340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103\\312\\155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-60\\-924\\-120\\-132\\-420\\-48\\-4620\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\110\\0\\-770\\-165\\-275\\-1155\\-165\\-18480\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\20\\0\\-30\\-110\\-630\\-110\\-13860\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\-770\\0\\308\\0\\-616\\-4620\\-924\\-124740\\-770
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-165\\-30\\0\\30\\0\\-210\\-60\\-9240\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-275\\-110\\-616\\0\\22\\0\\-22\\-4620\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-1155\\-630\\-4620\\-210\\0\\210\\0\\-4620\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-165\\-110\\-924\\-60\\-22\\0\\2\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-18480\\-13860\\-124740\\-9240\\-4620\\-4620\\0\\4620\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-55\\-70\\-770\\-75\\-55\\-105\\-5\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25728780, 8482320, 23100; 8482320, 2796470, 7615; 23100, 7615, 22],[4620, 5, 1],[-103, -69, -35, -661, -104, -207, -1033, -172, -21353, -103; 312, 209, 106, 2002, 315, 627, 3129, 521, 64680, 312; 156, 110, 60, 1078, 165, 319, 1575, 261, 32340, 155],117,10]
%
% END OF LATTICE L118.10
%
\hbox{}\par\smallskip%
%
% LATTICE L118.11
%
\leavevmode%
\llap{\spacer}%
${L_{118.11}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{165}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{2310}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7339842079185\\-221034660\\2443187522160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-221034660\\6657\\-73575033
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2443187522160\\-73575033\\813255272258
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3285479\\-170070\\9870245
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15992441\\-827871\\48044535
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6063194\\-313879\\18215064
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39222313\\-2030490\\117831780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2741310\\-141918\\8235451
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21042726\\-1089440\\63216615
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-215336\\-11150\\646913
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200584\\-10400\\602595
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43781\\2266\\-131527
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36405\\1891\\-109368
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-105\\-1155\\-140\\-2310\\-70\\-840\\-385\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\462\\0\\-1155\\-231\\-5775\\-231\\-3465\\-1848\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\0\\21\\0\\-21\\-1155\\-63\\-1155\\-693\\-147
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-1155\\0\\165\\0\\-4620\\-330\\-6930\\-4455\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-231\\-21\\0\\14\\0\\-14\\-420\\-308\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-5775\\-1155\\-4620\\0\\2310\\0\\-2310\\-2310\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-231\\-63\\-330\\-14\\0\\2\\0\\-22\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-3465\\-1155\\-6930\\-420\\-2310\\0\\210\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-1848\\-693\\-4455\\-308\\-2310\\-22\\0\\11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-231\\-147\\-1155\\-105\\-1155\\-21\\-105\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7339842079185, -221034660, 2443187522160; -221034660, 6657, -73575033; 2443187522160, -73575033, 813255272258],[1155, 21, 1],[-3285479, -15992441, -6063194, -39222313, -2741310, -21042726, -215336, -200584, 43781, 36405; -170070, -827871, -313879, -2030490, -141918, -1089440, -11150, -10400, 2266, 1891; 9870245, 48044535, 18215064, 117831780, 8235451, 63216615, 646913, 602595, -131527, -109368],117,11]
%
% END OF LATTICE L118.11
%
\hbox{}\par\smallskip%
%
% LATTICE L118.12
%
\leavevmode%
\llap{\spacer}%
${L_{118.12}}=\pdualwitharg{7}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{220}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{b}{2}{770}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{132}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132783420\\53102280\\87780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53102280\\-21236474\\-35105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
87780\\-35105\\-58
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
587\\1440\\16800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
305\\748\\8855
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
155\\380\\4578
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
763\\1870\\22880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\93\\1218
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67\\-165\\-1540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-27\\-306
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
215\\528\\5808
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\81\\917
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-420\\-4620\\-840\\-4620\\-420\\-1680\\-4620\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\154\\0\\-770\\-231\\-1925\\-231\\-1155\\-3696\\-77
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\28\\0\\-42\\-770\\-126\\-770\\-2772\\-98
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-770\\0\\220\\0\\-3080\\-660\\-4620\\-17820\\-770
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-231\\-42\\0\\42\\0\\-42\\-420\\-1848\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-1925\\-770\\-3080\\0\\770\\0\\-770\\-4620\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-231\\-126\\-660\\-42\\0\\6\\0\\-132\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-1155\\-770\\-4620\\-420\\-770\\0\\70\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-3696\\-2772\\-17820\\-1848\\-4620\\-132\\0\\132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-77\\-98\\-770\\-105\\-385\\-21\\-35\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-132783420, 53102280, 87780; 53102280, -21236474, -35105; 87780, -35105, -58],[4620, 7, 1],[587, 305, 155, 763, 38, -67, -11, 2, 215, 33; 1440, 748, 380, 1870, 93, -165, -27, 5, 528, 81; 16800, 8855, 4578, 22880, 1218, -1540, -306, 0, 5808, 917],117,12]
%
% END OF LATTICE L118.12
%
\hbox{}\par\smallskip%
%
% LATTICE L118.13
%
\leavevmode%
\llap{\spacer}%
${L_{118.13}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${55}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{s}{2}{154}\above{1pt}{1pt}{s}{2}{330}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3330773660445\\-186643380\\1108388893920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-186643380\\10461\\-62109729
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1108388893920\\-62109729\\368841015754
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4015407\\-170070\\12066505
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1776883\\-75261\\5339625
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7410410\\-313879\\22268664
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30505733\\-1292130\\91671300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3350462\\-141918\\10068311
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2338098\\-99040\\7026105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1842408\\-78050\\5536531
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-245272\\-10400\\737055
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34049\\1442\\-102319
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44541\\1891\\-133848
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\0\\-165\\-1155\\-220\\-330\\-770\\-1320\\-385\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-105\\-33\\-75\\-231\\-495\\-168\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\0\\33\\0\\-33\\-165\\-693\\-1815\\-693\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-105\\0\\105\\0\\-420\\-2310\\-6930\\-2835\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-33\\-33\\0\\22\\0\\-154\\-660\\-308\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-75\\-165\\-420\\0\\30\\0\\-330\\-210\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-770\\-231\\-693\\-2310\\-154\\0\\154\\0\\-154\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\-495\\-1815\\-6930\\-660\\-330\\0\\330\\0\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-168\\-693\\-2835\\-308\\-210\\-154\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-33\\-231\\-1155\\-165\\-165\\-231\\-165\\0\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3330773660445, -186643380, 1108388893920; -186643380, 10461, -62109729; 1108388893920, -62109729, 368841015754],[1155, 33, 1],[-4015407, -1776883, -7410410, -30505733, -3350462, -2338098, -1842408, -245272, 34049, 44541; -170070, -75261, -313879, -1292130, -141918, -99040, -78050, -10400, 1442, 1891; 12066505, 5339625, 22268664, 91671300, 10068311, 7026105, 5536531, 737055, -102319, -133848],117,13]
%
% END OF LATTICE L118.13
%
\hbox{}\par\smallskip%
%
% LATTICE L118.14
%
\leavevmode%
\llap{\spacer}%
${L_{118.14}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{770}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{s}{2}{154}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7525301124495\\-288937110\\1502372743410
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288937110\\11095\\-57684235
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1502372743410\\-57684235\\299938012154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1546001\\-102042\\7743813
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-3300592\\-217888\\16532439
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-506638\\-33450\\2537715
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-31462\\-2080\\157591
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\HardButStrongLineBreak\kern3pt%
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103007\\6798\\-515955
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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28551\\1891\\-143010
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1155\\-9240\\-3465\\-4455\\-4620\\-2310\\-330\\0\\165\\0
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0\\-385\\-245\\-385\\-525\\-385\\-105\\-35\\0\\70
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7525301124495, -288937110, 1502372743410; -288937110, 11095, -57684235; 1502372743410, -57684235, 299938012154],[1155, 35, 1],[-1546001, -12542225, -4755118, -6152095, -6449696, -3300592, -506638, -31462, 103007, 28551; -102042, -827871, -313879, -406098, -425754, -217888, -33450, -2080, 6798, 1891; 7743813, 62823145, 23818060, 30815422, 32306085, 16532439, 2537715, 157591, -515955, -143010],117,14]
%
% END OF LATTICE L118.14
%
\hbox{}\par\smallskip%
%
% LATTICE L118.15
%
\leavevmode%
\llap{\spacer}%
${L_{118.15}}=\pdualwitharg{11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${660}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{462}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1864224807060\\21996009420\\-24652320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21996009420\\259531162\\-290873
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24652320\\-290873\\326
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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569\\-48180\\39600
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\HardButStrongLineBreak\kern3pt%
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40\\-3387\\2779
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\HardButStrongLineBreak\kern3pt%
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323\\-27350\\22462
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1307\\-110670\\91000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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209\\-17697\\14586
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\-3895\\3240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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94\\-7959\\6930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\85\\220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\-84\\672
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-254\\231
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
660\\0\\-660\\-4620\\-1320\\-660\\-4620\\-2640\\-4620\\0
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-1320\\-33\\-66\\0\\66\\0\\-462\\-660\\-1848\\-165
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-660\\-25\\-110\\-280\\0\\10\\0\\-110\\-420\\-55
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\kern3pt%
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-4620\\-231\\-1386\\-4620\\-462\\0\\462\\0\\-924\\-231
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\kern3pt%
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-2640\\-165\\-1210\\-4620\\-660\\-110\\0\\110\\0\\-55
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-4620\\-336\\-2772\\-11340\\-1848\\-420\\-924\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-154\\-770\\-165\\-55\\-231\\-55\\0\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1864224807060, 21996009420, -24652320; 21996009420, 259531162, -290873; -24652320, -290873, 326],[4620, 11, 1],[569, 40, 323, 1307, 209, 46, 94, -1, 1, 3; -48180, -3387, -27350, -110670, -17697, -3895, -7959, 85, -84, -254; 39600, 2779, 22462, 91000, 14586, 3240, 6930, 220, 672, 231],117,15]
%
% END OF LATTICE L118.15
%
\hbox{}\par\smallskip%
%
% LATTICE L118.16
%
\leavevmode%
\llap{\spacer}%
${L_{118.16}}=\pdualwitharg{2.3}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{264}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{4620}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{1320}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{77}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
329055591480\\-177079980\\-82311569340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-177079980\\95304\\44295648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82311569340\\44295648\\20589817109
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-25686\\-13710\\-102655
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-60606\\-32351\\-242214
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2494073\\-1331330\\-9967650
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-20779\\-11092\\-83044
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-118903\\-63470\\-475200
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\HardButStrongLineBreak\kern3pt%
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-5773\\-3080\\-23072
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-13857\\-7390\\-55380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63214\\-33726\\-252637
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\HardButStrongLineBreak\kern3pt%
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-11545\\-6161\\-46140
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-30\\-2310\\-40\\-660\\-140\\-240\\-385\\0
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0\\264\\0\\-4620\\-132\\-3300\\-924\\-1980\\-3696\\-132
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\kern3pt%
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-30\\0\\12\\0\\-12\\-660\\-252\\-660\\-1386\\-84
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\kern3pt%
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-2310\\-4620\\0\\4620\\0\\-18480\\-9240\\-27720\\-62370\\-4620
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\kern3pt%
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-40\\-132\\-12\\0\\8\\0\\-56\\-240\\-616\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-3300\\-660\\-18480\\0\\1320\\0\\-1320\\-4620\\-660
\end{tabular}\endgroup%
\kern3pt%
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-140\\-924\\-252\\-9240\\-56\\0\\56\\0\\-308\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1980\\-660\\-27720\\-240\\-1320\\0\\120\\0\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-3696\\-1386\\-62370\\-616\\-4620\\-308\\0\\77\\0
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\kern3pt%
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0\\-132\\-84\\-4620\\-60\\-660\\-84\\-60\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[329055591480, -177079980, -82311569340; -177079980, 95304, 44295648; -82311569340, 44295648, 20589817109],[4620, 12, 1],[-25686, -191632, -60606, -2494073, -20779, -118903, -5773, -13857, -63214, -11545; -13710, -102289, -32351, -1331330, -11092, -63470, -3080, -7390, -33726, -6161; -102655, -765864, -242214, -9967650, -83044, -475200, -23072, -55380, -252637, -46140],117,16]
%
% END OF LATTICE L118.16
%
\hbox{}\par\smallskip%
%
% LATTICE L118.17
%
\leavevmode%
\llap{\spacer}%
${L_{118.17}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{2310}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3046121159235\\-230429430\\607839759450
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\kern3pt%
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\kern3pt%
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607839759450\\-45981155\\121291686658
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1784497\\-102042\\8942769
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-346360\\-19808\\1735737
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4093946\\-234150\\20516265
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36334\\-2080\\182083
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75659\\4326\\-379155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32991\\1891\\-165330
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\0\\-165\\-231\\-660\\-66\\-2310\\-264\\-1155\\0
\end{tabular}\endgroup%
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0\\10\\0\\-35\\-165\\-25\\-1155\\-165\\-840\\-55
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-66\\-25\\-55\\-28\\0\\2\\0\\-22\\-210\\-55
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-1155\\-3465\\-2310\\-2310\\0\\2310\\0\\-2310\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-165\\-605\\-462\\-660\\-22\\0\\22\\0\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-840\\-3465\\-2835\\-4620\\-210\\-2310\\0\\105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-55\\-385\\-385\\-825\\-55\\-1155\\-55\\0\\110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3046121159235, -230429430, 607839759450; -230429430, 17435, -45981155; 607839759450, -45981155, 121291686658],[1155, 55, 1],[-1784497, -1316115, -5488798, -4519043, -7444936, -346360, -4093946, -36334, 75659, 32991; -102042, -75261, -313879, -258426, -425754, -19808, -234150, -2080, 4326, 1891; 8942769, 6595535, 27506380, 22646582, 37309305, 1735737, 20516265, 182083, -379155, -165330],117,17]
%
% END OF LATTICE L118.17
%
\hbox{}\par\smallskip%
%
% LATTICE L118.18
%
\leavevmode%
\llap{\spacer}%
${L_{118.18}}=\pdualwitharg{3.5}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{924}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{1540}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2499420\\-1057980\\-18480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1057980\\-447630\\-8115
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18480\\-8115\\286
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
389\\-908\\-628
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
410\\-957\\-660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-371\\866\\600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2573\\6006\\4158
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-7\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1164\\-2717\\-1881
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2708\\-6321\\-4375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1547\\-3611\\-2499
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67691\\-158004\\-109340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1161\\-2710\\-1875
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-60\\-924\\-40\\-132\\-140\\-48\\-1540\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\330\\0\\-2310\\-165\\-825\\-1155\\-495\\-18480\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\60\\0\\-30\\-330\\-630\\-330\\-13860\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\-2310\\0\\924\\0\\-1848\\-4620\\-2772\\-124740\\-2310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-165\\-30\\0\\10\\0\\-70\\-60\\-3080\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-825\\-330\\-1848\\0\\66\\0\\-66\\-4620\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-1155\\-630\\-4620\\-70\\0\\70\\0\\-1540\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-495\\-330\\-2772\\-60\\-66\\0\\6\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1540\\-18480\\-13860\\-124740\\-3080\\-4620\\-1540\\0\\1540\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-210\\-2310\\-75\\-165\\-105\\-15\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2499420, -1057980, -18480; -1057980, -447630, -8115; -18480, -8115, 286],[4620, 15, 1],[389, 410, -371, -2573, 3, 1164, 2708, 1547, 67691, 1161; -908, -957, 866, 6006, -7, -2717, -6321, -3611, -158004, -2710; -628, -660, 600, 4158, -5, -1881, -4375, -2499, -109340, -1875],117,18]
%
% END OF LATTICE L118.18
%
\hbox{}\par\smallskip%
%
% LATTICE L118.19
%
\leavevmode%
\llap{\spacer}%
${L_{118.19}}=\pdualwitharg{7.11}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1155}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{77}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{s}{2}{770}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3081252186705\\-274215480\\879180946875
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-274215480\\24409\\-78242549
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
879180946875\\-78242549\\250858771211
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-15187766\\-510210\\53228175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-9342978\\-313879\\32744096
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-5494479\\-184590\\19256360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12672697\\-425754\\44413677
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2947853\\-99040\\10331265
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-995525\\-33450\\3488991
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-309237\\-10400\\1083775
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18398\\618\\-64479
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56157\\1891\\-196812
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1155\\0\\-1155\\-1155\\-4620\\-2310\\-2310\\-9240\\-1155\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-35\\-231\\-175\\-231\\-1155\\-168\\-77
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\kern3pt%
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-1155\\0\\77\\0\\-231\\-385\\-693\\-4235\\-693\\-539
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-35\\0\\5\\0\\-140\\-330\\-2310\\-405\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-231\\-231\\0\\462\\0\\-462\\-4620\\-924\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-175\\-385\\-140\\0\\70\\0\\-770\\-210\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-231\\-693\\-330\\-462\\0\\66\\0\\-66\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9240\\-1155\\-4235\\-2310\\-4620\\-770\\0\\770\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-168\\-693\\-405\\-924\\-210\\-66\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-77\\-539\\-385\\-1155\\-385\\-231\\-385\\0\\154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3081252186705, -274215480, 879180946875; -274215480, 24409, -78242549; 879180946875, -78242549, 250858771211],[1155, 77, 1],[-15187766, -2240278, -9342978, -5494479, -12672697, -2947853, -995525, -309237, 18398, 56157; -510210, -75261, -313879, -184590, -425754, -99040, -33450, -10400, 618, 1891; 53228175, 7851445, 32744096, 19256360, 44413677, 10331265, 3488991, 1083775, -64479, -196812],117,19]
%
% END OF LATTICE L118.19
%
\hbox{}\par\smallskip%
%
% LATTICE L118.20
%
\leavevmode%
\llap{\spacer}%
${L_{118.20}}=\pdualwitharg{2.5}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{440}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{308}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{s}{2}{840}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{1155}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25431168840\\59750460\\6359351460
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59750460\\-140360\\-14941280
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6359351460\\-14941280\\-1590227773
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1448\\381\\5787
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15083\\3960\\60280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
583\\152\\2330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27243\\-7161\\-108878
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-12761\\-3351\\-51000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-18573\\-4873\\-74228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-67363\\-17661\\-269220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8135\\-2131\\-32512
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-196808\\-51513\\-786555
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\1\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\end{tabular}\endgroup%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-18480\\-6930\\-62370\\-9240\\-4620\\-4620\\0\\1155\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-220\\-140\\-1540\\-300\\-220\\-420\\-20\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-25431168840, 59750460, 6359351460; 59750460, -140360, -14941280; 6359351460, -14941280, -1590227773],[4620, 20, 1],[1448, 15083, 583, -27243, -12761, -18573, -67363, -8135, -196808, -10; 381, 3960, 152, -7161, -3351, -4873, -17661, -2131, -51513, 1; 5787, 60280, 2330, -108878, -51000, -74228, -269220, -32512, -786555, -40],117,20]
%
% END OF LATTICE L118.20
%
\hbox{}\par\smallskip%
%
% LATTICE L118.21
%
\leavevmode%
\llap{\spacer}%
${L_{118.21}}=\pdualwitharg{3.7}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{660}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{b}{2}{2310}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78540\\-32340\\-4620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32340\\13314\\1869
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\1869\\-178
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
709\\1740\\-140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1152\\2827\\-231
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\HardButStrongLineBreak\kern3pt%
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617\\1514\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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3183\\7810\\-660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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64\\157\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-67\\-165\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-27\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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233\\572\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\270\\-21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-420\\-4620\\-280\\-4620\\-140\\-1680\\-1540\\0
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\end{tabular}\endgroup%
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-4620\\-5775\\-2310\\-9240\\0\\2310\\0\\-2310\\-4620\\-1155
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-231\\-126\\-660\\-14\\0\\2\\0\\-44\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-231\\-294\\-2310\\-105\\-1155\\-21\\-105\\0\\42
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[78540, -32340, -4620; -32340, 13314, 1869; -4620, 1869, -178],[4620, 21, 1],[709, 1152, 617, 3183, 64, -67, -11, 2, 233, 110; 1740, 2827, 1514, 7810, 157, -165, -27, 5, 572, 270; -140, -231, -126, -660, -14, 0, 2, 0, -44, -21],117,21]
%
% END OF LATTICE L118.21
%
\hbox{}\par\smallskip%
%
% LATTICE L118.22
%
\leavevmode%
\llap{\spacer}%
${L_{118.22}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{2310}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{s}{2}{462}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240795994770870\\-179916589545\\-80134943541345
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\kern3pt%
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-179916589545\\-134425830\\-59874773910
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-215363962\\-447\\647143147
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15914334\\33\\-47820685
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5152087\\11\\-15481410
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-240795994770870, -179916589545, -80134943541345; -179916589545, -134425830, -59874773910; -80134943541345, -59874773910, -26668255767647],[1155, 105, 1],[-215363962, -5203569293, -1962805689, -2532472388, -880833792, -1340013430, -66893344, -9073770, 15914334, 5152087; -447, -10802, -4075, -5258, -1829, -2783, -139, -19, 33, 11; 647143147, 15636108180, 5897998155, 7609778979, 2646801010, 4026581328, 201006560, 27265602, -47820685, -15481410],117,22]
%
% END OF LATTICE L118.22
%
\hbox{}\par\smallskip%
%
% LATTICE L118.23
%
\leavevmode%
\llap{\spacer}%
${L_{118.23}}=\pdualwitharg{2.7}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{r}{2}{616}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{220}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{*}{2}{3080}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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59771647320\\-37482060\\-14958030780
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\616\\0\\-1540\\-924\\-7700\\-924\\-4620\\-3696\\-308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\28\\0\\-84\\-1540\\-252\\-1540\\-1386\\-196
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-1540\\0\\220\\0\\-6160\\-1320\\-9240\\-8910\\-1540
\end{tabular}\endgroup%
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\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-7700\\-1540\\-6160\\0\\3080\\0\\-3080\\-4620\\-1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-924\\-252\\-1320\\-168\\0\\24\\0\\-132\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-4620\\-1540\\-9240\\-1680\\-3080\\0\\280\\0\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-3696\\-1386\\-8910\\-1848\\-4620\\-132\\0\\33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-308\\-196\\-1540\\-420\\-1540\\-84\\-140\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[59771647320, -37482060, -14958030780; -37482060, 23576, 9380000; -14958030780, 9380000, 3743291257],[4620, 28, 1],[-117308, -285918, -88196, -508999, -85313, -141803, -1153, -13837, -37760, -18447; 26850, 65439, 20185, 116490, 19524, 32450, 264, 3170, 8646, 4223; -468825, -1142680, -352478, -2034230, -340956, -566720, -4608, -55300, -150909, -73724],117,23]
%
% END OF LATTICE L118.23
%
\hbox{}\par\smallskip%
%
% LATTICE L118.24
%
\leavevmode%
\llap{\spacer}%
${L_{118.24}}=\pdualwitharg{3.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${220}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{154}\above{1pt}{1pt}{b}{2}{330}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56128380\\11189640\\-50820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11189640\\2230734\\-10131
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50820\\-10131\\46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-20\\-1100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-12\\-435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-136\\-4554
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107\\-630\\-20580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-41\\-1298
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-35\\-1080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-35\\-1078
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\28\\644
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\17\\429
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
220\\0\\-660\\-4620\\-440\\-660\\-1540\\-2640\\-1540\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-210\\-33\\-75\\-231\\-495\\-336\\-33
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\0\\132\\0\\-66\\-330\\-1386\\-3630\\-2772\\-462
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-210\\0\\420\\0\\-840\\-4620\\-13860\\-11340\\-2310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-440\\-33\\-66\\0\\22\\0\\-154\\-660\\-616\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-75\\-330\\-840\\0\\30\\0\\-330\\-420\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1540\\-231\\-1386\\-4620\\-154\\0\\154\\0\\-308\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2640\\-495\\-3630\\-13860\\-660\\-330\\0\\330\\0\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1540\\-336\\-2772\\-11340\\-616\\-420\\-308\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-33\\-462\\-2310\\-165\\-165\\-231\\-165\\0\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[56128380, 11189640, -50820; 11189640, 2230734, -10131; -50820, -10131, 46],[4620, 33, 1],[3, 2, 23, 107, 7, 6, 6, -1, -5, -3; -20, -12, -136, -630, -41, -35, -35, 5, 28, 17; -1100, -435, -4554, -20580, -1298, -1080, -1078, 0, 644, 429],117,24]
%
% END OF LATTICE L118.24
%
\hbox{}\par\smallskip%
%
% LATTICE L118.25
%
\leavevmode%
\llap{\spacer}%
${L_{118.25}}=\pdualwitharg{5.7}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{r}{2}{770}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{b}{2}{154}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{660}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30927790740\\9488934840\\23682120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9488934840\\2911293770\\7265895
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23682120\\7265895\\18134
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
683\\-2184\\-16884
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
485\\-1551\\-11935
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
267\\-854\\-6510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
681\\-2178\\-16676
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
727\\-2325\\-17850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1400\\-4477\\-34496
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
986\\-3153\\-24330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1143\\-3655\\-28224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20227\\-64680\\-499620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
678\\-2168\\-16765
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-420\\-924\\-840\\-924\\-420\\-336\\-4620\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\770\\0\\-770\\-1155\\-1925\\-1155\\-1155\\-18480\\-385
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\kern3pt%
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-420\\0\\140\\0\\-210\\-770\\-630\\-770\\-13860\\-490
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\kern3pt%
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-924\\-770\\0\\44\\0\\-616\\-660\\-924\\-17820\\-770
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-1155\\-210\\0\\210\\0\\-210\\-420\\-9240\\-525
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\-1925\\-770\\-616\\0\\154\\0\\-154\\-4620\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-1155\\-630\\-660\\-210\\0\\30\\0\\-660\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-1155\\-770\\-924\\-420\\-154\\0\\14\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-18480\\-13860\\-17820\\-9240\\-4620\\-660\\0\\660\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-385\\-490\\-770\\-525\\-385\\-105\\-35\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30927790740, 9488934840, 23682120; 9488934840, 2911293770, 7265895; 23682120, 7265895, 18134],[4620, 35, 1],[683, 485, 267, 681, 727, 1400, 986, 1143, 20227, 678; -2184, -1551, -854, -2178, -2325, -4477, -3153, -3655, -64680, -2168; -16884, -11935, -6510, -16676, -17850, -34496, -24330, -28224, -499620, -16765],117,25]
%
% END OF LATTICE L118.25
%
\hbox{}\par\smallskip%
%
% LATTICE L118.26
%
\leavevmode%
\llap{\spacer}%
${L_{118.26}}=\pdualwitharg{3.5.11}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{770}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-203944220100390\\-207561537645\\-67870953324405
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67870953324405\\-69074766750\\-22586893136251
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-96815458\\-253\\290919048
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\HardButStrongLineBreak\kern3pt%
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8054334\\21\\-24202325
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\HardButStrongLineBreak\kern3pt%
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4042631\\11\\-12147630
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-165\\-231\\-220\\-66\\-770\\-264\\-385\\0
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0\\30\\0\\-105\\-165\\-75\\-1155\\-495\\-840\\-165
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-770\\-1155\\-3465\\-2310\\-770\\0\\770\\0\\-770\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-840\\-3465\\-2835\\-1540\\-210\\-770\\0\\35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-1155\\-1155\\-825\\-165\\-1155\\-165\\0\\330
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-203944220100390, -207561537645, -67870953324405; -207561537645, -211240590, -69074766750; -67870953324405, -69074766750, -22586893136251],[1155, 165, 1],[-171207698, -376034759, -1560181617, -1280962316, -700100496, -96815458, -372053984, -7184442, 8054334, 4042631; -447, -982, -4075, -3346, -1829, -253, -973, -19, 21, 11; 514458967, 1129940160, 4688161995, 3849140883, 2103719530, 290919048, 1117978400, 21588402, -24202325, -12147630],117,26]
%
% END OF LATTICE L118.26
%
\hbox{}\par\smallskip%
%
% LATTICE L118.27
%
\leavevmode%
\llap{\spacer}%
${L_{118.27}}=\pdualwitharg{2.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${165}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}{1848}\above{1pt}{1pt}{*}{2}{440}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-75515664840\\-621708780\\18895929360
\end{tabular}\endgroup%
\kern3pt%
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-621708780\\-3863992\\155565916
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18895929360\\155565916\\-4728239459
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3159472\\-13545\\-12626955
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\HardButStrongLineBreak\kern3pt%
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-105191\\-451\\-420400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1830427\\7847\\7315374
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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12409573\\53200\\49595350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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6813101\\29208\\27228828
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2731483\\11710\\10916480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13088695\\56112\\52309488
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3123347\\13390\\12482580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396776\\-1701\\-1585731
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1665708\\-7141\\-6657068
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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0\\8\\0\\-140\\-132\\-100\\-924\\-660\\-336\\-44
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-140\\0\\140\\0\\-560\\-9240\\-9240\\-5670\\-1540
\end{tabular}\endgroup%
\kern3pt%
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-1320\\-132\\-132\\0\\264\\0\\-1848\\-2640\\-1848\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-100\\-220\\-560\\0\\40\\0\\-440\\-420\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-924\\-2772\\-9240\\-1848\\0\\1848\\0\\-924\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2640\\-660\\-2420\\-9240\\-2640\\-440\\0\\440\\0\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-336\\-1386\\-5670\\-1848\\-420\\-924\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-44\\-308\\-1540\\-660\\-220\\-924\\-220\\0\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-75515664840, -621708780, 18895929360; -621708780, -3863992, 155565916; 18895929360, 155565916, -4728239459],[4620, 44, 1],[-3159472, -105191, 1830427, 12409573, 6813101, 2731483, 13088695, 3123347, -396776, -1665708; -13545, -451, 7847, 53200, 29208, 11710, 56112, 13390, -1701, -7141; -12626955, -420400, 7315374, 49595350, 27228828, 10916480, 52309488, 12482580, -1585731, -6657068],117,27]
%
% END OF LATTICE L118.27
%
\hbox{}\par\smallskip%
%
% LATTICE L118.28
%
\leavevmode%
\llap{\spacer}%
${L_{118.28}}=\pdualwitharg{5.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{220}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{330}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{2310}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{420}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44301180\\-22203720\\50820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22203720\\-11128370\\25465
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50820\\25465\\-58
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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23\\-48\\-924
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-23\\-460
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
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-67\\140\\2758
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\HardButStrongLineBreak\kern3pt%
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-155\\324\\6435
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-22\\46\\919
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-964\\2016\\40425
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-131\\274\\5511
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\HardButStrongLineBreak\kern3pt%
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-1285\\2688\\54180
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-32\\67\\1375
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-660\\-924\\-1320\\-132\\-4620\\-528\\-4620\\0
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-1320\\-165\\-330\\0\\330\\0\\-2310\\-660\\-9240\\-825
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\kern3pt%
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-4620\\-1155\\-6930\\-4620\\-2310\\0\\2310\\0\\-4620\\-1155
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\kern3pt%
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0\\-55\\-770\\-770\\-825\\-55\\-1155\\-55\\0\\110
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-44301180, -22203720, 50820; -22203720, -11128370, 25465; 50820, 25465, -58],[4620, 55, 1],[23, 11, -1, -67, -155, -22, -964, -131, -1285, -32; -48, -23, 2, 140, 324, 46, 2016, 274, 2688, 67; -924, -460, 0, 2758, 6435, 919, 40425, 5511, 54180, 1375],117,28]
%
% END OF LATTICE L118.28
%
\hbox{}\par\smallskip%
%
% LATTICE L118.29
%
\leavevmode%
\llap{\spacer}%
${L_{118.29}}=\pdualwitharg{3.7.11}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${385}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{231}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{s}{2}{2310}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-231\\-1617\\-1155\\-1155\\-1155\\-231\\-1155\\0\\462
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-278251083895785, -286862399670, -92620560609885; -286862399670, -295736826, -95486982129; -92620560609885, -95486982129, -30830313857474],[1155, 231, 1],[-1182963807, -519644246, -2156022381, -1264405609, -967472122, -668948588, -73448992, -49640862, 1590047, 5586452; -2235, -982, -4075, -2390, -1829, -1265, -139, -95, 3, 11; 3553864930, 1561117467, 6477131661, 3798532740, 2906483888, 2009658210, 220655776, 149131290, -4776826, -16782843],117,29]
%
% END OF LATTICE L118.29
%
\hbox{}\par\smallskip%
%
% LATTICE L118.30
%
\leavevmode%
\llap{\spacer}%
${L_{118.30}}=\pdualwitharg{2.3.5}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{1320}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{924}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{385}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
470836110360\\-2541512820\\-117738353040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2541512820\\13718760\\635536500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117738353040\\635536500\\29441921449
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1422\\-166\\-5683
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19157\\-2266\\-76560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1689\\-200\\-6750
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7745\\-847\\-30954
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\3\\-260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8323\\-913\\-33264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22139\\-2499\\-88480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13881\\-1583\\-55476
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160010\\-18326\\-639485
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12596\\-1457\\-50340
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-30\\-462\\-40\\-132\\-140\\-48\\-385\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1320\\0\\-4620\\-660\\-3300\\-4620\\-1980\\-18480\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\60\\0\\-60\\-660\\-1260\\-660\\-6930\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\-4620\\0\\924\\0\\-3696\\-9240\\-5544\\-62370\\-4620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-660\\-60\\0\\40\\0\\-280\\-240\\-3080\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-3300\\-660\\-3696\\0\\264\\0\\-264\\-4620\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-4620\\-1260\\-9240\\-280\\0\\280\\0\\-1540\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-1980\\-660\\-5544\\-240\\-264\\0\\24\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-18480\\-6930\\-62370\\-3080\\-4620\\-1540\\0\\385\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-660\\-420\\-4620\\-300\\-660\\-420\\-60\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[470836110360, -2541512820, -117738353040; -2541512820, 13718760, 635536500; -117738353040, 635536500, 29441921449],[4620, 60, 1],[-1422, -19157, -1689, -7745, -65, -8323, -22139, -13881, -160010, -12596; -166, -2266, -200, -847, 3, -913, -2499, -1583, -18326, -1457; -5683, -76560, -6750, -30954, -260, -33264, -88480, -55476, -639485, -50340],117,30]
%
% END OF LATTICE L118.30
%
\hbox{}\par\smallskip%
%
% LATTICE L118.31
%
\leavevmode%
\llap{\spacer}%
${L_{118.31}}=\pdualwitharg{7.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4620}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{308}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{462}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{b}{2}{770}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47350380\\17842440\\83160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17842440\\6723178\\31339
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83160\\31339\\146
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-367\\780\\41580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\32\\1673
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67\\144\\7238
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\50\\2360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-9\\-924
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-25\\-1470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-21\\-1188
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\15\\770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\36\\1956
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\55\\3003
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4620\\0\\-4620\\-4620\\-9240\\-4620\\-4620\\-18480\\-4620\\0
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\kern3pt%
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0\\14\\0\\-70\\-231\\-175\\-231\\-1155\\-336\\-77
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\kern3pt%
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-4620\\0\\308\\0\\-462\\-770\\-1386\\-8470\\-2772\\-1078
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-70\\0\\20\\0\\-280\\-660\\-4620\\-1620\\-770
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9240\\-231\\-462\\0\\462\\0\\-462\\-4620\\-1848\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-175\\-770\\-280\\0\\70\\0\\-770\\-420\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-231\\-1386\\-660\\-462\\0\\66\\0\\-132\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18480\\-1155\\-8470\\-4620\\-4620\\-770\\0\\770\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-336\\-2772\\-1620\\-1848\\-420\\-132\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-77\\-1078\\-770\\-1155\\-385\\-231\\-385\\0\\154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[47350380, 17842440, 83160; 17842440, 6723178, 31339; 83160, 31339, 146],[4620, 77, 1],[-367, -15, -67, -23, 5, 12, 10, -7, -17, -26; 780, 32, 144, 50, -9, -25, -21, 15, 36, 55; 41580, 1673, 7238, 2360, -924, -1470, -1188, 770, 1956, 3003],117,31]
%
% END OF LATTICE L118.31
%
\hbox{}\par\smallskip%
%
% LATTICE L118.32
%
\leavevmode%
\llap{\spacer}%
${L_{118.32}}=\pdualwitharg{2.3.7}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{r}{2}{1848}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{660}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{*}{2}{9240}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5639735640\\60184740\\-1409987040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60184740\\642264\\-15046752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1409987040\\-15046752\\352510043
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4107\\-290\\16415
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\HardButStrongLineBreak\kern3pt%
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32597\\-2321\\130284
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\HardButStrongLineBreak\kern3pt%
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10603\\-759\\42378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62833\\-4510\\251130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3713\\-268\\14840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19651\\-1430\\78540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1051\\90\\-4200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
399\\-22\\1595
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1324\\-89\\5292
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-210\\-2310\\-280\\-4620\\-140\\-1680\\-385\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1848\\0\\-4620\\-924\\-23100\\-924\\-13860\\-3696\\-924
\end{tabular}\endgroup%
\kern3pt%
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-210\\0\\84\\0\\-84\\-4620\\-252\\-4620\\-1386\\-588
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-4620\\0\\660\\0\\-18480\\-1320\\-27720\\-8910\\-4620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-924\\-84\\0\\56\\0\\-56\\-1680\\-616\\-420
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-23100\\-4620\\-18480\\0\\9240\\0\\-9240\\-4620\\-4620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-924\\-252\\-1320\\-56\\0\\8\\0\\-44\\-84
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\kern3pt%
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-1680\\-13860\\-4620\\-27720\\-1680\\-9240\\0\\840\\0\\-420
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-924\\-588\\-4620\\-420\\-4620\\-84\\-420\\0\\168
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5639735640, 60184740, -1409987040; 60184740, 642264, -15046752; -1409987040, -15046752, 352510043],[4620, 84, 1],[4107, 32597, 10603, 62833, 3713, 19651, 1, -1051, 399, 1324; -290, -2321, -759, -4510, -268, -1430, 0, 90, -22, -89; 16415, 130284, 42378, 251130, 14840, 78540, 4, -4200, 1595, 5292],117,32]
%
% END OF LATTICE L118.32
%
\hbox{}\par\smallskip%
%
% LATTICE L118.33
%
\leavevmode%
\llap{\spacer}%
${L_{118.33}}=\pdualwitharg{5.7.11}(\pfillwitharg{2}({L_{118.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${231}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}{385}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{2310}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{s}{2}{330}\above{1pt}{1pt}{s}{2}{154}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{770}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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556594207\\-1669783962\\2787944544
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104915272\\-314746069\\525513842
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172791776\\-518375745\\865502880
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7785478\\-23356453\\38996958
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-3740651\\11221962\\-18736680
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-4380786\\13142369\\-21943075
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1155\\-840\\-3465\\-405\\-4620\\-210\\-330\\0\\15\\0
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0\\-385\\-2695\\-385\\-5775\\-385\\-1155\\-385\\0\\770
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-283153000860345, 373585639350, 56753326612290; 373585639350, -492894710, -74879046935; 56753326612290, -74879046935, -11375263801534],[1155, 385, 1],[556594207, 407494824, 1690710457, 198304415, 2276017978, 104915272, 172791776, 7785478, -3740651, -4380786; -1669783962, -1222485454, -5072135446, -594913723, -6828059421, -314746069, -518375745, -23356453, 11221962, 13142369; 2787944544, 2041115335, 8468659815, 993294046, 11400427500, 525513842, 865502880, 38996958, -18736680, -21943075],117,33]
%
% END OF LATTICE L118.33
%
\hbox{}\par\smallskip%
%
% LATTICE L118.34
%
\leavevmode%
\llap{\spacer}%
${L_{118.34}}=\pdualwitharg{3.5.7}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{r}{2}{2310}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{b}{2}{462}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{220}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-10288740\\-258720\\4620
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\kern3pt%
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-258720\\-4830\\105
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4620\\105\\-2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\-24\\-3584
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-1\\-22\\-3465
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1\\24\\3570
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1\\22\\3432
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0\\-1\\-70
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-3\\-77\\-11088
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-1\\-25\\-3650
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-4\\-99\\-14532
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-25\\-616\\-90640
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-3\\-73\\-10815
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10288740, -258720, 4620; -258720, -4830, 105; 4620, 105, -2],[4620, 105, 1],[-1, -1, 1, 1, 0, -3, -1, -4, -25, -3; -24, -22, 24, 22, -1, -77, -25, -99, -616, -73; -3584, -3465, 3570, 3432, -70, -11088, -3650, -14532, -90640, -10815],117,34]
%
% END OF LATTICE L118.34
%
\hbox{}\par\smallskip%
%
% LATTICE L118.35
%
\leavevmode%
\llap{\spacer}%
${L_{118.35}}=\pdualwitharg{2.3.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${55}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{s}{2}{616}\above{1pt}{1pt}{*}{2}{1320}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{264}\above{1pt}{1pt}{l}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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21886\\529\\87468
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74298\\1795\\296934
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266669\\6440\\1065750
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21887\\528\\87472
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4609\\110\\18420
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-20731\\-504\\-82852
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-44919\\-1090\\-179520
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9215\\223\\36828
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20557087320, -21422940, -5143607700; -21422940, 21912, 5360256; -5143607700, 5360256, 1286986807],[4620, 132, 1],[30813, 21886, 74298, 266669, 21887, 4609, -20731, -44919, -3743, 9215; 745, 529, 1795, 6440, 528, 110, -504, -1090, -91, 223; 123145, 87468, 296934, 1065750, 87472, 18420, -82852, -179520, -14959, 36828],117,35]
%
% END OF LATTICE L118.35
%
\hbox{}\par\smallskip%
%
% LATTICE L118.36
%
\leavevmode%
\llap{\spacer}%
${L_{118.36}}=\pdualwitharg{2.5.7}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{3080}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{*}{2}{616}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1154654169900\\-3913565460\\-288878502979
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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298583\\10428\\1193302
\end{tabular}\endgroup%
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1135189\\39648\\4536840
\end{tabular}\endgroup%
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1843661\\64394\\7368284
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1038941\\36288\\4152180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
958931\\33494\\3832416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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3579751\\125037\\14306655
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
162330\\5671\\648760
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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21\\0\\-210\\-462\\-840\\-924\\-420\\-336\\-1155\\0
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0\\-1540\\-980\\-1540\\-2100\\-1540\\-420\\-140\\0\\280
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4615179870840, 15642613140, 1154654169900; 15642613140, -53017720, -3913565460; 1154654169900, -3913565460, -288878502979],[4620, 140, 1],[-95953, -1387967, -103777, 298583, 1135189, 1843661, 1038941, 958931, 3579751, 162330; -3351, -48477, -3625, 10428, 39648, 64394, 36288, 33494, 125037, 5671; -383481, -5547080, -414750, 1193302, 4536840, 7368284, 4152180, 3832416, 14306655, 648760],117,36]
%
% END OF LATTICE L118.36
%
\hbox{}\par\smallskip%
%
% LATTICE L118.37
%
\leavevmode%
\llap{\spacer}%
${L_{118.37}}=\pdualwitharg{3.5.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{660}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{770}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
825727980\\412455120\\-332640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
412455120\\206023290\\-166155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-332640\\-166155\\134
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-16\\-2464
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-9\\-1230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-19\\42\\4914
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-14\\31\\3685
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-5\\11\\1227
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-61\\133\\13475
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-20\\43\\3663
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-53\\112\\7280
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-10\\-2475
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\-660\\-924\\-440\\-132\\-1540\\-528\\-1540\\0
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\kern3pt%
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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0\\-165\\-2310\\-2310\\-825\\-165\\-1155\\-165\\0\\330
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[825727980, 412455120, -332640; 412455120, 206023290, -166155; -332640, -166155, 134],[4620, 165, 1],[7, 4, -1, -19, -14, -5, -61, -20, -53, 4; -16, -9, 2, 42, 31, 11, 133, 43, 112, -10; -2464, -1230, 0, 4914, 3685, 1227, 13475, 3663, 7280, -2475],117,37]
%
% END OF LATTICE L118.37
%
\hbox{}\par\smallskip%
%
% LATTICE L118.38
%
\leavevmode%
\llap{\spacer}%
${L_{118.38}}=\pdualwitharg{2.5.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{220}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{1320}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{9240}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{440}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132635219640\\-288763860\\-33185002620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-6614\\-243\\-26433
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-13239\\-482\\-52910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-22444\\-819\\-89698
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-488990\\-17913\\-1954260
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33359\\-1225\\-133320
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\0\\-330\\-462\\-1320\\-132\\-4620\\-528\\-1155\\0
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\kern3pt%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\kern3pt%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-220\\-1540\\-1540\\-3300\\-220\\-4620\\-220\\0\\440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[132635219640, -288763860, -33185002620; -288763860, 628760, 72248000; -33185002620, 72248000, 8302805257],[4620, 220, 1],[-6614, -1151, -13239, -22444, -73654, -10356, -488990, -73632, -196442, -33359; -243, -42, -482, -819, -2691, -379, -17913, -2699, -7203, -1225; -26433, -4600, -52910, -89698, -294360, -41388, -1954260, -294272, -785085, -133320],117,38]
%
% END OF LATTICE L118.38
%
\hbox{}\par\smallskip%
%
% LATTICE L118.39
%
\leavevmode%
\llap{\spacer}%
${L_{118.39}}=\pdualwitharg{3.7.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1540}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{924}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{154}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{b}{2}{2310}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{l}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1182720\\473319\\562
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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3\\-7\\-418
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-2\\5\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\-231\\-3234\\-2310\\-1155\\-1155\\-231\\-1155\\0\\462
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2489011140, 996090480, 1182720; 996090480, 398630694, 473319; 1182720, 473319, 562],[4620, 231, 1],[-139, -19, -101, -43, -6, 7, 3, -2, -5, -25; 340, 47, 254, 110, 17, -15, -7, 5, 12, 60; 6160, 399, -1386, -2160, -1694, -2100, -418, 0, 416, 2079],117,39]
%
% END OF LATTICE L118.39
%
\hbox{}\par\smallskip%
%
% LATTICE L118.40
%
\leavevmode%
\llap{\spacer}%
${L_{118.40}}=\pdualwitharg{3.5.7.11}(\pfillwitharg{2}({L_{118.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${77}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}{1155}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{770}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{s}{2}{462}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{2310}\above{1pt}{1pt}{l}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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578856290648274870\\387628030253523675\\128372560854566490
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-34014\\-1307990620\\3949703219
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206\\7922308\\-23922775
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1891\\72543009\\-219056145
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[578856290648274870, 387628030253523675, 128372560854566490; 387628030253523675, 259573044753391215, 85964001266925705; 128372560854566490, 85964001266925705, 28469094396302717],[1155, 1155, 1],[-34014, -75261, -313879, -36918, -141918, -19808, -11150, -2080, 206, 1891; -1307990620, -2894034976, -12069439418, -1419574663, -5456943309, -761618199, -428678813, -79894811, 7922308, 72543009; 3949703219, 8739037650, 36445753575, 4286650488, 16478181335, 2299837491, 1294469605, 241256169, -23922775, -219056145],117,40]
%
% END OF LATTICE L118.40
%
\hbox{}\par\smallskip%
%
% LATTICE L118.41
%
\leavevmode%
\llap{\spacer}%
${L_{118.41}}=\pdualwitharg{2.7.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1155}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{308}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{1848}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{s}{2}{264}\above{1pt}{1pt}{*}{2}{3080}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{616}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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54717\\1060\\218680
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[122381794920, -140868420, -30621068940; -140868420, 162008, 35246596; -30621068940, 35246596, 7661677651],[4620, 308, 1],[-169642, -43066, -157716, -85551, -168775, -25607, 4657, 54717, 2038, -11637; -3315, -841, -3079, -1670, -3294, -500, 90, 1060, 39, -229; -677985, -172116, -630322, -341910, -674520, -102340, 18612, 218680, 8145, -46508],117,41]
%
% END OF LATTICE L118.41
%
\hbox{}\par\smallskip%
%
% LATTICE L118.42
%
\leavevmode%
\llap{\spacer}%
${L_{118.42}}=\pdualwitharg{5.7.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${924}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{1540}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{2310}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}{330}\above{1pt}{1pt}{b}{2}{154}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{770}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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155365980\\-1501500\\-198660
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-198660\\1925\\254
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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3\\8\\2286
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\HardButStrongLineBreak\kern3pt%
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47\\126\\35805
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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6\\16\\4571
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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34\\90\\25905
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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29\\76\\22099
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\96\\28200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\1\\770
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
924\\0\\-4620\\-924\\-9240\\-924\\-4620\\-3696\\-4620\\0
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0\\70\\0\\-70\\-1155\\-175\\-1155\\-1155\\-1680\\-385
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-4620\\0\\1540\\0\\-2310\\-770\\-6930\\-8470\\-13860\\-5390
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-924\\-175\\-770\\-56\\0\\14\\0\\-154\\-420\\-385
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-4620\\-1155\\-6930\\-660\\-2310\\0\\330\\0\\-660\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3696\\-1155\\-8470\\-924\\-4620\\-154\\0\\154\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-1680\\-13860\\-1620\\-9240\\-420\\-660\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-385\\-5390\\-770\\-5775\\-385\\-1155\\-385\\0\\770
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[155365980, -1501500, -198660; -1501500, 13090, 1925; -198660, 1925, 254],[4620, 385, 1],[-17, -4, 1, 3, 47, 6, 34, 29, 37, 1; -48, -11, 2, 8, 126, 16, 90, 76, 96, 1; -12936, -3045, 770, 2286, 35805, 4571, 25905, 22099, 28200, 770],117,42]
%
% END OF LATTICE L118.42
%
\hbox{}\par\smallskip%
%
% LATTICE L118.43
%
\leavevmode%
\llap{\spacer}%
${L_{118.43}}=\pdualwitharg{2.3.5.7}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{9240}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{*}{2}{1848}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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189229018440\\-114839340\\-47348259420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114839340\\69720\\28734720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47348259420\\28734720\\11847324943
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3750\\-4\\-14987
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-6936\\11\\-27720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-578\\1\\-2310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15573\\-22\\-62238
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-16149\\-24\\-64540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109589\\-154\\-437976
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26533\\-36\\-106040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-92291\\-122\\-368844
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135266\\-176\\-540595
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53071\\-65\\-212100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-210\\-462\\-280\\-924\\-140\\-336\\-385\\0
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[189229018440, -114839340, -47348259420; -114839340, 69720, 28734720; -47348259420, 28734720, 11847324943],[4620, 420, 1],[-3750, -6936, -578, -15573, -16149, -109589, -26533, -92291, -135266, -53071; -4, 11, 1, -22, -24, -154, -36, -122, -176, -65; -14987, -27720, -2310, -62238, -64540, -437976, -106040, -368844, -540595, -212100],117,43]
%
% END OF LATTICE L118.43
%
\hbox{}\par\smallskip%
%
% LATTICE L118.44
%
\leavevmode%
\llap{\spacer}%
${L_{118.44}}=\pdualwitharg{2.3.5.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{660}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{440}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{3080}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{1320}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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222524027880\\-241852380\\-55678516740
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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27411\\332\\109549
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\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-38663\\-469\\-154518
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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34625\\419\\138380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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54245\\657\\216792
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\HardButStrongLineBreak\kern3pt%
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1199159\\14525\\4792480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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637089\\7717\\2546148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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610255\\7392\\2438905
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\HardButStrongLineBreak\kern3pt%
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410876\\4977\\1642080
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-330\\-462\\-440\\-132\\-1540\\-528\\-385\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\120\\0\\-420\\-660\\-300\\-4620\\-1980\\-1680\\-660
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\kern3pt%
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-330\\0\\660\\0\\-660\\-660\\-13860\\-7260\\-6930\\-4620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\-420\\0\\84\\0\\-336\\-9240\\-5544\\-5670\\-4620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-660\\-4620\\-4620\\-3300\\-660\\-4620\\-660\\0\\1320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[222524027880, -241852380, -55678516740; -241852380, 260040, 60514740; -55678516740, 60514740, 13931516771],[4620, 660, 1],[27411, -9233, -102141, -38663, 34625, 54245, 1199159, 637089, 610255, 410876; 332, -112, -1238, -469, 419, 657, 14525, 7717, 7392, 4977; 109549, -36900, -408210, -154518, 138380, 216792, 4792480, 2546148, 2438905, 1642080],117,44]
%
% END OF LATTICE L118.44
%
\hbox{}\par\smallskip%
%
% LATTICE L118.45
%
\leavevmode%
\llap{\spacer}%
${L_{118.45}}=\pdualwitharg{2.3.7.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${385}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{924}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{616}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{s}{2}{88}\above{1pt}{1pt}{*}{2}{9240}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{1848}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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33427372440\\-135259740\\-8363830860
\end{tabular}\endgroup%
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-135259740\\467544\\33843348
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-8363830860\\33843348\\2092706113
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1016893\\-7028\\-4064060
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-499181\\-3450\\-1995000
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\HardButStrongLineBreak\kern3pt%
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5791\\40\\23144
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609211\\4210\\2434740
\end{tabular}\endgroup%
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2606\\18\\10415
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-244378\\-1689\\-976668
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[33427372440, -135259740, -8363830860; -135259740, 467544, 33843348; -8363830860, 33843348, 2092706113],[4620, 924, 1],[-1017182, -765565, -2806879, -1529393, -1016893, -499181, 5791, 609211, 2606, -244378; -7030, -5291, -19399, -10570, -7028, -3450, 40, 4210, 18, -1689; -4065215, -3059616, -11217822, -6112290, -4064060, -1995000, 23144, 2434740, 10415, -976668],117,45]
%
% END OF LATTICE L118.45
%
\hbox{}\par\smallskip%
%
% LATTICE L118.46
%
\leavevmode%
\llap{\spacer}%
${L_{118.46}}=\pdualwitharg{3.5.7.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${308}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}{4620}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{770}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{b}{2}{462}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{2310}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8791860\\-4291980\\877800
\end{tabular}\endgroup%
\kern3pt%
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-4291980\\2095170\\-428505
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
877800\\-428505\\87638
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\-128\\-616
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-1\\-45\\-210
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1\\2\\0
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1\\18\\78
\end{tabular}\endgroup%
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5\\89\\385
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2\\17\\63
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\HardButStrongLineBreak\kern3pt%
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4\\-3\\-55
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\HardButStrongLineBreak\kern3pt%
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11\\-119\\-693
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5\\-96\\-520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2\\-232\\-1155
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
308\\0\\-4620\\-924\\-3080\\-924\\-1540\\-3696\\-1540\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8791860, -4291980, 877800; -4291980, 2095170, -428505; 877800, -428505, 87638],[4620, 1155, 1],[-1, -1, 1, 1, 5, 2, 4, 11, 5, 2; -128, -45, 2, 18, 89, 17, -3, -119, -96, -232; -616, -210, 0, 78, 385, 63, -55, -693, -520, -1155],117,46]
%
% END OF LATTICE L118.46
%
\hbox{}\par\smallskip%
%
% LATTICE L118.47
%
\leavevmode%
\llap{\spacer}%
${L_{118.47}}=\pdualwitharg{2.5.7.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${231}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{*}{2}{1540}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{9240}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{s}{2}{1320}\above{1pt}{1pt}{*}{2}{616}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{3080}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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660077408760\\790200180\\164569814640
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790200180\\945560\\197012200
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6901\\-20702\\-27580
\end{tabular}\endgroup%
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75333\\-225990\\-301070
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6326\\-18977\\-25282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2312\\-6927\\-9240
\end{tabular}\endgroup%
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-24150\\72449\\96516
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-254156\\762453\\1015740
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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231\\0\\-2310\\-462\\-9240\\-924\\-4620\\-3696\\-1155\\0
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[660077408760, 790200180, 164569814640; 790200180, 945560, 197012200; 164569814640, 197012200, 41030375191],[4620, 1540, 1],[-44564, 6901, 75333, 6326, 2312, -24150, -254156, -326610, -135992, -223109; 133689, -20702, -225990, -18977, -6927, 72449, 762453, 979809, 407967, 669311; 178101, -27580, -301070, -25282, -9240, 96516, 1015740, 1305304, 543495, 891660],117,47]
%
% END OF LATTICE L118.47
%
\hbox{}\par\smallskip%
%
% LATTICE L118.48
%
\leavevmode%
\llap{\spacer}%
${L_{118.48}}=\pdualwitharg{2.3.5.7.11}({L_{118.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${77}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{*}{2}{4620}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{3080}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{s}{2}{440}\above{1pt}{1pt}{*}{2}{1848}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{9240}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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31813320\\-139491660\\34904100
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34904100\\-152321400\\38114453
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-12501\\-49973
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-4\\3257\\13020
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\HardButStrongLineBreak\kern3pt%
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-44\\36405\\145530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\815\\3258
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-25811\\-103180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\-27443\\-109704
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99\\-82331\\-329120
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\HardButStrongLineBreak\kern3pt%
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361\\-300255\\-1200276
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\HardButStrongLineBreak\kern3pt%
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49\\-40759\\-162935
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\HardButStrongLineBreak\kern3pt%
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225\\-187226\\-748440
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31813320, -139491660, 34904100; -139491660, 608740440, -152321400; 34904100, -152321400, 38114453],[4620, 4620, 1],[15, -4, -44, -1, 31, 33, 99, 361, 49, 225; -12501, 3257, 36405, 815, -25811, -27443, -82331, -300255, -40759, -187226; -49973, 13020, 145530, 3258, -103180, -109704, -329120, -1200276, -162935, -748440],117,48]
%
% END OF LATTICE L118.48
%

\medskip%
%------------------------------WEYL-GROUP-W119--------------------------------
%
\leavevmode\llap{}%
$W_{119}$%
\qquad\llap{48} lattices, $\chi=90$%
\hfill%
$422222222422222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L119.1
%
\leavevmode%
\llap{\spacer}%
${L_{119.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{119}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{220}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{770}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1671382020\\328020\\475860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
328020\\-62\\-97
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
475860\\-97\\-130
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
352351\\918317400\\604503900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-187651\\-123525
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-250200\\-164701
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2607\\1715
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2606\\-1716
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5215\\-3430
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\65142\\42900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\44300\\29170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
85\\221508\\145845
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
453\\1180520\\777260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\72969\\48042
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
368\\959035\\631400
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-35\\-198\\-90\\-357\\-1760\\-99\\-1155\\-17\\-28\\-70\\-132\\-30\\-63\\-220\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-35\\-198\\-90\\-357\\-1760\\-99\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-132\\-80\\-378\\-1980\\-120\\-1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\70\\0\\-70\\-525\\-3080\\-210\\-3080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-1980\\-10780\\-40260\\-15620\\-55440\\-261580\\-13860\\-149380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-770\\-5005\\-20790\\-8470\\-31185\\-149380\\-8085\\-89705
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1671382020, 328020, 475860; 328020, -62, -97; 475860, -97, -130],[4620, 1, 1],[1, -1, -2, 25, 17, 85, 453, 28, 368, 7, 17, 58, 151, 47, 139, 603, 28, 248; 2607, -2606, -5215, 65142, 44300, 221508, 1180520, 72969, 959035, 18243, 44306, 151165, 393558, 122500, 362292, 1571680, 72981, 646415; 1715, -1716, -3430, 42900, 29170, 145845, 777260, 48042, 631400, 12010, 29166, 99505, 259050, 80630, 238455, 1034440, 48033, 425425],118,1]
%
% END OF LATTICE L119.1
%
\hbox{}\par\smallskip%
%
% LATTICE L119.2
%
\leavevmode%
\llap{\spacer}%
${L_{119.2}}=\pfillwitharg{2}({L_{119.1}})$%
{${}={}$}Nikulin $119$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{770}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\1155\\2310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-30\\-61
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-33\\-132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-20\\-65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-189\\-546
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-43\\-495\\-1375
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-60\\-159
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\-770\\-1925
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1155, 0, 0; 0, 1, 0; 0, 0, 1],[1155, 1, 1],[0, 0, -1, -4, -2, -17, -43, -5, -61, -1, -1, -6, -7, -2, -11, -23, -2, -16; -1, 1, 0, -33, -20, -189, -495, -60, -770, -14, -16, -105, -132, -40, -231, -495, -45, -385; 1, 0, -35, -132, -65, -546, -1375, -159, -1925, -31, -30, -175, -198, -55, -294, -605, -51, -385],118,2]
%
% END OF LATTICE L119.2
%
\hbox{}\par\smallskip%
%
% LATTICE L119.3
%
\leavevmode%
\llap{\spacer}%
${L_{119.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{119.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{4}{3}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{2310}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2239968885\\10117800\\-747025125
\end{tabular}\endgroup%
\kern3pt%
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10117800\\-43782\\3374085
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\kern3pt%
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-747025125\\3374085\\-249131362
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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0\\0\\0
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}}\right.$}%
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39526\\-11189\\-118671
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2239968885, 10117800, -747025125; 10117800, -43782, 3374085; -747025125, 3374085, -249131362],[1155, 3, 1],[39526, -19125, -61202, 183175, 362101, 1191280, 9487319, 389302, 15277138, 288152, 346803, 2354948, 1018311, 948611, 1867048, 12139429, 375280, 9954068; -11189, 5414, 17325, -51854, -102505, -337232, -2685705, -110205, -4324705, -81571, -98174, -666645, -288266, -268535, -528528, -3436455, -106235, -2817815; -118671, 57420, 183750, -549956, -1087155, -3576643, -28484280, -1168822, -45867360, -865134, -1041225, -7070385, -3057329, -2848065, -5605537, -36446850, -1126723, -29885625],118,3]
%
% END OF LATTICE L119.3
%
\hbox{}\par\smallskip%
%
% LATTICE L119.4
%
\leavevmode%
\llap{\spacer}%
${L_{119.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{119.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{4}{5}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{154}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4316813655\\18006450\\-863751735
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4316813655, 18006450, -863751735; 18006450, -72970, 3602775; -863751735, 3602775, -172828174],[1155, 5, 1],[34752, -16815, -10762, 483151, 63673, 3142178, 1668281, 1026842, 2686382, 253348, 304915, 414102, 2685949, 166807, 4924622, 2134639, 989858, 1750358; -11189, 5414, 3465, -155562, -20501, -1011696, -537141, -330615, -864941, -81571, -98174, -133329, -864798, -53707, -1585584, -687291, -318705, -563563; -173915, 84150, 53858, -2417910, -318649, -15724905, -8348846, -5138790, -13443892, -1267870, -1525935, -2072357, -13441725, -834779, -24645075, -10682716, -4953705, -8759597],118,4]
%
% END OF LATTICE L119.4
%
\hbox{}\par\smallskip%
%
% LATTICE L119.5
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6406973265, 25895100, -1830961440; 25895100, -102158, 7400085; -1830961440, 7400085, -523245473],[1155, 7, 1],[65443, -31665, -14476, 909841, 599526, 845309, 15708039, 1933688, 3613454, 477090, 574198, 557009, 5058022, 1570606, 1324823, 20099139, 1864043, 2354409; -11189, 5414, 2475, -155562, -102505, -144528, -2685705, -330615, -617815, -81571, -98174, -95235, -864798, -268535, -226512, -3436455, -318705, -402545; -229159, 110880, 50690, -3185952, -2099335, -2959983, -55004180, -6771114, -12653080, -1670606, -2010645, -1950455, -17711463, -5499725, -4639077, -70380310, -6527241, -8244335],118,5]
%
% END OF LATTICE L119.5
%
\hbox{}\par\smallskip%
%
% LATTICE L119.6
%
\leavevmode%
\llap{\spacer}%
${L_{119.6}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{119.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10598186445, 41672400, -1927348500; 41672400, -160534, 7578285; -1927348500, 7578285, -350500741],[1155, 11, 1],[61723, -29865, -95572, 78011, 565446, 5580803, 1346829, 1823768, 2168758, 449970, 541558, 3677423, 433682, 1481326, 8746601, 1723329, 1758083, 1413093; -11189, 5414, 17325, -14142, -102505, -1011696, -244155, -330615, -393155, -81571, -98174, -666645, -78618, -268535, -1585584, -312405, -318705, -256165; -339647, 164340, 525910, -429276, -3111515, -30709833, -7411280, -10035762, -11934160, -2476078, -2980065, -20235985, -2386449, -8151385, -48130467, -9483070, -9674313, -7775915],118,6]
%
% END OF LATTICE L119.6
%
\hbox{}\par\smallskip%
%
% LATTICE L119.7
%
\leavevmode%
\llap{\spacer}%
${L_{119.7}}=\pdualwitharg{3}({L_{119.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{*}{4}{12}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{660}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{2310}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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249480\\-186\\-19149
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25008060\\-19149\\-1744702
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-332607660, 249480, 25008060; 249480, -186, -19149; 25008060, -19149, -1744702],[4620, 3, 1],[176, -177, -352, 1475, 3007, 5010, 80093, 1650, 65048, 1237, 3003, 10243, 8887, 8297, 8178, 106423, 1647, 43753; 182017, -183050, -364035, 1525414, 3109780, 5181246, 82830660, 1706399, 67271435, 1279283, 3105650, 10593135, 9190786, 8580620, 8457554, 110060940, 1703301, 45248665; 525, -528, -1050, 4400, 8970, 14945, 238920, 4922, 194040, 3690, 8958, 30555, 26510, 24750, 24395, 317460, 4913, 130515],118,7]
%
% END OF LATTICE L119.7
%
\hbox{}\par\smallskip%
%
% LATTICE L119.8
%
\leavevmode%
\llap{\spacer}%
${L_{119.8}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{119.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1819058361624735, -950262771150, 605368119169275; -950262771150, 496410315, -316239873675; 605368119169275, -316239873675, 201461683383937],[1155, 15, 1],[-364727, -15375, 20903, -705127, -276796, -4647456, -7461698, -1545519, -12321451, -1200643, -1549995, -2212421, -5034563, -975500, -9962664, -13201186, -2107011, -11785247; -479, -16, 0, -1144, -431, -7077, -11286, -2320, -18326, -1756, -2219, -3129, -7051, -1357, -13783, -18216, -2895, -16093; 1095960, 46200, -62811, 2118820, 831738, 13965035, 22421487, 4644095, 37024449, 3607785, 4657545, 6648054, 15128245, 2931258, 29936585, 39667947, 6331310, 35413224],118,8]
%
% END OF LATTICE L119.8
%
\hbox{}\par\smallskip%
%
% LATTICE L119.9
%
\leavevmode%
\llap{\spacer}%
${L_{119.9}}=\pdualwitharg{2}({L_{119.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4565645450520, -6018395460, -1142401906800; -6018395460, 7933400, 1505904592; -1142401906800, 1505904592, 285848327647],[4620, 4, 1],[-33493, -1733, -14993, -284591, -171456, -1653418, -2183122, -534601, -6977623, -131635, -163395, -1140903, -1531817, -489056, -2970166, -3266102, -621295, -5737233; 31, 2, 0, 198, 125, 1239, 1650, 408, 5390, 104, 133, 945, 1287, 415, 2541, 2805, 537, 5005; -133856, -6926, -59920, -1137378, -685230, -6607944, -8724925, -2136552, -27886320, -526084, -653014, -4559660, -6121962, -1954530, -11870376, -13053095, -2483028, -22929060],118,9]
%
% END OF LATTICE L119.9
%
\hbox{}\par\smallskip%
%
% LATTICE L119.10
%
\leavevmode%
\llap{\spacer}%
${L_{119.10}}=\pdualwitharg{5}({L_{119.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-138373620, 39939900, -101640; 39939900, -11528170, 29335; -101640, 29335, -74],[4620, 5, 1],[421, 933, 597, 7291, 431, 6064, 5125, 1135, 1851, 4, -83, -2, 2059, 249, 5836, 6095, 1840, 4694; 1471, 3260, 2086, 25476, 1506, 21189, 17908, 3966, 6468, 14, -290, -7, 7194, 870, 20391, 21296, 6429, 16401; 4880, 10830, 6937, 84810, 5018, 70665, 59752, 13245, 21637, 55, -960, -28, 23760, 2878, 67515, 70532, 21300, 54362],118,10]
%
% END OF LATTICE L119.10
%
\hbox{}\par\smallskip%
%
% LATTICE L119.11
%
\leavevmode%
\llap{\spacer}%
${L_{119.11}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{119.1}}))$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3039962517263595, -1453511696175, 1011903297369195; -1453511696175, 694974441, -483826122780; 1011903297369195, -483826122780, 336829246219901],[1155, 21, 1],[-601934, -24990, 22843, -1183706, -2315018, -1108469, -62254327, -2577293, -14665397, -1997935, -2574879, -2622712, -8349147, -8084458, -2358023, -109337027, -3489068, -13933162; -479, -16, 0, -1144, -2155, -1011, -56430, -2320, -13090, -1756, -2219, -2235, -7051, -6785, -1969, -91080, -2895, -11495; 1808331, 75075, -68625, 3556091, 6954780, 3330064, 187024530, 7742707, 44057805, 6002199, 7735455, 7879155, 25082519, 24287340, 7083976, 328470450, 10481863, 41858025],118,11]
%
% END OF LATTICE L119.11
%
\hbox{}\par\smallskip%
%
% LATTICE L119.12
%
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${L_{119.12}}=\pdualwitharg{7}({L_{119.1}})$%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-565104540, -113263920, 244860; -113263920, -22701490, 49077; 244860, 49077, -106],[4620, 7, 1],[-230, -511, -234, -4009, -1187, -478, -14153, -628, -734, -3, 45, 1, -1117, -677, -454, -16603, -1003, -1829; 1155, 2566, 1175, 20130, 5960, 2400, 71060, 3153, 3685, 15, -226, -5, 5610, 3400, 2280, 83380, 5037, 9185; 3451, 7616, 3470, 59136, 17430, 6987, 206360, 9114, 10560, 14, -686, -5, 17094, 10290, 6873, 251020, 15141, 27555],118,12]
%
% END OF LATTICE L119.12
%
\hbox{}\par\smallskip%
%
% LATTICE L119.13
%
\leavevmode%
\llap{\spacer}%
${L_{119.13}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{119.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2943145648735215, -1792823830875, 979399436299515; -1792823830875, 1092102693, -596603382540; 979399436299515, -596603382540, 325917698376913],[1155, 33, 1],[-797774, -32670, 197173, -144750, -3104474, -10388395, -7573421, -3447013, -12473617, -2667223, -3432327, -24452152, -1010249, -10755514, -21951577, -13216561, -4637980, -11781362; -479, -16, 0, -104, -2155, -7077, -5130, -2320, -8330, -1756, -2219, -15645, -641, -6785, -13783, -8280, -2895, -7315; 2397351, 98175, -592515, 434981, 9329100, 31217648, 22758510, 10358447, 37483845, 8015139, 10314315, 73479945, 3035849, 32320860, 65965592, 39716430, 13937363, 35403585],118,13]
%
% END OF LATTICE L119.13
%
\hbox{}\par\smallskip%
%
% LATTICE L119.14
%
\leavevmode%
\llap{\spacer}%
${L_{119.14}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{119.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3118735177014765, -1900632015975, 622634877097695; -1900632015975, 1158290735, -379448627250; 622634877097695, -379448627250, 124304940360323],[1155, 35, 1],[-472112, -19600, 3583, -2785276, -363150, -2608229, -9765629, -6064367, -2300509, -1567043, -2019555, -411414, -19645429, -1268174, -5548391, -17151217, -8209718, -2185632; -479, -16, 0, -3432, -431, -3033, -11286, -6960, -2618, -1756, -2219, -447, -21153, -1357, -5907, -18216, -8685, -2299; 2364775, 98175, -17947, 13951245, 1818992, 13064430, 48915328, 30375975, 11523083, 7849205, 10115805, 2060743, 98402535, 6352192, 27791490, 85909208, 41121885, 10947673],118,14]
%
% END OF LATTICE L119.14
%
\hbox{}\par\smallskip%
%
% LATTICE L119.15
%
\leavevmode%
\llap{\spacer}%
${L_{119.15}}=\pdualwitharg{11}({L_{119.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{*}{4}{44}\above{1pt}{1pt}{b}{2}{770}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{220}\above{1pt}{1pt}{b}{2}{462}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13495020\\97020\\-2989140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
97020\\-682\\21791
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2989140\\21791\\-656246
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1362929\\-73287480\\3774540
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\kern3pt%
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10188\\547829\\-28215
\end{tabular}\endgroup%
\kern3pt%
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-294320\\-15826200\\815099
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-139\\-7473\\385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143\\7690\\-396
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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278\\14945\\-770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-325\\-17478\\900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2423\\-130300\\6710
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12095\\-650412\\33495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5857\\-314960\\16220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-3980\\-214023\\11022
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4752\\-255535\\13160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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\kern3pt%
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\kern3pt%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-1980\\-10780\\-3660\\-15620\\-55440\\-23780\\-13860\\-13580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-770\\-5005\\-1890\\-8470\\-31185\\-13580\\-8085\\-8155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13495020, 97020, -2989140; 97020, -682, 21791; -2989140, 21791, -656246],[4620, 11, 1],[-139, 143, 278, -325, -2423, -12095, -5857, -3980, -4752, -993, -2407, -8202, -1939, -6633, -19601, -7727, -3944, -3172; -7473, 7690, 14945, -17478, -130300, -650412, -314960, -214023, -255535, -53397, -129430, -441035, -104262, -356660, -1053948, -415480, -212067, -170555; 385, -396, -770, 900, 6710, 33495, 16220, 11022, 13160, 2750, 6666, 22715, 5370, 18370, 54285, 21400, 10923, 8785],118,15]
%
% END OF LATTICE L119.15
%
\hbox{}\par\smallskip%
%
% LATTICE L119.16
%
\leavevmode%
\llap{\spacer}%
${L_{119.16}}=\pdualwitharg{2.3}({L_{119.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{4}{12}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{9240}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16790937240\\44351071380\\-4187073660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44351071380\\117147572328\\-11059609128
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4187073660\\-11059609128\\1044110021
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3527268821\\-4979205\\14092256640
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9316664280\\-13151701\\37222233600
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16790937240, 44351071380, -4187073660; 44351071380, 117147572328, -11059609128; -4187073660, -11059609128, 1044110021],[4620, 12, 1],[-22423, -1491, 1571, -45275, -86608, -287760, -1151611, -95103, -3778039, -73211, -94143, -671009, -305383, -295928, -604824, -2004311, -128043, -3585779; 31, 2, 0, 66, 125, 413, 1650, 136, 5390, 104, 133, 945, 429, 415, 847, 2805, 179, 5005; -89592, -5958, 6300, -180862, -345990, -1149596, -4600695, -379940, -15093540, -292488, -376122, -2680860, -1220098, -1182330, -2416484, -8007945, -511580, -14326620],118,16]
%
% END OF LATTICE L119.16
%
\hbox{}\par\smallskip%
%
% LATTICE L119.17
%
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0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2694325115719545, -2214527682675, 537642814822575; -2214527682675, 1820171155, -441900975450; 537642814822575, -441900975450, 107284675722271],[1155, 55, 1],[-590952, -24200, 29209, -321676, -459934, -23085883, -1122015, -7660207, -1847985, -1975763, -2542515, -3622610, -2245039, -1593438, -48782137, -1958043, -10306798, -1745416; -479, -16, 0, -312, -431, -21231, -1026, -6960, -1666, -1756, -2219, -3129, -1923, -1357, -41349, -1656, -8685, -1463; 2961475, 121275, -146377, 1612035, 2304896, 115691730, 5622824, 38388075, 9260923, 9901265, 12741465, 18154213, 11250705, 7985296, 244464990, 9812464, 51651105, 8746913],118,17]
%
% END OF LATTICE L119.17
%
\hbox{}\par\smallskip%
%
% LATTICE L119.18
%
\leavevmode%
\llap{\spacer}%
${L_{119.18}}=\pdualwitharg{3.5}({L_{119.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{*}{4}{60}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{b}{2}{220}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{132}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{462}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-233819479740, -31129384440, 149590980; -31129384440, -4144387650, 19915695; 149590980, 19915695, -95702],[4620, 15, 1],[14432, 32029, 20516, 83609, 14841, 69666, 176723, 13058, 63996, 163, -2839, -83, 23421, 8511, 66554, 208585, 20997, 160767; -105889, -235000, -150528, -613448, -108890, -511147, -1296636, -95808, -469546, -1196, 20830, 609, -171842, -62446, -488313, -1530408, -154057, -1179563; 522915, 1160490, 743337, 3029290, 537708, 2524060, 6402792, 473095, 2318547, 5895, -102870, -3003, 848650, 308388, 2411500, 7557792, 760795, 5825127],118,18]
%
% END OF LATTICE L119.18
%
\hbox{}\par\smallskip%
%
% LATTICE L119.19
%
\leavevmode%
\llap{\spacer}%
${L_{119.19}}=\pdualwitharg{7.11}(\pfillwitharg{2}({L_{119.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${154}\above{1pt}{1pt}{}{4}{77}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{385}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2727261854690835, -2636231534475, 778178423715000; -2636231534475, 2548239617, -752204448765; 778178423715000, -752204448765, 222040160204642],[1155, 77, 1],[-1005971, -41195, 35514, -547591, -3914734, -5614159, -9550031, -13039966, -2247016, -3363336, -4328112, -4404821, -3821716, -13562494, -11863069, -16665811, -17545183, -2122291; -479, -16, 0, -312, -2155, -3033, -5130, -6960, -1190, -1756, -2219, -2235, -1923, -6785, -5907, -8280, -8685, -1045; 3525599, 144375, -124465, 1919127, 13719860, 19675788, 33469730, 45700809, 7875055, 11787391, 15168615, 15437455, 13393863, 47532100, 41576172, 58408210, 61490121, 7437935],118,19]
%
% END OF LATTICE L119.19
%
\hbox{}\par\smallskip%
%
% LATTICE L119.20
%
\leavevmode%
\llap{\spacer}%
${L_{119.20}}=\pdualwitharg{2.5}({L_{119.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{*}{4}{20}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{*}{2}{660}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{s}{2}{616}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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98544258120\\-197066100\\-24668674800
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\kern3pt%
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-197066100\\394120\\49331740
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\kern3pt%
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-24668674800\\49331740\\6175332059
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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91026\\-20945\\363790
\end{tabular}\endgroup%
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126396\\-29085\\505148
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845942\\-194667\\3380850
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[98544258120, -197066100, -24668674800; -197066100, 394120, 49331740; -24668674800, 49331740, 6175332059],[4620, 20, 1],[74209, 91026, 126396, 845942, 53923, 1636261, 359782, 342100, 632022, 18556, 1739, 3475, 174473, 20289, 961159, 252467, 307255, 796559; -17074, -20945, -29085, -194667, -12409, -376551, -82797, -78729, -145453, -4271, -400, -798, -40128, -4667, -221109, -58080, -70686, -183260; 296580, 363790, 505148, 3380850, 215506, 6539400, 1437887, 1367220, 2525908, 74160, 6950, 13888, 697290, 81086, 3841320, 1008997, 1227960, 3183488],118,20]
%
% END OF LATTICE L119.20
%
\hbox{}\par\smallskip%
%
% LATTICE L119.21
%
\leavevmode%
\llap{\spacer}%
${L_{119.21}}=\pdualwitharg{3.7}({L_{119.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{*}{4}{84}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{b}{2}{308}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{4620}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{330}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-309350580\\123427920\\-22808940
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\kern3pt%
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123427920\\-49246554\\9100581
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22808940\\9100581\\-1681510
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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27669552\\67606199\\-9430575
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3329\\8134\\-1134
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43\\105\\-15
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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-1220399\\-2981880\\415800
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\HardButStrongLineBreak\kern3pt%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-309350580, 123427920, -22808940; 123427920, -49246554, 9100581; -22808940, 9100581, -1681510],[4620, 21, 1],[-120, 3329, 43, -27579, -49931, -11133, -1220399, -24551, -134137, -16824, -37217, -16987, -96677, -85621, -11459, -1016209, -14985, -52247; -293, 8134, 105, -67386, -122000, -27202, -2981880, -59987, -327745, -41107, -90934, -41505, -236214, -209200, -27998, -2482920, -36613, -127655; 42, -1134, -15, 9394, 17010, 3793, 415800, 8365, 45705, 5733, 12684, 5790, 32956, 29190, 3907, 346500, 5110, 17820],118,21]
%
% END OF LATTICE L119.21
%
\hbox{}\par\smallskip%
%
% LATTICE L119.22
%
\leavevmode%
\llap{\spacer}%
${L_{119.22}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{119.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8594044582510695, -116153244900, -2860027960202490; -116153244900, -1532370, -38654852775; -2860027960202490, -38654852775, -951793984148771],[1155, 105, 1],[-341940851, 165450195, 15127472, -1584645201, -626506828, -1472251579, -16414957289, -3367851532, -3776072434, -2492803834, -3000194880, -582076409, -8809418644, -1641288920, -2307408001, -21003672553, -3246552733, -2460365873; -11189, 5414, 495, -51854, -20501, -48176, -537141, -110205, -123563, -81571, -98174, -19047, -288266, -53707, -75504, -687291, -106235, -80509; 1027491675, -497158200, -45456258, 4761670760, 1882578663, 4423941265, 49324998492, 10119994150, 11346649512, 7490579670, 9015229545, 1749070527, 26471257505, 4931878413, 6933487195, 63113543262, 9755505655, 7393107447],118,22]
%
% END OF LATTICE L119.22
%
\hbox{}\par\smallskip%
%
% LATTICE L119.23
%
\leavevmode%
\llap{\spacer}%
${L_{119.23}}=\pdualwitharg{2.7}({L_{119.1}})$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[134082647160, -212016420, -33543653220; -212016420, 335384, 53040456; -33543653220, 53040456, 8391665105],[4620, 28, 1],[-192723, -242925, -244657, -2319055, -744938, -649730, -5014061, -957863, -1271767, -54241, -4039, -2307, -397549, -242918, -338126, -3135751, -770885, -1444837; -22089, -27842, -28040, -265782, -85375, -74463, -574640, -109776, -145750, -6216, -463, -265, -45573, -27845, -38757, -359425, -88359, -165605; -770224, -970858, -977780, -9268182, -2977170, -2596668, -20038865, -3828132, -5082660, -216776, -16142, -9220, -1588818, -970830, -1351332, -12532135, -3080868, -5774340],118,23]
%
% END OF LATTICE L119.23
%
\hbox{}\par\smallskip%
%
% LATTICE L119.24
%
\leavevmode%
\llap{\spacer}%
${L_{119.24}}=\pdualwitharg{3.11}({L_{119.1}})$%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30166507140, 8075760, -19348560; 8075760, -2046, 5181; -19348560, 5181, -12410],[4620, 33, 1],[-229, -555, -1891, -149, -1529, -1506, -1781, -303, -731, -32, 33, 64, -25, -559, -930, -1351, -306, -1096; -3897, -9442, -32165, -2534, -26000, -25606, -30280, -5151, -12425, -543, 562, 1085, -426, -9520, -15834, -23000, -5209, -18655; 355410, 861366, 2934855, 231250, 2373030, 2337335, 2764140, 470261, 1134525, 49665, -51216, -99330, 38800, 867570, 1443365, 2096760, 474914, 1701000],118,24]
%
% END OF LATTICE L119.24
%
\hbox{}\par\smallskip%
%
% LATTICE L119.25
%
\leavevmode%
\llap{\spacer}%
${L_{119.25}}=\pdualwitharg{5.7}({L_{119.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-20813100, 212520, 106260; 212520, -2170, -1085; 106260, -1085, -542],[4620, 35, 1],[0, -1, 0, 25, 3, 10, 73, 22, 8, 5, 11, 1, 85, 5, 10, 59, 13, 3; 17, -98, -1, 2442, 300, 1014, 7436, 2253, 825, 523, 1178, 109, 9438, 564, 1146, 6820, 1527, 363; -35, 0, 2, 0, -14, -75, -616, -210, -88, -70, -210, -23, -2310, -154, -345, -2156, -525, -143],118,25]
%
% END OF LATTICE L119.25
%
\hbox{}\par\smallskip%
%
% LATTICE L119.26
%
\leavevmode%
\llap{\spacer}%
${L_{119.26}}=\pdualwitharg{3.5.11}(\pfillwitharg{2}({L_{119.1}}))$%
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{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8781033812989515, -146908931400, -2922255585777990; -146908931400, -2408010, -48890079975; -2922255585777990, -48890079975, -972502542465823],[1155, 165, 1],[-398341483, 192739635, 123358528, -167819763, -729843332, -12005602229, -1738405019, -3923348156, -2799301018, -2903969642, -3495050760, -4746594103, -932950612, -1912005568, -18815968271, -2224368331, -3782044469, -1823934557; -11189, 5414, 3465, -4714, -20501, -337232, -48831, -110205, -78631, -81571, -98174, -133329, -26206, -53707, -528528, -62481, -106235, -51233; 1196969235, -579159900, -370677846, 504278620, 2193093243, 36075420535, 5223702312, 11789199070, 8411569824, 8726086710, 10502215845, 14262956169, 2803406695, 5745351513, 56539768285, 6683964822, 11364598135, 5480708499],118,26]
%
% END OF LATTICE L119.26
%
\hbox{}\par\smallskip%
%
% LATTICE L119.27
%
\leavevmode%
\llap{\spacer}%
${L_{119.27}}=\pdualwitharg{2.11}({L_{119.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[131295992520, 9933069300, -32835624360; 9933069300, 751476616, -2484146896; -32835624360, -2484146896, 8211813677],[4620, 44, 1],[-18797, -985, -8473, -14513, -96156, -927194, -111292, -299777, -355693, -73811, -91623, -639783, -78095, -274276, -1665830, -166532, -348479, -292563; 31, 2, 0, 18, 125, 1239, 150, 408, 490, 104, 133, 945, 117, 415, 2541, 255, 537, 455; -75152, -3938, -33880, -58026, -384450, -3707088, -444965, -1198560, -1422120, -295108, -366322, -2557940, -312234, -1096590, -6660192, -665815, -1393260, -1169700],118,27]
%
% END OF LATTICE L119.27
%
\hbox{}\par\smallskip%
%
% LATTICE L119.28
%
\leavevmode%
\llap{\spacer}%
${L_{119.28}}=\pdualwitharg{5.11}({L_{119.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L119.28
%
\hbox{}\par\smallskip%
%
% LATTICE L119.29
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11981997831331605, -203050230075, -3988409824877940; -203050230075, -3371214, -67588689630; -3988409824877940, -67588689630, -1327609398291389],[1155, 231, 1],[-550659865, 266439690, 121806098, -231990922, -5044609303, -2370903063, -12015693982, -5423563590, -2764072652, -4014393674, -4831493229, -4686859637, -1289693591, -13215604853, -3715834989, -15374627252, -5228228001, -1800980887; -11189, 5414, 2475, -4714, -102505, -48176, -244155, -110205, -56165, -81571, -98174, -95235, -26206, -268535, -75504, -312405, -106235, -36595; 1654294719, -800439255, -365930400, 696948119, 15155036820, 7122677111, 36097599195, 16293492908, 8303838870, 12060058626, 14514792600, 14080283745, 3874502996, 39702376560, 11163127349, 46188520875, 15706664897, 5410514475],118,29]
%
% END OF LATTICE L119.29
%
\hbox{}\par\smallskip%
%
% LATTICE L119.30
%
\leavevmode%
\llap{\spacer}%
${L_{119.30}}=\pdualwitharg{2.3.5}({L_{119.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{*}{4}{60}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{*}{2}{220}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{s}{2}{1848}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[697705395756600, 2062453151220, -174596107394940; 2062453151220, 6096717960, -516115102560; -174596107394940, -516115102560, 43691507766553],[4620, 60, 1],[6156059, 7885026, 11215558, 25477994, 4932097, 50355227, 33359815, 10648250, 59580604, 1851766, 122799, -4603, 3884031, 1474163, 24384473, 19491830, 8036675, 63739901; 218913, 280393, 398825, 905993, 175384, 1790614, 1186262, 378647, 2118655, 65847, 4367, -161, 138127, 52424, 867146, 693154, 285793, 2266649; 24602880, 31512750, 44823324, 101823590, 19711278, 201246220, 133323531, 42556060, 238115724, 7400640, 490770, -18396, 15522650, 5891538, 97453300, 77899701, 32118820, 254738484],118,30]
%
% END OF LATTICE L119.30
%
\hbox{}\par\smallskip%
%
% LATTICE L119.31
%
\leavevmode%
\llap{\spacer}%
${L_{119.31}}=\pdualwitharg{7.11}({L_{119.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${154}\above{1pt}{1pt}{*}{4}{308}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{1540}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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448140\\1033109\\-27542
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5761140, -16553460, 448140; -16553460, -38709286, 1033109; 448140, 1033109, -27542],[4620, 77, 1],[5376, 11981, 5499, 8585, 28017, 11305, 30463, 14899, 1589, 92, -1045, -31, 2351, 15727, 10567, 35153, 23377, 3879; -32091, -71518, -32825, -51246, -167240, -67482, -181840, -88935, -9485, -549, 6238, 185, -14034, -93880, -63078, -209840, -139545, -23155; -1116269, -2487716, -1141800, -1782564, -5817350, -2347323, -6325200, -3093552, -329930, -19096, 216986, 6435, -488166, -3265570, -2194137, -7299180, -4854003, -805435],118,31]
%
% END OF LATTICE L119.31
%
\hbox{}\par\smallskip%
%
% LATTICE L119.32
%
\leavevmode%
\llap{\spacer}%
${L_{119.32}}=\pdualwitharg{2.3.7}({L_{119.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12436753532280, 127286021940, -3109399639500; 127286021940, 1302729960, -31823667624; -3109399639500, -31823667624, 777402727571],[4620, 84, 1],[-92525, -6735, -3077, -219327, -404678, -188428, -5248076, -430585, -2424647, -324289, -410079, -413737, -1308991, -1262578, -367324, -8506786, -541981, -2160307; 5444, 395, 175, 12859, 23745, 11061, 308110, 25283, 142395, 19051, 24100, 24320, 76956, 74235, 21599, 500225, 31872, 127050; -369852, -26922, -12300, -876722, -1617630, -753208, -20978265, -1721188, -9692100, -1296288, -1639218, -1653840, -5232458, -5046930, -1468312, -34004355, -2166472, -8635440],118,32]
%
% END OF LATTICE L119.32
%
\hbox{}\par\smallskip%
%
% LATTICE L119.33
%
\leavevmode%
\llap{\spacer}%
${L_{119.33}}=\pdualwitharg{5.7.11}(\pfillwitharg{2}({L_{119.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L119.33
%
\hbox{}\par\smallskip%
%
% LATTICE L119.34
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1582305255300, 108008859420, 1029622440; 108008859420, -7372732710, -70282485; 1029622440, -70282485, -669986],[4620, 105, 1],[35, -3, -2, 25, 23, 35, 835, 92, 112, 85, 243, 26, 855, 169, 125, 2333, 188, 152; 787, -98, -45, 814, 608, 888, 20988, 2291, 2761, 2063, 5798, 615, 20086, 3952, 2912, 54252, 4359, 3509; -28770, 5670, 1647, -46970, -28434, -39365, -918456, -98945, -117513, -85785, -234780, -24558, -793100, -154854, -113375, -2105796, -168350, -134508],118,34]
%
% END OF LATTICE L119.34
%
\hbox{}\par\smallskip%
%
% LATTICE L119.35
%
\leavevmode%
\llap{\spacer}%
${L_{119.35}}=\pdualwitharg{2.3.11}({L_{119.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[123034662449160, 38445417780, -30785228032020; 38445417780, 12013320, -9619654572; -30785228032020, -9619654572, 7702953347599],[4620, 132, 1],[-3213805, -3963073, -27563603, -1117819, -11750136, -23742492, -7113997, -4954561, -12450723, -788093, -38825, -449683, -216147, -4257456, -13608556, -4891697, -4383201, -15562563; 104, 133, 945, 39, 415, 847, 255, 179, 455, 31, 2, 0, 6, 125, 413, 150, 136, 490; -12844128, -15838614, -110159280, -4467418, -46959990, -94888024, -28431435, -19801144, -49759920, -3149652, -155166, -1797180, -863842, -17015130, -54387256, -19549905, -17517676, -62196540],118,35]
%
% END OF LATTICE L119.35
%
\hbox{}\par\smallskip%
%
% LATTICE L119.36
%
\leavevmode%
\llap{\spacer}%
${L_{119.36}}=\pdualwitharg{2.5.7}({L_{119.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11245080, 586740, -2813580; 586740, 317800, -146720; -2813580, -146720, 703973],[4620, 140, 1],[-26063, -1699, 1, -165889, -20952, -148376, -276655, -342077, -129133, -87227, -111591, -22659, -1080301, -69680, -304784, -471059, -450953, -120105; 31, 2, 0, 198, 25, 177, 330, 408, 154, 104, 133, 27, 1287, 83, 363, 561, 537, 143; -104160, -6790, 4, -662970, -83734, -592980, -1105643, -1367100, -516076, -348600, -445970, -90556, -4317390, -278474, -1218060, -1882573, -1802220, -479996],118,36]
%
% END OF LATTICE L119.36
%
\hbox{}\par\smallskip%
%
% LATTICE L119.37
%
\leavevmode%
\llap{\spacer}%
${L_{119.37}}=\pdualwitharg{3.5.11}({L_{119.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[448140, -697620, 4620; -697620, 877470, -5775; 4620, -5775, 38],[4620, 165, 1],[3, 7, 4, 1, 1, -1, -1, -2, -2, -3, -7, -4, -1, -1, 1, 1, 2, 2; 73, 160, 84, 16, 2, -161, -60, -84, -70, -88, -190, -105, -26, -26, 21, 24, 49, 49; 10725, 23430, 12243, 2290, 132, -24640, -9072, -12595, -10437, -13035, -28050, -15477, -3830, -3828, 3080, 3528, 7205, 7203],118,37]
%
% END OF LATTICE L119.37
%
\hbox{}\par\smallskip%
%
% LATTICE L119.38
%
\leavevmode%
\llap{\spacer}%
${L_{119.38}}=\pdualwitharg{2.5.11}({L_{119.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[104095406488920, 236524942500, -26045673455340; 236524942500, 537430520, -59180819060; -26045673455340, -59180819060, 6516878396689],[4620, 220, 1],[63914, 3771, -3391, 31747, 45819, 2315611, 56394, 771925, 186835, 200991, 261134, 374258, 233158, 166105, 5101729, 102549, 1082410, 184032; -2401, -150, 70, -1278, -1811, -90699, -2203, -30066, -7252, -7754, -10005, -14287, -8877, -6313, -193641, -3890, -41019, -6965; 255420, 15070, -13552, 126870, 183106, 9253860, 225367, 3084840, 746648, 803220, 1043570, 1495648, 931770, 663806, 20388060, 409817, 4325640, 735448],118,38]
%
% END OF LATTICE L119.38
%
\hbox{}\par\smallskip%
%
% LATTICE L119.39
%
\leavevmode%
\llap{\spacer}%
${L_{119.39}}=\pdualwitharg{3.7.11}({L_{119.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-204522780, 13718706540, -138581520; 13718706540, -917494103526, 9268196091; -138581520, 9268196091, -93623990],[4620, 231, 1],[-246, 2657, 82, -1993, -40019, -8966, -89491, -19840, -9878, -13695, -30539, -14023, -7301, -71509, -9622, -77801, -12689, -4063; -21166, 228534, 7055, -171422, -3442130, -771189, -7697360, -1706493, -849635, -1177949, -2626764, -1206170, -627988, -6150790, -827631, -6692020, -1091442, -349480; -2094939, 22619520, 698280, -16966768, -340690350, -76329671, -761858580, -168902888, -84093990, -116589396, -259988190, -119382615, -62156122, -608784330, -81916109, -662353440, -108027227, -34590345],118,39]
%
% END OF LATTICE L119.39
%
\hbox{}\par\smallskip%
%
% LATTICE L119.40
%
\leavevmode%
\llap{\spacer}%
${L_{119.40}}=\pdualwitharg{3.5.7.11}(\pfillwitharg{2}({L_{119.1}}))$%
\nopagebreak\par%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[512364134824524440535, 343099219085878781295, 113625749281021060545; 343099219085878781295, 229752760071030031605, 76088280183229085310; 113625749281021060545, 76088280183229085310, 25198506339822517154],[1155, 1155, 1],[-479, -16, 0, -104, -431, -1011, -1026, -2320, -238, -1756, -2219, -447, -641, -1357, -1969, -1656, -2895, -209; -1312326125, -53741781, 9267164, -238110955, -1021360486, -2441246105, -2491628443, -5670275312, -586253750, -4387535950, -5646120294, -1149239579, -1661843070, -3538529174, -5158569995, -4348206047, -7629402863, -553718665; 3962643300, 162276345, -27982713, 718989635, 3084056283, 7371481865, 7523614182, 17121719615, 1770226617, 13248414795, 17048781750, 3470194332, 5018029730, 10684790193, 15576595595, 13129655532, 23037415940, 1671985002],118,40]
%
% END OF LATTICE L119.40
%
\hbox{}\par\smallskip%
%
% LATTICE L119.41
%
\leavevmode%
\llap{\spacer}%
${L_{119.41}}=\pdualwitharg{2.7.11}({L_{119.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[794873373099960, -1494630267900, -198887142059760; -1494630267900, 2810409448, 373974965672; -198887142059760, 373974965672, 49764021057131],[4620, 308, 1],[82247, 110805, 115567, 102157, 367098, 324268, 228816, 485377, 59327, 29907, 1349, -3413, 9997, 81518, 124348, 107646, 299701, 52827; 9685, 12486, 12730, 11070, 39355, 34485, 24245, 51126, 6200, 2990, 189, -55, 1605, 11345, 16215, 13780, 37599, 6475; 328636, 442750, 461780, 408198, 1466850, 1295712, 914305, 1939476, 237060, 119504, 5390, -13640, 39942, 325710, 496848, 430115, 1197504, 211080],118,41]
%
% END OF LATTICE L119.41
%
\hbox{}\par\smallskip%
%
% LATTICE L119.42
%
\leavevmode%
\llap{\spacer}%
${L_{119.42}}=\pdualwitharg{5.7.11}({L_{119.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-610030136100, 111762563220, -289978920; 111762563220, -20475825370, 53126535; -289978920, 53126535, -137842],[4620, 385, 1],[125, 331, 34, 295, 209, 455, 255, 668, 16, 35, -11, -2, 25, 47, 185, 129, 452, 16; 943, 2438, 247, 2118, 1488, 3216, 1796, 4677, 111, 227, -98, -13, 222, 384, 1464, 1012, 3513, 123; 100485, 243320, 23672, 195720, 133826, 282315, 155764, 397320, 9122, 13860, -14630, -803, 32970, 49126, 175065, 118664, 403095, 13747],118,42]
%
% END OF LATTICE L119.42
%
\hbox{}\par\smallskip%
%
% LATTICE L119.43
%
\leavevmode%
\llap{\spacer}%
${L_{119.43}}=\pdualwitharg{2.3.5.7}({L_{119.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2820347570040, 6624345420, -705694632720; 6624345420, 15829800, -1657513620; -705694632720, -1657513620, 176575724191],[4620, 420, 1],[12685237, 845721, -1153, 26725817, 10131932, 23928078, 133865738, 55183431, 62509553, 42243383, 54082899, 10986877, 174679123, 33810964, 49310082, 228663154, 72983349, 58333411; -33044, -2203, 3, -69619, -26393, -62331, -348711, -143749, -162833, -110041, -140882, -28620, -455026, -88075, -128449, -595650, -190116, -151954; 50696940, 3379950, -4608, 106810550, 40492578, 95629300, 534998541, 220542280, 249821352, 168826980, 216143970, 43909392, 698110490, 135126558, 197069260, 913859331, 291680200, 233131272],118,43]
%
% END OF LATTICE L119.43
%
\hbox{}\par\smallskip%
%
% LATTICE L119.44
%
\leavevmode%
\llap{\spacer}%
${L_{119.44}}=\pdualwitharg{2.3.5.11}({L_{119.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[652701837480, -634312140, -163316889120; -634312140, 616440, 158715480; -163316889120, 158715480, 40864610363],[4620, 660, 1],[-8257, -30138, -62654, -17142, -43151, -503621, -32215, -124130, -72112, -44918, -23037, -11791, -583, 611, 7321, 310, 55, -2333; 155, 617, 1295, 355, 894, 10430, 667, 2569, 1491, 925, 463, 217, 5, -30, -350, -19, -49, 21; -33000, -120450, -250404, -68510, -172458, -2012780, -128751, -496100, -288204, -179520, -92070, -47124, -2330, 2442, 29260, 1239, 220, -9324],118,44]
%
% END OF LATTICE L119.44
%
\hbox{}\par\smallskip%
%
% LATTICE L119.45
%
\leavevmode%
\llap{\spacer}%
${L_{119.45}}=\pdualwitharg{2.3.7.11}({L_{119.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1848}\above{1pt}{1pt}{*}{4}{924}\above{1pt}{1pt}{*}{2}{1320}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{4620}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{616}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L119.45
%
\hbox{}\par\smallskip%
%
% LATTICE L119.46
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-82462380, 278170200, 1136520; 278170200, -936952170, -3828825; 1136520, -3828825, -15646],[4620, 1155, 1],[-3323, -7805, -744, -2011, -4063, -2817, -4629, -3886, -262, -307, 545, 18, -409, -1745, -2023, -4083, -4584, -464; 6839, 16062, 1531, 4138, 8360, 5796, 9524, 7995, 539, 631, -1122, -37, 842, 3592, 4164, 8404, 9435, 955; -1914990, -4497570, -428703, -1158710, -2340954, -1622995, -2666916, -2238775, -150933, -176715, 314160, 10362, -235760, -1005774, -1165945, -2353176, -2641870, -267408],118,46]
%
% END OF LATTICE L119.46
%
\hbox{}\par\smallskip%
%
% LATTICE L119.47
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L119.47
%
\hbox{}\par\smallskip%
%
% LATTICE L119.48
%
\leavevmode%
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${L_{119.48}}=\pdualwitharg{2.3.5.7.11}({L_{119.1}})$%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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0\\-4620\\-1716\\-3780\\-10164\\-17820\\-8148\\-32340\\-2796
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1471312920, 409114559700, -102367811700; 409114559700, 113758753203480, -28464483482820; -102367811700, -28464483482820, 7122325070621],[4620, 4620, 1],[-12016, -15459, -3149, -4561, -9725, -14199, -5989, -21045, -1531, -3689, -246, 8, -674, -2839, -6741, -3434, -15600, -1610; 957217, 1224845, 248751, 359447, 765196, 1115818, 470364, 1651239, 119963, 286703, 19075, 33, 55193, 229940, 542742, 275988, 1251241, 128821; 3825360, 4894890, 994092, 1436470, 3057978, 4459180, 1879731, 6598900, 479412, 1145760, 76230, 132, 220570, 918918, 2168980, 1102941, 5000380, 514812],118,48]
%
% END OF LATTICE L119.48
%

\medskip%
%------------------------------WEYL-GROUP-W120--------------------------------
%
\leavevmode\llap{}%
$W_{120}$%
\qquad\llap{48} lattices, $\chi=40$%
\hfill%
$2222262222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L120.1
%
\leavevmode%
\llap{\spacer}%
${L_{120.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{120}}%
\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{s}{2}{2310}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-929927460\\291060\\161700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
291060\\-82\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
161700\\-55\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\17815\\36939
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\63840\\132360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\57904\\120043
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\2970\\6155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1484\\-3080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1485\\-3078
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1484\\3079
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
263\\390390\\809655
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\13360\\27706
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\99462\\206250
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-77\\-35\\-140\\-63\\-21\\-1155\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-60\\-420\\-240\\-120\\-9240\\-180\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\0\\154\\0\\-308\\-231\\-154\\-13860\\-308\\-1386
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-60\\0\\10\\0\\-15\\-20\\-2310\\-60\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-420\\-308\\0\\28\\0\\-28\\-4620\\-140\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-240\\-231\\-15\\0\\6\\-3\\-1155\\-42\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-120\\-154\\-20\\-28\\-3\\2\\0\\-4\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-9240\\-13860\\-2310\\-4620\\-1155\\0\\2310\\0\\-2310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-180\\-308\\-60\\-140\\-42\\-4\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-660\\-1386\\-330\\-924\\-330\\-66\\-2310\\0\\132
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-929927460, 291060, 161700; 291060, -82, -55; 161700, -55, -26],[4620, 1, 1],[12, 43, 39, 2, -1, -1, 1, 263, 9, 67; 17815, 63840, 57904, 2970, -1484, -1485, 1484, 390390, 13360, 99462; 36939, 132360, 120043, 6155, -3080, -3078, 3079, 809655, 27706, 206250],119,1]
%
% END OF LATTICE L120.1
%
\hbox{}\par\smallskip%
%
% LATTICE L120.2
%
\leavevmode%
\llap{\spacer}%
${L_{120.2}}=\pfillwitharg{2}({L_{120.1}})$%
{${}={}$}Nikulin $120$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{s}{2}{2310}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
123585\\-57750\\1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57750\\26986\\-539
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1155\\-539\\38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-308\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\15\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1548\\3311\\-77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
886\\1895\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2183\\4669\\-112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2212\\4731\\-114
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
944\\2019\\-49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66424\\142065\\-3465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
591\\1264\\-31
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1867\\3993\\-99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-77\\-35\\-70\\-63\\-21\\-1155\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-30\\-105\\-120\\-60\\-4620\\-45\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\0\\154\\0\\-154\\-231\\-154\\-13860\\-154\\-693
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-30\\0\\10\\0\\-15\\-20\\-2310\\-30\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-105\\-154\\0\\7\\0\\-14\\-2310\\-35\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-120\\-231\\-15\\0\\6\\-3\\-1155\\-21\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-60\\-154\\-20\\-14\\-3\\2\\0\\-2\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-4620\\-13860\\-2310\\-2310\\-1155\\0\\2310\\0\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-45\\-154\\-30\\-35\\-21\\-2\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-693\\-165\\-231\\-165\\-33\\-1155\\0\\33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[123585, -57750, 1155; -57750, 26986, -539; 1155, -539, 38],[1155, 1, 1],[-144, 7, 1548, 886, 2183, 2212, 944, 66424, 591, 1867; -308, 15, 3311, 1895, 4669, 4731, 2019, 142065, 1264, 3993; 7, 0, -77, -45, -112, -114, -49, -3465, -31, -99],119,2]
%
% END OF LATTICE L120.2
%
\hbox{}\par\smallskip%
%
% LATTICE L120.3
%
\leavevmode%
\llap{\spacer}%
${L_{120.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{s}{2}{770}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-262884930\\292215\\-87580185
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
292215\\-309\\97350
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87580185\\97350\\-29177362
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19597\\5600\\58842
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11142\\3185\\33455
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57623\\16478\\173019
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2303\\660\\6915
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1154\\-329\\-3465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
384\\-110\\-1153
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2307\\658\\6927
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-164509\\46970\\493955
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8071\\2305\\24234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19215\\5489\\57695
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-231\\-105\\-210\\-63\\-63\\-1155\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-30\\-105\\-40\\-60\\-1540\\-45\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\0\\462\\0\\-462\\-231\\-462\\-13860\\-462\\-693
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-30\\0\\30\\0\\-15\\-60\\-2310\\-90\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-105\\-462\\0\\21\\0\\-42\\-2310\\-105\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-40\\-231\\-15\\0\\2\\-3\\-385\\-21\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-60\\-462\\-60\\-42\\-3\\6\\0\\-6\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-1540\\-13860\\-2310\\-2310\\-385\\0\\770\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-45\\-462\\-90\\-105\\-21\\-6\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-55\\-693\\-165\\-231\\-55\\-33\\-385\\0\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-262884930, 292215, -87580185; 292215, -309, 97350; -87580185, 97350, -29177362],[1155, 3, 1],[-19597, -11142, -57623, -2303, 1154, 384, -2307, -164509, -8071, -19215; 5600, 3185, 16478, 660, -329, -110, 658, 46970, 2305, 5489; 58842, 33455, 173019, 6915, -3465, -1153, 6927, 493955, 24234, 57695],119,3]
%
% END OF LATTICE L120.3
%
\hbox{}\par\smallskip%
%
% LATTICE L120.4
%
\leavevmode%
\llap{\spacer}%
${L_{120.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{770}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{s}{2}{462}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{165}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-367112130\\444675\\-73378305
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
444675\\-515\\88880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73378305\\88880\\-14666842
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16979\\5600\\84980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5792\\1911\\28989
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49923\\16478\\249865
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-399\\132\\1997
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1000\\-329\\-5005
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
998\\-330\\-4995
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1999\\658\\10005
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85523\\28182\\428043
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6993\\2305\\35000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49945\\16467\\249975
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\0\\-385\\-35\\-350\\-315\\-105\\-1155\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-6\\-105\\-120\\-60\\-924\\-45\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\0\\770\\0\\-770\\-1155\\-770\\-13860\\-770\\-3465
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-6\\0\\2\\0\\-15\\-20\\-462\\-30\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-350\\-105\\-770\\0\\35\\0\\-70\\-2310\\-175\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-120\\-1155\\-15\\0\\30\\-15\\-1155\\-105\\-825
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-60\\-770\\-20\\-70\\-15\\10\\0\\-10\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-924\\-13860\\-462\\-2310\\-1155\\0\\462\\0\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-45\\-770\\-30\\-175\\-105\\-10\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-3465\\-165\\-1155\\-825\\-165\\-1155\\0\\165
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-367112130, 444675, -73378305; 444675, -515, 88880; -73378305, 88880, -14666842],[1155, 5, 1],[-16979, -5792, -49923, -399, 1000, 998, -1999, -85523, -6993, -49945; 5600, 1911, 16478, 132, -329, -330, 658, 28182, 2305, 16467; 84980, 28989, 249865, 1997, -5005, -4995, 10005, 428043, 35000, 249975],119,4]
%
% END OF LATTICE L120.4
%
\hbox{}\par\smallskip%
%
% LATTICE L120.5
%
\leavevmode%
\llap{\spacer}%
${L_{120.5}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{}{6}{14}\above{1pt}{1pt}{s}{2}{330}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{231}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-473981970\\597135\\-135381015
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
597135\\-721\\170555
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135381015\\170555\\-38668178
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4533\\800\\15874
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54121\\9555\\189525
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13328\\2354\\46673
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3728\\660\\13055
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
267\\-47\\-935
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1865\\-330\\-6531
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3736\\658\\13083
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114166\\20130\\399795
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13069\\2305\\45766
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93340\\16467\\326865
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-11\\-35\\-10\\-63\\-21\\-165\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\0\\-210\\-105\\-840\\-420\\-4620\\-315\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\22\\0\\-22\\-231\\-154\\-1980\\-154\\-693
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-210\\0\\70\\0\\-105\\-140\\-2310\\-210\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-105\\-22\\0\\1\\0\\-14\\-330\\-35\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-840\\-231\\-105\\0\\42\\-21\\-1155\\-147\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-420\\-154\\-140\\-14\\-21\\14\\0\\-14\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-4620\\-1980\\-2310\\-330\\-1155\\0\\330\\0\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-315\\-154\\-210\\-35\\-147\\-14\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1155\\-693\\-1155\\-231\\-1155\\-231\\-1155\\0\\231
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-473981970, 597135, -135381015; 597135, -721, 170555; -135381015, 170555, -38668178],[1155, 7, 1],[-4533, -54121, -13328, -3728, 267, 1865, -3736, -114166, -13069, -93340; 800, 9555, 2354, 660, -47, -330, 658, 20130, 2305, 16467; 15874, 189525, 46673, 13055, -935, -6531, 13083, 399795, 45766, 326865],119,5]
%
% END OF LATTICE L120.5
%
\hbox{}\par\smallskip%
%
% LATTICE L120.6
%
\leavevmode%
\llap{\spacer}%
${L_{120.6}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${154}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{l}{2}{77}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{}{6}{22}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-689883810\\902055\\-125392575
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
902055\\-1133\\163955
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-125392575\\163955\\-22791226
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29691\\5600\\163394
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50641\\9555\\278685
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7936\\1498\\43673
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3488\\660\\19195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1749\\-329\\-9625
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1745\\-330\\-9603
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3496\\658\\19239
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67982\\12810\\374115
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12229\\2305\\67298
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7940\\1497\\43695
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
154\\0\\-77\\-385\\-770\\-693\\-231\\-1155\\-77\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\165\\0\\-330\\-1155\\-1320\\-660\\-4620\\-495\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\0\\14\\0\\-154\\-231\\-154\\-1260\\-154\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-330\\0\\110\\0\\-165\\-220\\-2310\\-330\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-770\\-1155\\-154\\0\\77\\0\\-154\\-2310\\-385\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693\\-1320\\-231\\-165\\0\\66\\-33\\-1155\\-231\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-660\\-154\\-220\\-154\\-33\\22\\0\\-22\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-4620\\-1260\\-2310\\-2310\\-1155\\0\\210\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\-495\\-154\\-330\\-385\\-231\\-22\\0\\11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-63\\-165\\-231\\-165\\-33\\-105\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-689883810, 902055, -125392575; 902055, -1133, 163955; -125392575, 163955, -22791226],[1155, 11, 1],[-29691, -50641, -7936, -3488, 1749, 1745, -3496, -67982, -12229, -7940; 5600, 9555, 1498, 660, -329, -330, 658, 12810, 2305, 1497; 163394, 278685, 43673, 19195, -9625, -9603, 19239, 374115, 67298, 43695],119,6]
%
% END OF LATTICE L120.6
%
\hbox{}\par\smallskip%
%
% LATTICE L120.7
%
\leavevmode%
\llap{\spacer}%
${L_{120.7}}=\pdualwitharg{3}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{}{6}{6}\above{1pt}{1pt}{s}{2}{770}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-541505580\\364980\\244860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
364980\\-246\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
244860\\-165\\-82
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\49000\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\60880\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
116\\172249\\-231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\10395\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1484\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1485\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\2969\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\320705\\-385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\34150\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\87604\\-110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-231\\-105\\-420\\-63\\-63\\-1155\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-60\\-420\\-80\\-120\\-3080\\-180\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\0\\462\\0\\-924\\-231\\-462\\-13860\\-924\\-1386
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-60\\0\\30\\0\\-15\\-60\\-2310\\-180\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-420\\-924\\0\\84\\0\\-84\\-4620\\-420\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-80\\-231\\-15\\0\\2\\-3\\-385\\-42\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-120\\-462\\-60\\-84\\-3\\6\\0\\-12\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-3080\\-13860\\-2310\\-4620\\-385\\0\\770\\0\\-770
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-180\\-924\\-180\\-420\\-42\\-12\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-220\\-1386\\-330\\-924\\-110\\-66\\-770\\0\\44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-541505580, 364980, 244860; 364980, -246, -165; 244860, -165, -82],[4620, 3, 1],[33, 41, 116, 7, -1, -1, 2, 216, 23, 59; 49000, 60880, 172249, 10395, -1484, -1485, 2969, 320705, 34150, 87604; -63, -80, -231, -15, 0, 2, -3, -385, -42, -110],119,7]
%
% END OF LATTICE L120.7
%
\hbox{}\par\smallskip%
%
% LATTICE L120.8
%
\leavevmode%
\llap{\spacer}%
${L_{120.8}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${210}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{2310}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{s}{2}{154}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{55}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1708361050935\\-914315325\\568528899015
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-914315325\\489345\\-304276830
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
568528899015\\-304276830\\189201872074
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
704629\\315\\-2117325
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80342\\37\\-241418
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2107143\\1001\\-6331710
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20291\\11\\-60972
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6150\\7\\-18480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2664\\0\\-8005
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105139\\41\\-315930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1292217\\539\\-3882956
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
305583\\130\\-918240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
709135\\308\\-2130865
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210\\0\\-1155\\-105\\-1050\\-315\\-315\\-1155\\-105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-6\\-105\\-40\\-60\\-308\\-45\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\0\\2310\\0\\-2310\\-1155\\-2310\\-13860\\-2310\\-3465
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-6\\0\\6\\0\\-15\\-60\\-462\\-90\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1050\\-105\\-2310\\0\\105\\0\\-210\\-2310\\-525\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-40\\-1155\\-15\\0\\10\\-15\\-385\\-105\\-275
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-60\\-2310\\-60\\-210\\-15\\30\\0\\-30\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-308\\-13860\\-462\\-2310\\-385\\0\\154\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-45\\-2310\\-90\\-525\\-105\\-30\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-55\\-3465\\-165\\-1155\\-275\\-165\\-385\\0\\55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1708361050935, -914315325, 568528899015; -914315325, 489345, -304276830; 568528899015, -304276830, 189201872074],[1155, 15, 1],[704629, 80342, 2107143, 20291, 6150, 2664, 105139, 1292217, 305583, 709135; 315, 37, 1001, 11, 7, 0, 41, 539, 130, 308; -2117325, -241418, -6331710, -60972, -18480, -8005, -315930, -3882956, -918240, -2130865],119,8]
%
% END OF LATTICE L120.8
%
\hbox{}\par\smallskip%
%
% LATTICE L120.9
%
\leavevmode%
\llap{\spacer}%
${L_{120.9}}=\pdualwitharg{2}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{616}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{s}{2}{9240}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9748078688040\\-4076304540\\-2439132391080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4076304540\\1704568\\1019959600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2439132391080\\1019959600\\610311735431
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-214855\\245\\-858676
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187426\\225\\-749055
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-670018\\847\\-2677752
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35816\\55\\-143140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2601\\7\\-10395
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2309\\0\\-9228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30030\\27\\-120016
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5699084\\5775\\-22776600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90674\\95\\-362382
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-638195\\693\\-2550570
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-308\\-140\\-140\\-252\\-84\\-4620\\-28\\0
\end{tabular}\endgroup%
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0\\15\\0\\-60\\-105\\-240\\-120\\-9240\\-90\\-330
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-252\\-240\\-924\\-60\\0\\24\\-12\\-4620\\-84\\-660
\end{tabular}\endgroup%
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-84\\-120\\-616\\-80\\-28\\-12\\8\\0\\-8\\-132
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-9240\\-55440\\-9240\\-4620\\-4620\\0\\9240\\0\\-4620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-90\\-616\\-120\\-70\\-84\\-8\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-330\\-2772\\-660\\-462\\-660\\-132\\-4620\\0\\132
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9748078688040, -4076304540, -2439132391080; -4076304540, 1704568, 1019959600; -2439132391080, 1019959600, 610311735431],[4620, 4, 1],[-214855, -187426, -670018, -35816, -2601, -2309, -30030, -5699084, -90674, -638195; 245, 225, 847, 55, 7, 0, 27, 5775, 95, 693; -858676, -749055, -2677752, -143140, -10395, -9228, -120016, -22776600, -362382, -2550570],119,9]
%
% END OF LATTICE L120.9
%
\hbox{}\par\smallskip%
%
% LATTICE L120.10
%
\leavevmode%
\llap{\spacer}%
${L_{120.10}}=\pdualwitharg{5}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{770}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{}{6}{10}\above{1pt}{1pt}{s}{2}{462}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{660}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-235615380\\314160\\-30621360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
314160\\-410\\42625
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30621360\\42625\\-3616858
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7338\\-3712373\\18375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5257\\-2659572\\13164
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23831\\-12056352\\59675
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-244\\-123442\\611
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
615\\311136\\-1540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
611\\309111\\-1530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-613\\-310124\\1535
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32195\\-16287810\\80619
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5507\\-2786052\\13790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40985\\-20734758\\102630
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\0\\-385\\-35\\-700\\-315\\-105\\-1155\\-70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12\\-420\\-240\\-120\\-1848\\-180\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\0\\770\\0\\-1540\\-1155\\-770\\-13860\\-1540\\-6930
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-12\\0\\2\\0\\-15\\-20\\-462\\-60\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-700\\-420\\-1540\\0\\140\\0\\-140\\-4620\\-700\\-4620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-240\\-1155\\-15\\0\\30\\-15\\-1155\\-210\\-1650
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-120\\-770\\-20\\-140\\-15\\10\\0\\-20\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-1848\\-13860\\-462\\-4620\\-1155\\0\\462\\0\\-2310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-180\\-1540\\-60\\-700\\-210\\-20\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-660\\-6930\\-330\\-4620\\-1650\\-330\\-2310\\0\\660
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-235615380, 314160, -30621360; 314160, -410, 42625; -30621360, 42625, -3616858],[4620, 5, 1],[-7338, -5257, -23831, -244, 615, 611, -613, -32195, -5507, -40985; -3712373, -2659572, -12056352, -123442, 311136, 309111, -310124, -16287810, -2786052, -20734758; 18375, 13164, 59675, 611, -1540, -1530, 1535, 80619, 13790, 102630],119,10]
%
% END OF LATTICE L120.10
%
\hbox{}\par\smallskip%
%
% LATTICE L120.11
%
\leavevmode%
\llap{\spacer}%
${L_{120.11}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{77}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2811619893390\\-1387868790\\935895528645
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1387868790\\685083\\-461975745
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
935895528645\\-461975745\\311528753441
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
157057\\45\\-471831
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
627278\\185\\-1884470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
470460\\143\\-1413357
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
159202\\55\\-478275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1428\\1\\-4290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4045\\0\\-12152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
163480\\41\\-491127
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1437538\\385\\-4318655
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
476166\\130\\-1430499
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1105583\\308\\-3321395
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-33\\-105\\-30\\-63\\-63\\-165\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\35\\0\\-210\\-105\\-280\\-420\\-1540\\-315\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\66\\0\\-66\\-231\\-462\\-1980\\-462\\-693
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-210\\0\\210\\0\\-105\\-420\\-2310\\-630\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-105\\-66\\0\\3\\0\\-42\\-330\\-105\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-280\\-231\\-105\\0\\14\\-21\\-385\\-147\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-420\\-462\\-420\\-42\\-21\\42\\0\\-42\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-1540\\-1980\\-2310\\-330\\-385\\0\\110\\0\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-315\\-462\\-630\\-105\\-147\\-42\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-385\\-693\\-1155\\-231\\-385\\-231\\-385\\0\\77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2811619893390, -1387868790, 935895528645; -1387868790, 685083, -461975745; 935895528645, -461975745, 311528753441],[1155, 21, 1],[157057, 627278, 470460, 159202, 1428, 4045, 163480, 1437538, 476166, 1105583; 45, 185, 143, 55, 1, 0, 41, 385, 130, 308; -471831, -1884470, -1413357, -478275, -4290, -12152, -491127, -4318655, -1430499, -3321395],119,11]
%
% END OF LATTICE L120.11
%
\hbox{}\par\smallskip%
%
% LATTICE L120.12
%
\leavevmode%
\llap{\spacer}%
${L_{120.12}}=\pdualwitharg{7}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{420}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{}{6}{14}\above{1pt}{1pt}{s}{2}{330}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{924}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10400271420\\1413720\\2822820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1413720\\-182\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2822820\\-385\\-766
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\2637\\20785
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
133\\58440\\460740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\6589\\51964
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\875\\6930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-440\\-3464
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-438\\-3465
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\3079\\24248
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
179\\78705\\620070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\17146\\135100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
265\\116490\\917994
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-11\\-35\\-20\\-63\\-21\\-165\\-14\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-420\\-420\\-1680\\-840\\-9240\\-1260\\-4620
\end{tabular}\endgroup%
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-11\\0\\22\\0\\-44\\-231\\-154\\-1980\\-308\\-1386
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-20\\-420\\-44\\0\\4\\0\\-28\\-660\\-140\\-924
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-1680\\-231\\-105\\0\\42\\-21\\-1155\\-294\\-2310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-840\\-154\\-140\\-28\\-21\\14\\0\\-28\\-462
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-9240\\-1980\\-2310\\-660\\-1155\\0\\330\\0\\-2310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-1260\\-308\\-420\\-140\\-294\\-28\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4620\\-1386\\-2310\\-924\\-2310\\-462\\-2310\\0\\924
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10400271420, 1413720, 2822820; 1413720, -182, -385; 2822820, -385, -766],[4620, 7, 1],[6, 133, 15, 2, -1, -1, 7, 179, 39, 265; 2637, 58440, 6589, 875, -440, -438, 3079, 78705, 17146, 116490; 20785, 460740, 51964, 6930, -3464, -3465, 24248, 620070, 135100, 917994],119,12]
%
% END OF LATTICE L120.12
%
\hbox{}\par\smallskip%
%
% LATTICE L120.13
%
\leavevmode%
\llap{\spacer}%
${L_{120.13}}=\pdualwitharg{3.11}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${462}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{330}\above{1pt}{1pt}{l}{2}{231}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{}{6}{66}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2679381365430\\-1698378990\\891625764645
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1698378990\\1076559\\-565174665
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
891625764645\\-565174665\\296708977093
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1382671\\315\\-4154997
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
789534\\185\\-2372590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
377148\\91\\-1133349
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
201346\\55\\-605055
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13068\\7\\-39270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4949\\0\\-14872
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
204904\\41\\-615747
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1148458\\245\\-3451175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
598086\\130\\-1797279
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
126309\\28\\-379565
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
462\\0\\-231\\-1155\\-2310\\-693\\-693\\-1155\\-231\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\55\\0\\-330\\-1155\\-440\\-660\\-1540\\-495\\-55
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\0\\42\\0\\-462\\-231\\-462\\-1260\\-462\\-63
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-330\\0\\330\\0\\-165\\-660\\-2310\\-990\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-1155\\-462\\0\\231\\0\\-462\\-2310\\-1155\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693\\-440\\-231\\-165\\0\\22\\-33\\-385\\-231\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693\\-660\\-462\\-660\\-462\\-33\\66\\0\\-66\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-1540\\-1260\\-2310\\-2310\\-385\\0\\70\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-495\\-462\\-990\\-1155\\-231\\-66\\0\\33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-55\\-63\\-165\\-231\\-55\\-33\\-35\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2679381365430, -1698378990, 891625764645; -1698378990, 1076559, -565174665; 891625764645, -565174665, 296708977093],[1155, 33, 1],[1382671, 789534, 377148, 201346, 13068, 4949, 204904, 1148458, 598086, 126309; 315, 185, 91, 55, 7, 0, 41, 245, 130, 28; -4154997, -2372590, -1133349, -605055, -39270, -14872, -615747, -3451175, -1797279, -379565],119,13]
%
% END OF LATTICE L120.13
%
\hbox{}\par\smallskip%
%
% LATTICE L120.14
%
\leavevmode%
\llap{\spacer}%
${L_{120.14}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{}{6}{70}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{1155}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2884430377290\\-1814782200\\575857434075
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1814782200\\1141805\\-362309255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
575857434075\\-362309255\\114966125371
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
123177\\45\\-616985
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
295178\\111\\-1478526
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
368974\\143\\-1848165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24972\\11\\-125083
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1120\\1\\-5610
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9517\\0\\-47670
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
128214\\41\\-642215
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
676460\\231\\-3388341
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
373448\\130\\-1870575
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2601265\\924\\-13029555
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-55\\-35\\-50\\-315\\-105\\-165\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-42\\-105\\-840\\-420\\-924\\-315\\-1155
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\0\\110\\0\\-110\\-1155\\-770\\-1980\\-770\\-3465
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-42\\0\\14\\0\\-105\\-140\\-462\\-210\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-105\\-110\\0\\5\\0\\-70\\-330\\-175\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315\\-840\\-1155\\-105\\0\\210\\-105\\-1155\\-735\\-5775
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-420\\-770\\-140\\-70\\-105\\70\\0\\-70\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-924\\-1980\\-462\\-330\\-1155\\0\\66\\0\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-315\\-770\\-210\\-175\\-735\\-70\\0\\35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1155\\-3465\\-1155\\-1155\\-5775\\-1155\\-1155\\0\\1155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2884430377290, -1814782200, 575857434075; -1814782200, 1141805, -362309255; 575857434075, -362309255, 114966125371],[1155, 35, 1],[123177, 295178, 368974, 24972, 1120, 9517, 128214, 676460, 373448, 2601265; 45, 111, 143, 11, 1, 0, 41, 231, 130, 924; -616985, -1478526, -1848165, -125083, -5610, -47670, -642215, -3388341, -1870575, -13029555],119,14]
%
% END OF LATTICE L120.14
%
\hbox{}\par\smallskip%
%
% LATTICE L120.15
%
\leavevmode%
\llap{\spacer}%
${L_{120.15}}=\pdualwitharg{11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${154}\above{1pt}{1pt}{l}{2}{660}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{110}\above{1pt}{1pt}{l}{2}{308}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{}{6}{22}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88671660\\286440\\14553000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
286440\\-902\\-48455
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14553000\\-48455\\-2299006
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6648\\1130815\\18249
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23803\\4048860\\65340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1961\\333564\\5383
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1102\\187450\\3025
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-561\\-95424\\-1540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-553\\-94065\\-1518
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
557\\94744\\1529
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13273\\2257710\\36435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4993\\849300\\13706
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3377\\574422\\9270
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
154\\0\\-77\\-385\\-1540\\-693\\-231\\-1155\\-154\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\660\\0\\-660\\-4620\\-2640\\-1320\\-9240\\-1980\\-660
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\0\\14\\0\\-308\\-231\\-154\\-1260\\-308\\-126
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-660\\0\\110\\0\\-165\\-220\\-2310\\-660\\-330
\end{tabular}\endgroup%
\kern3pt%
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-1540\\-4620\\-308\\0\\308\\0\\-308\\-4620\\-1540\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693\\-2640\\-231\\-165\\0\\66\\-33\\-1155\\-462\\-330
\end{tabular}\endgroup%
\kern3pt%
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-231\\-1320\\-154\\-220\\-308\\-33\\22\\0\\-44\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-9240\\-1260\\-2310\\-4620\\-1155\\0\\210\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-154\\-1980\\-308\\-660\\-1540\\-462\\-44\\0\\44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-660\\-126\\-330\\-924\\-330\\-66\\-210\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-88671660, 286440, 14553000; 286440, -902, -48455; 14553000, -48455, -2299006],[4620, 11, 1],[6648, 23803, 1961, 1102, -561, -553, 557, 13273, 4993, 3377; 1130815, 4048860, 333564, 187450, -95424, -94065, 94744, 2257710, 849300, 574422; 18249, 65340, 5383, 3025, -1540, -1518, 1529, 36435, 13706, 9270],119,15]
%
% END OF LATTICE L120.15
%
\hbox{}\par\smallskip%
%
% LATTICE L120.16
%
\leavevmode%
\llap{\spacer}%
${L_{120.16}}=\pdualwitharg{2.3}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{1848}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{}{6}{24}\above{1pt}{1pt}{s}{2}{3080}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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96807480\\62476260\\-24208800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62476260\\40397304\\-15623520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24208800\\-15623520\\6053933
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-272270\\245\\-1088136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-83366\\75\\-333175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-941679\\847\\-3763452
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61193\\55\\-244560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7803\\7\\-31185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29975\\27\\-119796
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2138219\\1925\\-8545460
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105537\\95\\-421782
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-256660\\231\\-1025750
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-924\\-420\\-420\\-252\\-252\\-4620\\-84\\0
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0\\5\\0\\-60\\-105\\-80\\-120\\-3080\\-90\\-110
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-924\\0\\1848\\0\\-924\\-924\\-1848\\-55440\\-1848\\-2772
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-60\\0\\120\\0\\-60\\-240\\-9240\\-360\\-660
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-105\\-924\\0\\21\\0\\-84\\-4620\\-210\\-462
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-80\\-924\\-60\\0\\8\\-12\\-1540\\-84\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-120\\-1848\\-240\\-84\\-12\\24\\0\\-24\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-3080\\-55440\\-9240\\-4620\\-1540\\0\\3080\\0\\-1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-90\\-1848\\-360\\-210\\-84\\-24\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-110\\-2772\\-660\\-462\\-220\\-132\\-1540\\0\\44
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[96807480, 62476260, -24208800; 62476260, 40397304, -15623520; -24208800, -15623520, 6053933],[4620, 12, 1],[-272270, -83366, -941679, -61193, -7803, 1, -29975, -2138219, -105537, -256660; 245, 75, 847, 55, 7, 0, 27, 1925, 95, 231; -1088136, -333175, -3763452, -244560, -31185, 4, -119796, -8545460, -421782, -1025750],119,16]
%
% END OF LATTICE L120.16
%
\hbox{}\par\smallskip%
%
% LATTICE L120.17
%
\leavevmode%
\llap{\spacer}%
${L_{120.17}}=\pdualwitharg{5.11}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${770}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{l}{2}{385}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{}{6}{110}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2452817660370\\-2097849600\\489450839955
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2097849600\\1794265\\-418618255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
489450839955\\-418618255\\97668134327
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1024159\\315\\-5132435
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\HardButStrongLineBreak\kern3pt%
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350890\\111\\-1758438
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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279358\\91\\-1399965
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29828\\11\\-149479
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9680\\7\\-48510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10997\\0\\-55110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
151774\\41\\-760595
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
510404\\147\\-2557821
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
443008\\130\\-2220075
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280675\\84\\-1406565
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
770\\0\\-385\\-385\\-3850\\-3465\\-1155\\-1155\\-385\\0
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0\\33\\0\\-66\\-1155\\-1320\\-660\\-924\\-495\\-165
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-385\\-66\\0\\22\\0\\-165\\-220\\-462\\-330\\-165
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\kern3pt%
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-3850\\-1155\\-770\\0\\385\\0\\-770\\-2310\\-1925\\-1155
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\kern3pt%
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\kern3pt%
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-1155\\-660\\-770\\-220\\-770\\-165\\110\\0\\-110\\-165
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-924\\-1260\\-462\\-2310\\-1155\\0\\42\\0\\-105
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-495\\-770\\-330\\-1925\\-1155\\-110\\0\\55\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-165\\-315\\-165\\-1155\\-825\\-165\\-105\\0\\15
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2452817660370, -2097849600, 489450839955; -2097849600, 1794265, -418618255; 489450839955, -418618255, 97668134327],[1155, 55, 1],[1024159, 350890, 279358, 29828, 9680, 10997, 151774, 510404, 443008, 280675; 315, 111, 91, 11, 7, 0, 41, 147, 130, 84; -5132435, -1758438, -1399965, -149479, -48510, -55110, -760595, -2557821, -2220075, -1406565],119,17]
%
% END OF LATTICE L120.17
%
\hbox{}\par\smallskip%
%
% LATTICE L120.18
%
\leavevmode%
\llap{\spacer}%
${L_{120.18}}=\pdualwitharg{3.5}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${210}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{2310}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{420}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{}{6}{30}\above{1pt}{1pt}{s}{2}{154}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{220}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-105803493180\\-1708258860\\-114645300
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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228\\-3206\\-162645
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811\\-11473\\-577500
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293\\-4081\\-209594
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157\\-2194\\-112200
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405\\-5676\\-289190
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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-210\\-180\\-4620\\-180\\-2100\\-210\\-60\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-220\\-6930\\-330\\-4620\\-550\\-330\\-770\\0\\220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-105803493180, -1708258860, -114645300; -1708258860, -27580830, -1851015; -114645300, -1851015, -124226],[4620, 15, 1],[228, 57, 811, 10, -5, -7, 13, 293, 157, 405; -3206, -804, -11473, -143, 56, 99, -177, -4081, -2194, -5676; -162645, -40624, -577500, -7098, 3780, 4985, -9360, -209594, -112200, -289190],119,18]
%
% END OF LATTICE L120.18
%
\hbox{}\par\smallskip%
%
% LATTICE L120.19
%
\leavevmode%
\llap{\spacer}%
${L_{120.19}}=\pdualwitharg{7.11}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{l}{2}{1155}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{770}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{}{6}{154}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{77}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2482772885670\\-2497320210\\708417560235
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-2497320210\\2511971\\-712568395
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
708417560235\\-712568395\\202135057357
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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249050\\45\\-872839
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2986472\\555\\-10466610
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67933\\13\\-238083
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
253871\\55\\-889735
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2354\\1\\-8250
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18719\\0\\-65604
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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258353\\41\\-905443
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
620587\\105\\-2174955
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
754099\\130\\-2642871
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
477772\\84\\-1674435
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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22\\0\\-11\\-385\\-110\\-693\\-231\\-165\\-77\\0
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\end{tabular}\endgroup%
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-693\\-9240\\-231\\-1155\\0\\462\\-231\\-1155\\-1617\\-1155
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-165\\-4620\\-180\\-2310\\-330\\-1155\\0\\30\\0\\-105
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-77\\-3465\\-154\\-2310\\-385\\-1617\\-154\\0\\77\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1155\\-63\\-1155\\-231\\-1155\\-231\\-105\\0\\21
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2482772885670, -2497320210, 708417560235; -2497320210, 2511971, -712568395; 708417560235, -712568395, 202135057357],[1155, 77, 1],[249050, 2986472, 67933, 253871, 2354, 18719, 258353, 620587, 754099, 477772; 45, 555, 13, 55, 1, 0, 41, 105, 130, 84; -872839, -10466610, -238083, -889735, -8250, -65604, -905443, -2174955, -2642871, -1674435],119,19]
%
% END OF LATTICE L120.19
%
\hbox{}\par\smallskip%
%
% LATTICE L120.20
%
\leavevmode%
\llap{\spacer}%
${L_{120.20}}=\pdualwitharg{2.5}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${280}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{3080}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{}{6}{40}\above{1pt}{1pt}{s}{2}{1848}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{660}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3263651280120\\1098952557780\\-816184792500
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\kern3pt%
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1098952557780\\370044658760\\-274829719340
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-816184792500\\-274829719340\\204114213907
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-358707\\245\\-1434020
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-63328\\45\\-253167
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\HardButStrongLineBreak\kern3pt%
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-1146042\\847\\-4581500
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\HardButStrongLineBreak\kern3pt%
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-12896\\11\\-51552
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-5683\\7\\-22715
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-3001\\0\\-12000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-47750\\27\\-190900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1855244\\1155\\-7416948
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\HardButStrongLineBreak\kern3pt%
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-148614\\95\\-594130
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1054027\\693\\-4213770
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\0\\-1540\\-140\\-700\\-1260\\-420\\-4620\\-140\\0
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0\\3\\0\\-12\\-105\\-240\\-120\\-1848\\-90\\-330
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3263651280120, 1098952557780, -816184792500; 1098952557780, 370044658760, -274829719340; -816184792500, -274829719340, 204114213907],[4620, 20, 1],[-358707, -63328, -1146042, -12896, -5683, -3001, -47750, -1855244, -148614, -1054027; 245, 45, 847, 11, 7, 0, 27, 1155, 95, 693; -1434020, -253167, -4581500, -51552, -22715, -12000, -190900, -7416948, -594130, -4213770],119,20]
%
% END OF LATTICE L120.20
%
\hbox{}\par\smallskip%
%
% LATTICE L120.21
%
\leavevmode%
\llap{\spacer}%
${L_{120.21}}=\pdualwitharg{3.7}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{}{6}{42}\above{1pt}{1pt}{s}{2}{110}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{308}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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1325940\\34545\\-5182
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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9\\-3469\\-20823
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71\\-27380\\-164360
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26\\-10032\\-60225
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7\\-2710\\-16275
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\HardButStrongLineBreak\kern3pt%
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-1\\384\\2304
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-1\\387\\2324
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8\\-3076\\-18459
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78\\-30030\\-180235
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53\\-20412\\-122514
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125\\-48158\\-289058
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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6\\0\\-33\\-105\\-60\\-63\\-63\\-165\\-42\\0
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-60\\-420\\-132\\0\\12\\0\\-84\\-660\\-420\\-924
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0\\-1540\\-1386\\-2310\\-924\\-770\\-462\\-770\\0\\308
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-338844660, -8838060, 1325940; -8838060, -230286, 34545; 1325940, 34545, -5182],[4620, 21, 1],[9, 71, 26, 7, -1, -1, 8, 78, 53, 125; -3469, -27380, -10032, -2710, 384, 387, -3076, -30030, -20412, -48158; -20823, -164360, -60225, -16275, 2304, 2324, -18459, -180235, -122514, -289058],119,21]
%
% END OF LATTICE L120.21
%
\hbox{}\par\smallskip%
%
% LATTICE L120.22
%
\leavevmode%
\llap{\spacer}%
${L_{120.22}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{}{6}{210}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{}{2}{385}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-2677642275\\-10815\\-891097515
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-211450162521675\\-891097515\\-70368889873658
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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23685611\\800\\-71172450
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18852695\\637\\-56650111
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69640822\\2354\\-209262405
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3895874\\132\\-11706639
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-1395115\\-47\\4192155
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-3248303\\-110\\9760765
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19521162\\658\\-58658775
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39768972\\1342\\-119501041
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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68287499\\2305\\-205195830
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162571870\\5489\\-488509175
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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-105\\-315\\-2310\\-630\\-525\\-735\\-210\\0\\105\\0
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0\\-385\\-3465\\-1155\\-1155\\-1925\\-1155\\-385\\0\\385
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-635382642965070, -2677642275, -211450162521675; -2677642275, -10815, -891097515; -211450162521675, -891097515, -70368889873658],[1155, 105, 1],[23685611, 18852695, 69640822, 3895874, -1395115, -3248303, 19521162, 39768972, 68287499, 162571870; 800, 637, 2354, 132, -47, -110, 658, 1342, 2305, 5489; -71172450, -56650111, -209262405, -11706639, 4192155, 9760765, -58658775, -119501041, -205195830, -488509175],119,22]
%
% END OF LATTICE L120.22
%
\hbox{}\par\smallskip%
%
% LATTICE L120.23
%
\leavevmode%
\llap{\spacer}%
${L_{120.23}}=\pdualwitharg{2.7}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{}{6}{56}\above{1pt}{1pt}{s}{2}{1320}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}{924}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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17409273632520\\-13411901580\\-4356091005420
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\kern3pt%
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-4356091005420\\3355881760\\1089966718201
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-333719\\23\\-1333720
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-2012992\\135\\-8044995
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-1012701\\66\\-4047296
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-348727\\20\\-1393700
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-866\\0\\-3461
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-2312\\3\\-9240
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-329103\\26\\-1315272
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\HardButStrongLineBreak\kern3pt%
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-8936549\\660\\-35715240
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-994809\\72\\-3975790
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-6980975\\495\\-27899718
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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0\\-2310\\-2772\\-4620\\-462\\-4620\\-924\\-4620\\0\\924
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17409273632520, -13411901580, -4356091005420; -13411901580, 10332392, 3355881760; -4356091005420, 3355881760, 1089966718201],[4620, 28, 1],[-333719, -2012992, -1012701, -348727, -866, -2312, -329103, -8936549, -994809, -6980975; 23, 135, 66, 20, 0, 3, 26, 660, 72, 495; -1333720, -8044995, -4047296, -1393700, -3461, -9240, -1315272, -35715240, -3975790, -27899718],119,23]
%
% END OF LATTICE L120.23
%
\hbox{}\par\smallskip%
%
% LATTICE L120.24
%
\leavevmode%
\llap{\spacer}%
${L_{120.24}}=\pdualwitharg{3.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${462}\above{1pt}{1pt}{l}{2}{220}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{330}\above{1pt}{1pt}{l}{2}{924}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{}{6}{66}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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426\\-2450\\-490413
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531\\-3060\\-610280
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137\\-791\\-157206
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92\\-535\\-104940
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-13\\75\\14927
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\HardButStrongLineBreak\kern3pt%
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25\\-141\\-29238
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\HardButStrongLineBreak\kern3pt%
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251\\-1435\\-290360
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\HardButStrongLineBreak\kern3pt%
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295\\-1690\\-340692
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69\\-396\\-79570
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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462\\0\\-231\\-1155\\-4620\\-693\\-693\\-1155\\-462\\0
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0\\-220\\-126\\-330\\-924\\-110\\-66\\-70\\0\\4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-209062294980, -16412055660, -99611820; -16412055660, -1288398606, -7819845; -99611820, -7819845, -47462],[4620, 33, 1],[426, 531, 137, 92, -11, -13, 25, 251, 295, 69; -2450, -3060, -791, -535, 56, 75, -141, -1435, -1690, -396; -490413, -610280, -157206, -104940, 13860, 14927, -29238, -290360, -340692, -79570],119,24]
%
% END OF LATTICE L120.24
%
\hbox{}\par\smallskip%
%
% LATTICE L120.25
%
\leavevmode%
\llap{\spacer}%
${L_{120.25}}=\pdualwitharg{5.7}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{}{6}{70}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}{4620}\above{1pt}{1pt}{b}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-554\\-24377\\1386
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1383\\60852\\-3460
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1385\\60942\\-3465
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-9681\\-425965\\24220
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-49517\\-2178759\\123882
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-366563\\-16128882\\917070
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1953007980, 1372140, -756506520; 1372140, -910, 532455; -756506520, 532455, -293019526],[4620, 35, 1],[-8300, -36799, -20753, -554, 1383, 1385, -9681, -49517, -53945, -366563; -365203, -1619172, -913143, -24377, 60852, 60942, -425965, -2178759, -2373594, -16128882; 20765, 92064, 51920, 1386, -3460, -3465, 24220, 123882, 134960, 917070],119,25]
%
% END OF LATTICE L120.25
%
\hbox{}\par\smallskip%
%
% LATTICE L120.26
%
\leavevmode%
\llap{\spacer}%
${L_{120.26}}=\pdualwitharg{3.5.11}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2310}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{l}{2}{1155}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{}{6}{330}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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191623741\\5600\\-575806770
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\HardButStrongLineBreak\kern3pt%
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21788911\\637\\-65473111
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\HardButStrongLineBreak\kern3pt%
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51218422\\1498\\-153905325
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4502266\\132\\-13528779
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-11287925\\-329\\33918885
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-3754039\\-110\\11280445
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22562946\\658\\-67798995
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29250084\\854\\-87893057
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\HardButStrongLineBreak\kern3pt%
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78925147\\2305\\-237160770
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17081410\\499\\-51327625
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-571305132896190, -3179235675, -190125633354195; -3179235675, -16995, -1058023395; -190125633354195, -1058023395, -63272241709234],[1155, 165, 1],[191623741, 21788911, 51218422, 4502266, -11287925, -3754039, 22562946, 29250084, 78925147, 17081410; 5600, 637, 1498, 132, -329, -110, 658, 854, 2305, 499; -575806770, -65473111, -153905325, -13528779, 33918885, 11280445, -67798995, -87893057, -237160770, -51327625],119,26]
%
% END OF LATTICE L120.26
%
\hbox{}\par\smallskip%
%
% LATTICE L120.27
%
\leavevmode%
\llap{\spacer}%
${L_{120.27}}=\pdualwitharg{2.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${616}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{440}\above{1pt}{1pt}{l}{2}{77}\above{1pt}{1pt}{r}{2}{264}\above{1pt}{1pt}{}{6}{88}\above{1pt}{1pt}{s}{2}{840}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-70023\\245\\-279356
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-66536\\225\\-265485
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\HardButStrongLineBreak\kern3pt%
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-23498\\77\\-93772
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-18516\\55\\-73920
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-2795\\7\\-11165
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527\\0\\2112
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\HardButStrongLineBreak\kern3pt%
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-6282\\27\\-25036
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-17381\\63\\-69330
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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0\\-330\\-252\\-660\\-462\\-660\\-132\\-420\\0\\12
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[45341151240, 58529380140, -11313820980; 58529380140, 75553625048, -14604634220; -11313820980, -14604634220, 2823098701],[4620, 44, 1],[-70023, -66536, -23498, -18516, -2795, 527, -6282, -137024, -25490, -17381; 245, 225, 77, 55, 7, 0, 27, 525, 95, 63; -279356, -265485, -93772, -73920, -11165, 2112, -25036, -546420, -101662, -69330],119,27]
%
% END OF LATTICE L120.27
%
\hbox{}\par\smallskip%
%
% LATTICE L120.28
%
\leavevmode%
\llap{\spacer}%
${L_{120.28}}=\pdualwitharg{5.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${770}\above{1pt}{1pt}{l}{2}{132}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{1540}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{}{6}{110}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{b}{2}{220}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-37426620\\420420\\-660660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420420\\-4510\\7425
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660660\\7425\\-11662
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-639\\-623\\35805
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\HardButStrongLineBreak\kern3pt%
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-457\\-444\\25608
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\HardButStrongLineBreak\kern3pt%
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-188\\-182\\10535
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-21\\-20\\1177
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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55\\56\\-3080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\51\\-2970
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-54\\3025
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-256\\-252\\14343
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-481\\-472\\26950
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-325\\-318\\18210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
770\\0\\-385\\-385\\-7700\\-3465\\-1155\\-1155\\-770\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\132\\0\\-132\\-4620\\-2640\\-1320\\-1848\\-1980\\-660
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\0\\70\\0\\-1540\\-1155\\-770\\-1260\\-1540\\-630
\end{tabular}\endgroup%
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-385\\-132\\0\\22\\0\\-165\\-220\\-462\\-660\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7700\\-4620\\-1540\\0\\1540\\0\\-1540\\-4620\\-7700\\-4620
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3465\\-2640\\-1155\\-165\\0\\330\\-165\\-1155\\-2310\\-1650
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-1320\\-770\\-220\\-1540\\-165\\110\\0\\-220\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-1848\\-1260\\-462\\-4620\\-1155\\0\\42\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-770\\-1980\\-1540\\-660\\-7700\\-2310\\-220\\0\\220\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-660\\-630\\-330\\-4620\\-1650\\-330\\-210\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-37426620, 420420, -660660; 420420, -4510, 7425; -660660, 7425, -11662],[4620, 55, 1],[-639, -457, -188, -21, 55, 53, -54, -256, -481, -325; -623, -444, -182, -20, 56, 51, -54, -252, -472, -318; 35805, 25608, 10535, 1177, -3080, -2970, 3025, 14343, 26950, 18210],119,28]
%
% END OF LATTICE L120.28
%
\hbox{}\par\smallskip%
%
% LATTICE L120.29
%
\leavevmode%
\llap{\spacer}%
${L_{120.29}}=\pdualwitharg{3.7.11}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{l}{2}{385}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{2310}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{}{6}{462}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{231}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-779493889872390\\-4393986135\\-259467672399705
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4393986135\\-23793\\-1462612305
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-259467672399705\\-1462612305\\-86368185684614
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37842220\\800\\-113685759
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
150602041\\3185\\-452439295
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10114711\\214\\-30386658
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31119059\\660\\-93488010
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2229161\\-47\\6696855
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5189483\\-110\\15590267
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31190415\\658\\-93702378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28881811\\610\\-86766860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
109104016\\2305\\-327770751
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23612885\\499\\-70937930
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\0\\-33\\-1155\\-330\\-693\\-693\\-165\\-231\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\385\\0\\-2310\\-1155\\-3080\\-4620\\-1540\\-3465\\-385
\end{tabular}\endgroup%
\kern3pt%
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-33\\0\\6\\0\\-66\\-231\\-462\\-180\\-462\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-2310\\0\\2310\\0\\-1155\\-4620\\-2310\\-6930\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-330\\-1155\\-66\\0\\33\\0\\-462\\-330\\-1155\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693\\-3080\\-231\\-1155\\0\\154\\-231\\-385\\-1617\\-385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693\\-4620\\-462\\-4620\\-462\\-231\\462\\0\\-462\\-231
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-1540\\-180\\-2310\\-330\\-385\\0\\10\\0\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-3465\\-462\\-6930\\-1155\\-1617\\-462\\0\\231\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-385\\-63\\-1155\\-231\\-385\\-231\\-35\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-779493889872390, -4393986135, -259467672399705; -4393986135, -23793, -1462612305; -259467672399705, -1462612305, -86368185684614],[1155, 231, 1],[37842220, 150602041, 10114711, 31119059, -2229161, -5189483, 31190415, 28881811, 109104016, 23612885; 800, 3185, 214, 660, -47, -110, 658, 610, 2305, 499; -113685759, -452439295, -30386658, -93488010, 6696855, 15590267, -93702378, -86766860, -327770751, -70937930],119,29]
%
% END OF LATTICE L120.29
%
\hbox{}\par\smallskip%
%
% LATTICE L120.30
%
\leavevmode%
\llap{\spacer}%
${L_{120.30}}=\pdualwitharg{2.3.5}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${840}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{9240}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{}{6}{120}\above{1pt}{1pt}{s}{2}{616}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{220}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13546101316440\\-3465568260\\-3389441018040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3465568260\\887160\\867138000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3389441018040\\867138000\\848089804321
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8474184\\-36295\\-33867540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-522743\\-2239\\-2089171
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29733365\\-127358\\-118831020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-395903\\-1696\\-1582248
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-267902\\-1148\\-1070685
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1161\\5\\4640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900941\\-3858\\-3600660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13089185\\-56056\\-52311644
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3245593\\-13900\\-12971190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7933736\\-33979\\-31707610
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\0\\-4620\\-420\\-2100\\-1260\\-1260\\-4620\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-12\\-105\\-80\\-120\\-616\\-90\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\0\\9240\\0\\-4620\\-4620\\-9240\\-55440\\-9240\\-13860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-12\\0\\24\\0\\-60\\-240\\-1848\\-360\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2100\\-105\\-4620\\0\\105\\0\\-420\\-4620\\-1050\\-2310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-80\\-4620\\-60\\0\\40\\-60\\-1540\\-420\\-1100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-120\\-9240\\-240\\-420\\-60\\120\\0\\-120\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\-616\\-55440\\-1848\\-4620\\-1540\\0\\616\\0\\-1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-90\\-9240\\-360\\-1050\\-420\\-120\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-110\\-13860\\-660\\-2310\\-1100\\-660\\-1540\\0\\220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13546101316440, -3465568260, -3389441018040; -3465568260, 887160, 867138000; -3389441018040, 867138000, 848089804321],[4620, 60, 1],[-8474184, -522743, -29733365, -395903, -267902, 1161, -900941, -13089185, -3245593, -7933736; -36295, -2239, -127358, -1696, -1148, 5, -3858, -56056, -13900, -33979; -33867540, -2089171, -118831020, -1582248, -1070685, 4640, -3600660, -52311644, -12971190, -31707610],119,30]
%
% END OF LATTICE L120.30
%
\hbox{}\par\smallskip%
%
% LATTICE L120.31
%
\leavevmode%
\llap{\spacer}%
${L_{120.31}}=\pdualwitharg{7.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{l}{2}{4620}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{770}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{r}{2}{462}\above{1pt}{1pt}{}{6}{154}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}{308}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109544820\\480480\\-18082680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
480480\\-2002\\79695
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18082680\\79695\\-2983546
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1244\\-24883\\6875
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-27587\\-551820\\152460
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-283\\-5661\\1564
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-418\\-8365\\2310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
207\\4140\\-1144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
209\\4182\\-1155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1449\\-28981\\8008
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3371\\-67425\\18630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8081\\-161634\\44660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4993\\-99870\\27594
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\0\\-11\\-385\\-220\\-693\\-231\\-165\\-154\\0
\end{tabular}\endgroup%
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0\\4620\\0\\-4620\\-4620\\-18480\\-9240\\-9240\\-13860\\-4620
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-11\\0\\2\\0\\-44\\-231\\-154\\-180\\-308\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-4620\\0\\770\\0\\-1155\\-1540\\-2310\\-4620\\-2310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-220\\-4620\\-44\\0\\44\\0\\-308\\-660\\-1540\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-693\\-18480\\-231\\-1155\\0\\462\\-231\\-1155\\-3234\\-2310
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\-9240\\-154\\-1540\\-308\\-231\\154\\0\\-308\\-462
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-9240\\-180\\-2310\\-660\\-1155\\0\\30\\0\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-154\\-13860\\-308\\-4620\\-1540\\-3234\\-308\\0\\308\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4620\\-126\\-2310\\-924\\-2310\\-462\\-210\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-109544820, 480480, -18082680; 480480, -2002, 79695; -18082680, 79695, -2983546],[4620, 77, 1],[-1244, -27587, -283, -418, 207, 209, -1449, -3371, -8081, -4993; -24883, -551820, -5661, -8365, 4140, 4182, -28981, -67425, -161634, -99870; 6875, 152460, 1564, 2310, -1144, -1155, 8008, 18630, 44660, 27594],119,31]
%
% END OF LATTICE L120.31
%
\hbox{}\par\smallskip%
%
% LATTICE L120.32
%
\leavevmode%
\llap{\spacer}%
${L_{120.32}}=\pdualwitharg{2.3.7}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{264}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{}{6}{168}\above{1pt}{1pt}{s}{2}{440}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{308}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4422028574040\\-13371578220\\-1106462640360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13371578220\\40433736\\3345783840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1106462640360\\3345783840\\276854740763
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103555\\-521\\-413856
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-208547\\-1045\\-833455
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315262\\-1573\\-1259940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109296\\-535\\-436800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-286\\-1\\-1143
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-2\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-101246\\-519\\-404628
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-920464\\-4675\\-3678620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-307784\\-1559\\-1230054
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720853\\-3641\\-2880878
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-132\\-420\\-60\\-252\\-252\\-660\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\35\\0\\-420\\-105\\-560\\-840\\-3080\\-630\\-770
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\264\\0\\-132\\-924\\-1848\\-7920\\-1848\\-2772
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-420\\0\\840\\0\\-420\\-1680\\-9240\\-2520\\-4620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-105\\-132\\0\\3\\0\\-84\\-660\\-210\\-462
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-560\\-924\\-420\\0\\56\\-84\\-1540\\-588\\-1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-840\\-1848\\-1680\\-84\\-84\\168\\0\\-168\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\-3080\\-7920\\-9240\\-660\\-1540\\0\\440\\0\\-1540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-630\\-1848\\-2520\\-210\\-588\\-168\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-770\\-2772\\-4620\\-462\\-1540\\-924\\-1540\\0\\308
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4422028574040, -13371578220, -1106462640360; -13371578220, 40433736, 3345783840; -1106462640360, 3345783840, 276854740763],[4620, 84, 1],[-103555, -208547, -315262, -109296, -286, 7, -101246, -920464, -307784, -720853; -521, -1045, -1573, -535, -1, -2, -519, -4675, -1559, -3641; -413856, -833455, -1259940, -436800, -1143, 28, -404628, -3678620, -1230054, -2880878],119,32]
%
% END OF LATTICE L120.32
%
\hbox{}\par\smallskip%
%
% LATTICE L120.33
%
\leavevmode%
\llap{\spacer}%
${L_{120.33}}=\pdualwitharg{5.7.11}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{l}{2}{231}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{154}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{2310}\above{1pt}{1pt}{}{6}{770}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{385}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-799483527682455\\5744891460\\159614901694710
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5744891460\\-39655\\-1146952345
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
159614901694710\\-1146952345\\-31866718900886
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29675138\\89026214\\-148640855
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70859527\\212580492\\-354930807
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7931761\\23795497\\-39729680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4880593\\14641911\\-24446576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1748065\\-5244242\\8755945
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12208477\\-36625761\\61151475
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24458921\\73377421\\-122513160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13589135\\40767771\\-68067102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85557262\\256674091\\-428550815
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55550305\\166652412\\-278247900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\0\\-55\\-385\\-550\\-3465\\-1155\\-165\\-385\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\231\\0\\-462\\-1155\\-9240\\-4620\\-924\\-3465\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\0\\10\\0\\-110\\-1155\\-770\\-180\\-770\\-315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-462\\0\\154\\0\\-1155\\-1540\\-462\\-2310\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-550\\-1155\\-110\\0\\55\\0\\-770\\-330\\-1925\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3465\\-9240\\-1155\\-1155\\0\\2310\\-1155\\-1155\\-8085\\-5775
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1155\\-4620\\-770\\-1540\\-770\\-1155\\770\\0\\-770\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-924\\-180\\-462\\-330\\-1155\\0\\6\\0\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-385\\-3465\\-770\\-2310\\-1925\\-8085\\-770\\0\\385\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1155\\-315\\-1155\\-1155\\-5775\\-1155\\-105\\0\\105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-799483527682455, 5744891460, 159614901694710; 5744891460, -39655, -1146952345; 159614901694710, -1146952345, -31866718900886],[1155, 385, 1],[-29675138, -70859527, -7931761, -4880593, 1748065, 12208477, -24458921, -13589135, -85557262, -55550305; 89026214, 212580492, 23795497, 14641911, -5244242, -36625761, 73377421, 40767771, 256674091, 166652412; -148640855, -354930807, -39729680, -24446576, 8755945, 61151475, -122513160, -68067102, -428550815, -278247900],119,33]
%
% END OF LATTICE L120.33
%
\hbox{}\par\smallskip%
%
% LATTICE L120.34
%
\leavevmode%
\llap{\spacer}%
${L_{120.34}}=\pdualwitharg{3.5.7}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{330}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{}{6}{210}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}{1540}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27156360\\-389155515\\-11113598
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4008\\-9988\\359535
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217\\-542\\19509
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-155\\387\\-13930
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1226\\-3048\\109725
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\HardButStrongLineBreak\kern3pt%
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2396\\-5962\\214621
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\HardButStrongLineBreak\kern3pt%
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8145\\-20272\\729750
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\HardButStrongLineBreak\kern3pt%
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19221\\-47850\\1722490
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\28\\0\\-84\\-420\\-560\\-840\\-616\\-1260\\-1540
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-165\\-616\\-1980\\-462\\-660\\-385\\0\\22\\0\\-770
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-1260\\-4620\\-1260\\-2100\\-1470\\-420\\0\\420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1540\\-6930\\-2310\\-4620\\-3850\\-2310\\-770\\0\\1540
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-66357060, 950911500, 27156360; 950911500, -13626731370, -389155515; 27156360, -389155515, -11113598],[4620, 105, 1],[1385, 2187, 4008, 217, -153, -155, 1226, 2396, 8145, 19221; -3449, -5448, -9988, -542, 380, 387, -3048, -5962, -20272, -47850; 124155, 196112, 359535, 19509, -13680, -13930, 109725, 214621, 729750, 1722490],119,34]
%
% END OF LATTICE L120.34
%
\hbox{}\par\smallskip%
%
% LATTICE L120.35
%
\leavevmode%
\llap{\spacer}%
${L_{120.35}}=\pdualwitharg{2.3.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1848}\above{1pt}{1pt}{l}{2}{55}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{1320}\above{1pt}{1pt}{l}{2}{231}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{}{6}{264}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55246513448280\\-7881308820\\-13823545021440
\end{tabular}\endgroup%
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-7881308820\\1124904\\1972027200
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13823545021440\\1972027200\\3458867990623
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-17501580\\-166075\\-69945876
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-4254073\\-40370\\-17001600
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5791\\55\\23144
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\HardButStrongLineBreak\kern3pt%
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-1788263\\-16968\\-7146876
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\HardButStrongLineBreak\kern3pt%
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-12061509\\-114450\\-48204380
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-6615067\\-62770\\-26437422
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-1478444\\-14029\\-5908670
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\55\\0\\-660\\-1155\\-880\\-1320\\-3080\\-990\\-110
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\kern3pt%
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\kern3pt%
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\kern3pt%
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0\\-110\\-252\\-660\\-462\\-220\\-132\\-140\\0\\4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[55246513448280, -7881308820, -13823545021440; -7881308820, 1124904, 1972027200; -13823545021440, 1972027200, 3458867990623],[4620, 132, 1],[-17501580, -5440361, -5668237, -4254073, -607766, 5791, -1788263, -12061509, -6615067, -1478444; -166075, -51625, -53788, -40370, -5768, 55, -16968, -114450, -62770, -14029; -69945876, -21742655, -22653372, -17001600, -2428965, 23144, -7146876, -48204380, -26437422, -5908670],119,35]
%
% END OF LATTICE L120.35
%
\hbox{}\par\smallskip%
%
% LATTICE L120.36
%
\leavevmode%
\llap{\spacer}%
${L_{120.36}}=\pdualwitharg{2.5.7}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{440}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{}{6}{280}\above{1pt}{1pt}{s}{2}{264}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}{4620}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23520908916120\\4823474441940\\-5882186217600
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5882186217600\\-1206270343740\\1471036464701
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-829591\\23\\-3317240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1002034\\27\\-4006779
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2523649\\66\\-10091180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-174695\\4\\-698544
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2252\\0\\-9005
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17331451\\495\\-69302310
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-220\\-140\\-100\\-1260\\-420\\-660\\-140\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\21\\0\\-84\\-105\\-1680\\-840\\-1848\\-630\\-2310
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\kern3pt%
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\kern3pt%
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\end{tabular}\endgroup%
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\kern3pt%
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2310\\-13860\\-4620\\-2310\\-23100\\-4620\\-4620\\0\\4620
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[23520908916120, 4823474441940, -5882186217600; 4823474441940, 989158445240, -1206270343740; -5882186217600, -1206270343740, 1471036464701],[4620, 140, 1],[-829591, -1002034, -2523649, -174695, -2252, -1156, -812731, -4428749, -2467409, -17331451; 23, 27, 66, 4, 0, 3, 26, 132, 72, 495; -3317240, -4006779, -10091180, -698544, -9005, -4620, -3249820, -17708988, -9866290, -69302310],119,36]
%
% END OF LATTICE L120.36
%
\hbox{}\par\smallskip%
%
% LATTICE L120.37
%
\leavevmode%
\llap{\spacer}%
${L_{120.37}}=\pdualwitharg{3.5.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${2310}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{l}{2}{4620}\above{1pt}{1pt}{r}{2}{110}\above{1pt}{1pt}{}{6}{330}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{660}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
226380\\122925\\434
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2159\\4395\\-118668
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1513\\-3080\\83160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1542\\-3139\\84755
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3056\\6221\\-167970
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-6028\\12271\\-331324
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-35327\\71914\\-1941720
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8243\\16780\\-453070
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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0\\44\\0\\-132\\-4620\\-880\\-1320\\-616\\-1980\\-220
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-3465\\-880\\-1155\\-165\\0\\110\\-165\\-385\\-2310\\-550
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-3465\\-1320\\-2310\\-660\\-4620\\-165\\330\\0\\-660\\-330
\end{tabular}\endgroup%
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-1155\\-616\\-1260\\-462\\-4620\\-385\\0\\14\\0\\-70
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2310\\-1980\\-4620\\-1980\\-23100\\-2310\\-660\\0\\660\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-220\\-630\\-330\\-4620\\-550\\-330\\-70\\0\\20
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-41908020, -14474460, 226380; -14474460, -3791370, 122925; 226380, 122925, 434],[4620, 165, 1],[-50748, -12617, -16234, -2159, 1513, 1542, -3056, -6028, -35327, -8243; 103306, 25684, 33047, 4395, -3080, -3139, 6221, 12271, 71914, 16780; -2789325, -693484, -892290, -118668, 83160, 84755, -167970, -331324, -1941720, -453070],119,37]
%
% END OF LATTICE L120.37
%
\hbox{}\par\smallskip%
%
% LATTICE L120.38
%
\leavevmode%
\llap{\spacer}%
${L_{120.38}}=\pdualwitharg{2.5.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3080}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{l}{2}{385}\above{1pt}{1pt}{r}{2}{1320}\above{1pt}{1pt}{}{6}{440}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{*}{2}{220}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
191852125080\\147826140\\-48001790760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
147826140\\113960\\-36986400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48001790760\\-36986400\\12010145393
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-332909\\245\\-1330560
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\HardButStrongLineBreak\kern3pt%
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-59093\\45\\-236181
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-97764\\77\\-390740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12374\\11\\-49456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-5876\\7\\-23485
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2477\\0\\-9900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-43320\\27\\-173140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-154706\\105\\-618324
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-136758\\95\\-546590
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-88489\\63\\-353670
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3080\\0\\-1540\\-1540\\-7700\\-13860\\-4620\\-4620\\-1540\\0
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0\\33\\0\\-132\\-1155\\-2640\\-1320\\-1848\\-990\\-330
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\kern3pt%
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0\\-330\\-1260\\-660\\-2310\\-3300\\-660\\-420\\0\\60
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[191852125080, 147826140, -48001790760; 147826140, 113960, -36986400; -48001790760, -36986400, 12010145393],[4620, 220, 1],[-332909, -59093, -97764, -12374, -5876, -2477, -43320, -154706, -136758, -88489; 245, 45, 77, 11, 7, 0, 27, 105, 95, 63; -1330560, -236181, -390740, -49456, -23485, -9900, -173140, -618324, -546590, -353670],119,38]
%
% END OF LATTICE L120.38
%
\hbox{}\par\smallskip%
%
% LATTICE L120.39
%
\leavevmode%
\llap{\spacer}%
${L_{120.39}}=\pdualwitharg{3.7.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{l}{2}{1540}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{2310}\above{1pt}{1pt}{l}{2}{132}\above{1pt}{1pt}{r}{2}{154}\above{1pt}{1pt}{}{6}{462}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{924}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-273379260\\-3705845220\\-48145020
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48145020\\-652633443\\-8478782
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1871\\-10349\\807213
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-14769\\-81700\\6372520
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\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1463\\-8100\\631785
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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207\\1144\\-89232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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209\\1157\\-90244
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1658\\-9166\\714945
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1472\\-8140\\634915
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11007\\-60872\\4747974
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2361\\-13058\\1018514
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\0\\-33\\-1155\\-660\\-693\\-693\\-165\\-462\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\1540\\0\\-4620\\-4620\\-6160\\-9240\\-3080\\-13860\\-1540
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\kern3pt%
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-33\\0\\6\\0\\-132\\-231\\-462\\-180\\-924\\-126
\end{tabular}\endgroup%
\kern3pt%
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-1155\\-4620\\0\\2310\\0\\-1155\\-4620\\-2310\\-13860\\-2310
\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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-693\\-9240\\-462\\-4620\\-924\\-231\\462\\0\\-924\\-462
\end{tabular}\endgroup%
\kern3pt%
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-165\\-3080\\-180\\-2310\\-660\\-385\\0\\10\\0\\-70
\end{tabular}\endgroup%
\kern3pt%
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-462\\-13860\\-924\\-13860\\-4620\\-3234\\-924\\0\\924\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\-1540\\-126\\-2310\\-924\\-770\\-462\\-70\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-273379260, -3705845220, -48145020; -3705845220, -50234858058, -652633443; -48145020, -652633443, -8478782],[4620, 231, 1],[-1871, -14769, -492, -1463, 207, 209, -1658, -1472, -11007, -2361; -10349, -81700, -2722, -8100, 1144, 1157, -9166, -8140, -60872, -13058; 807213, 6372520, 212313, 631785, -89232, -90244, 714945, 634915, 4747974, 1018514],119,39]
%
% END OF LATTICE L120.39
%
\hbox{}\par\smallskip%
%
% LATTICE L120.40
%
\leavevmode%
\llap{\spacer}%
${L_{120.40}}=\pdualwitharg{3.5.7.11}(\pfillwitharg{2}({L_{120.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${330}\above{1pt}{1pt}{l}{2}{77}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{462}\above{1pt}{1pt}{l}{2}{165}\above{1pt}{1pt}{r}{2}{770}\above{1pt}{1pt}{}{6}{2310}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{1155}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[466432877615828970, 312341729450853795, 103439649717532590; 312341729450853795, 209156259427968930, 69267242162923035; 103439649717532590, 69267242162923035, 22939551749391383],[1155, 1155, 1],[45, 37, 13, 11, 1, 0, 41, 7, 130, 28; 324923176, 259753060, 88628535, 66241877, 3070926, 8141017, 337062523, 53976787, 983837949, 207775438; -981123660, -784338863, -267618810, -200021052, -9272835, -24582250, -1017779070, -162985918, -2970753555, -627389525],119,40]
%
% END OF LATTICE L120.40
%
\hbox{}\par\smallskip%
%
% LATTICE L120.41
%
\leavevmode%
\llap{\spacer}%
${L_{120.41}}=\pdualwitharg{2.7.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${88}\above{1pt}{1pt}{l}{2}{1155}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{3080}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{1848}\above{1pt}{1pt}{}{6}{616}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{*}{2}{308}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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514900044120\\72068345580\\-128789665620
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72068345580\\10087096712\\-18026135820
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-128789665620\\-18026135820\\32213588171
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-188803\\23\\-754820
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-1139132\\135\\-4554165
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\HardButStrongLineBreak\kern3pt%
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-52111\\6\\-208336
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\HardButStrongLineBreak\kern3pt%
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-197607\\20\\-790020
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-498\\0\\-1991
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-1156\\3\\-4620
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\HardButStrongLineBreak\kern3pt%
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-358961\\45\\-1435098
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[514900044120, 72068345580, -128789665620; 72068345580, 10087096712, -18026135820; -128789665620, -18026135820, 32213588171],[4620, 308, 1],[-188803, -1139132, -52111, -197607, -498, -1156, -185975, -459359, -562585, -358961; 23, 135, 6, 20, 0, 3, 26, 60, 72, 45; -754820, -4554165, -208336, -790020, -1991, -4620, -743512, -1836480, -2249170, -1435098],119,41]
%
% END OF LATTICE L120.41
%
\hbox{}\par\smallskip%
%
% LATTICE L120.42
%
\leavevmode%
\llap{\spacer}%
${L_{120.42}}=\pdualwitharg{5.7.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${110}\above{1pt}{1pt}{l}{2}{924}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{154}\above{1pt}{1pt}{l}{2}{220}\above{1pt}{1pt}{r}{2}{2310}\above{1pt}{1pt}{}{6}{770}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{1540}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-76772956260\\13392650040\\168537600
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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-47\\-189\\-6391
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163\\652\\22440
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127\\510\\17325
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\HardButStrongLineBreak\kern3pt%
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-1122\\-4489\\-154385
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\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3743\\-14982\\-514500
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\0\\-55\\-385\\-1100\\-3465\\-1155\\-165\\-770\\0
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\kern3pt%
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\kern3pt%
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-55\\0\\10\\0\\-220\\-1155\\-770\\-180\\-1540\\-630
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\kern3pt%
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\kern3pt%
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0\\-4620\\-630\\-2310\\-4620\\-11550\\-2310\\-210\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-76772956260, 13392650040, 168537600; 13392650040, -2336279330, -29400525; 168537600, -29400525, -369986],[4620, 385, 1],[-923, -4043, -204, -47, 163, 127, -1122, -512, -6099, -3743; -3695, -16188, -817, -189, 652, 510, -4489, -2049, -24410, -14982; -126830, -555324, -28005, -6391, 22440, 17325, -154385, -70407, -838530, -514500],119,42]
%
% END OF LATTICE L120.42
%
\hbox{}\par\smallskip%
%
% LATTICE L120.43
%
\leavevmode%
\llap{\spacer}%
${L_{120.43}}=\pdualwitharg{2.3.5.7}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{1320}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{}{6}{840}\above{1pt}{1pt}{s}{2}{88}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{b}{2}{1540}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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201026\\172667\\805200
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13904\\11961\\55692
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35\\12\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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64802\\55537\\259560
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\HardButStrongLineBreak\kern3pt%
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117672\\100925\\471328
\end{tabular}\endgroup%
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196660\\168709\\787710
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460411\\395065\\1844150
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\-770\\-13860\\-4620\\-2310\\-7700\\-4620\\-1540\\0\\1540
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28611285327240, 75630947700, -7159303545240; 75630947700, 199922520, -18924871980; -7159303545240, -18924871980, 1791447908287],[4620, 420, 1],[66105, 26611, 201026, 13904, 176, 35, 64802, 117672, 196660, 460411; 56741, 22849, 172667, 11961, 155, 12, 55537, 100925, 168709, 395065; 264780, 106589, 805200, 55692, 705, 140, 259560, 471328, 787710, 1844150],119,43]
%
% END OF LATTICE L120.43
%
\hbox{}\par\smallskip%
%
% LATTICE L120.44
%
\leavevmode%
\llap{\spacer}%
${L_{120.44}}=\pdualwitharg{2.3.5.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${9240}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{l}{2}{1155}\above{1pt}{1pt}{r}{2}{440}\above{1pt}{1pt}{}{6}{1320}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{2}{660}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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304649921997960\\1738376395140\\-76233597547800
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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551420\\5159\\2203740
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33687\\294\\134629
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\HardButStrongLineBreak\kern3pt%
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172563\\1400\\689640
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\HardButStrongLineBreak\kern3pt%
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24475\\146\\97812
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\HardButStrongLineBreak\kern3pt%
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15318\\7\\61215
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55\\9\\220
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61433\\756\\245520
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79183\\854\\316456
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214605\\2228\\857670
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47364\\471\\189290
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[304649921997960, 1738376395140, -76233597547800; 1738376395140, 9919426440, -434999902920; -76233597547800, -434999902920, 19076195250491],[4620, 660, 1],[551420, 33687, 172563, 24475, 15318, 55, 61433, 79183, 214605, 47364; 5159, 294, 1400, 146, 7, 9, 756, 854, 2228, 471; 2203740, 134629, 689640, 97812, 61215, 220, 245520, 316456, 857670, 189290],119,44]
%
% END OF LATTICE L120.44
%
\hbox{}\par\smallskip%
%
% LATTICE L120.45
%
\leavevmode%
\llap{\spacer}%
${L_{120.45}}=\pdualwitharg{2.3.7.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${264}\above{1pt}{1pt}{l}{2}{385}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{9240}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{r}{2}{616}\above{1pt}{1pt}{}{6}{1848}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}{924}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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613584428420040\\1558856084940\\-153527697486240
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
264\\0\\-132\\-4620\\-660\\-2772\\-2772\\-660\\-924\\0
\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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\kern3pt%
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\kern3pt%
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\kern3pt%
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\end{tabular}\endgroup%
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0\\-770\\-252\\-4620\\-462\\-1540\\-924\\-140\\0\\28
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[613584428420040, 1558856084940, -153527697486240; 1558856084940, 3960387816, -390048336216; -153527697486240, -390048336216, 38414850187969],[4620, 924, 1],[-533815, -1075177, -147780, -563926, -1494, 77, -521632, -431234, -1586274, -337771; -73487, -148045, -20353, -77745, -209, 26, -71737, -59335, -218293, -46489; -2134176, -4298525, -590820, -2254560, -5973, 308, -2085468, -1724060, -6341874, -1350398],119,45]
%
% END OF LATTICE L120.45
%
\hbox{}\par\smallskip%
%
% LATTICE L120.46
%
\leavevmode%
\llap{\spacer}%
${L_{120.46}}=\pdualwitharg{3.5.7.11}({L_{120.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30093169260, -5763091058340, 15047312280; -5763091058340, -1103679597250230, 2881684740165; 15047312280, 2881684740165, -7524019618],[4620, 1155, 1],[-122024, -192749, -32125, -19176, 13447, 13697, -107827, -19175, -717197, -153899; -243549, -384708, -64118, -38272, 26840, 27337, -215218, -38272, -1431472, -307170; -93522825, -147727888, -24621315, -14696451, 10306560, 10497410, -82643715, -14696449, -549685290, -117953290],119,46]
%
% END OF LATTICE L120.46
%
\hbox{}\par\smallskip%
%
% LATTICE L120.47
%
\leavevmode%
\llap{\spacer}%
${L_{120.47}}=\pdualwitharg{2.5.7.11}({L_{120.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12842199779640, 124465317900, 3119980636080; 124465317900, 1206315880, 30238532940; 3119980636080, 30238532940, 757991575591],[4620, 1540, 1],[6715861, 8114662, 1858577, 1417421, 18620, 1156, 6568057, 3256525, 19962539, 12750515; -20135315, -24329163, -5572336, -4249674, -55826, -3465, -19692172, -9763626, -59851150, -38228253; -26840000, -32430321, -7427820, -5664736, -74415, -4620, -26249300, -13014732, -79780470, -50957550],119,47]
%
% END OF LATTICE L120.47
%
\hbox{}\par\smallskip%
%
% LATTICE L120.48
%
\leavevmode%
\llap{\spacer}%
${L_{120.48}}=\pdualwitharg{2.3.5.7.11}({L_{120.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10971804255720, -4191184697935620, 1048708116737820; -4191184697935620, 1601015545191853560, -400602244537950840; 1048708116737820, -400602244537950840, 100237726492289117],[4620, 4620, 1],[-4868069, -1960771, -1347357, -1027813, -13569, -128, -4759901, -786755, -14468901, -3080639; 7334165, 2953990, 2029796, 1548206, 20401, 385, 7172112, 1185390, 21799640, 4641371; 29362080, 11826199, 8126220, 6198192, 81675, 1540, 28713300, 4745668, 87274110, 18581570],119,48]
%
% END OF LATTICE L120.48
%

\medskip%
%------------------------------WEYL-GROUP-W121--------------------------------
%
\leavevmode\llap{}%
$W_{121}$%
\qquad\llap{48} lattices, $\chi=84$%
\hfill%
$222222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L121.1
%
\leavevmode%
\llap{\spacer}%
${L_{121.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N_{121}}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{1820}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-606294780\\55970460\\125580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55970460\\-5166946\\-11593
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
125580\\-11593\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}\right.$}%
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-491401\\-5307120\\-7174440
\end{tabular}\endgroup%
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45365\\489941\\662329
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100\\1080\\1459
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-95\\-1026\\-1386
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-24\\-259\\-441
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\HardButStrongLineBreak\kern3pt%
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-7\\-75\\-375
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\HardButStrongLineBreak\kern3pt%
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7\\76\\-80
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108\\1170\\-65
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75\\812\\182
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589\\6370\\4550
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-606294780, 55970460, 125580; 55970460, -5166946, -11593; 125580, -11593, -26],[5460, 1, 1],[-95, -24, -7, 7, 108, 75, 209, 11, 589, 5, -11, -68, -67, -758, -495, -1379, -76, -5139; -1026, -259, -75, 76, 1170, 812, 2262, 119, 6370, 54, -119, -735, -724, -8190, -5348, -14898, -821, -55510; -1386, -441, -375, -80, -65, 182, 858, 69, 4550, 72, -70, -720, -796, -9425, -6314, -17940, -1018, -70980],120,1]
%
% END OF LATTICE L121.1
%
\hbox{}\par\smallskip%
%
% LATTICE L121.2
%
\leavevmode%
\llap{\spacer}%
${L_{121.2}}=\pfillwitharg{2}({L_{121.1}})$%
{${}={}$}Nikulin $121$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{455}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58695\\8190\\9555
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8190\\-1142\\-1365
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9555\\-1365\\-281
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-10767121\\-74987640\\-1878240
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\kern3pt%
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1532244\\10671317\\267288
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\kern3pt%
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549202\\3824919\\95803
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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}}\!\right]$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-58695, 8190, 9555; 8190, -1142, -1365; 9555, -1365, -281],[1365, 1, 1],[-8168, -4533, -3179, -309, -1484, 1, 224, -29, -13066, -706, -2369, -11611, -5607, -126696, -41413, -115586, -12789, -435564; -56886, -31570, -22140, -2152, -10335, 7, 1560, -202, -91000, -4917, -16499, -80865, -39050, -882375, -288421, -804999, -89069, -3033485; -1425, -791, -555, -54, -260, 0, 39, -5, -2275, -123, -413, -2025, -978, -22100, -7224, -20163, -2231, -75985],120,2]
%
% END OF LATTICE L121.2
%
\hbox{}\par\smallskip%
%
% LATTICE L121.3
%
\leavevmode%
\llap{\spacer}%
${L_{121.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{121.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{1365}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-999652290, 625170, 333317985; 625170, -381, -208455; 333317985, -208455, -111139523],[1365, 3, 1],[-2382, -121444, -325266, -531470, -12562373, -4207453, -3986480, -1358668, -47484732, -310497, -608607, -196199, -94288, -995717, -172853, -80947, 2859, 31417; -1516, -77329, -207125, -338437, -7999680, -2679299, -2538588, -865201, -30238390, -197726, -387569, -124945, -60047, -634140, -110089, -51558, 1819, 20020; -7141, -364077, -975115, -1593294, -37660740, -12613524, -11951069, -4073151, -142354485, -930839, -1824543, -588185, -282666, -2985060, -518196, -242671, 8571, 94185],120,3]
%
% END OF LATTICE L121.3
%
\hbox{}\par\smallskip%
%
% LATTICE L121.4
%
\leavevmode%
\llap{\spacer}%
${L_{121.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{121.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{195}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{91}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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982800\\-635\\393055
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-595056735\\393055\\-237984589
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1487880030, 982800, -595056735; 982800, -635, 393055; -595056735, 393055, -237984589],[1365, 5, 1],[12457, 211693, 340187, 926416, 4379543, 7334091, 20846723, 2368319, 16554297, 1623698, 1060872, 205198, 164354, 347127, 301299, 423292, -4984, -10952; -4548, -77329, -124275, -338437, -1599936, -2679299, -7615764, -865201, -6047678, -593178, -387569, -74967, -60047, -126828, -110089, -154674, 1819, 4004; -31155, -529445, -850809, -2316970, -10953254, -18342590, -52137735, -5923175, -41402361, -4060875, -2653245, -513201, -411050, -868166, -753550, -1058655, 12465, 27391],120,4]
%
% END OF LATTICE L121.4
%
\hbox{}\par\smallskip%
%
% LATTICE L121.5
%
\leavevmode%
\llap{\spacer}%
${L_{121.5}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{121.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{910}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{273}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{65}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1980224610\\1340430\\282982245
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1340430\\-889\\-191555
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
282982245\\-191555\\-40439327
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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400828627289\\312034888620\\2803399544400
\end{tabular}\endgroup%
\kern3pt%
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-306444296\\-238559089\\-2143274560
\end{tabular}\endgroup%
\kern3pt%
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-57276162245\\-44588035110\\-400590068201
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-5846\\-4548\\-40887
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-14192\\-11047\\-99259
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-798218\\-621375\\-5582745
\end{tabular}\endgroup%
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-10276193\\-7999680\\-71871800
\end{tabular}\endgroup%
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-491679\\-382757\\-3438808
\end{tabular}\endgroup%
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-5549016\\-4319770\\-38809875
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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0\\-65\\-4095\\-2275\\-54145\\-2600\\-51870\\-5915\\-29640
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1980224610, 1340430, 282982245; 1340430, -889, -191555; 282982245, -191555, -40439327],[1365, 7, 1],[-5846, -14192, -798218, -434750, -10276193, -491679, -9782980, -1111408, -5549016, -761971, -71121, -481477, -77128, -814497, -20199, -198641, 2339, 3671; -4548, -11047, -621375, -338437, -7999680, -382757, -7615764, -865201, -4319770, -593178, -55367, -374835, -60047, -634140, -15727, -154674, 1819, 2860; -40887, -99259, -5582745, -3040646, -71871800, -3438808, -68422263, -7773199, -38809875, -5329233, -497421, -3367455, -539434, -5696600, -141272, -1389297, 16359, 25675],120,5]
%
% END OF LATTICE L121.5
%
\hbox{}\par\smallskip%
%
% LATTICE L121.6
%
\leavevmode%
\llap{\spacer}%
${L_{121.6}}=\pdualwitharg{3}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{5460}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-115975860\\76440\\16380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76440\\6\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16380\\-15\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-178361\\-70095480\\-937192620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\10217\\136617
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\12576\\168143
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-4322\\-57800
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-115975860, 76440, 16380; 76440, 6, -15; 16380, -15, -2],[5460, 3, 1],[-11, -9, -2, -1, -1, 1, 1, 0, -53, -1, -5, -8, -23, -259, -169, -157, -26, -1767; -4322, -3535, -785, -392, -390, 392, 390, -1, -20930, -394, -1967, -3145, -9040, -101790, -66416, -61698, -10217, -694330; -57800, -47292, -10510, -5256, -5265, 5250, 5252, 0, -278460, -5254, -26271, -42035, -120852, -1360905, -888006, -824954, -136617, -9284730],120,6]
%
% END OF LATTICE L121.6
%
\hbox{}\par\smallskip%
%
% LATTICE L121.7
%
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${L_{121.7}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{121.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3462958590, 2413320, -266291025; 2413320, -1651, 185575; -266291025, 185575, -20476973],[1365, 13, 1],[5386, 91524, 735382, 400526, 728249, 3170813, 693296, 1023916, 2752716, 701987, 458655, 443573, 71056, 57721, 130261, 14077, -2155, -1821; -4548, -77329, -621375, -338437, -615360, -2679299, -585828, -865201, -2326030, -593178, -387569, -374835, -60047, -48780, -110089, -11898, 1819, 1540; -70083, -1190917, -9568845, -5211674, -9476030, -41258854, -9021219, -13323271, -35818545, -9134307, -5968053, -5771805, -924586, -751070, -1694966, -183171, 28041, 23695],120,7]
%
% END OF LATTICE L121.7
%
\hbox{}\par\smallskip%
%
% LATTICE L121.8
%
\leavevmode%
\llap{\spacer}%
${L_{121.8}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{121.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9285708767790, -6761247675, -3098329076115; -6761247675, 4923105, 2256001215; -3098329076115, 2256001215, 1033808328902],[1365, 15, 1],[310932, 19767079, 10866601, 89209863, 422453861, 708092148, 671353038, 229028227, 1602372004, 52470753, 103390186, 6727294, 16353507, 34987409, 30851442, 14808867, -307592, -1327559; -63, -4032, -2218, -18211, -86242, -144557, -137059, -46758, -327145, -10713, -21112, -1374, -3341, -7150, -6307, -3029, 62, 273; 931865, 59242050, 32567266, 267362475, 1266096663, 2122156260, 2012048935, 686398920, 4802318157, 157255150, 309860985, 20161739, 49011555, 104857467, 92461950, 44382260, -921855, -3978702],120,8]
%
% END OF LATTICE L121.8
%
\hbox{}\par\smallskip%
%
% LATTICE L121.9
%
\leavevmode%
\llap{\spacer}%
${L_{121.9}}=\pdualwitharg{2}({L_{121.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2006233320, -2549820, 500955000; -2549820, 4888, -636536; 500955000, -636536, 125088113],[5460, 4, 1],[217381, 163547, 203071, 22147, 639196, 154688, 334589, 23169, 549919, 3407, 8177, 164909, 51449, 2549964, 875656, 2535655, 295747, 10612141; 80397, 60487, 75105, 8191, 236405, 57211, 123747, 8569, 203385, 1260, 3024, 60990, 19028, 943085, 323855, 937794, 109380, 3924830; -870162, -654668, -812880, -88653, -2558660, -619206, -1339338, -92744, -2201290, -13638, -32732, -660120, -205947, -10207340, -3505194, -10150062, -1183856, -42479710],120,9]
%
% END OF LATTICE L121.9
%
\hbox{}\par\smallskip%
%
% LATTICE L121.10
%
\leavevmode%
\llap{\spacer}%
${L_{121.10}}=\pdualwitharg{5}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}{780}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{364}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2265900\\-180590\\2175
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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5\\-66\\-273
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-28430220, -2265900, 27300; -2265900, -180590, 2175; 27300, 2175, -26],[5460, 5, 1],[113, 34, 5, 3, -5, -31, -107, -7, -83, -5, 8, 13, 69, 161, 535, 1511, 85, 1175; -1488, -448, -66, -40, 65, 406, 1404, 92, 1092, 66, -105, -171, -908, -2119, -7042, -19890, -1119, -15470; -5850, -1785, -273, -200, 182, 1400, 5070, 345, 4186, 270, -385, -657, -3520, -8242, -27440, -77610, -4375, -60606],120,10]
%
% END OF LATTICE L121.10
%
\hbox{}\par\smallskip%
%
% LATTICE L121.11
%
\leavevmode%
\llap{\spacer}%
${L_{121.11}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{121.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{2730}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{91}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{195}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[770158768470, -2303952105, -257707606065; -2303952105, 6892347, 770939715; -257707606065, 770939715, 86233141714],[1365, 21, 1],[111077, 1009090, 19416094, 31879635, 754832467, 36148764, 239912563, 81844876, 409014059, 18750850, 5278219, 12020451, 5844219, 62517703, 1575090, 5292488, -109859, -338974; -63, -576, -11090, -18211, -431210, -20651, -137059, -46758, -233675, -10713, -3016, -6870, -3341, -35750, -901, -3029, 62, 195; 331954, 3015669, 58025065, 95272401, 2255819475, 108030708, 716979536, 244593699, 1222339950, 56036981, 15773976, 35923160, 17465469, 186834375, 4707162, 15816619, -328314, -1013025],120,11]
%
% END OF LATTICE L121.11
%
\hbox{}\par\smallskip%
%
% LATTICE L121.12
%
\leavevmode%
\llap{\spacer}%
${L_{121.12}}=\pdualwitharg{7}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{910}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{1092}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{b}{2}{260}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
322140\\-54600\\-5460
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54600\\9254\\917
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5460\\917\\-206
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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629\\3720\\-126
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\HardButStrongLineBreak\kern3pt%
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47\\278\\-9
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568\\3360\\-105
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309\\1828\\-56
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2538\\15015\\-455
\end{tabular}\endgroup%
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191\\1130\\-34
\end{tabular}\endgroup%
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3085\\18252\\-546
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120\\710\\-21
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747\\4420\\-130
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[322140, -54600, -5460; -54600, 9254, 917; -5460, 917, -206],[5460, 7, 1],[629, 47, 568, 309, 2538, 191, 3085, 120, 747, 1, -12, -43, 111, 2012, 229, 5105, 335, 3803; 3720, 278, 3360, 1828, 15015, 1130, 18252, 710, 4420, 6, -71, -255, 656, 11895, 1354, 30186, 1981, 22490; -126, -9, -105, -56, -455, -34, -546, -21, -130, 0, 2, 0, -28, -455, -50, -1092, -70, -780],120,12]
%
% END OF LATTICE L121.12
%
\hbox{}\par\smallskip%
%
% LATTICE L121.13
%
\leavevmode%
\llap{\spacer}%
${L_{121.13}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{121.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{182}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{1365}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115178162190\\-1150251375\\45775303665
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1150251375\\11487245\\-457144870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45775303665\\-457144870\\18192497482
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[115178162190, -1150251375, 45775303665; -1150251375, 11487245, -457144870; 45775303665, -457144870, 18192497482],[1365, 35, 1],[-197803, -598991, -6915185, -18923614, -89613064, -21457752, -427233227, -48582771, -48557803, -33391277, -3133129, -4281175, -3469106, -7422056, -934968, -9424813, 65211, 40243; -189, -576, -6654, -18211, -86242, -20651, -411177, -46758, -46735, -32139, -3016, -4122, -3341, -7150, -901, -9087, 62, 39; 497700, 1507145, 17399571, 47614455, 225478981, 53990700, 1074978450, 122241035, 122178212, 84017115, 7883390, 10772034, 8728755, 18674929, 2352510, 23714145, -164080, -101257],120,13]
%
% END OF LATTICE L121.13
%
\hbox{}\par\smallskip%
%
% LATTICE L121.14
%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5658675206730, -8510661705, -1889593913025; -8510661705, 12800073, 2841953985; -1889593913025, 2841953985, 630989591326],[1365, 39, 1],[432883, 27537550, 75696246, 124287825, 226372381, 986525292, 71949329, 319087024, 858639959, 73103810, 144048067, 46864909, 22785561, 18749929, 42988410, 1587364, -428021, -711874; -63, -4032, -11090, -18211, -33170, -144557, -10543, -46758, -125825, -10713, -21112, -6870, -3341, -2750, -6307, -233, 62, 105; 1296334, 82465383, 226683925, 372198801, 677906535, 2954300076, 215463212, 955554639, 2571327990, 218920481, 431373852, 140344100, 68234829, 56149515, 128735334, 4753603, -1281774, -2131815],120,14]
%
% END OF LATTICE L121.14
%
\hbox{}\par\smallskip%
%
% LATTICE L121.15
%
\leavevmode%
\llap{\spacer}%
${L_{121.15}}=\pdualwitharg{2.3}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{1560}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{*}{2}{5460}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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742126241220\\-121705560\\185395536163
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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2046\\7\\-8190
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17037\\0\\-68198
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2970682947960, -487179420, 742126241220; -487179420, 79896, -121705560; 742126241220, -121705560, 185395536163],[5460, 12, 1],[175149, 268519, 51797, 5112, 19096, 2046, 17037, 19081, 1790249, 25215, 198997, 282143, 195252, 8663344, 2803050, 2587695, 849163, 28598291; 38, 70, 20, 4, 65, 7, 0, -2, 0, 1, 21, 45, 35, 1625, 539, 507, 171, 5915; -701110, -1074864, -207340, -20463, -76440, -8190, -68198, -76380, -7166250, -100934, -796572, -1129400, -781581, -34678800, -11220426, -10358374, -3399144, -114477090],120,15]
%
% END OF LATTICE L121.15
%
\hbox{}\par\smallskip%
%
% LATTICE L121.16
%
\leavevmode%
\llap{\spacer}%
${L_{121.16}}=\pdualwitharg{13}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{b}{2}{182}\above{1pt}{1pt}{b}{2}{390}\above{1pt}{1pt}{l}{2}{52}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{364}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21452340\\4045860\\21840
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\kern3pt%
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4045860\\763022\\4121
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\kern3pt%
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21840\\4121\\22
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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7139\\-36120\\-322140
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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169\\-855\\-7605
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185\\-936\\-8320
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1461\\-7392\\-65702
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319\\-1614\\-14346
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235\\-1189\\-10569
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1259\\-6370\\-56630
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[21452340, 4045860, 21840; 4045860, 763022, 4121; 21840, 4121, 22],[5460, 13, 1],[-13, 18, 169, 185, 168, 1461, 319, 235, 1259, 319, 101, 86, 19, 2, -33, -13, -14, -69; 66, -91, -855, -936, -850, -7392, -1614, -1189, -6370, -1614, -511, -435, -96, -10, 168, 66, 71, 350; 546, -819, -7605, -8320, -7555, -65702, -14346, -10569, -56630, -14352, -4550, -3900, -884, -115, 1274, 540, 598, 2940],120,16]
%
% END OF LATTICE L121.16
%
\hbox{}\par\smallskip%
%
% LATTICE L121.17
%
\leavevmode%
\llap{\spacer}%
${L_{121.17}}=\pdualwitharg{3.5}({L_{121.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[81173820, -327600, -70980; -327600, 1290, 285; -70980, 285, 62],[5460, 15, 1],[3, 9, 2, 25, 53, 167, 151, 24, 313, 9, 5, 0, -1, -1, 1, 5, 2, 51; -204, -616, -138, -1732, -3679, -11606, -10504, -1672, -21840, -630, -357, -1, 64, 65, -70, -338, -135, -3458; 4370, 13125, 2921, 36540, 77493, 244230, 220870, 35115, 458094, 13180, 7350, 4, -1440, -1443, 1470, 7280, 2910, 74256],120,17]
%
% END OF LATTICE L121.17
%
\hbox{}\par\smallskip%
%
% LATTICE L121.18
%
\leavevmode%
\llap{\spacer}%
${L_{121.18}}=\pdualwitharg{5.13}(\pfillwitharg{2}({L_{121.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4296337764810, -9573693675, 1716261326325; -9573693675, 21333455, -3824410720; 1716261326325, -3824410720, 685596222058],[1365, 65, 1],[-1747693, -37059437, -61122221, -167263864, -60929410, -1327644384, -290483549, -429420411, -231107815, -295144577, -193856773, -37841869, -30664316, -5046650, -57852876, -6408721, 576021, 191605; -189, -4032, -6654, -18211, -6634, -144557, -31629, -46758, -25165, -32139, -21112, -4122, -3341, -550, -6307, -699, 62, 21; 4375020, 92771315, 153007959, 418713555, 152525293, 3323507460, 727170810, 1074973055, 578534852, 738838815, 485283890, 94729986, 76762335, 12633337, 144823770, 16043025, -1441960, -479647],120,18]
%
% END OF LATTICE L121.18
%
\hbox{}\par\smallskip%
%
% LATTICE L121.19
%
\leavevmode%
\llap{\spacer}%
${L_{121.19}}=\pdualwitharg{2.5}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{b}{2}{780}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{*}{2}{364}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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28430427480\\-25197900\\7102782960
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-25197900\\22360\\-6295200
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\kern3pt%
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7102782960\\-6295200\\1774490581
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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58442\\-3731\\-233940
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7136\\-455\\-28565
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40777\\-2600\\-163228
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49611\\-3164\\-198590
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8154\\-521\\-32640
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0\\0\\0
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\GramMatrix{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28430427480, -25197900, 7102782960; -25197900, 22360, -6295200; 7102782960, -6295200, 1774490581],[5460, 20, 1],[82222, 58442, 13592, 7136, 40777, 49611, 109412, 8154, 46876, 3395, 8149, 14947, 21403, 206561, 349935, 1003609, 115515, 818807; -5253, -3731, -867, -455, -2600, -3164, -6981, -521, -3003, -219, -525, -957, -1369, -13208, -22372, -64155, -7383, -52325; -329130, -233940, -54408, -28565, -163228, -198590, -437970, -32640, -187642, -13590, -32620, -59832, -85675, -826852, -1400770, -4017390, -462400, -3277638],120,19]
%
% END OF LATTICE L121.19
%
\hbox{}\par\smallskip%
%
% LATTICE L121.20
%
\leavevmode%
\llap{\spacer}%
${L_{121.20}}=\pdualwitharg{3.7}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{2730}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{364}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{b}{2}{780}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3812734380, 59574060, -87360; 59574060, -930846, 1365; -87360, 1365, -2],[5460, 21, 1],[3, 1, 6, 13, 128, 11, 67, 10, 87, 3, 0, -1, -1, 2, 1, 11, 3, 43; 190, 63, 375, 808, 7930, 680, 4134, 615, 5330, 182, -1, -65, -64, 130, 64, 702, 191, 2730; -1414, -705, -6335, -16884, -184275, -16878, -108290, -17577, -167310, -7028, -702, -700, 0, 1365, -6, -1456, -714, -15600],120,20]
%
% END OF LATTICE L121.20
%
\hbox{}\par\smallskip%
%
% LATTICE L121.21
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3788150645370, -10636725645, -542968844145; -10636725645, 29866837, 1524598985; -542968844145, 1524598985, 77825618174],[1365, 91, 1],[696749, 2110630, 121837358, 66682745, 121453021, 75612756, 115806487, 171196164, 65810917, 117664690, 11040641, 75431707, 12224881, 10059689, 3294870, 2554952, -229641, -54562; -189, -576, -33270, -18211, -33170, -20651, -31629, -46758, -17975, -32139, -3016, -20610, -3341, -2750, -901, -699, 62, 15; 4861038, 14725321, 850027815, 465228309, 847346395, 527530692, 807951984, 1194391471, 459145790, 820916187, 77027704, 526267560, 85289841, 70183855, 22987458, 17825241, -1602146, -380665],120,21]
%
% END OF LATTICE L121.21
%
\hbox{}\par\smallskip%
%
% LATTICE L121.22
%
\leavevmode%
\llap{\spacer}%
${L_{121.22}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{121.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-96347093929905, -1145091675, 32147894258025; -1145091675, -13335, 382079880; 32147894258025, 382079880, -10726707605494],[1365, 105, 1],[3879809, 28256339, 105950371, 865589146, 4091988974, 978935018, 6492651821, 2212818119, 2209622943, 505695841, 141602211, 63908179, 153562154, 324333326, 40216282, 131831629, -4656969, -1461793; -1516, -11047, -41425, -338437, -1599936, -382757, -2538588, -865201, -863954, -197726, -55367, -24989, -60047, -126828, -15727, -51558, 1819, 572; 11627770, 84684120, 317532782, 2594166750, 12263672421, 2933863815, 19458448100, 6631805880, 6622229952, 1515568145, 424381185, 191532523, 460225080, 972025509, 120528015, 395098795, -13956915, -4380987],120,22]
%
% END OF LATTICE L121.22
%
\hbox{}\par\smallskip%
%
% LATTICE L121.23
%
\leavevmode%
\llap{\spacer}%
${L_{121.23}}=\pdualwitharg{2.7}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9301688760, 67731300, 2325353940; 67731300, 493192, 16932328; 2325353940, 16932328, 581321423],[5460, 28, 1],[-26662, -2888, -24658, -2555, -69113, -2223, -30692, -1560, -2696, 661, -1, -18677, -6112, -306457, -15111, -307321, -35997, -185089; -2433, -263, -2235, -230, -6175, -197, -2691, -131, -195, 66, 0, -1710, -559, -28015, -1381, -28080, -3288, -16900; 106722, 11560, 98700, 10227, 276640, 8898, 122850, 6244, 10790, -2646, 4, 74760, 24465, 1226680, 60486, 1230138, 144088, 740870],120,23]
%
% END OF LATTICE L121.23
%
\hbox{}\par\smallskip%
%
% LATTICE L121.24
%
\leavevmode%
\llap{\spacer}%
${L_{121.24}}=\pdualwitharg{5.7}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{182}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{5460}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81151980\\13486200\\-70980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13486200\\2241190\\-11795
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70980\\-11795\\62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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14351\\-89700\\-627900
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-20\\126\\1071
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-59\\372\\3220
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-39\\246\\2150
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-26\\164\\1435
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[81151980, 13486200, -70980; 13486200, 2241190, -11795; -70980, -11795, 62],[5460, 35, 1],[-71, -7, -20, -59, -101, -39, -643, -26, -33, -1, 3, 8, 11, -3, -9, -293, -26, -71; 444, 44, 126, 372, 637, 246, 4056, 164, 208, 6, -19, -51, -72, 13, 54, 1794, 161, 442; 3150, 355, 1071, 3220, 5551, 2150, 35490, 1435, 1794, 0, -180, -546, -1120, -1001, -50, 5460, 840, 2756],120,24]
%
% END OF LATTICE L121.24
%
\hbox{}\par\smallskip%
%
% LATTICE L121.25
%
\leavevmode%
\llap{\spacer}%
${L_{121.25}}=\pdualwitharg{3.13}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{b}{2}{546}\above{1pt}{1pt}{b}{2}{130}\above{1pt}{1pt}{l}{2}{156}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{1092}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
259912380\\2598960\\-196560
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2598960\\21606\\-1599
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-196560\\-1599\\118
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-243601\\254781240\\3046778280
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-1070\\1119112\\13382811
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70\\-73213\\-875512
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-11504\\-137566
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\HardButStrongLineBreak\kern3pt%
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37\\-38696\\-462735
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[259912380, 2598960, -196560; 2598960, 21606, -1599; -196560, -1599, 118],[5460, 39, 1],[11, 37, 46, 121, 101, 839, 59, 124, 633, 49, 33, 4, -1, -1, 1, 1, 6, 67; -11504, -38696, -48110, -126552, -105635, -877506, -61708, -129692, -662060, -51250, -34517, -4185, 1044, 1045, -1050, -1046, -6275, -70070; -137566, -462735, -575315, -1513356, -1263225, -10493574, -737930, -1550913, -7917210, -612872, -412776, -50050, 12480, 12495, -12558, -12508, -75036, -837900],120,25]
%
% END OF LATTICE L121.25
%
\hbox{}\par\smallskip%
%
% LATTICE L121.26
%
\leavevmode%
\llap{\spacer}%
${L_{121.26}}=\pdualwitharg{3.5.13}(\pfillwitharg{2}({L_{121.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3984888261494595, -10026368625, 1329624014473755; -10026368625, -24765, 3345464070; 1329624014473755, 3345464070, -443651089780426],[1365, 195, 1],[32378981, 1650643157, 884178511, 7223520934, 2626806704, 57185885984, 4167884393, 18466413431, 9929094141, 4220132059, 8271882603, 533325169, 1281501146, 208200656, 2349268576, 84626647, -38865591, -6568381; -1516, -77329, -41425, -338437, -123072, -2679299, -195276, -865201, -465206, -197726, -387569, -24989, -60047, -9756, -110089, -3966, 1819, 308; 97039930, 4946983860, 2649886382, 21648919890, 7872549747, 171386319195, 12491165480, 55343911740, 29757533148, 12647751935, 24790863825, 1598377573, 3840663840, 623978163, 7040766915, 253626385, -116480325, -19685463],120,26]
%
% END OF LATTICE L121.26
%
\hbox{}\par\smallskip%
%
% LATTICE L121.27
%
\leavevmode%
\llap{\spacer}%
${L_{121.27}}=\pdualwitharg{2.13}({L_{121.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[46999603560, -47485620, 11742013920; -47485620, 47944, -11863436; 11742013920, -11863436, 2933533061],[5460, 52, 1],[-8905, -20551, -35659, -3095, -2788, 7320, 4747, 13627, 53913, 21181, 30099, 56119, 7187, 16428, 49968, 7529, 4105, -6173; 513, 1183, 2025, 169, 125, -581, -309, -839, -3255, -1260, -1764, -3270, -418, -955, -2905, -438, -240, 350; 35646, 82264, 142740, 12389, 11160, -29302, -19002, -54548, -215810, -84786, -120484, -224640, -28769, -65760, -200018, -30138, -16432, 24710],120,27]
%
% END OF LATTICE L121.27
%
\hbox{}\par\smallskip%
%
% LATTICE L121.28
%
\leavevmode%
\llap{\spacer}%
${L_{121.28}}=\pdualwitharg{2.3.5}({L_{121.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[235975532520, 118181700, 58950544380; 118181700, 59160, 29523720; 58950544380, 29523720, 14726809367],[5460, 60, 1],[-21072, -187602, -55736, -195415, -1742759, -2826883, -2614814, -860502, -5810762, -178761, -278675, -10875, -4418, 10873, 29221, 16985, -5441, -250839; -17, -161, -49, -173, -1547, -2513, -2327, -767, -5187, -160, -252, -10, -4, 13, 35, 26, 0, -182; 84350, 750960, 223108, 782235, 6976164, 11315850, 10466950, 3444540, 23260146, 715570, 1115520, 43532, 17685, -43524, -116970, -67990, 21780, 1004094],120,28]
%
% END OF LATTICE L121.28
%
\hbox{}\par\smallskip%
%
% LATTICE L121.29
%
\leavevmode%
\llap{\spacer}%
${L_{121.29}}=\pdualwitharg{5.13}({L_{121.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1268669220, 15288000, -447720; 15288000, 184210, -5395; -447720, -5395, 158],[5460, 65, 1],[-19, -6, -1, -1, 0, 3, 1, 1, 1, 1, -1, -2, -11, -2, -87, -19, -14, -15; -1404, -448, -78, -92, -3, 98, 48, 56, 56, 54, -77, -147, -808, -147, -6398, -1398, -1031, -1106; -101790, -32305, -5499, -5980, -103, 11830, 4470, 4745, 4746, 4680, -5460, -10686, -58760, -10687, -465010, -101580, -74880, -80276],120,29]
%
% END OF LATTICE L121.29
%
\hbox{}\par\smallskip%
%
% LATTICE L121.30
%
\leavevmode%
\llap{\spacer}%
${L_{121.30}}=\pdualwitharg{3.7.13}(\pfillwitharg{2}({L_{121.1}}))$%
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{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-236217933115710, -2888389140, 79042913639235; -2888389140, -34671, 966508725; 79042913639235, 966508725, -26449228956377],[1365, 273, 1],[9353934, 68121824, 1277147538, 2086796242, 3794278723, 2360053913, 1204056244, 5334744992, 2048861976, 1219150023, 341379495, 770359057, 370211672, 300734467, 96954001, 24447713, -11227845, -1355381; -1516, -11047, -207125, -338437, -615360, -382757, -195276, -865201, -332290, -197726, -55367, -124945, -60047, -48780, -15727, -3966, 1819, 220; 27954017, 203580507, 3816726095, 6236342658, 11339114850, 7052966934, 3598294441, 15942762927, 6122976975, 3643401853, 1020204783, 2302200355, 1106369082, 898738050, 289744806, 73061429, -33554157, -4050525],120,30]
%
% END OF LATTICE L121.30
%
\hbox{}\par\smallskip%
%
% LATTICE L121.31
%
\leavevmode%
\llap{\spacer}%
${L_{121.31}}=\pdualwitharg{2.3.7}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\GramMatrix{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5024848920, -7059780, 1255286760; -7059780, 96936, -1763580; 1255286760, -1763580, 313590493],[5460, 84, 1],[-549059, -504725, -4320173, -2828165, -122560864, -5589488, -35748851, -11550017, -54721053, -2278765, -437883, -392867, 341, 21824, -66160, -1012861, -703887, -6548467; -1635, -1503, -12865, -8422, -364975, -16645, -106457, -34395, -162955, -6786, -1304, -1170, 1, 65, -197, -3016, -2096, -19500; 2197846, 2020380, 17293360, 11320953, 490602840, 22374342, 143100230, 46233936, 219044670, 9121742, 1752816, 1572620, -1365, -87360, 264834, 4054414, 2817612, 26213070],120,31]
%
% END OF LATTICE L121.31
%
\hbox{}\par\smallskip%
%
% LATTICE L121.32
%
\leavevmode%
\llap{\spacer}%
${L_{121.32}}=\pdualwitharg{7.13}({L_{121.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[423384780, 70679700, 333060; 70679700, 11799242, 55601; 333060, 55601, 262],[5460, 91, 1],[-1, 2, 49, 37, 28, 31, 43, 27, 17, 19, -1, -34, -25, -18, -19, -25, -14, -7; 6, -11, -255, -184, -135, -146, -198, -119, -70, -60, 8, 210, 148, 105, 110, 144, 80, 40; 0, -208, -8190, -8008, -6965, -8450, -12684, -9100, -6780, -11466, -429, -1365, 364, 595, 806, 1218, 819, 410],120,32]
%
% END OF LATTICE L121.32
%
\hbox{}\par\smallskip%
%
% LATTICE L121.33
%
\leavevmode%
\llap{\spacer}%
${L_{121.33}}=\pdualwitharg{3.5.7}({L_{121.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12812889180, 1067741220, -1097460; 1067741220, -88978470, 91455; -1097460, 91455, -94],[5460, 105, 1],[19, 0, -1, -1, 16, 13, 115, 27, 71, 23, 7, 8, 85, 166, 71, 431, 64, 111; 234, -1, -13, -12, 208, 168, 1482, 347, 910, 294, 89, 101, 1068, 2080, 888, 5382, 797, 1378; 5810, -975, -973, 0, 15561, 11670, 99190, 22365, 56394, 17500, 4860, 4858, 46620, 85449, 34950, 203840, 28140, 44616],120,33]
%
% END OF LATTICE L121.33
%
\hbox{}\par\smallskip%
%
% LATTICE L121.34
%
\leavevmode%
\llap{\spacer}%
${L_{121.34}}=\pdualwitharg{5.7.13}(\pfillwitharg{2}({L_{121.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35010821253135, 1435574595, 13915514247180; 1435574595, -57785, -570586835; 13915514247180, -570586835, -5530905296671],[1365, 455, 1],[16566509, 40216263, 452384551, 1231956832, 447996551, 1393276681, 2132472667, 3149409323, 241912407, 2159204866, 201536112, 272872574, 218557418, 35508199, 57237569, 43298708, -6628448, -160032; -49704075, -120659836, -1357277928, -3696208933, -1344112725, -4180212800, -6398003829, -9449093170, -725803679, -6478207776, -604663703, -818692689, -655732301, -106534353, -171728434, -129908022, 19887163, 480140; 41685735, 101194795, 1138319637, 3099930470, 1127278264, 3505854040, 5365867395, 7924750015, 608715843, 5433132705, 507118365, 686619843, 549948490, 89348056, 144024920, 108951045, -16678935, -402683],120,34]
%
% END OF LATTICE L121.34
%
\hbox{}\par\smallskip%
%
% LATTICE L121.35
%
\leavevmode%
\llap{\spacer}%
${L_{121.35}}=\pdualwitharg{2.5.7}({L_{121.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{r}{2}{728}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{5460}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25996359480, -67130700, 6494631780; -67130700, 172760, -16771160; 6494631780, -16771160, 1622544187],[5460, 140, 1],[-310624, -32626, -50300, -21413, -87023, -8841, -27962, 17662, 27858, 29221, 1359, -43501, -72388, -725919, -178761, -3630277, -424133, -434329; 5913, 621, 957, 407, 1651, 167, 507, -339, -533, -558, -26, 828, 1378, 13819, 3403, 69108, 8074, 8268; 1243410, 130600, 201348, 85715, 348348, 35390, 111930, -70700, -111514, -116970, -5440, 174132, 289765, 2905812, 715570, 14531790, 1697780, 1738594],120,35]
%
% END OF LATTICE L121.35
%
\hbox{}\par\smallskip%
%
% LATTICE L121.36
%
\leavevmode%
\llap{\spacer}%
${L_{121.36}}=\pdualwitharg{2.3.13}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[39887939789880, -95282170620, 9964681801620; -95282170620, 227604936, -23803097292; 9964681801620, -23803097292, 2489346001087],[5460, 156, 1],[-71584, -647354, -978312, -690427, -2374707, -10030711, -714574, -3062606, -7969058, -639605, -1012087, -207003, -20762, 4077, 74065, 3385, -19213, -328147; 15, 231, 415, 307, 1075, 4585, 329, 1425, 3745, 306, 518, 120, 14, -5, -91, -8, -34, 0; 286546, 2591316, 3916120, 2763735, 9505800, 40152294, 2860394, 12259416, 31899630, 2560298, 4051320, 828620, 83109, -16320, -296478, -13550, 76908, 1313550],120,36]
%
% END OF LATTICE L121.36
%
\hbox{}\par\smallskip%
%
% LATTICE L121.37
%
\leavevmode%
\llap{\spacer}%
${L_{121.37}}=\pdualwitharg{3.5.13}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${260}\above{1pt}{1pt}{b}{2}{2730}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{l}{2}{780}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{5460}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{390}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4405471980, 157695720, -2424240; 157695720, 5644470, -86775; -2424240, -86775, 1334],[5460, 195, 1],[309, 215, 6, -1, -1, 1, 5, 32, 75, 63, 219, 56, 745, 125, 5207, 367, 774, 793; 4526, 3143, 87, -20, -15, 14, 74, 473, 1106, 928, 3220, 822, 10928, 1833, 76342, 5380, 11344, 11620; 855920, 595140, 16562, -3120, -2793, 2730, 13900, 88920, 208236, 174850, 607425, 155233, 2064660, 346383, 14428050, 1016870, 2144415, 2196894],120,37]
%
% END OF LATTICE L121.37
%
\hbox{}\par\smallskip%
%
% LATTICE L121.38
%
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${L_{121.38}}=\pdualwitharg{2.5.13}({L_{121.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[448411465320, -272699700, 112020674220; -272699700, 164840, -68124940; 112020674220, -68124940, 27984635593],[5460, 260, 1],[-542774, -637442, -219292, -137228, -62655, -960937, -147784, -91258, 10216, 138251, 322813, 84451, 2387, -27239, -926837, -256739, -493053, -324845; 417, 483, 165, 103, 47, 721, 111, 69, -7, -102, -238, -60, 2, 23, 749, 204, 386, 252; 2172690, 2551640, 877812, 549315, 250804, 3846570, 591570, 365300, -40894, -553410, -1292200, -338052, -9555, 109036, 3710070, 1027710, 1973660, 1300334],120,38]
%
% END OF LATTICE L121.38
%
\hbox{}\par\smallskip%
%
% LATTICE L121.39
%
\leavevmode%
\llap{\spacer}%
${L_{121.39}}=\pdualwitharg{3.7.13}({L_{121.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-977340, -245700, -38220; -245700, -61698, -9555; -38220, -9555, -1454],[5460, 273, 1],[-1459, -144, -139, 1, 16, -13, -31, -171, -271, -311, -151, -1336, -3541, -2966, -3527, -1739, -3664, -2679; 7282, 719, 695, -4, -80, 64, 154, 851, 1350, 1550, 753, 6665, 17668, 14800, 17600, 8678, 18285, 13370; -9464, -936, -910, 0, 105, -78, -196, -1092, -1740, -2002, -975, -8645, -22932, -19215, -22854, -11270, -23751, -17370],120,39]
%
% END OF LATTICE L121.39
%
\hbox{}\par\smallskip%
%
% LATTICE L121.40
%
\leavevmode%
\llap{\spacer}%
${L_{121.40}}=\pdualwitharg{3.5.7.13}(\pfillwitharg{2}({L_{121.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[53703572974138815, -8658632820479460, 2853214577430945; -8658632820479460, 1396032296696310, -460024091803185; 2853214577430945, -460024091803185, 151588301746418],[1365, 1365, 1],[-63, -576, -2218, -18211, -6634, -20651, -10543, -46758, -3595, -10713, -3016, -1374, -3341, -550, -901, -233, 62, 3; -39727172, -361032725, -1389389037, -11406388467, -4155019045, -12933908244, -6603076750, -29283888767, -2251451852, -6709028509, -1888552502, -860196190, -2091122895, -344151409, -563603118, -145678931, 39280816, 1866625; -120558620, -1095612960, -4216328662, -34614554520, -12609086007, -39250063515, -20038118170, -88866757980, -6832399494, -20359646125, -5731121955, -2610406253, -6345854970, -1044383823, -1710345975, -442086575, 119204085, 5664579],120,40]
%
% END OF LATTICE L121.40
%
\hbox{}\par\smallskip%
%
% LATTICE L121.41
%
\leavevmode%
\llap{\spacer}%
${L_{121.41}}=\pdualwitharg{2.7.13}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8498151480, -37821420, 2123055480; -37821420, 143416, -9448712; 2123055480, -9448712, 530393531],[5460, 364, 1],[-86887, -4105, 1310807, 436822, 1691189, 1085727, 1700083, 2591515, 1025523, 1921744, 207978, 1747288, 174797, 347541, 137511, 134774, 58834, -6158; 636, 30, -9600, -3199, -12385, -7951, -12450, -18978, -7510, -14073, -1523, -12795, -1280, -2545, -1007, -987, -431, 45; 347802, 16432, -5247060, -1748565, -6769700, -4346082, -6805302, -10373636, -4105090, -7692594, -832520, -6994260, -699699, -1391180, -550446, -539490, -235508, 24650],120,41]
%
% END OF LATTICE L121.41
%
\hbox{}\par\smallskip%
%
% LATTICE L121.42
%
\leavevmode%
\llap{\spacer}%
${L_{121.42}}=\pdualwitharg{2.3.5.7}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[481838480760, -497225820, 120371007120; -497225820, 528360, -124215000; 120371007120, -124215000, 30070614809],[5460, 420, 1],[-3787077, -570347, -20467, 341, -1592334, -700654, -6644273, -3374327, -4900483, -1739013, -1271683, -1821563, -5526431, -46300446, -10351526, -65194897, -20572063, -19045907; -12025, -1811, -65, 1, -5057, -2225, -21099, -10715, -15561, -5522, -4038, -5784, -17548, -147017, -32869, -207012, -65322, -60476; 15159410, 2283060, 81928, -1365, 6374004, 2804670, 26596570, 13507200, 19616298, 6961150, 5090460, 7291592, 22121925, 185337516, 41436450, 260970710, 82348560, 76239462],120,42]
%
% END OF LATTICE L121.42
%
\hbox{}\par\smallskip%
%
% LATTICE L121.43
%
\leavevmode%
\llap{\spacer}%
${L_{121.43}}=\pdualwitharg{5.7.13}({L_{121.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[38236380, -6295380, -2473380; -6295380, 1036490, 407225; -2473380, 407225, 159994],[5460, 455, 1],[1, -1, -2, 1, 1, 9, 17, 16, 3, 35, 3, 8, 23, 3, 15, 19, 10, 1; 6, 45, 417, 2152, 383, 2354, 3570, 2601, 394, 3432, 146, 156, 140, -1, -62, -132, -118, -12; 0, -130, -1092, -5460, -959, -5850, -8820, -6370, -956, -8190, -325, -273, 0, 49, 390, 630, 455, 46],120,43]
%
% END OF LATTICE L121.43
%
\hbox{}\par\smallskip%
%
% LATTICE L121.44
%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L121.44
%
\hbox{}\par\smallskip%
%
% LATTICE L121.45
%
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${L_{121.45}}=\pdualwitharg{2.3.7.13}({L_{121.1}})$%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[52601640, 20393100, 13125840; 20393100, 7943208, 5088720; 13125840, 5088720, 3275329],[5460, 1092, 1],[219579, 42249, 37741, -341, -1574, 5398, 6971, 65205, 47437, 52155, 48329, 415149, 272075, 907354, 538070, 264763, 1112309, 405453; -696, -134, -120, 1, 5, -17, -22, -206, -150, -165, -153, -1315, -862, -2875, -1705, -839, -3525, -1285; -878878, -169104, -151060, 1365, 6300, -21606, -27902, -260988, -189870, -208754, -193440, -1661660, -1088997, -3631740, -2153658, -1059730, -4452084, -1622850],120,45]
%
% END OF LATTICE L121.45
%
\hbox{}\par\smallskip%
%
% LATTICE L121.46
%
\leavevmode%
\llap{\spacer}%
${L_{121.46}}=\pdualwitharg{3.5.7.13}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5460, 0, 0; 0, -37349130, 28665; 0, 28665, -22],[5460, 1365, 1],[-13, -1, 0, 1, 0, -1, -1, -4, -1, -5, -2, -3, -37, -6, -35, -17, -35, -5; -122, -13, -3, 0, 1, 6, 2, -1, -2, -16, -10, -20, -276, -47, -282, -140, -298, -44; -159250, -16965, -3913, 0, 1302, 7800, 2590, -1365, -2622, -20930, -13065, -26117, -360360, -61362, -368160, -182770, -389025, -57438],120,46]
%
% END OF LATTICE L121.46
%
\hbox{}\par\smallskip%
%
% LATTICE L121.47
%
\leavevmode%
\llap{\spacer}%
${L_{121.47}}=\pdualwitharg{2.5.7.13}({L_{121.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[222298440, -36467340, -82860960; -36467340, 5747560, 13417040; -82860960, 13417040, 30754151],[5460, 1820, 1],[42955, 2013, -132547, -220798, -170967, -548805, -859363, -1310015, -103687, -971602, -105196, -177002, -88751, -35399, -70293, -69284, -31364, 504; -128949, -6043, 397899, 662824, 513234, 1647484, 2579763, 3932597, 311263, 2916699, 315793, 531351, 266426, 106266, 211016, 207987, 94153, -1513; 171990, 8060, -530712, -884065, -684544, -2197390, -3440850, -5245240, -415158, -3890250, -421200, -708708, -355355, -141736, -281450, -277410, -125580, 2018],120,47]
%
% END OF LATTICE L121.47
%
\hbox{}\par\smallskip%
%
% LATTICE L121.48
%
\leavevmode%
\llap{\spacer}%
${L_{121.48}}=\pdualwitharg{2.3.5.7.13}({L_{121.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120120, -156510900, -39099060; -156510900, 203926271640, 50944218780; -39099060, 50944218780, 12726724253],[5460, 5460, 1],[-3043, -559, -87, 0, -20, -186, -167, -1303, -169, -869, -745, -1217, -3912, -2588, -7638, -3745, -15649, -1135; 5681, 779, -91, -341, -21, 487, 507, 4091, 505, 2500, 1948, 2818, 8522, 5475, 15875, 7674, 31360, 2222; -22750, -3120, 364, 1365, 84, -1950, -2030, -16380, -2022, -10010, -7800, -11284, -34125, -21924, -63570, -30730, -125580, -8898],120,48]
%
% END OF LATTICE L121.48
%

\medskip%
%------------------------------WEYL-GROUP-W122--------------------------------
%
\leavevmode\llap{}%
$W_{122}$%
\qquad\llap{48} lattices, $\chi=108$%
\hfill%
$2222222222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L122.1
%
\leavevmode%
\llap{\spacer}%
${L_{122.1}}$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-573936958140, 420774900, 139639500; 420774900, -308486, -102375; 139639500, -102375, -33974],[5460, 1, 1],[5, -151, -1, 244, 423, 939, 18151, 943, 59737, 689, 1039, 523, 8159, 353, 1604, 721, 909, 13881, 641, 36359, 279, 193; 6822, -206115, -1364, 333060, 577382, 1281700, 24775400, 1287157, 81538548, 940455, 1418186, 713868, 11136580, 481824, 2189355, 984113, 1240715, 18946460, 874913, 49627032, 380810, 263424; -6, 455, 0, -735, -1235, -2723, -52520, -2726, -172536, -1985, -2982, -1494, -23205, -1000, -4515, -2015, -2527, -38480, -1774, -100464, -765, -518],121,1]
%
% END OF LATTICE L122.1
%
\hbox{}\par\smallskip%
%
% LATTICE L122.2
%
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\llap{\spacer}%
${L_{122.2}}=\pfillwitharg{2}({L_{122.1}})$%
{${}={}$}Nikulin $122$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-148785, 70980, 1365; 70980, -33862, -651; 1365, -651, -10],[1365, 1, 1],[587, 12748, 179, 548, 6, -30, 31, 20, 1429, 88, 193, 211, 11458, 353, 5038, 3452, 5616, 48381, 4768, 143807, 2838, 1669; 1236, 26845, 377, 1155, 13, -63, 65, 42, 3003, 185, 406, 444, 24115, 743, 10605, 7267, 11823, 101855, 10038, 302757, 5975, 3514; -450, -10010, -146, -525, -39, 7, 0, -5, -546, -40, -112, -135, -7735, -244, -3570, -2496, -4102, -35490, -3505, -105924, -2105, -1253],121,2]
%
% END OF LATTICE L122.2
%
\hbox{}\par\smallskip%
%
% LATTICE L122.3
%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-4095\\-150\\-840\\-1950\\-3360\\-29640\\-2955\\-30030\\-1845\\-1155
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1365\\-63\\-420\\-1092\\-1974\\-17745\\-1785\\-18291\\-1155\\-756
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-304395, -307125, 101010; -307125, -101922, 102375; 101010, 102375, -33518],[1365, 3, 1],[151, -68856, -907, 21198, 122012, 275500, 2676253, 278674, 2947785, 205104, 155755, 26337, 2479264, 53883, 164218, 222332, 280748, 2144563, 198110, 1873031, 86264, 29661; -1, 455, 6, -140, -806, -1820, -17680, -1841, -19474, -1355, -1029, -174, -16380, -356, -1085, -1469, -1855, -14170, -1309, -12376, -570, -196; 452, -206115, -2715, 63455, 365235, 824691, 8011185, 834192, 8823997, 613965, 466242, 78838, 7421505, 161295, 491575, 665535, 840399, 6419595, 593028, 5606783, 258225, 88788],121,3]
%
% END OF LATTICE L122.3
%
\hbox{}\par\smallskip%
%
% LATTICE L122.4
%
\leavevmode%
\llap{\spacer}%
${L_{122.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{122.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{182}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{130}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{1365}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1169805\\-511875\\-468195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-511875\\-169870\\-204750
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468195\\-204750\\-187387
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77862329\\429975\\-195012090
\end{tabular}\endgroup%
\kern3pt%
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48802600\\269499\\-122229800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31195636\\172270\\-78131829
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-539\\-3\\1350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16459\\91\\-41223
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1080\\6\\-2705
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15235\\-84\\38157
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-146035\\-806\\365755
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-329699\\-1820\\825755
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-640524\\-3536\\1604239
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-333478\\-1841\\835220
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\HardButStrongLineBreak\kern3pt%
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-10582324\\-58422\\26504205
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\HardButStrongLineBreak\kern3pt%
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-49085\\-271\\122937
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-186363\\-1029\\466760
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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}}\!\right]$}%
}}{}{\hbox{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1169805, -511875, -468195; -511875, -169870, -204750; -468195, -204750, -187387],[1365, 5, 1],[-539, 16459, 1080, -15235, -146035, -329699, -640524, -333478, -10582324, -49085, -186363, -94531, -593217, -64460, -117863, -265935, -335791, -512992, -236942, -6720416, -20633, -35467; -3, 91, 6, -84, -806, -1820, -3536, -1841, -58422, -271, -1029, -522, -3276, -356, -651, -1469, -1855, -2834, -1309, -37128, -114, -196; 1350, -41223, -2705, 38157, 365755, 825755, 1604239, 835220, 26504205, 122937, 466760, 236760, 1485757, 161445, 295197, 666055, 841015, 1284829, 593440, 16831815, 51677, 88830],121,4]
%
% END OF LATTICE L122.4
%
\hbox{}\par\smallskip%
%
% LATTICE L122.5
%
\leavevmode%
\llap{\spacer}%
${L_{122.5}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{122.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{182}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{455}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-716625\\-237818\\102375
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
270270\\102375\\-38582
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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-35148049\\539175\\-244785450
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5008112\\-76825\\34878549
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
193\\-3\\1344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-4228\\65\\-29445
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-387\\6\\-2695
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\HardButStrongLineBreak\kern3pt%
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3922\\-60\\27315
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\HardButStrongLineBreak\kern3pt%
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52592\\-806\\366275
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\HardButStrongLineBreak\kern3pt%
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16960\\-260\\118117
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\HardButStrongLineBreak\kern3pt%
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1153173\\-17680\\8031205
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120074\\-1841\\836248
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\HardButStrongLineBreak\kern3pt%
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544325\\-8346\\3790917
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88364\\-1355\\615405
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9585\\-147\\66754
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{$\left.\llap{\phantom{%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1893255, -716625, 270270; -716625, -237818, 102375; 270270, 102375, -38582],[1365, 7, 1],[193, -4228, -387, 3922, 52592, 16960, 1153173, 120074, 544325, 88364, 9585, 34031, 152532, 23203, 30302, 95712, 17264, 923083, 85270, 345499, 37124, 1823; -3, 65, 6, -60, -806, -260, -17680, -1841, -8346, -1355, -147, -522, -2340, -356, -465, -1469, -265, -14170, -1309, -5304, -570, -28; 1344, -29445, -2695, 27315, 366275, 118117, 8031205, 836248, 3790917, 615405, 66754, 237006, 1062295, 161595, 211035, 666575, 120233, 6428695, 593852, 2406183, 258545, 12696],121,5]
%
% END OF LATTICE L122.5
%
\hbox{}\par\smallskip%
%
% LATTICE L122.6
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-193571376180, -424498620, 622440; -424498620, -930918, 1365; 622440, 1365, -2],[5460, 3, 1],[7, 362, 17, 29, 44, 60, 953, 45, 869, 22, 19, 1, 2, -1, -1, 8, 24, 503, 27, 587, 22, 37; -3194, -165165, -7756, -13230, -20072, -27370, -434720, -20527, -396396, -10035, -8666, -456, -910, 456, 455, -3653, -10955, -229580, -12323, -267904, -10040, -16884; -1408, -65520, -2820, -4235, -5655, -7077, -107640, -4959, -93548, -2130, -1428, -2, 1365, 0, -700, -3510, -7728, -148980, -7731, -163072, -5625, -8442],121,6]
%
% END OF LATTICE L122.6
%
\hbox{}\par\smallskip%
%
% LATTICE L122.7
%
\leavevmode%
\llap{\spacer}%
${L_{122.7}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{122.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${39}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{2730}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{182}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{130}\above{1pt}{1pt}{l}{2}{91}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3867045, -1330875, -297570; -1330875, -441662, -102375; -297570, -102375, -22898],[1365, 13, 1],[-101, 1208, 203, -14666, -2156, -63244, -47249, -63954, -156103, -47056, -35723, -18115, -43712, -12347, -112846, -3916, -64268, -37759, -45342, -98921, -19736, -6781; -3, 35, 6, -420, -62, -1820, -1360, -1841, -4494, -1355, -1029, -522, -1260, -356, -3255, -113, -1855, -1090, -1309, -2856, -570, -196; 1326, -15855, -2665, 192465, 28295, 830011, 620095, 839332, 2048697, 617565, 468832, 237744, 573685, 162045, 1481025, 51395, 843479, 495565, 595088, 1298283, 259025, 88998],121,7]
%
% END OF LATTICE L122.7
%
\hbox{}\par\smallskip%
%
% LATTICE L122.8
%
\leavevmode%
\llap{\spacer}%
${L_{122.8}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{122.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[27824721030015, 307125000, -9284181917010; 307125000, 3390, -102477375; -9284181917010, -102477375, 3097822032974],[1365, 15, 1],[-669, 1002, 5015, 14710, 280820, 579688, 1094183, 562636, 5884771, 79230, 286831, 45799, 820364, 85245, 48472, 305870, 368042, 548549, 249704, 2328889, 20056, 29079; -151, 91, 906, 2597, 48919, 100555, 189514, 97384, 1017926, 13679, 49371, 7851, 140049, 14494, 8183, 51181, 61145, 90766, 41216, 383474, 3261, 4529; -2005, 3003, 15030, 44086, 841620, 1737330, 3279276, 1686225, 17636710, 237453, 859635, 137260, 2458638, 255480, 145271, 916695, 1103025, 1644006, 748365, 6979700, 60108, 87150],121,8]
%
% END OF LATTICE L122.8
%
\hbox{}\par\smallskip%
%
% LATTICE L122.9
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9153875640, 1308407100, 2287750920; 1308407100, 187358536, 326999400; 2287750920, 326999400, 571758289],[5460, 4, 1],[-466717, 464669, 341, -8828746, -16281212, -36756738, -178677426, -37225501, -590969833, -27456501, -20943607, -10705844, -339365686, -3724528, -69379181, -32133219, -41451189, -160223591, -29809865, -425733218, -13509556, -5125702; -1371, 1365, 1, -25935, -47827, -107975, -524875, -109352, -1736007, -80655, -61523, -31449, -996905, -10941, -203805, -94393, -121765, -470665, -87568, -1250613, -39685, -15057; 1868238, -1860040, -1365, 35340900, 65172640, 147134848, 715234195, 149011276, 2365614069, 109906600, 83835906, 42854802, 1358458920, 14909045, 277720380, 128627200, 165926432, 641364685, 119326964, 1704182571, 54077880, 20517854],121,9]
%
% END OF LATTICE L122.9
%
\hbox{}\par\smallskip%
%
% LATTICE L122.10
%
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${L_{122.10}}=\pdualwitharg{5}({L_{122.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6259775340, 23013900, -371280; 23013900, -84610, 1365; -371280, 1365, -22],[5460, 5, 1],[5, 3, -1, -1, 8, 26, 111, 30, 1969, 5, 43, 25, 88, 21, 22, 57, 79, 253, 60, 3491, 6, 27; 1362, 819, -272, -273, 2171, 7063, 30160, 8152, 535080, 1359, 11690, 6798, 23933, 5712, 5985, 15509, 21497, 68848, 16328, 950040, 1633, 7350; 120, 182, 0, -63, -325, -595, -2132, -535, -32760, -68, -420, -150, -273, -20, 42, 260, 490, 1768, 445, 27300, 57, 350],121,10]
%
% END OF LATTICE L122.10
%
\hbox{}\par\smallskip%
%
% LATTICE L122.11
%
\leavevmode%
\llap{\spacer}%
${L_{122.11}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{122.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{546}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{1365}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1333222207590, 79545375, -446123261220; 79545375, 4746, -26617500; -446123261220, -26617500, 149281914949],[1365, 21, 1],[-349, 261, 2354, 4882, 129716, 38182, 2520784, 259165, 387134, 182277, 18829, 21008, 268306, 38964, 15777, 138851, 23795, 1239154, 112697, 150000, 44972, 1830; -151, 65, 906, 1855, 48919, 14365, 947570, 97384, 145418, 68395, 7053, 7851, 100035, 14494, 5845, 51181, 8735, 453830, 41216, 54782, 16305, 647; -1043, 780, 7035, 14590, 387660, 114108, 7533435, 774522, 1156961, 544740, 56271, 62783, 801840, 116445, 47150, 414960, 71112, 3703245, 336798, 448279, 134400, 5469],121,11]
%
% END OF LATTICE L122.11
%
\hbox{}\par\smallskip%
%
% LATTICE L122.12
%
\leavevmode%
\llap{\spacer}%
${L_{122.12}}=\pdualwitharg{7}({L_{122.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{130}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{182}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{1820}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{l}{2}{156}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-47088115620, -240245460, 1337700; -240245460, -1225742, 6825; 1337700, 6825, -38],[5460, 7, 1],[29, 74, 25, 19, 70, 14, 1577, 75, 625, 38, 5, 7, 4, -1, -1, 8, 4, 607, 33, 311, 28, 7; -5682, -14495, -4896, -3720, -13702, -2740, -308620, -14677, -122304, -7435, -978, -1368, -780, 196, 195, -1573, -785, -119080, -6473, -60996, -5490, -1372; 336, 1560, 700, 705, 3185, 709, 83720, 4081, 34788, 2310, 356, 714, 715, 0, -180, -910, -184, -20020, -931, -7488, -385, -6],121,12]
%
% END OF LATTICE L122.12
%
\hbox{}\par\smallskip%
%
% LATTICE L122.13
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[196396262790, 39414375, 78048384960; 39414375, 7910, 15663375; 78048384960, 15663375, 31016630909],[1365, 35, 1],[626, -31, -1405, -1748, -77400, -22782, -300811, -154633, -692957, -21751, -11234, -37601, -32014, -23245, -5647, -82825, -14193, -147819, -67217, -268393, -5364, -1091; -453, 13, 906, 1113, 48919, 14365, 189514, 97384, 436254, 13679, 7053, 23553, 20007, 14494, 3507, 51181, 8735, 90766, 41216, 164346, 3261, 647; -1575, 78, 3535, 4398, 194740, 57320, 756847, 389060, 1743495, 54726, 28265, 94605, 80548, 58485, 14208, 208390, 35710, 371917, 169120, 675285, 13496, 2745],121,13]
%
% END OF LATTICE L122.13
%
\hbox{}\par\smallskip%
%
% LATTICE L122.14
%
\leavevmode%
\llap{\spacer}%
${L_{122.14}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{122.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${13}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{910}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{546}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{l}{2}{273}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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171682875\\8814\\-57330000
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1871\\-151\\-5603
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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561\\35\\1680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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11786\\906\\35295
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\HardButStrongLineBreak\kern3pt%
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169866\\12985\\508690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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49388\\3763\\147900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1321126\\100555\\3956316
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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958012\\72890\\2868915
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\HardButStrongLineBreak\kern3pt%
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1280155\\97384\\3833622
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1029474\\78302\\3082919
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899643\\68395\\2694120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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649897\\49371\\1946217
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3344124521010, 171682875, -1116702284880; 171682875, 8814, -57330000; -1116702284880, -57330000, 372899987791],[1365, 39, 1],[-1871, 561, 11786, 169866, 49388, 1321126, 958012, 1280155, 1029474, 899643, 649897, 103452, 710506, 191376, 541201, 52193, 812075, 464122, 548303, 392660, 217748, 61170; -151, 35, 906, 12985, 3763, 100555, 72890, 97384, 78302, 68395, 49371, 7851, 53865, 14494, 40915, 3937, 61145, 34910, 41216, 29498, 16305, 4529; -5603, 1680, 35295, 508690, 147900, 3956316, 2868915, 3833622, 3082919, 2694120, 1946217, 309803, 2127720, 573105, 1620710, 156300, 2431884, 1389885, 1641978, 1175881, 652080, 183183],121,14]
%
% END OF LATTICE L122.14
%
\hbox{}\par\smallskip%
%
% LATTICE L122.15
%
\leavevmode%
\llap{\spacer}%
${L_{122.15}}=\pdualwitharg{2.3}({L_{122.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{10920}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{l}{2}{195}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}{91}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L122.15
%
\hbox{}\par\smallskip%
%
% LATTICE L122.16
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10068365580, -96811260, 141960; -96811260, -930878, 1365; 141960, 1365, -2],[5460, 13, 1],[-1, 1, 1, -2, -1, -33, -51, -35, -173, -27, -43, -23, -29, -17, -82, -3, -51, -61, -37, -163, -17, -13; 102, -105, -104, 210, 104, 3430, 5300, 3637, 17976, 2805, 4466, 2388, 3010, 1764, 8505, 311, 5285, 6320, 3833, 16884, 1760, 1344; -1404, -700, 0, 1365, -5, -1547, -3020, -2249, -11928, -2210, -4368, -2886, -4305, -2860, -16380, -710, -13468, -17000, -10621, -48132, -5655, -5642],121,16]
%
% END OF LATTICE L122.16
%
\hbox{}\par\smallskip%
%
% LATTICE L122.17
%
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\llap{\spacer}%
${L_{122.17}}=\pdualwitharg{3.5}({L_{122.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-753116828100, -8186052420, 8413860; -8186052420, -88978830, 91455; 8413860, 91455, -94],[5460, 15, 1],[5, 16, -1, -1, 8, 36, 163, 45, 1003, 8, 73, 15, 166, 41, 15, 122, 174, 565, 135, 2637, 14, 67; -456, -1456, 92, 91, -741, -3311, -14976, -4133, -92092, -734, -6692, -1374, -15197, -3752, -1372, -11154, -15904, -51636, -12337, -240968, -1279, -6118; 3890, 15561, 0, -973, -4875, 945, 19344, 6780, 178360, 1941, 23310, 5830, 72891, 19440, 7777, 68055, 101115, 334464, 80700, 1590680, 8751, 44730],121,17]
%
% END OF LATTICE L122.17
%
\hbox{}\par\smallskip%
%
% LATTICE L122.18
%
\leavevmode%
\llap{\spacer}%
${L_{122.18}}=\pdualwitharg{5.13}(\pfillwitharg{2}({L_{122.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2535071876610, 192976875, 1012683884940; 192976875, 14690, 77088375; 1012683884940, 77088375, 404536321151],[1365, 65, 1],[7556, -151, -15865, -137192, -66480, -1778334, -257911, -1723183, -4157243, -242197, -874808, -417761, -191278, -257605, -437095, -70255, -1093101, -124947, -738047, -1585627, -58620, -82337; -453, 7, 906, 7791, 3763, 100555, 14578, 97384, 234906, 13679, 49371, 23553, 10773, 14494, 24549, 3937, 61145, 6982, 41216, 88494, 3261, 4529; -18915, 378, 39715, 343434, 166420, 4451720, 645631, 4313660, 10406865, 606294, 2189915, 1045785, 478828, 644865, 1094184, 175870, 2736370, 312781, 1847560, 3969315, 146744, 206115],121,18]
%
% END OF LATTICE L122.18
%
\hbox{}\par\smallskip%
%
% LATTICE L122.19
%
\leavevmode%
\llap{\spacer}%
${L_{122.19}}=\pdualwitharg{2.5}({L_{122.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L122.19
%
\hbox{}\par\smallskip%
%
% LATTICE L122.20
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-652060276140, 10188441900, -9347520; 10188441900, -159194406, 146055; -9347520, 146055, -134],[5460, 21, 1],[7, 17, -1, -1, 8, 6, 971, 54, 173, 49, 13, 19, 152, 53, 14, 161, 33, 3761, 180, 503, 94, 13; 454, 1105, -64, -65, 507, 385, 62400, 3472, 11128, 3155, 838, 1226, 9815, 3424, 905, 10413, 2135, 243360, 11648, 32552, 6085, 842; 6538, 18525, 0, -1090, -5460, 1086, 278460, 17409, 60944, 20685, 6534, 10892, 94770, 34860, 9805, 118755, 25059, 2893800, 139461, 392236, 75180, 10896],121,20]
%
% END OF LATTICE L122.20
%
\hbox{}\par\smallskip%
%
% LATTICE L122.21
%
\leavevmode%
\llap{\spacer}%
${L_{122.21}}=\pdualwitharg{7.13}(\pfillwitharg{2}({L_{122.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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43\\5\\300
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49831\\7053\\347659
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2232430726890, 214270875, -319985292900; 214270875, 20566, -30712500; -319985292900, -30712500, 45865068259],[1365, 91, 1],[-3013, 43, 6326, 39074, 26508, 101298, 514192, 687095, 236806, 482863, 49831, 166576, 54478, 102716, 124489, 28013, 62265, 249102, 294283, 90320, 116868, 4690; -453, 5, 906, 5565, 3763, 14365, 72890, 97384, 33558, 68395, 7053, 23553, 7695, 14494, 17535, 3937, 8735, 34910, 41216, 12642, 16305, 647; -21021, 300, 44135, 272610, 184940, 706732, 3587395, 4793698, 1652139, 3368820, 347659, 1162161, 380080, 716625, 868530, 195440, 434408, 1737925, 2053142, 630141, 815360, 32721],121,21]
%
% END OF LATTICE L122.21
%
\hbox{}\par\smallskip%
%
% LATTICE L122.22
%
\leavevmode%
\llap{\spacer}%
${L_{122.22}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{122.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{2730}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{273}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}{65}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L122.22
%
\hbox{}\par\smallskip%
%
% LATTICE L122.23
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6402166680, 35364420, 1599343200; 35364420, 476392, 8834280; 1599343200, 8834280, 399536407],[5460, 28, 1],[3894461, 18934267, 1544214, 4497962, 16042912, 3154574, 88189299, 16711195, 34696562, 8320819, 536673, 737158, 1136082, -341, 133657, 4027437, 1477045, 52303144, 11078519, 25514485, 8661784, 1007200; -11364, -55250, -4506, -13125, -46813, -9205, -257335, -48763, -101244, -24280, -1566, -2151, -3315, 1, -390, -11752, -4310, -152620, -32327, -74451, -25275, -2939; -15589266, -75792600, -6181385, -18005040, -64218700, -12627548, -353015845, -66893792, -138888009, -33307680, -2148266, -2950794, -4547660, 1365, -535020, -16121560, -5912512, -209365975, -44346568, -102132771, -34672540, -4031754],121,23]
%
% END OF LATTICE L122.23
%
\hbox{}\par\smallskip%
%
% LATTICE L122.24
%
\leavevmode%
\llap{\spacer}%
${L_{122.24}}=\pdualwitharg{5.7}({L_{122.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{910}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{364}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{l}{2}{780}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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-1854974940\\-46374370\\102375
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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2249519\\-88293660\\763712040
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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35\\-1368\\14490
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6\\-234\\2717
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-1\\40\\0
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8\\-325\\-2275
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8\\-317\\1355
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267\\-10556\\56056
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75\\-2963\\16730
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14\\-552\\3619
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-74199003060, -1854974940, 4095000; -1854974940, -46374370, 102375; 4095000, 102375, -226],[5460, 35, 1],[35, 6, -1, -1, 8, 8, 267, 75, 727, 14, 19, 85, 46, 81, 13, 252, 52, 1189, 285, 2393, 30, 21; -1368, -234, 40, 39, -325, -317, -10556, -2963, -28704, -552, -748, -3342, -1807, -3180, -510, -9880, -2038, -46592, -11167, -93756, -1175, -822; 14490, 2717, 0, -453, -2275, 1355, 56056, 16730, 170040, 3619, 5430, 26250, 14937, 27160, 4527, 90545, 19015, 438256, 105490, 889200, 11319, 8150],121,24]
%
% END OF LATTICE L122.24
%
\hbox{}\par\smallskip%
%
% LATTICE L122.25
%
\leavevmode%
\llap{\spacer}%
${L_{122.25}}=\pdualwitharg{3.13}({L_{122.1}})$%
\nopagebreak\par%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-81184740, 2254980, 343980; 2254980, -62634, -9555; 343980, -9555, -1454],[5460, 39, 1],[-5, 16, 1, -139, -56, -1620, -2411, -1629, -2647, -1192, -1801, -303, -1094, -617, -939, -98, -1614, -1901, -1143, -1665, -502, -355; -176, 560, 36, -4865, -1961, -56735, -84440, -57053, -92708, -41750, -63084, -10614, -38325, -21616, -32900, -3434, -56560, -66620, -40057, -58352, -17595, -12446; -26, 105, 0, -910, -360, -10374, -15420, -10413, -16912, -7605, -11466, -1924, -6930, -3900, -5915, -615, -10101, -11880, -7137, -10388, -3120, -2184],121,25]
%
% END OF LATTICE L122.25
%
\hbox{}\par\smallskip%
%
% LATTICE L122.26
%
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\llap{\spacer}%
${L_{122.26}}=\pdualwitharg{3.5.13}(\pfillwitharg{2}({L_{122.1}}))$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L122.26
%
\hbox{}\par\smallskip%
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5144968920, -8064420, 1285305840; -8064420, 151528, -2014740; 1285305840, -2014740, 321092533],[5460, 52, 1],[711287, 638317, -341, -35464, 178674, 6537288, 2607482, 7250621, 9071801, 5856691, 4927177, 2819914, 7601152, 1177408, 24753871, 998393, 18181119, 5631987, 13936585, 15646606, 7091046, 3312292; -2106, -1890, 1, 105, -529, -19355, -7720, -21467, -26859, -17340, -14588, -8349, -22505, -3486, -73290, -2956, -53830, -16675, -41263, -46326, -20995, -9807; -2847234, -2555140, 1365, 141960, -715220, -26168324, -10437575, -29023748, -36313809, -23443940, -19723158, -11287926, -30426900, -4713085, -99088080, -3996500, -72777796, -22544465, -55787212, -62632311, -28384980, -13258882],121,27]
%
% END OF LATTICE L122.27
%
\hbox{}\par\smallskip%
%
% LATTICE L122.28
%
\leavevmode%
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${L_{122.28}}=\pdualwitharg{2.3.5}({L_{122.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[481840184280, 496789020, 120371433000; 496789020, 640200, 124105800; 120371433000, 124105800, 30070721279],[5460, 60, 1],[1788806, 12203162, -341, 156853, 21820297, 58742127, 60217568, 64110591, 346171481, 10203628, 42289612, 7942899, 164913869, 9732046, 13467534, 104691868, 145623678, 116899463, 111060099, 539413674, 11206123, 25832323; -5189, -35399, 1, -455, -63297, -170401, -174681, -185974, -1004185, -29599, -122675, -23041, -478387, -28231, -39067, -303693, -422429, -339105, -322166, -1564745, -32507, -74935; -7160470, -48848436, 1365, -627872, -87345180, -235140780, -241046871, -256630380, -1385701135, -40844436, -169282470, -31794890, -660139116, -38956725, -53909632, -419074500, -582921780, -467940681, -444566100, -2159236625, -44857356, -103405050],121,28]
%
% END OF LATTICE L122.28
%
\hbox{}\par\smallskip%
%
% LATTICE L122.29
%
\leavevmode%
\llap{\spacer}%
${L_{122.29}}=\pdualwitharg{5.13}({L_{122.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4076321340, 203816340, -68113500; 203816340, -10190830, 3405675; -68113500, 3405675, -1138114],[5460, 65, 1],[785, 131, -1, -1186, -659, -18059, -5273, -17665, -85427, -2505, -18197, -8705, -1979, -5279, -4358, -661, -9661, -2119, -6095, -25453, -399, -283; 16482, 2751, -20, -24906, -13840, -379274, -110744, -371003, -1794156, -52611, -382186, -182832, -41566, -110880, -91539, -13885, -202951, -44516, -128047, -534744, -8384, -5964; 2340, 392, 0, -3549, -1975, -54145, -15812, -52975, -256200, -7514, -54600, -26130, -5943, -15860, -13104, -1990, -29120, -6392, -18395, -76860, -1209, -910],121,29]
%
% END OF LATTICE L122.29
%
\hbox{}\par\smallskip%
%
% LATTICE L122.30
%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L122.30
%
\hbox{}\par\smallskip%
%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[862835358840, -794184300, 215550229440; -794184300, 817656, -198400020; 215550229440, -198400020, 53847933949],[5460, 84, 1],[-2074081, -10192393, 341, -87301, -23807257, -9260727, -333275341, -71049378, -54864931, -56764751, -6748777, -8910841, -132611593, -10985263, -10897621, -118996729, -23694039, -666352561, -126679218, -87939055, -64069391, -4236145; -6177, -30355, 1, -260, -70902, -27580, -992550, -211597, -163397, -169055, -20099, -26538, -394940, -32716, -32455, -354393, -70565, -1984515, -377273, -261898, -190810, -12616; 8302406, 40799460, -1365, 349460, 95298840, 37070064, 1334078655, 284405856, 219620609, 227225460, 27014898, 35669494, 530835240, 43973265, 43622440, 476335860, 94845636, 2667364245, 507088344, 352014091, 256465440, 16957002],121,31]
%
% END OF LATTICE L122.31
%
\hbox{}\par\smallskip%
%
% LATTICE L122.32
%
\leavevmode%
\llap{\spacer}%
${L_{122.32}}=\pdualwitharg{7.13}({L_{122.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1092}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{364}\above{1pt}{1pt}{r}{2}{390}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{r}{2}{182}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{910}\above{1pt}{1pt}{b}{2}{52}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-3619980\\-226226\\4095
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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13\\-198\\546
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\HardButStrongLineBreak\kern3pt%
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1\\-15\\55
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-7\\105\\-390
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\HardButStrongLineBreak\kern3pt%
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-4\\59\\-280
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\HardButStrongLineBreak\kern3pt%
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-14\\205\\-1066
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-137\\2000\\-10780
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-90\\1312\\-7189
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-57925140, -3619980, 65520; -3619980, -226226, 4095; 65520, 4095, -74],[5460, 91, 1],[13, 1, -1, -7, -4, -14, -137, -90, -61, -59, -11, -31, -4, -11, -2, 1, 5, 53, 36, 25, 26, 5; -198, -15, 16, 105, 59, 205, 2000, 1312, 888, 855, 158, 438, 55, 144, 15, -19, -85, -880, -592, -408, -415, -78; 546, 55, 0, -390, -280, -1066, -10780, -7189, -4944, -5005, -1014, -3276, -510, -1820, -975, -175, -299, -1960, -1001, -516, 0, 104],121,32]
%
% END OF LATTICE L122.32
%
\hbox{}\par\smallskip%
%
% LATTICE L122.33
%
\leavevmode%
\llap{\spacer}%
${L_{122.33}}=\pdualwitharg{3.5.7}({L_{122.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{l}{2}{420}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{2730}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{1092}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{l}{2}{260}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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221\\2852\\-420
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12\\155\\-23
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147\\1901\\-285
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3373\\43628\\-6552
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-322140, 27300, 16380; 27300, -2310, -1365; 16380, -1365, -682],[5460, 105, 1],[75, 124, 221, 12, 707, 147, 3373, 810, 2271, 86, 61, 35, 19, -1, -1, 8, 18, 631, 180, 589, 35, 49; 966, 1599, 2852, 155, 9139, 1901, 43628, 10478, 29380, 1113, 790, 454, 247, -12, -13, 91, 229, 8060, 2302, 7540, 449, 630; -140, -234, -420, -23, -1365, -285, -6552, -1575, -4420, -168, -120, -70, -39, 0, 2, 0, -30, -1092, -315, -1040, -63, -90],121,33]
%
% END OF LATTICE L122.33
%
\hbox{}\par\smallskip%
%
% LATTICE L122.34
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L122.34
%
\hbox{}\par\smallskip%
%
% LATTICE L122.35
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[248753679720, 553813260, 62142454920; 553813260, 1289960, 138350940; 62142454920, 138350940, 15524130971],[5460, 140, 1],[2852176, 940952, -341, 13637, 10313657, 4057005, 29286462, 31253257, 72477541, 5012958, 2991274, 11896229, 11840999, 4916476, 2936044, 53596098, 10691400, 60187195, 57237173, 119251724, 5802539, 1924861; -8157, -2691, 1, -39, -29497, -11603, -83759, -89384, -207285, -14337, -8555, -34023, -33865, -14061, -8397, -153283, -30577, -172133, -163696, -341055, -16595, -5505; -11417070, -3766568, 1365, -54588, -41284880, -16239920, -117231751, -125104700, -290122755, -20066536, -11973870, -47619810, -47398728, -19680325, -11752788, -214541600, -42796960, -240925321, -229116580, -477356685, -23227176, -7705090],121,35]
%
% END OF LATTICE L122.35
%
\hbox{}\par\smallskip%
%
% LATTICE L122.36
%
\leavevmode%
\llap{\spacer}%
${L_{122.36}}=\pdualwitharg{2.3.13}({L_{122.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[53256840, -20218380, 13289640; -20218380, 7989384, -5045040; 13289640, -5045040, 3316279],[5460, 156, 1],[-12750, 12066, 341, -289619, -122039, -3571453, -1334114, -3611749, -1469585, -2659572, -2024560, -344029, -2509519, -357120, -2207834, -234740, -3920906, -1163339, -2810273, -1027868, -1264997, -472893; -37, 35, 1, -840, -354, -10360, -3870, -10477, -4263, -7715, -5873, -998, -7280, -1036, -6405, -681, -11375, -3375, -8153, -2982, -3670, -1372; 51038, -48300, -1365, 1159340, 488520, 14296464, 5340435, 14457768, 5882723, 10646220, 8104278, 1377142, 10045560, 1429545, 8837920, 939660, 15695316, 4656825, 11249472, 4114537, 5063760, 1892982],121,36]
%
% END OF LATTICE L122.36
%
\hbox{}\par\smallskip%
%
% LATTICE L122.37
%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2390306100, -298787580, 229320; -298787580, -37348350, 28665; 229320, 28665, -22],[5460, 195, 1],[3, 1, -1, -3, -4, -102, -29, -96, -153, -13, -89, -13, -8, -19, -4, -1, -3, 1, 6, 13, 2, 19; -22, -7, 8, 21, 27, 679, 192, 634, 1008, 85, 574, 82, 49, 112, 21, 3, -49, -24, -94, -168, -19, -154; 2600, 1302, 0, -3913, -6525, -178815, -52212, -174915, -281960, -24804, -180180, -28730, -19593, -52260, -14378, -6540, -95550, -20952, -60255, -83860, -3939, -2730],121,37]
%
% END OF LATTICE L122.37
%
\hbox{}\par\smallskip%
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L122.38
%
\hbox{}\par\smallskip%
%
% LATTICE L122.39
%
\leavevmode%
\llap{\spacer}%
${L_{122.39}}=\pdualwitharg{3.7.13}({L_{122.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1789149180, 447288660, -333060; 447288660, -111822438, 83265; -333060, 83265, -62],[5460, 273, 1],[9, 2, -1, -3, -4, -12, -109, -69, -15, -38, -5, -1, 2, 17, 7, 8, 24, 221, 141, 31, 82, 13; 44, 10, -4, -15, -21, -65, -600, -383, -84, -220, -32, -14, 5, 64, 30, 36, 110, 1020, 653, 144, 385, 62; 10738, 2685, 0, -4030, -6720, -22854, -220500, -143871, -32272, -91455, -16146, -13468, -4050, -5460, 2665, 5355, 18759, 182280, 119301, 26812, 76440, 13416],121,39]
%
% END OF LATTICE L122.39
%
\hbox{}\par\smallskip%
%
% LATTICE L122.40
%
\leavevmode%
\llap{\spacer}%
${L_{122.40}}=\pdualwitharg{3.5.7.13}(\pfillwitharg{2}({L_{122.1}}))$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L122.40
%
\hbox{}\par\smallskip%
%
% LATTICE L122.41
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1065104040, 20655180, 266065800; 20655180, 427336, 5159700; 266065800, 5159700, 66463939],[5460, 364, 1],[61789, 5431, -341, 43453, 55679, 245805, 654237, 1784470, 313369, 1349901, 151423, 561557, 202351, 208123, 579893, 152103, 377541, 800237, 1955606, 309977, 935701, 56359; -171, -15, 1, -120, -154, -680, -1810, -4937, -867, -3735, -419, -1554, -560, -576, -1605, -421, -1045, -2215, -5413, -858, -2590, -156; -247338, -21740, 1365, -173940, -222880, -983944, -2618875, -7143136, -1254399, -5403580, -606138, -2247882, -810000, -833105, -2321280, -608860, -1511276, -3203305, -7828184, -1240821, -3745560, -225602],121,41]
%
% END OF LATTICE L122.41
%
\hbox{}\par\smallskip%
%
% LATTICE L122.42
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L122.42
%
\hbox{}\par\smallskip%
%
% LATTICE L122.43
%
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${L_{122.43}}=\pdualwitharg{5.7.13}({L_{122.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L122.43
%
\hbox{}\par\smallskip%
%
% LATTICE L122.44
%
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${L_{122.44}}=\pdualwitharg{2.3.5.13}({L_{122.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[202676237400, 155986740, 50631824880; 155986740, 132600, 38968020; 50631824880, 38968020, 12648654443],[5460, 780, 1],[169753, 54539, -341, 237245, 587659, 17830203, 1348139, 18341508, 7498789, 2751499, 10701259, 1868647, 2799221, 2038741, 2616515, 1450741, 24976197, 1505621, 18349692, 6770011, 1732981, 3567541; -436, -140, 1, -609, -1509, -45787, -3462, -47101, -19257, -7066, -27482, -4799, -7189, -5236, -6720, -3726, -64148, -3867, -47129, -17388, -4451, -9163; -679510, -218316, 1365, -949676, -2352360, -71373120, -5396511, -73419840, -30017155, -11014068, -42836430, -7480070, -11205096, -8160945, -10473736, -5807220, -99978060, -6026901, -73452600, -27099905, -6937008, -14280630],121,44]
%
% END OF LATTICE L122.44
%
\hbox{}\par\smallskip%
%
% LATTICE L122.45
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L122.45
%
\hbox{}\par\smallskip%
%
% LATTICE L122.46
%
\leavevmode%
\llap{\spacer}%
${L_{122.46}}=\pdualwitharg{3.5.7.13}({L_{122.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5460, 0, 0; 0, -1984710, 9555; 0, 9555, -46],[5460, 1365, 1],[-1, 0, 1, 0, -1, -5, -11, -38, -9, -6, -7, -9, -1, -21, -1, -4, -10, -17, -52, -11, -5, -3; 22, 1, 0, -1, -5, -11, -16, -44, -8, -1, 6, 18, 3, 80, 5, 25, 71, 128, 404, 88, 45, 34; 4550, 207, 0, -208, -1050, -2340, -3444, -9555, -1760, -273, 1170, 3640, 612, 16380, 1027, 5145, 14625, 26376, 83265, 18140, 9282, 7020],121,46]
%
% END OF LATTICE L122.46
%
\hbox{}\par\smallskip%
%
% LATTICE L122.47
%
\leavevmode%
\llap{\spacer}%
${L_{122.47}}=\pdualwitharg{2.5.7.13}({L_{122.1}})$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L122.47
%
\hbox{}\par\smallskip%
%
% LATTICE L122.48
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\end{tabular}\endgroup%
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\end{tabular}\endgroup%
\kern3pt%
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-398580\\-60060\\-207480\\-16380\\-16380\\-5460\\0\\10920\\0\\-5460\\-27300\\-92820\\-38220\\-283920\\-81900\\-475020\\-1468740\\-687960\\-8856120\\-491400\\-1075620\\-464100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1820\\0\\-2730\\-1820\\-16380\\-60060\\-30030\\-398580\\-22750\\-54600\\-27300
\end{tabular}\endgroup%
\kern3pt%
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0\\24\\0\\-156\\-2100\\-8580\\-4452\\-60060\\-3480\\-8736\\-4680
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\end{tabular}\endgroup%
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-1820\\-156\\0\\104\\0\\-1560\\-1092\\-16380\\-1040\\-3276\\-2340
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-16380\\-2100\\-5460\\0\\840\\0\\-840\\-16380\\-1260\\-5460\\-5460
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-60060\\-8580\\-27300\\-1560\\0\\1560\\0\\-5460\\-780\\-5460\\-8580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30030\\-4452\\-15015\\-1092\\-840\\0\\21\\0\\-105\\-1092\\-2730
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-398580\\-60060\\-207480\\-16380\\-16380\\-5460\\0\\10920\\0\\-5460\\-27300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22750\\-3480\\-12285\\-1040\\-1260\\-780\\-105\\0\\5\\0\\-1170
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54600\\-8736\\-32760\\-3276\\-5460\\-5460\\-1092\\-5460\\0\\2184\\0
\end{tabular}\endgroup%
\kern3pt%
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-27300\\-4680\\-19110\\-2340\\-5460\\-8580\\-2730\\-27300\\-1170\\0\\780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29120\\-5460\\-24570\\-3640\\-10920\\-21840\\-8190\\-92820\\-4550\\-5460\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5460\\-1116\\-5460\\-936\\-3360\\-7800\\-3192\\-38220\\-1980\\-3276\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24570\\-5460\\-28665\\-5460\\-21840\\-54600\\-23205\\-283920\\-15015\\-27300\\-8190
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3640\\-936\\-5460\\-1196\\-5460\\-14820\\-6552\\-81900\\-4420\\-8736\\-3120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10920\\-3360\\-21840\\-5460\\-28140\\-81900\\-37380\\-475020\\-26040\\-54600\\-21840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21840\\-7800\\-54600\\-14820\\-81900\\-247260\\-114660\\-1468740\\-81120\\-174720\\-73320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8190\\-3192\\-23205\\-6552\\-37380\\-114660\\-53529\\-687960\\-38115\\-82992\\-35490
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92820\\-38220\\-283920\\-81900\\-475020\\-1468740\\-687960\\-8856120\\-491400\\-1075620\\-464100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4550\\-1980\\-15015\\-4420\\-26040\\-81120\\-38115\\-491400\\-27305\\-60060\\-26130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5460\\-3276\\-27300\\-8736\\-54600\\-174720\\-82992\\-1075620\\-60060\\-134316\\-60060
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-780\\-8190\\-3120\\-21840\\-73320\\-35490\\-464100\\-26130\\-60060\\-28080
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[251160, -52170300, -13033020; -52170300, 10836538440, 2707149900; -13033020, 2707149900, 676291661],[5460, 5460, 1],[-69, -5, 0, -2, -80, -386, -211, -2909, -172, -457, -267, -346, -78, -415, -81, -335, -859, -370, -4561, -243, -456, -148; 227, -33, -341, -13, 839, 4285, 2365, 32722, 1939, 5181, 3019, 3863, 851, 4431, 831, 3251, 7985, 3361, 40898, 2151, 3817, 1071; -910, 132, 1365, 52, -3360, -17160, -9471, -131040, -7765, -20748, -12090, -15470, -3408, -17745, -3328, -13020, -31980, -13461, -163800, -8615, -15288, -4290],121,48]
%
% END OF LATTICE L122.48
%

\medskip%
%------------------------------WEYL-GROUP-W123--------------------------------
%
\leavevmode\llap{}%
$W_{123}$%
\qquad\llap{16} lattices, $\chi=3$%
\hfill%
$4222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L123.1
%
\leavevmode%
\llap{\spacer}%
${L_{123.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{3}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1176\\-552\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-552\\-259\\56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\56\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-7\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\9\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\12\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-20\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1176, -552, 120; -552, -259, 56; 120, 56, -11],[24, 1, 1],[3, -4, -5, 9; -7, 9, 12, -20; -3, 2, 6, -4],122,1]
%
% END OF LATTICE L123.1
%
\hbox{}\par\smallskip%
%
% LATTICE L123.2
%
\leavevmode%
\llap{\spacer}%
${L_{123.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2616\\120\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-5\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-4\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\15\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2616, 120, 72; 120, -5, -4; 72, -4, -1],[24, 1, 1],[1, -1, -1, 1; 15, -16, -15, 16; 11, -10, -12, 8],122,2]
%
% END OF LATTICE L123.2
%
\hbox{}\par\smallskip%
%
% LATTICE L123.3
%
\leavevmode%
\llap{\spacer}%
${L_{123.3}}=\pfillwitharg{2}({L_{123.1}})$%
{${}={}$}Nikulin $3'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-6, 0, 0; 0, 1, 0; 0, 0, 1],[6, 1, 1],[0, 0, -1, -1; -1, 1, 0, -2; 1, 0, -3, -2],122,3]
%
% END OF LATTICE L123.3
%
\hbox{}\par\smallskip%
%
% LATTICE L123.4
%
\leavevmode%
\llap{\spacer}%
${L_{123.4}}=\main({L_{123.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-588\\36\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\-2\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-5\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-21\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-588, 36, 72; 36, -2, -5; 72, -5, -7],[12, 1, 1],[1, -1, -2, 1; 10, -9, -21, 8; 3, -4, -6, 4],122,4]
%
% END OF LATTICE L123.4
%
\hbox{}\par\smallskip%
%
% LATTICE L123.5
%
\leavevmode%
\llap{\spacer}%
${L_{123.5}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{123.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\30\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-8\\12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\12\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\3\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-102, 30, -42; 30, -8, 12; -42, 12, -17],[6, 2, 1],[0, -1, 5, 3; -3, 2, 3, -1; -2, 4, -12, -9],122,5]
%
% END OF LATTICE L123.5
%
\hbox{}\par\smallskip%
%
% LATTICE L123.6
%
\leavevmode%
\llap{\spacer}%
${L_{123.6}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{123.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{4}{3}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138\\-36\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-3\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-15\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\2\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-1\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-138, -36, -60; -36, -3, -15; -60, -15, -26],[6, 3, 1],[6, -5, 0, 11; 2, -1, -1, 0; -15, 12, 1, -24],122,6]
%
% END OF LATTICE L123.6
%
\hbox{}\par\smallskip%
%
% LATTICE L123.7
%
\leavevmode%
\llap{\spacer}%
${L_{123.7}}=\pdualwitharg{3}(\main({L_{123.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{4}{6}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-3\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\6\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12, 0, 0; 0, -6, -3; 0, -3, -1],[12, 3, 1],[0, -1, 0, 1; 1, -1, -1, 0; -3, 0, 2, 0],122,7]
%
% END OF LATTICE L123.7
%
\hbox{}\par\smallskip%
%
% LATTICE L123.8
%
\leavevmode%
\llap{\spacer}%
${L_{123.8}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{123.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{4}{6}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\162\\66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
162\\2910\\1188
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\1188\\485
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\19\\-46
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\31\\-75
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12, 162, 66; 162, 2910, 1188; 66, 1188, 485],[6, 6, 1],[1, 1, -4, -6; 0, -5, 19, 31; 0, 12, -46, -75],122,8]
%
% END OF LATTICE L123.8
%
\hbox{}\par\smallskip%
%
% LATTICE L123.9
%
\leavevmode%
\llap{\spacer}%
${L_{123.9}}=\pdualwitharg{2}(\main({L_{123.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{4}{8}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1140\\204\\444
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
204\\40\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
444\\80\\173
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\1\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\3\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1140, 204, 444; 204, 40, 80; 444, 80, 173],[12, 4, 1],[3, 6, 4, 1; 1, 1, 3, 1; -8, -16, -12, -3],122,9]
%
% END OF LATTICE L123.9
%
\hbox{}\par\smallskip%
%
% LATTICE L123.10
%
\leavevmode%
\llap{\spacer}%
${L_{123.10}}=\pdualwitharg{3}({L_{123.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{4}{3}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-96\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-33\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-42\\-53
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-264, -96, -120; -96, -33, -42; -120, -42, -53],[24, 3, 1],[-1, 0, 1, 1; -1, -4, 2, 12; 3, 3, -4, -12],122,10]
%
% END OF LATTICE L123.10
%
\hbox{}\par\smallskip%
%
% LATTICE L123.11
%
\leavevmode%
\llap{\spacer}%
${L_{123.11}}=\pdualwitharg{3}({L_{123.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{*}{4}{12}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14376\\288\\576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\-3\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
576\\-12\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\11\\69
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\12\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14376, 288, 576; 288, -3, -12; 576, -12, -23],[24, 3, 1],[3, -1, -1, 1; 11, -2, -4, 0; 69, -24, -23, 24],122,11]
%
% END OF LATTICE L123.11
%
\hbox{}\par\smallskip%
%
% LATTICE L123.12
%
\leavevmode%
\llap{\spacer}%
${L_{123.12}}=\pdualwitharg{2.3}(\main({L_{123.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{4}{24}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-36\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\120\\36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\36\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12, -36, -12; -36, 120, 36; -12, 36, 11],[12, 12, 1],[1, 2, 0, 0; 0, 1, -1, -1; 0, 0, 4, 3],122,12]
%
% END OF LATTICE L123.12
%
\hbox{}\par\smallskip%
%
% LATTICE L123.13
%
\leavevmode%
\llap{\spacer}%
${L_{123.13}}=\pdualwitharg{2}({L_{123.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{4}{8}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\72\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-88\\32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\32\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-48, 72, -24; 72, -88, 32; -24, 32, -11],[24, 8, 1],[-1, 2, 1, -1; 2, -1, -3, 0; 8, -8, -12, 2],122,13]
%
% END OF LATTICE L123.13
%
\hbox{}\par\smallskip%
%
% LATTICE L123.14
%
\leavevmode%
\llap{\spacer}%
${L_{123.14}}=\pdualwitharg{2}({L_{123.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{4}{8}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24816\\552\\1080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
552\\-8\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1080\\-24\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\92
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24816, 552, 1080; 552, -8, -24; 1080, -24, -47],[24, 8, 1],[-1, 4, 1, -1; 2, -1, -3, 0; -24, 92, 24, -23],122,14]
%
% END OF LATTICE L123.14
%
\hbox{}\par\smallskip%
%
% LATTICE L123.15
%
\leavevmode%
\llap{\spacer}%
${L_{123.15}}=\pdualwitharg{2.3}({L_{123.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{}{4}{24}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\1416\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1416\\-6864\\-816
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-816\\-97
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-11\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\14\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-264, 1416, 168; 1416, -6864, -816; 168, -816, -97],[24, 24, 1],[2, -1, -1, 0; -11, 14, 5, -5; 96, -120, -44, 42],122,15]
%
% END OF LATTICE L123.15
%
\hbox{}\par\smallskip%
%
% LATTICE L123.16
%
\leavevmode%
\llap{\spacer}%
${L_{123.16}}=\pdualwitharg{2.3}({L_{123.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{4}{24}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\504\\-192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\-6864\\2616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\2616\\-997
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2\\-9\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\32\\84
\end{tabular}\endgroup%
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-1\\3\\8
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0\\-8\\-21
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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48\\-24\\-24\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24, 504, -192; 504, -6864, 2616; -192, 2616, -997],[24, 24, 1],[2, -1, -1, 0; -9, 32, 3, -8; -24, 84, 8, -21],122,16]
%
% END OF LATTICE L123.16
%

\medskip%
%------------------------------WEYL-GROUP-W124--------------------------------
%
\leavevmode\llap{}%
$W_{124}$%
\qquad\llap{46} lattices, $\chi=12$%
\hfill%
$222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L124.1
%
\leavevmode%
\llap{\spacer}%
${L_{124.1}}$%
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{$1\above{1pt}{1pt}{-2}{4}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{4}}%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3320\\120\\120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-4\\-5
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-5\\-3
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-1\\-18\\-10
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\5
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3\\56\\24
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1\\18\\9
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3\\50\\30
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\-8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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4\\0\\-8\\-4\\-20\\0
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0\\5\\0\\-5\\-60\\-20
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-8\\0\\8\\0\\-40\\-24
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-4\\-5\\0\\1\\0\\-4
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-20\\-60\\-40\\0\\20\\0
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0\\-20\\-24\\-4\\0\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3320, 120, 120; 120, -4, -5; 120, -5, -3],[40, 1, 1],[-1, 1, 3, 1, 3, -1; -18, 20, 56, 18, 50, -20; -10, 5, 24, 9, 30, -8],123,1]
%
% END OF LATTICE L124.1
%
\hbox{}\par\smallskip%
%
% LATTICE L124.2
%
\leavevmode%
\llap{\spacer}%
${L_{124.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-2120\\280\\80
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\-19\\-10
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-10\\-3
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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79\\-80\\2320
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-9\\8\\-261
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-3\\3\\-88
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-1\\1\\-30
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1\\0\\25
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5\\-4\\144
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2120, 280, 80; 280, -19, -10; 80, -10, -3],[40, 1, 1],[-1, 1, 5, 2, 4, -1; 1, 0, -4, -2, -5, 0; -30, 25, 144, 59, 120, -28],123,2]
%
% END OF LATTICE L124.2
%
\hbox{}\par\smallskip%
%
% LATTICE L124.3
%
\leavevmode%
\llap{\spacer}%
${L_{124.3}}$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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920\\200\\0
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200\\43\\1
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0\\1\\-2
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0\\-1\\-1
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0\\0\\1
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[920, 200, 0; 200, 43, 1; 0, 1, -2],[40, 1, 1],[3, 11, 1, -3, -11, -1; -14, -50, -4, 14, 50, 4; -8, -30, -4, 6, 20, 0],123,3]
%
% END OF LATTICE L124.3
%
\hbox{}\par\smallskip%
%
% LATTICE L124.4
%
\leavevmode%
\llap{\spacer}%
${L_{124.4}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[5680, 2720, -80; 2720, 1302, -38; -80, -38, 1],[80, 2, 1],[0, -9, -1, 5, 17, 3; 0, 20, 2, -12, -40, -7; -1, 40, 0, -48, -150, -26],123,4]
%
% END OF LATTICE L124.4
%
\hbox{}\par\smallskip%
%
% LATTICE L124.5
%
\leavevmode%
\llap{\spacer}%
${L_{124.5}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[4080, -160, 80; -160, 6, -2; 80, -2, -3],[80, 2, 1],[1, 1, -1, -1, 1, 1; 28, 20, -30, -28, 30, 29; 6, 0, -8, -8, 5, 6],123,5]
%
% END OF LATTICE L124.5
%
\hbox{}\par\smallskip%
%
% LATTICE L124.6
%
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${L_{124.6}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2480, 80, 80; 80, -2, -4; 80, -4, 1],[80, 2, 1],[0, -7, -1, -1, 1, 1; 0, -160, -23, -24, 20, 22; -1, -80, -10, -8, 10, 8],123,6]
%
% END OF LATTICE L124.6
%
\hbox{}\par\smallskip%
%
% LATTICE L124.7
%
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${L_{124.7}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-64880, 1120, 1120; 1120, -18, -20; 1120, -20, -19],[80, 2, 1],[-1, 7, 2, 5, 2, -1; -20, 120, 37, 96, 40, -18; -38, 280, 78, 192, 75, -40],123,7]
%
% END OF LATTICE L124.7
%
\hbox{}\par\smallskip%
%
% LATTICE L124.8
%
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{${}={}$}Nikulin $4'$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-30, -10, 0; -10, -3, 0; 0, 0, 1],[10, 1, 1],[0, -2, 1, 4, 22, 7; 0, 5, -2, -9, -50, -16; 1, 0, -2, -2, -5, 0],123,8]
%
% END OF LATTICE L124.8
%
\hbox{}\par\smallskip%
%
% LATTICE L124.9
%
\leavevmode%
\llap{\spacer}%
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\HardButStrongLineBreak\kern3pt%
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1\\-2\\-4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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0\\-20\\-12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[60, 20, -20; 20, 6, -9; -20, -9, -1],[20, 1, 1],[3, 11, 1, -3, -11, -1; -7, -25, -2, 7, 25, 2; 2, 0, -4, -4, -10, 0],123,9]
%
% END OF LATTICE L124.9
%
\hbox{}\par\smallskip%
%
% LATTICE L124.10
%
\leavevmode%
\llap{\spacer}%
${L_{124.10}}=\pfillwitharg{2}({L_{124.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\0\\-4
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-3\\-5\\-15
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\HardButStrongLineBreak\kern3pt%
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-1\\-3\\-6
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\HardButStrongLineBreak\kern3pt%
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0\\-1\\-1
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\HardButStrongLineBreak\kern3pt%
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1\\0\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\0\\-4\\-4\\-20\\0
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0\\5\\0\\-5\\-60\\-10
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-4\\0\\2\\0\\-20\\-6
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-4\\-5\\0\\1\\0\\-2
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-60\\-20\\0\\20\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-6\\-2\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[20, 0, 0; 0, 2, 0; 0, 0, -1],[20, 2, 1],[-1, -3, -1, 0, 1, 0; 0, -5, -3, -1, 0, 1; -4, -15, -6, -1, 0, 0],123,10]
%
% END OF LATTICE L124.10
%
\hbox{}\par\smallskip%
%
% LATTICE L124.11
%
\leavevmode%
\llap{\spacer}%
${L_{124.11}}=\main({L_{124.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1960\\360\\-120
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\kern3pt%
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360\\-66\\22
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\22\\-7
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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-1\\-4\\4
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2\\15\\10
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\HardButStrongLineBreak\kern3pt%
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1\\6\\1
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1\\5\\-2
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\HardButStrongLineBreak\kern3pt%
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1\\0\\-20
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-1\\-6\\-2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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8\\0\\-4\\-8\\-40\\0
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\kern3pt%
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0\\10\\0\\-10\\-120\\-20
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\kern3pt%
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-4\\0\\1\\0\\-20\\-6
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-8\\-10\\0\\2\\0\\-4
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-40\\-120\\-20\\0\\40\\0
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0\\-20\\-6\\-4\\0\\4
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1960, 360, -120; 360, -66, 22; -120, 22, -7],[40, 2, 1],[-1, 2, 1, 1, 1, -1; -4, 15, 6, 5, 0, -6; 4, 10, 1, -2, -20, -2],123,11]
%
% END OF LATTICE L124.11
%
\hbox{}\par\smallskip%
%
% LATTICE L124.12
%
\leavevmode%
\llap{\spacer}%
${L_{124.12}}=\main({L_{124.6}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2840\\0\\160
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\kern3pt%
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0\\2\\0
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\kern3pt%
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160\\0\\-9
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\1\\0
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\HardButStrongLineBreak\kern3pt%
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7\\0\\120
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1\\-1\\17
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\HardButStrongLineBreak\kern3pt%
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1\\-4\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-5\\-20
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-1\\0\\-18
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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2\\0\\-2\\-8\\-10\\0
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\kern3pt%
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0\\40\\0\\-40\\-120\\-40
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-2\\0\\1\\0\\-10\\-6
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-8\\-40\\0\\8\\0\\-8
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-10\\-120\\-10\\0\\10\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-6\\-8\\0\\4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-2840, 0, 160; 0, 2, 0; 160, 0, -9],[40, 2, 1],[0, 7, 1, 1, -1, -1; 1, 0, -1, -4, -5, 0; 0, 120, 17, 16, -20, -18],123,12]
%
% END OF LATTICE L124.12
%
\hbox{}\par\smallskip%
%
% LATTICE L124.13
%
\leavevmode%
\llap{\spacer}%
${L_{124.13}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{124.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\70\\-290
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\-6\\30
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-290\\30\\-127
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{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-39\\52\\103
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-6\\9\\16
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-7\\15\\20
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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2\\0\\-2
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0\\10\\0
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-2\\0\\1
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-4\\-10\\0
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0\\-10\\-3
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-660, 70, -290; 70, -6, 30; -290, 30, -127],[10, 2, 1],[-6, -7, 2, 3, -8, -5; 9, 15, -2, -5, 5, 6; 16, 20, -5, -8, 20, 13],123,13]
%
% END OF LATTICE L124.13
%
\hbox{}\par\smallskip%
%
% LATTICE L124.14
%
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\llap{\spacer}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-410, -150, -180; -150, -20, -65; -180, -65, -79],[10, 5, 1],[-28, -9, 13, 15, -4, -39; -2, -1, 0, 1, 0, -2; 65, 21, -30, -35, 9, 90],123,14]
%
% END OF LATTICE L124.14
%
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%
% LATTICE L124.15
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4760, -660, 1240; -660, 92, -172; 1240, -172, 323],[20, 4, 1],[-1, -37, -11, -9, -13, 1; 0, 70, 22, 18, 20, -4; 4, 180, 54, 44, 60, -6],123,15]
%
% END OF LATTICE L124.15
%
\hbox{}\par\smallskip%
%
% LATTICE L124.16
%
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${L_{124.16}}=\pdualwitharg{5}(\main({L_{124.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[460, 180, -40; 180, 70, -15; -40, -15, 3],[20, 5, 1],[-1, -1, -1, 1, 1, 1; 3, 3, 2, -5, -5, -6; 0, 2, 0, -10, -12, -20],123,16]
%
% END OF LATTICE L124.16
%
\hbox{}\par\smallskip%
%
% LATTICE L124.17
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10, 650, 290; 650, 49480, 22080; 290, 22080, 9853],[10, 10, 1],[3, 1, 0, 1, 3, 4; -67, -50, -29, 0, -17, -38; 150, 112, 65, 0, 38, 85],123,17]
%
% END OF LATTICE L124.17
%
\hbox{}\par\smallskip%
%
% LATTICE L124.18
%
\leavevmode%
\llap{\spacer}%
${L_{124.18}}=\pdualwitharg{2}(\main({L_{124.6}}))$%
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{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\12\\-4
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80\\-4\\-3
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\-1\\27
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3\\-5\\80
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-1\\0\\-28
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1\\0\\25
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3\\-2\\80
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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1\\0\\-4\\-4\\-5\\0
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0\\20\\0\\-20\\-60\\-40
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-4\\0\\8\\0\\-20\\-24
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-4\\-20\\0\\4\\0\\-8
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-5\\-60\\-20\\0\\5\\0
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0\\-40\\-24\\-8\\0\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2120, 120, 80; 120, 12, -4; 80, -4, -3],[40, 4, 1],[1, 3, -1, -1, 1, 3; -1, -5, 0, 1, 0, -2; 27, 80, -28, -28, 25, 80],123,18]
%
% END OF LATTICE L124.18
%
\hbox{}\par\smallskip%
%
% LATTICE L124.19
%
\leavevmode%
\llap{\spacer}%
${L_{124.19}}=\pdualwitharg{2}(\main({L_{124.5}}))$%
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{$1\above{1pt}{1pt}{-}{3}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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840\\-212\\-80
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{$\left.\llap{\phantom{%
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-1\\-4\\0
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3\\15\\-8
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17\\80\\-30
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5\\22\\-4
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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0\\0\\0\\0\\0\\0
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4\\0\\-8\\-8\\-20\\0
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0\\20\\0\\-20\\-120\\-40
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-8\\0\\8\\0\\-40\\-24
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-8\\-20\\0\\4\\0\\-8
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-20\\-120\\-40\\0\\20\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3320, 840, 320; 840, -212, -80; 320, -80, -29],[40, 4, 1],[-1, -7, -1, 3, 17, 5; -6, -35, -4, 15, 80, 22; 6, 20, 0, -8, -30, -4],123,19]
%
% END OF LATTICE L124.19
%
\hbox{}\par\smallskip%
%
% LATTICE L124.20
%
\leavevmode%
\llap{\spacer}%
${L_{124.20}}=\pdualwitharg{5}({L_{124.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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680\\280\\-80
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280\\95\\-30
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-80\\-30\\9
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\2\\15
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1\\2\\16
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-1\\-4\\-20
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-3\\-10\\-60
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\HardButStrongLineBreak\kern3pt%
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-1\\-4\\-23
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\HardButStrongLineBreak\kern3pt%
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-1\\-8\\-40
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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5\\0\\-20\\-20\\-5\\0
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0\\4\\0\\-20\\-12\\-40
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-20\\0\\40\\0\\-20\\-120
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-20\\-20\\0\\20\\0\\-40
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-5\\-12\\-20\\0\\1\\0
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0\\-40\\-120\\-40\\0\\40
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[680, 280, -80; 280, 95, -30; -80, -30, 9],[40, 5, 1],[1, 1, -1, -3, -1, -1; 2, 2, -4, -10, -4, -8; 15, 16, -20, -60, -23, -40],123,20]
%
% END OF LATTICE L124.20
%
\hbox{}\par\smallskip%
%
% LATTICE L124.21
%
\leavevmode%
\llap{\spacer}%
${L_{124.21}}=\pdualwitharg{5}({L_{124.2}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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4440\\880\\-200
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880\\165\\-40
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-1\\24\\120
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0\\6\\35
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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2\\-2\\35
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\-20\\-120
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4440, 880, -200; 880, 165, -40; -200, -40, 9],[40, 5, 1],[2, 1, -1, -2, -1, 1; -2, -1, 0, 1, 0, -4; 35, 18, -20, -40, -23, 0],123,21]
%
% END OF LATTICE L124.21
%
\hbox{}\par\smallskip%
%
% LATTICE L124.22
%
\leavevmode%
\llap{\spacer}%
${L_{124.22}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{124.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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-20\\30\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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0\\-1\\-5
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20, -20, 0; -20, 30, 0; 0, 0, -1],[20, 10, 1],[0, 1, 1, -1, -1, -2; -1, 0, 1, 0, -1, -3; -5, -4, 0, 0, -3, -10],123,22]
%
% END OF LATTICE L124.22
%
\hbox{}\par\smallskip%
%
% LATTICE L124.23
%
\leavevmode%
\llap{\spacer}%
${L_{124.23}}=\pdualwitharg{5}({L_{124.3}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120, 120, -40; 120, 35, -15; -40, -15, 6],[40, 5, 1],[1, 1, 1, -1, -1, -1; 6, 6, 4, -10, -10, -12; 20, 22, 20, -30, -32, -40],123,23]
%
% END OF LATTICE L124.23
%
\hbox{}\par\smallskip%
%
% LATTICE L124.24
%
\leavevmode%
\llap{\spacer}%
${L_{124.24}}=\pdualwitharg{2}({L_{124.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2040, -560, 120; -560, -136, 32; 120, 32, -7],[40, 8, 1],[0, 7, 1, 1, -2, -1; 1, 10, 1, 0, -5, -1; 4, 160, 21, 16, -60, -22],123,24]
%
% END OF LATTICE L124.24
%
\hbox{}\par\smallskip%
%
% LATTICE L124.25
%
\leavevmode%
\llap{\spacer}%
${L_{124.25}}=\pdualwitharg{2}({L_{124.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2840, 0, 160; 0, 8, 0; 160, 0, -9],[40, 8, 1],[0, 7, 2, 1, -2, -1; 1, 0, -1, -2, -5, 0; 0, 120, 34, 16, -40, -18],123,25]
%
% END OF LATTICE L124.25
%
\hbox{}\par\smallskip%
%
% LATTICE L124.26
%
\leavevmode%
\llap{\spacer}%
${L_{124.26}}=\pdualwitharg{2}({L_{124.3}})$%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-240\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-40, 160, -40; 160, -616, 152; -40, 152, -37],[40, 8, 1],[0, 17, 5, 3, -2, -2; 1, 10, 2, 0, -5, -1; 4, 20, 2, -4, -20, -2],123,26]
%
% END OF LATTICE L124.26
%
\hbox{}\par\smallskip%
%
% LATTICE L124.27
%
\leavevmode%
\llap{\spacer}%
${L_{124.27}}=\pdualwitharg{5}(\main({L_{124.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2040\\1040\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1040\\510\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-60\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-20\\-40\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-10\\-24\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\5\\0\\-20\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-10\\0\\10\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-24\\-20\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-30\\-20\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2040, 1040, -120; 1040, 510, -60; -120, -60, 7],[40, 10, 1],[1, 1, 1, 0, -1, -1; 0, 1, 1, -1, -4, -4; 20, 26, 25, -10, -52, -50],123,27]
%
% END OF LATTICE L124.27
%
\hbox{}\par\smallskip%
%
% LATTICE L124.28
%
\leavevmode%
\llap{\spacer}%
${L_{124.28}}=\pdualwitharg{5}(\main({L_{124.6}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-10\\-40\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-40\\-24\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\5\\0\\-10\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\0\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-24\\-10\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-30\\-40\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-120, 0, 40; 0, 10, 0; 40, 0, -13],[40, 10, 1],[0, 3, 2, 1, -1, -3; 1, 0, -1, -4, -1, 0; 0, 8, 5, 0, -4, -10],123,28]
%
% END OF LATTICE L124.28
%
\hbox{}\par\smallskip%
%
% LATTICE L124.29
%
\leavevmode%
\llap{\spacer}%
${L_{124.29}}=\pdualwitharg{2.5}(\main({L_{124.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
220\\-540\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\1240\\280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\280\\63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-8\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\39\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\8\\-41
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-11\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\5\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-40\\-80\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-40\\-48\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\20\\0\\-40\\-60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-48\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[220, -540, -120; -540, 1240, 280; -120, 280, 63],[20, 20, 1],[6, 0, -4, -4, 2, 6; -11, -1, 5, 3, -7, -13; 60, 4, -30, -20, 36, 70],123,29]
%
% END OF LATTICE L124.29
%
\hbox{}\par\smallskip%
%
% LATTICE L124.30
%
\leavevmode%
\llap{\spacer}%
${L_{124.30}}=\pdualwitharg{2}({L_{124.7}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5680\\80\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-20\\-230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\-8\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-10\\-240\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\8\\0\\-80\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-10\\0\\1\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-240\\-80\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-24\\-4\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5680, 80, -80; 80, -8, 0; -80, 0, 1],[80, 8, 1],[-1, -3, -1, 0, 1, 0; -6, -20, -7, 0, 10, 1; -72, -230, -80, -1, 80, 4],123,30]
%
% END OF LATTICE L124.30
%
\hbox{}\par\smallskip%
%
% LATTICE L124.31
%
\leavevmode%
\llap{\spacer}%
${L_{124.31}}=\pdualwitharg{2}({L_{124.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{80}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1520\\0\\-720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\0\\-341
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\0\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\0\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-5\\-150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-3\\-52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-8\\-16\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-80\\-240\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\8\\0\\-40\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-80\\0\\16\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-240\\-40\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-24\\-16\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1520, 0, -720; 0, 8, 0; -720, 0, -341],[80, 8, 1],[1, -19, 0, 23, 72, 25; -1, 0, 1, 0, -5, -3; -2, 40, 0, -48, -150, -52],123,31]
%
% END OF LATTICE L124.31
%
\hbox{}\par\smallskip%
%
% LATTICE L124.32
%
\leavevmode%
\llap{\spacer}%
${L_{124.32}}=\pdualwitharg{2}({L_{124.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{80}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-8\\-16\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-80\\-240\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\8\\0\\-40\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-80\\0\\16\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-240\\-40\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-24\\-16\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[80, 0, 0; 0, 8, 0; 0, 0, -1],[80, 8, 1],[0, 1, 0, -1, -3, -1; -1, 0, 1, 0, -5, -3; -2, 0, 0, -8, -30, -12],123,32]
%
% END OF LATTICE L124.32
%
\hbox{}\par\smallskip%
%
% LATTICE L124.33
%
\leavevmode%
\llap{\spacer}%
${L_{124.33}}=\pdualwitharg{2}({L_{124.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1520\\240\\160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\72\\32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
160\\32\\13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\24\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-20\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-43\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-22\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-20\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\23\\-44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\-16\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-10\\-120\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\8\\0\\-80\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-10\\0\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-120\\-80\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-24\\-8\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1520, 240, 160; 240, 72, 32; 160, 32, 13],[80, 8, 1],[-1, 1, 2, 1, 1, -1; 24, -20, -43, -22, -20, 23; -48, 35, 80, 42, 40, -44],123,33]
%
% END OF LATTICE L124.33
%
\hbox{}\par\smallskip%
%
% LATTICE L124.34
%
\leavevmode%
\llap{\spacer}%
${L_{124.34}}=\pdualwitharg{5}({L_{124.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-40\\-40\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-10\\-48\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\10\\0\\-40\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-10\\0\\5\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-48\\-40\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-60\\-20\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240, -80, 0; -80, 30, 0; 0, 0, -1],[80, 10, 1],[-1, -1, -1, 0, 1, 1; -4, -2, -1, 1, 4, 2; 0, -6, -10, -5, -8, 0],123,34]
%
% END OF LATTICE L124.34
%
\hbox{}\par\smallskip%
%
% LATTICE L124.35
%
\leavevmode%
\llap{\spacer}%
${L_{124.35}}=\pdualwitharg{2.5}(\main({L_{124.6}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4440\\1320\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1320\\380\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-60\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\1\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-40\\-20\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-5\\-12\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\0\\-40\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-5\\0\\5\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-12\\-40\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-120\\-20\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4440, 1320, -200; 1320, 380, -60; -200, -60, 9],[40, 20, 1],[-3, -1, -1, 1, 1, -1; 1, 0, -2, -1, -1, 0; -60, -23, -40, 15, 16, -20],123,35]
%
% END OF LATTICE L124.35
%
\hbox{}\par\smallskip%
%
% LATTICE L124.36
%
\leavevmode%
\llap{\spacer}%
${L_{124.36}}=\pdualwitharg{5}({L_{124.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8240\\2160\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2160\\510\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-100\\19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\37
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\9\\110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-12\\-128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-18\\-200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-40\\-80\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-10\\-24\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\10\\0\\-40\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-10\\0\\20\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-24\\-40\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-60\\-40\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8240, 2160, -400; 2160, 510, -100; -400, -100, 19],[80, 10, 1],[-1, 1, 3, 1, -3, -5; -4, 3, 9, 2, -12, -18; -40, 37, 110, 30, -128, -200],123,36]
%
% END OF LATTICE L124.36
%
\hbox{}\par\smallskip%
%
% LATTICE L124.37
%
\leavevmode%
\llap{\spacer}%
${L_{124.37}}=\pdualwitharg{5}({L_{124.7}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5040\\240\\400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-10\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\-20\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\34\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-80\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-10\\-24\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\40\\0\\-80\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-10\\0\\20\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-24\\-80\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-60\\-20\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5040, 240, 400; 240, -10, -20; 400, -20, -31],[80, 10, 1],[1, 1, 5, 1, -1, -1; 16, 8, 34, 4, -8, -5; 0, 7, 40, 10, -8, -10],123,37]
%
% END OF LATTICE L124.37
%
\hbox{}\par\smallskip%
%
% LATTICE L124.38
%
\leavevmode%
\llap{\spacer}%
${L_{124.38}}=\pdualwitharg{5}({L_{124.6}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2320\\0\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\0\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-6\\100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-40\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-10\\-48\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\40\\0\\-80\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-10\\0\\5\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-48\\-80\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-60\\-10\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2320, 0, -160; 0, 10, 0; -160, 0, 11],[80, 10, 1],[3, 3, 7, 1, 1, 0; 0, -2, -6, -1, 0, 1; 40, 42, 100, 15, 16, 0],123,38]
%
% END OF LATTICE L124.38
%
\hbox{}\par\smallskip%
%
% LATTICE L124.39
%
\leavevmode%
\llap{\spacer}%
${L_{124.39}}=\pdualwitharg{2.5}(\main({L_{124.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-620\\-260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-260\\-109
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\17\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\18\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\26\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-40\\-40\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20\\-24\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\0\\-40\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-20\\0\\20\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-24\\-40\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-120\\-40\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40, 0, 0; 0, -620, -260; 0, -260, -109],[40, 20, 1],[0, -1, -3, -1, 0, 1; 17, 18, 26, -4, -5, 0; -40, -42, -60, 10, 12, 0],123,39]
%
% END OF LATTICE L124.39
%
\hbox{}\par\smallskip%
%
% LATTICE L124.40
%
\leavevmode%
\llap{\spacer}%
${L_{124.40}}=\pdualwitharg{2.5}({L_{124.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-1960\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\2\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-20\\-80\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-40\\-48\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\5\\0\\-20\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-40\\0\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-48\\-20\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-30\\-40\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40, -80, 0; -80, -1960, 80; 0, 80, -3],[40, 40, 1],[0, 3, 2, 1, -2, -3; 1, 2, 1, 0, -1, -1; 20, 48, 25, 0, -28, -30],123,40]
%
% END OF LATTICE L124.40
%
\hbox{}\par\smallskip%
%
% LATTICE L124.41
%
\leavevmode%
\llap{\spacer}%
${L_{124.41}}=\pdualwitharg{2.5}({L_{124.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2040, 2080, -120; 2080, 2040, -120; -120, -120, 7],[40, 40, 1],[0, 1, 1, -1, -2, -2; 1, 2, 2, 0, -1, -1; 20, 52, 50, -20, -52, -50],123,41]
%
% END OF LATTICE L124.41
%
\hbox{}\par\smallskip%
%
% LATTICE L124.42
%
\leavevmode%
\llap{\spacer}%
${L_{124.42}}=\pdualwitharg{2.5}({L_{124.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\40\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
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0\\1\\0
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3\\0\\8
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-120, 0, 40; 0, 40, 0; 40, 0, -13],[40, 40, 1],[0, 3, 4, 1, -2, -3; 1, 0, -1, -2, -1, 0; 0, 8, 10, 0, -8, -10],123,42]
%
% END OF LATTICE L124.42
%
\hbox{}\par\smallskip%
%
% LATTICE L124.43
%
\leavevmode%
\llap{\spacer}%
${L_{124.43}}=\pdualwitharg{2.5}({L_{124.7}})$%
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{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\6\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\13\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-5
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\20
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{$\left.\llap{\phantom{%
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0\\0\\0
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\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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80\\0\\-80\\-40\\-80\\0
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0\\4\\0\\-10\\-48\\-40
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-80\\0\\40\\0\\-80\\-120
\end{tabular}\endgroup%
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-40\\-10\\0\\5\\0\\-20
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0\\-40\\-120\\-20\\0\\40
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240, -480, -80; -480, 680, 120; -80, 120, 21],[80, 40, 1],[-1, -1, -2, 0, 1, 1; 6, 6, 13, 1, -2, -3; -40, -38, -80, -5, 16, 20],123,43]
%
% END OF LATTICE L124.43
%
\hbox{}\par\smallskip%
%
% LATTICE L124.44
%
\leavevmode%
\llap{\spacer}%
${L_{124.44}}=\pdualwitharg{2.5}({L_{124.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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320\\-40\\-31
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\6\\0
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1\\2\\7
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\7\\40
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1\\0\\10
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\HardButStrongLineBreak\kern3pt%
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-1\\-2\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-20
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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80\\0\\-80\\-80\\-80\\0
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0\\1\\0\\-10\\-24\\-20
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-80\\0\\40\\0\\-80\\-120
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-80\\-10\\0\\20\\0\\-40
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-80\\-24\\-80\\0\\16\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-120\\-40\\0\\40
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3280, 400, 320; 400, -40, -40; 320, -40, -31],[80, 40, 1],[1, 1, 5, 1, -1, -2; 6, 2, 7, 0, -2, -1; 0, 7, 40, 10, -8, -20],123,44]
%
% END OF LATTICE L124.44
%
\hbox{}\par\smallskip%
%
% LATTICE L124.45
%
\leavevmode%
\llap{\spacer}%
${L_{124.45}}=\pdualwitharg{2.5}({L_{124.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\120\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-8
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\HardButStrongLineBreak\kern3pt%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[80, -80, 0; -80, 120, 0; 0, 0, -1],[80, 40, 1],[0, 1, 1, -1, -1, -2; -1, 0, 1, 0, -1, -3; -10, -8, 0, 0, -6, -20],123,45]
%
% END OF LATTICE L124.45
%
\hbox{}\par\smallskip%
%
% LATTICE L124.46
%
\leavevmode%
\llap{\spacer}%
${L_{124.46}}=\pdualwitharg{2.5}({L_{124.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2320, -2320, 160; -2320, 2360, -160; 160, -160, 11],[80, 40, 1],[-3, -1, 1, -3, -4, -10; -1, 0, 1, 0, -1, -3; 30, 16, 0, 40, 42, 100],123,46]
%
% END OF LATTICE L124.46
%

\medskip%
%------------------------------WEYL-GROUP-W125--------------------------------
%
\leavevmode\llap{}%
$W_{125}$%
\qquad\llap{20} lattices, $\chi=18$%
\hfill%
$422422\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L125.1
%
\leavevmode%
\llap{\spacer}%
${L_{125.1}}$%
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{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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392\\-5\\-7
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-28056, 392, 448; 392, -5, -7; 448, -7, -6],[56, 1, 1],[1, -1, -1, 1, 9, 9; 40, -42, -40, 42, 370, 368; 27, -26, -28, 25, 234, 236],124,1]
%
% END OF LATTICE L125.1
%
\hbox{}\par\smallskip%
%
% LATTICE L125.2
%
\leavevmode%
\llap{\spacer}%
${L_{125.2}}$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-57232\\-28311\\886
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1792\\886\\-27
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-115696, -57232, 1792; -57232, -28311, 886; 1792, 886, -27],[112, 1, 1],[-17, 16, 35, -31, -146, -295; 35, -33, -72, 64, 301, 608; 20, -21, -40, 42, 185, 368],124,2]
%
% END OF LATTICE L125.2
%
\hbox{}\par\smallskip%
%
% LATTICE L125.3
%
\leavevmode%
\llap{\spacer}%
${L_{125.3}}=\pfillwitharg{2}({L_{125.1}})$%
{${}={}$}Nikulin $6'$%
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0\\1\\0
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0\\0\\1
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-14, 0, 0; 0, 1, 0; 0, 0, 1],[14, 1, 1],[0, 0, -1, -2, -4, -3; -1, 1, 0, -3, -9, -8; 1, 0, -4, -7, -12, -8],124,3]
%
% END OF LATTICE L125.3
%
\hbox{}\par\smallskip%
%
% LATTICE L125.4
%
\leavevmode%
\llap{\spacer}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3836, 196, 140; 196, -10, -7; 140, -7, -3],[28, 1, 1],[1, 9, 9, 1, -1, -1; 21, 185, 184, 20, -21, -20; -3, -18, -16, -1, 2, 0],124,4]
%
% END OF LATTICE L125.4
%
\hbox{}\par\smallskip%
%
% LATTICE L125.5
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-67256, 2632, 2632; 2632, -102, -103; 2632, -103, -103],[56, 1, 1],[-2, 0, 5, 3, 2, -1; -1, 1, 0, -3, -9, -16; -50, -1, 128, 80, 61, -8],124,5]
%
% END OF LATTICE L125.5
%
\hbox{}\par\smallskip%
%
% LATTICE L125.6
%
\leavevmode%
\llap{\spacer}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1582, -574, -728; -574, -204, -264; -728, -264, -335],[14, 2, 1],[-18, 7, 38, 120, 197, 64; -1, 1, 0, -3, -9, -4; 40, -16, -83, -260, -424, -137],124,6]
%
% END OF LATTICE L125.6
%
\hbox{}\par\smallskip%
%
% LATTICE L125.7
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-406, 126, -252; 126, -35, 77; -252, 77, -156],[14, 7, 1],[-11, 4, 49, 79, 132, 87; 6, -1, -34, -60, -107, -74; 21, -7, -98, -161, -273, -182],124,7]
%
% END OF LATTICE L125.7
%
\hbox{}\par\smallskip%
%
% LATTICE L125.8
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4088, 2156, -1064; 2156, 1156, -560; -1064, -560, 277],[28, 4, 1],[-38, -205, -45, -13, 1, -6; 7, 40, 9, 3, 0, 1; -132, -708, -155, -44, 4, -21],124,8]
%
% END OF LATTICE L125.8
%
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%
% LATTICE L125.9
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28, 0, 0; 0, -21, -7; 0, -7, -2],[28, 7, 1],[-1, 0, 1, 0, -4, -5; -1, 2, 0, -3, -18, -16; 0, -7, 0, 7, 35, 28],124,9]
%
% END OF LATTICE L125.9
%
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%
% LATTICE L125.10
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L125.10
%
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%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-616, 168, -56; 168, -35, 21; -56, 21, -2],[56, 7, 1],[-5, 3, 7, -1, -27, -31; -12, 6, 16, -2, -62, -72; 21, -14, -28, 7, 126, 140],124,11]
%
% END OF LATTICE L125.11
%
\hbox{}\par\smallskip%
%
% LATTICE L125.12
%
\leavevmode%
\llap{\spacer}%
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-1736\\1120\\-336
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1736, 1120, -336; 1120, -714, 217; -336, 217, -65],[56, 7, 1],[2, 2, -11, -13, -24, -33; -1, 1, 0, -3, -9, -16; -14, -7, 56, 56, 91, 112],124,12]
%
% END OF LATTICE L125.12
%
\hbox{}\par\smallskip%
%
% LATTICE L125.13
%
\leavevmode%
\llap{\spacer}%
${L_{125.13}}=\pdualwitharg{2}({L_{125.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8680, -7952, -3640; -7952, -7272, -3328; -3640, -3328, -1523],[56, 8, 1],[5, -2, -1, 5, 22, 11; -53, 15, 13, -31, -183, -97; 104, -28, -26, 56, 348, 186],124,13]
%
% END OF LATTICE L125.13
%
\hbox{}\par\smallskip%
%
% LATTICE L125.14
%
\leavevmode%
\llap{\spacer}%
${L_{125.14}}=\pdualwitharg{2}(\main({L_{125.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-73528, 0, 3976; 0, 8, 0; 3976, 0, -215],[56, 8, 1],[-13, 0, 2, 3, -20, -7; -1, 1, 0, -3, -9, -2; -240, 0, 37, 56, -368, -129],124,14]
%
% END OF LATTICE L125.14
%
\hbox{}\par\smallskip%
%
% LATTICE L125.15
%
\leavevmode%
\llap{\spacer}%
${L_{125.15}}=\pdualwitharg{7}({L_{125.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6160, -1120, -1232; -1120, -189, -210; -1232, -210, -233],[112, 7, 1],[3, 7, 11, 0, -1, 1; 106, 213, 304, -8, -33, 56; -112, -231, -336, 7, 35, -56],124,15]
%
% END OF LATTICE L125.15
%
\hbox{}\par\smallskip%
%
% LATTICE L125.16
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[56, 196, -56; 196, 700, -196; -56, -196, 55],[28, 28, 1],[3, 1, 0, 2, 19, 5; -1, 0, 2, 7, 24, 4; 0, 0, 7, 28, 112, 21],124,16]
%
% END OF LATTICE L125.16
%
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%
% LATTICE L125.17
%
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${L_{125.17}}=\pdualwitharg{2}({L_{125.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-224, 784, -112; 784, -2672, 384; -112, 384, -55],[112, 16, 1],[1, 21, 3, 4, -1, -1; -3, -18, -2, -1, 2, 0; -24, -176, -21, -16, 16, 2],124,17]
%
% END OF LATTICE L125.17
%
\hbox{}\par\smallskip%
%
% LATTICE L125.18
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2651600, -1280888, 89264; -1280888, -618744, 43120; 89264, 43120, -3005],[56, 56, 1],[17, 100, 53, 29, -8, -7; -4, 29, 20, 18, -1, -6; 448, 3388, 1862, 1120, -252, -294],124,18]
%
% END OF LATTICE L125.18
%
\hbox{}\par\smallskip%
%
% LATTICE L125.19
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[56, 0, 0; 0, -9352, -3472; 0, -3472, -1289],[56, 56, 1],[-3, -9, -2, -1, 1, 0; -21, 124, 44, 83, 0, -13; 56, -336, -119, -224, 0, 35],124,19]
%
% END OF LATTICE L125.19
%
\hbox{}\par\smallskip%
%
% LATTICE L125.20
%
\leavevmode%
\llap{\spacer}%
${L_{125.20}}=\pdualwitharg{2.7}({L_{125.2}})$%
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167440\\-1490720\\-30912
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0\\0\\0
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18704, 167440, 3472; 167440, -1490720, -30912; 3472, -30912, -641],[112, 112, 1],[-1, 2, 0, -3, -18, -2; 44, -23, -9, 31, 323, 42; -2128, 1120, 434, -1512, -15680, -2037],124,20]
%
% END OF LATTICE L125.20
%

\medskip%
%------------------------------WEYL-GROUP-W126--------------------------------
%
\leavevmode\llap{}%
$W_{126}$%
\qquad\llap{16} lattices, $\chi=30$%
\hfill%
$42224222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L126.1
%
\leavevmode%
\llap{\spacer}%
${L_{126.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{9}}%
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{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{r}{2}{44}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
616\\-176\\-88
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\50\\23
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\23\\-3
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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109\\440\\440
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\kern3pt%
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-29\\-117\\-116
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2\\8\\7
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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0\\0\\0
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3\\11\\-6
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4\\15\\-3
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35\\132\\-22
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1\\4\\0
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-6\\-3\\-22\\0
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-3\\-5\\-88\\-20
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-22\\-88\\-1980\\-528
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[616, -176, -88; -176, 50, 23; -88, 23, -3],[88, 1, 1],[3, 4, 35, 1, -4, -5, -57, -7; 11, 15, 132, 4, -15, -19, -220, -28; -6, -3, -22, 0, 2, -1, -66, -24],125,1]
%
% END OF LATTICE L126.1
%
\hbox{}\par\smallskip%
%
% LATTICE L126.2
%
\leavevmode%
\llap{\spacer}%
${L_{126.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
440\\-14\\-11
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
352\\-11\\-5
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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461\\14784\\-616
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-481\\20
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-15\\-480\\19
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-1\\-32\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-13816, 440, 352; 440, -14, -11; 352, -11, -5],[88, 1, 1],[4, 3, 4, -1, -1, 3, 29, 19; 129, 98, 132, -32, -33, 94, 924, 608; -6, -6, -11, 0, 2, -2, -33, -24],125,2]
%
% END OF LATTICE L126.2
%
\hbox{}\par\smallskip%
%
% LATTICE L126.3
%
\leavevmode%
\llap{\spacer}%
${L_{126.3}}=\pfillwitharg{2}({L_{126.1}})$%
{${}={}$}Nikulin $9'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-2\\0\\2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-3\\-11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5\\-44\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-44\\-495\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-132\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-22, 0, 0; 0, 1, 0; 0, 0, 1],[22, 1, 1],[0, 0, -7, -3, -2, -2, -15, -3; -1, 1, 0, -2, -3, -5, -44, -10; 1, 0, -33, -14, -9, -8, -55, -10],125,3]
%
% END OF LATTICE L126.3
%
\hbox{}\par\smallskip%
%
% LATTICE L126.4
%
\leavevmode%
\llap{\spacer}%
${L_{126.4}}=\main({L_{126.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-176\\44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-78\\21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\21\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
461\\-924\\-616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\-433\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-42\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-17\\-1
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-13\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\11\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\24\\-4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-33\\-12\\-3\\-3\\-11\\0
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\-1\\-33\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-4\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-3\\-11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-88\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-88\\-990\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-264\\-76
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-396, -176, 44; -176, -78, 21; 44, 21, 5],[44, 1, 1],[8, 6, -5, -11, -5, 0, 71, 29; -17, -13, 11, 24, 11, 1, -143, -60; -1, 2, 0, -4, -3, -10, -88, -20],125,4]
%
% END OF LATTICE L126.4
%
\hbox{}\par\smallskip%
%
% LATTICE L126.5
%
\leavevmode%
\llap{\spacer}%
${L_{126.5}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{126.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{}{2}{22}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9966\\638\\-4818
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
638\\-40\\308
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4818\\308\\-2329
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-550353\\752840\\1240888
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39008\\-53361\\-87952
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-267756\\366270\\603713
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
152\\-203\\-342
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
441\\-601\\-994
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4693\\-6424\\-10582
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
607\\-833\\-1369
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-66\\-12\\-12\\-6\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-2\\-6\\-10\\-88\\-10
\end{tabular}\endgroup%
\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-66\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-2
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\0\\22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-2\\0\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-6\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-10\\-88\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-88\\-990\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-132\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9966, 638, -4818; 638, -40, 308; -4818, 308, -2329],[22, 2, 1],[152, 441, 4693, 607, 272, -17, -29, 29; -203, -601, -6424, -833, -377, 21, 44, -37; -342, -994, -10582, -1369, -614, 38, 66, -65],125,5]
%
% END OF LATTICE L126.5
%
\hbox{}\par\smallskip%
%
% LATTICE L126.6
%
\leavevmode%
\llap{\spacer}%
${L_{126.6}}=\pdualwitharg{2}(\main({L_{126.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{4}{8}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
123992\\-6644\\-30756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6644\\356\\1648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30756\\1648\\7629
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8623\\12672\\32032
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-294\\-433\\-1092
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2205\\-3240\\-8191
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-70\\-103\\-260
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\22\\88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-132\\-12\\-12\\-12\\-44\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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4\\-4\\-132\\-12
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-4
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-132\\0\\88\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-4\\0\\1
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\-44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-40\\-352\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-352\\-3960\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-264\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[123992, -6644, -30756; -6644, 356, 1648; -30756, 1648, 7629],[44, 4, 1],[-70, 1, 23, -3, -28, -197, -2179, -144; -103, 0, 22, -5, -41, -288, -3190, -211; -260, 4, 88, -11, -104, -732, -8096, -535],125,6]
%
% END OF LATTICE L126.6
%
\hbox{}\par\smallskip%
%
% LATTICE L126.7
%
\leavevmode%
\llap{\spacer}%
${L_{126.7}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{126.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{}{4}{11}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{22}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31130\\-1452\\-14388
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1452\\-55\\-671
\end{tabular}\endgroup%
\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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3518\\117\\-7620
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-31130, -1452, -14388; -1452, -55, -671; -14388, -671, -6650],[22, 11, 1],[1224, 3621, 3518, 10035, 2270, -127, -24, 447; 38, 119, 117, 336, 78, -3, -1, 12; -2651, -7843, -7620, -21736, -4917, 275, 52, -968],125,7]
%
% END OF LATTICE L126.7
%
\hbox{}\par\smallskip%
%
% LATTICE L126.8
%
\leavevmode%
\llap{\spacer}%
${L_{126.8}}=\pdualwitharg{11}(\main({L_{126.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{}{4}{22}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{44}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-12\\-132
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-3\\-22
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3\\2\\21
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\1\\0
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-5\\-1\\-33
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-3\\-41
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-12\\-132
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-33\\-11\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-110\\-88\\-220
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[44, 0, 0; 0, 11, 0; 0, 0, -1],[44, 11, 1],[0, -5, -6, -19, -3, -1, 0, 1; 1, -1, -3, -12, -3, -3, -1, 0; 0, -33, -41, -132, -22, -11, -3, 0],125,8]
%
% END OF LATTICE L126.8
%
\hbox{}\par\smallskip%
%
% LATTICE L126.9
%
\leavevmode%
\llap{\spacer}%
${L_{126.9}}=\pdualwitharg{2.11}(\pfillwitharg{2}({L_{126.1}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{}{4}{22}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\6050\\2882
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6050\\1465926\\698324
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2882\\698324\\332661
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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4059\\-1283192\\2693636
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
509460\\-161018473\\338004876
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242690\\-76703908\\161014413
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-144\\45317\\-95128
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-202\\64103\\-134563
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-132\\-66\\-22\\0
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-66\\-110\\-88\\-110
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[44, 6050, 2882; 6050, 1465926, 698324; 2882, 698324, 332661],[22, 22, 1],[-47, -144, -141, -202, -93, 4, 1, -8; 14400, 45317, 44611, 64103, 29848, -1069, -363, 2269; -30228, -95128, -93646, -134563, -62656, 2244, 762, -4763],125,9]
%
% END OF LATTICE L126.9
%
\hbox{}\par\smallskip%
%
% LATTICE L126.10
%
\leavevmode%
\llap{\spacer}%
${L_{126.10}}=\pdualwitharg{2}({L_{126.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3432\\-704\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-704\\-136\\-32
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-32\\-7
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\88\\0\\-88\\-352\\-3960\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-264\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-4\\0\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-24\\-88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-40\\-352\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-352\\-3960\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-264\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3432, -704, -176; -704, -136, -32; -176, -32, -7],[88, 8, 1],[-1, 0, 19, 2, 5, 4, 25, 1; 11, -1, -176, -18, -43, -31, -176, -6; -24, 4, 352, 35, 80, 52, 264, 7],125,10]
%
% END OF LATTICE L126.10
%
\hbox{}\par\smallskip%
%
% LATTICE L126.11
%
\leavevmode%
\llap{\spacer}%
${L_{126.11}}=\pdualwitharg{2}({L_{126.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{4}{8}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36344\\0\\-17600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17600\\0\\-8523
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13201\\-440\\27280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-5\\248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6390\\-213\\13205
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-1\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-575\\0\\1188
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\-1\\250
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-264\\-48\\-48\\-24\\-88\\0
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\kern3pt%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
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16\\-8\\-264\\-48
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\88\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-8\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-24\\-88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-40\\-352\\-40
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-352\\-3960\\-528
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-528\\-76
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36344, 0, -17600; 0, 8, 0; -17600, 0, -8523],[88, 8, 1],[31, 0, -575, -121, -151, -120, -745, -59; -1, 1, 0, -1, -3, -5, -44, -5; -64, 0, 1188, 250, 312, 248, 1540, 122],125,11]
%
% END OF LATTICE L126.11
%
\hbox{}\par\smallskip%
%
% LATTICE L126.12
%
\leavevmode%
\llap{\spacer}%
${L_{126.12}}=\pdualwitharg{11}({L_{126.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{}{4}{11}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{88}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1848\\1320\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1320\\803\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-110\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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5\\96\\792
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\127\\1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-133
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-1\\-9\\-77
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-2\\-22\\-187
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-38\\-320
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-100\\-836
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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22\\-11\\-66\\-264\\-66\\-33\\-22\\0
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-11\\11\\0\\-44\\-33\\-55\\-88\\-220
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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22\\-11\\-66\\-264
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\11\\0\\-44
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-66\\0\\4\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-44\\0\\88
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-33\\-22\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-55\\-88\\-220
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-88\\-180\\-528
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-220\\-528\\-1672
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1848, 1320, -176; 1320, 803, -110; -176, -110, 15],[88, 11, 1],[-1, -2, -3, -7, 0, 1, 1, 1; -9, -22, -38, -100, -7, 6, 6, 4; -77, -187, -320, -836, -55, 55, 56, 44],125,12]
%
% END OF LATTICE L126.12
%
\hbox{}\par\smallskip%
%
% LATTICE L126.13
%
\leavevmode%
\llap{\spacer}%
${L_{126.13}}=\pdualwitharg{11}({L_{126.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${22}\above{1pt}{1pt}{*}{4}{44}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{88}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-275\\44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\44\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-72\\-528
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-157\\-1144
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\24\\175
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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0\\-5\\-33
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-3\\-13\\-94
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-19\\-72\\-528
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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22\\-22\\-33\\-264\\-66\\-66\\-11\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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22\\-22\\-33\\-264
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-22\\44\\0\\-88
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-264\\-88\\0\\88
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-66\\-11\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-220\\-88\\-440
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-88\\-45\\-264
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-440\\-264\\-1672
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[88, 0, 0; 0, -275, 44; 0, 44, -7],[88, 11, 1],[0, -5, -3, -19, -3, -1, 0, 1; -5, -28, -13, -72, -7, 4, 1, 0; -33, -198, -94, -528, -55, 22, 6, 0],125,13]
%
% END OF LATTICE L126.13
%
\hbox{}\par\smallskip%
%
% LATTICE L126.14
%
\leavevmode%
\llap{\spacer}%
${L_{126.14}}=\pdualwitharg{2.11}(\main({L_{126.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{}{4}{88}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-308\\88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\88\\-25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-14\\-44
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\69\\220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-21\\-67
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-27\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-31\\-100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-24\\-77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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44\\-44\\-132\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\88\\0\\-44
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-44\\0\\11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-132\\-44\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-440\\-352\\-220
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-352\\-360\\-264
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[44, 0, 0; 0, -308, 88; 0, 88, -25],[44, 44, 1],[1, -1, -3, -3, -3, -3, -1, 0; 0, -27, -31, -24, -14, -1, 3, 3; 0, -88, -100, -77, -44, 0, 12, 11],125,14]
%
% END OF LATTICE L126.14
%
\hbox{}\par\smallskip%
%
% LATTICE L126.15
%
\leavevmode%
\llap{\spacer}%
${L_{126.15}}=\pdualwitharg{2.11}({L_{126.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{*}{4}{88}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10296\\-7744\\-3696
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7744\\-5720\\-2728
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3696\\-2728\\-1301
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\3240\\-6600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\2591\\-5280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\1242\\-2531
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\41\\-88
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\237\\-484
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\244\\-496
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\179\\-363
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
176\\-88\\-264\\-264\\-528\\-264\\-88\\0
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\kern3pt%
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-88\\88\\0\\-44\\-264\\-440\\-352\\-220
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
176\\-88\\-264\\-264
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\88\\0\\-44
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-264\\0\\8\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-44\\0\\11
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-264\\-88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-440\\-352\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-352\\-360\\-264
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0\\-220\\-264\\-209
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10296, -7744, -3696; -7744, -5720, -2728; -3696, -2728, -1301],[88, 88, 1],[1, -4, -5, -4, -5, 0, 1, 1; 41, 237, 244, 179, 175, -21, -28, -17; -88, -484, -496, -363, -352, 44, 56, 33],125,15]
%
% END OF LATTICE L126.15
%
\hbox{}\par\smallskip%
%
% LATTICE L126.16
%
\leavevmode%
\llap{\spacer}%
${L_{126.16}}=\pdualwitharg{2.11}({L_{126.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{}{4}{88}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-616\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-264\\-113
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\40\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\159\\-352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\70\\-155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\1\\0
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\HardButStrongLineBreak\kern3pt%
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-5\\40\\-88
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\45\\-100
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\69\\-154
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
176\\-88\\-264\\-528\\-528\\-264\\-88\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
176\\-88\\-264\\-528
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\88\\0\\-88
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\8\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-88\\0\\44
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-264\\-88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-440\\-352\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-352\\-360\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-440\\-528\\-836
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[88, 0, 0; 0, -616, -264; 0, -264, -113],[88, 88, 1],[-3, -5, -4, -5, -1, 1, 0, -1; 1, 40, 45, 69, 39, 0, -5, -9; 0, -88, -100, -154, -88, 0, 12, 22],125,16]
%
% END OF LATTICE L126.16
%

\medskip%
%------------------------------WEYL-GROUP-W127--------------------------------
%
\leavevmode\llap{}%
$W_{127}$%
\qquad\llap{32} lattices, $\chi=9$%
\hfill%
$42222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L127.1
%
\leavevmode%
\llap{\spacer}%
${L_{127.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{11}}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27480\\600\\1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
600\\-13\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\-25\\-38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-110\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\54\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\276\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-50\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-272\\20
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-12\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-20\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-20\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-16\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-27480, 600, 1200; 600, -13, -25; 1200, -25, -38],[120, 1, 1],[-2, 1, 5, -1, -5; -110, 54, 276, -50, -272; 9, -4, -24, 0, 20],126,1]
%
% END OF LATTICE L127.1
%
\hbox{}\par\smallskip%
%
% LATTICE L127.2
%
\leavevmode%
\llap{\spacer}%
${L_{127.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-236280\\2280\\1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2280\\-22\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1560\\-15\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-105\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\104\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\732\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\210\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-104\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-12\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-5\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\24\\0\\-24
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-5\\0\\5\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-236280, 2280, 1560; 2280, -22, -15; 1560, -15, -7],[120, 1, 1],[-1, 1, 7, 2, -1; -105, 104, 732, 210, -104; 2, -1, -12, -5, 0],126,2]
%
% END OF LATTICE L127.2
%
\hbox{}\par\smallskip%
%
% LATTICE L127.3
%
\leavevmode%
\llap{\spacer}%
${L_{127.3}}=\pfillwitharg{2}({L_{127.1}})$%
{${}={}$}Nikulin $11'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-6\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0\\-5\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-5\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-6\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-30, 0, 0; 0, 1, 0; 0, 0, 1],[30, 1, 1],[0, 0, -1, -2, -1; -1, 1, 0, -5, -4; 1, 0, -6, -10, -4],126,3]
%
% END OF LATTICE L127.3
%
\hbox{}\par\smallskip%
%
% LATTICE L127.4
%
\leavevmode%
\llap{\spacer}%
${L_{127.4}}=\main({L_{127.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-154140\\5580\\1740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5580\\-202\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1740\\-63\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-55\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\27\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\138\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-25\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-136\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-6\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0\\-10\\-8
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-154140, 5580, 1740; 5580, -202, -63; 1740, -63, -19],[60, 1, 1],[-2, 1, 5, -1, -5; -55, 27, 138, -25, -136; -1, 2, 0, -10, -8],126,4]
%
% END OF LATTICE L127.4
%
\hbox{}\par\smallskip%
%
% LATTICE L127.5
%
\leavevmode%
\llap{\spacer}%
${L_{127.5}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{127.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26610\\1230\\-12990
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1230\\-56\\600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12990\\600\\-6341
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-15\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\28\\58
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\36\\69
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\30\\50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-7\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-6\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-10\\-4
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\3\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-10\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-3\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26610, 1230, -12990; 1230, -56, 600; -12990, 600, -6341],[30, 2, 1],[12, -27, -32, -23, 7; -15, 28, 36, 30, -7; -26, 58, 69, 50, -15],126,5]
%
% END OF LATTICE L127.5
%
\hbox{}\par\smallskip%
%
% LATTICE L127.6
%
\leavevmode%
\llap{\spacer}%
${L_{127.6}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{127.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{4}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30990\\1620\\-10050
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1620\\-84\\525
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10050\\525\\-3259
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-15\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\28\\57
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\24\\44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\30\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-14\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\-6\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\0\\-15\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-15\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30990, 1620, -10050; 1620, -84, 525; -10050, 525, -3259],[30, 3, 1],[7, -17, -13, -13, 9; -15, 28, 24, 30, -14; -24, 57, 44, 45, -30],126,6]
%
% END OF LATTICE L127.6
%
\hbox{}\par\smallskip%
%
% LATTICE L127.7
%
\leavevmode%
\llap{\spacer}%
${L_{127.7}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{127.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{4}{5}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41010\\2400\\-7950
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2400\\-140\\465
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7950\\465\\-1541
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-15\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\28\\55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\72\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-14\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-5\\-30\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\5\\0\\-5\\-20
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\30\\0\\-30
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-5\\0\\1\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-30\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-41010, 2400, -7950; 2400, -140, 465; -7950, 465, -1541],[30, 5, 1],[3, -9, -19, -1, 5; -15, 28, 72, 6, -14; -20, 55, 120, 7, -30],126,7]
%
% END OF LATTICE L127.7
%
\hbox{}\par\smallskip%
%
% LATTICE L127.8
%
\leavevmode%
\llap{\spacer}%
${L_{127.8}}=\pdualwitharg{3}(\main({L_{127.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{4}{6}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6420\\-3000\\180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3000\\-1401\\84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\84\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-3\\-6\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\6\\0\\-30\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4\\0\\-12
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6420, -3000, 180; -3000, -1401, 84; 180, 84, -5],[60, 3, 1],[-1, 1, 1, -1, -3; 2, -3, -2, 5, 8; -3, -15, 2, 45, 24],126,8]
%
% END OF LATTICE L127.8
%
\hbox{}\par\smallskip%
%
% LATTICE L127.9
%
\leavevmode%
\llap{\spacer}%
${L_{127.9}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{127.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{4}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
269580\\-17910\\129150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17910\\1218\\-8580
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
129150\\-8580\\61873
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\0\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
521\\11\\-1086
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
178\\4\\-371
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
619\\15\\-1290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\1\\-75
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-6\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\6\\0\\-30\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\1\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-3\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[269580, -17910, 129150; -17910, 1218, -8580; 129150, -8580, 61873],[30, 6, 1],[23, 521, 178, 619, 36; 0, 11, 4, 15, 1; -48, -1086, -371, -1290, -75],126,9]
%
% END OF LATTICE L127.9
%
\hbox{}\par\smallskip%
%
% LATTICE L127.10
%
\leavevmode%
\llap{\spacer}%
${L_{127.10}}=\pdualwitharg{2}(\main({L_{127.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{4}{8}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26040\\12780\\-6480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12780\\6596\\-3168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6480\\-3168\\1613
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\13\\-344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85\\12\\-318
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-711\\100\\-2660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\19\\-505
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-12\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-40\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-40\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-6\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26040, 12780, -6480; 12780, 6596, -3168; -6480, -3168, 1613],[60, 4, 1],[-92, 1, -85, -711, -135; 13, 0, 12, 100, 19; -344, 4, -318, -2660, -505],126,10]
%
% END OF LATTICE L127.10
%
\hbox{}\par\smallskip%
%
% LATTICE L127.11
%
\leavevmode%
\llap{\spacer}%
${L_{127.11}}=\pdualwitharg{5}(\main({L_{127.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{4}{10}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1740\\420\\60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\-95\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-15\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-5\\-30\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\10\\0\\-10\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\60\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-60\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1740, 420, 60; 420, -95, -15; 60, -15, -2],[60, 5, 1],[0, 1, -1, -1, -3; -1, 2, 0, -2, -8; 5, 15, -30, -17, -40],126,11]
%
% END OF LATTICE L127.11
%
\hbox{}\par\smallskip%
%
% LATTICE L127.12
%
\leavevmode%
\llap{\spacer}%
${L_{127.12}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{127.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{4}{10}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
97620\\-13650\\46350
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13650\\2030\\-6480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46350\\-6480\\22007
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\0\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
633\\11\\-1330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
664\\12\\-1395
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
159\\3\\-334
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\1\\-105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-10\\-30\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\10\\0\\-10\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\15\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-10\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-15\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[97620, -13650, 46350; -13650, 2030, -6480; 46350, -6480, 22007],[30, 10, 1],[19, 633, 664, 159, 50; 0, 11, 12, 3, 1; -40, -1330, -1395, -334, -105],126,12]
%
% END OF LATTICE L127.12
%
\hbox{}\par\smallskip%
%
% LATTICE L127.13
%
\leavevmode%
\llap{\spacer}%
${L_{127.13}}=\pdualwitharg{3}({L_{127.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{4}{3}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67560\\-3480\\2040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3480\\-165\\108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2040\\108\\-61
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\25\\-123
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-21\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-56\\268
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-55\\270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\16\\-72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\-12\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\0\\-15\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-15\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-67560, -3480, 2040; -3480, -165, 108; 2040, 108, -61],[120, 3, 1],[-5, 4, 11, 11, -3; 25, -21, -56, -55, 16; -123, 96, 268, 270, -72],126,13]
%
% END OF LATTICE L127.13
%
\hbox{}\par\smallskip%
%
% LATTICE L127.14
%
\leavevmode%
\llap{\spacer}%
${L_{127.14}}=\pdualwitharg{3}({L_{127.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{*}{4}{12}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5640\\-1200\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1200\\-255\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-51\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\16\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-6\\-12\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\12\\0\\-60\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-60\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5640, -1200, -240; -1200, -255, -51; -240, -51, -10],[120, 3, 1],[-1, 1, 1, -1, -3; 4, -6, -4, 10, 16; 3, 6, -4, -30, -12],126,14]
%
% END OF LATTICE L127.14
%
\hbox{}\par\smallskip%
%
% LATTICE L127.15
%
\leavevmode%
\llap{\spacer}%
${L_{127.15}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{127.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{}{4}{15}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
155310\\21390\\-34980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21390\\3045\\-4755
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34980\\-4755\\7918
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\28\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-473\\957\\-1515
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-331\\670\\-1060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119\\241\\-381
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\152\\-240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-15\\-30\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\15\\0\\-15\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\10\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-15\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-30\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[155310, 21390, -34980; 21390, 3045, -4755; -34980, -4755, 7918],[30, 15, 1],[-14, -473, -331, -119, -75; 28, 957, 670, 241, 152; -45, -1515, -1060, -381, -240],126,15]
%
% END OF LATTICE L127.15
%
\hbox{}\par\smallskip%
%
% LATTICE L127.16
%
\leavevmode%
\llap{\spacer}%
${L_{127.16}}=\pdualwitharg{5}({L_{127.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{*}{4}{20}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51000\\3120\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3120\\-190\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-15\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\14\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-44\\-80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-10\\-60\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\20\\0\\-20\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-20\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-120\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-51000, 3120, 240; 3120, -190, -15; 240, -15, -1],[120, 5, 1],[0, 1, -1, -1, -3; -1, 14, -12, -14, -44; 10, 30, -60, -34, -80],126,16]
%
% END OF LATTICE L127.16
%
\hbox{}\par\smallskip%
%
% LATTICE L127.17
%
\leavevmode%
\llap{\spacer}%
${L_{127.17}}=\pdualwitharg{5}({L_{127.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{4}{5}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58200\\-13080\\-2040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13080\\-2935\\-460
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2040\\-460\\-71
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\21\\65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-19\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\-144\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-9\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\16\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-5\\-60\\-5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\5\\0\\-5\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\120\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-5\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-120\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-58200, -13080, -2040; -13080, -2935, -460; -2040, -460, -71],[120, 5, 1],[-7, 6, 47, 3, -5; 21, -19, -144, -9, 16; 65, -50, -420, -28, 40],126,17]
%
% END OF LATTICE L127.17
%
\hbox{}\par\smallskip%
%
% LATTICE L127.18
%
\leavevmode%
\llap{\spacer}%
${L_{127.18}}=\pdualwitharg{2.3}(\main({L_{127.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{4}{24}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39180\\-1200\\-10320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1200\\60\\312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10320\\312\\2719
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-18\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-19\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-5\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-113\\-75\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-16\\-93
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-12\\-12\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-120\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[39180, -1200, -10320; -1200, 60, 312; -10320, 312, 2719],[60, 12, 1],[-29, -29, -7, -113, -25; -18, -19, -5, -75, -16; -108, -108, -26, -420, -93],126,18]
%
% END OF LATTICE L127.18
%
\hbox{}\par\smallskip%
%
% LATTICE L127.19
%
\leavevmode%
\llap{\spacer}%
${L_{127.19}}=\pdualwitharg{3.5}(\main({L_{127.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{4}{30}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2100\\-2040\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2040\\-1950\\345
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\345\\-61
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\7\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\12\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-15\\-30\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\30\\0\\-30\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\20\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-30\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-60\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2100, -2040, 360; -2040, -1950, 345; 360, 345, -61],[60, 15, 1],[1, 1, -1, -1, -1; 7, -1, -6, -1, 12; 45, 0, -40, -12, 60],126,19]
%
% END OF LATTICE L127.19
%
\hbox{}\par\smallskip%
%
% LATTICE L127.20
%
\leavevmode%
\llap{\spacer}%
${L_{127.20}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{127.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{}{4}{30}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6144180\\-3650910\\-1501350
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3650910\\-2169390\\-892110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1501350\\-892110\\-366859
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\346\\-780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-775\\1770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-306\\695
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-131\\294
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\203\\-465
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-30\\-30\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\30\\0\\-30\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\5\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-30\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-15\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6144180, -3650910, -1501350; -3650910, -2169390, -892110; -1501350, -892110, -366859],[30, 30, 1],[-15, 28, 12, 6, -7; 346, -775, -306, -131, 203; -780, 1770, 695, 294, -465],126,20]
%
% END OF LATTICE L127.20
%
\hbox{}\par\smallskip%
%
% LATTICE L127.21
%
\leavevmode%
\llap{\spacer}%
${L_{127.21}}=\pdualwitharg{2}({L_{127.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{4}{8}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\360\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-520\\176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\176\\-59
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-24\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-40\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-16\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-240, 360, -120; 360, -520, 176; -120, 176, -59],[120, 8, 1],[-1, 4, 4, 2, -1; 2, -1, -3, -5, 0; 8, -12, -18, -20, 2],126,21]
%
% END OF LATTICE L127.21
%
\hbox{}\par\smallskip%
%
% LATTICE L127.22
%
\leavevmode%
\llap{\spacer}%
${L_{127.22}}=\pdualwitharg{2}({L_{127.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{4}{8}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-120\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-56\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-120\\-239
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-3\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-24\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-40\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\12\\0\\-6
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\0\\40\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-6\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-240, -120, -240; -120, -56, -120; -240, -120, -239],[120, 8, 1],[-1, 8, 7, 2, -1; 2, -1, -3, -5, 0; 0, -8, -6, 0, 1],126,22]
%
% END OF LATTICE L127.22
%
\hbox{}\par\smallskip%
%
% LATTICE L127.23
%
\leavevmode%
\llap{\spacer}%
${L_{127.23}}=\pdualwitharg{2.5}(\main({L_{127.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{4}{40}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24900\\780\\6120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
780\\40\\180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6120\\180\\1513
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-77\\-100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-21\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-79\\-104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-88\\-115
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-20\\-60\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\40\\0\\-40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\60\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-40\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-30\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24900, 780, 6120; 780, 40, 180; 6120, 180, 1513],[60, 20, 1],[27, 0, 8, 28, 31; -77, 1, -21, -79, -88; -100, 0, -30, -104, -115],126,23]
%
% END OF LATTICE L127.23
%
\hbox{}\par\smallskip%
%
% LATTICE L127.24
%
\leavevmode%
\llap{\spacer}%
${L_{127.24}}=\pdualwitharg{3.5}({L_{127.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{}{4}{15}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-15\\-60\\-15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\15\\0\\-15\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-15\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-120\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120, 0, 0; 0, -105, -15; 0, -15, -2],[120, 15, 1],[0, -1, -1, 0, 1; 2, -1, -4, -1, 0; -15, 0, 20, 6, 0],126,24]
%
% END OF LATTICE L127.24
%
\hbox{}\par\smallskip%
%
% LATTICE L127.25
%
\leavevmode%
\llap{\spacer}%
${L_{127.25}}=\pdualwitharg{3.5}({L_{127.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{*}{4}{60}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4200\\-3480\\600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3480\\-2775\\480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
600\\480\\-83
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\17\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\12\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-30\\-60\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\60\\0\\-60\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\40\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-60\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-240\\-120\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4200, -3480, 600; -3480, -2775, 480; 600, 480, -83],[120, 15, 1],[1, 1, -1, -1, -1; 17, 4, -16, -8, 12; 105, 30, -100, -54, 60],126,25]
%
% END OF LATTICE L127.25
%
\hbox{}\par\smallskip%
%
% LATTICE L127.26
%
\leavevmode%
\llap{\spacer}%
${L_{127.26}}=\pdualwitharg{2.3}({L_{127.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{4}{24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-779280\\-35160\\19680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35160\\-1560\\888
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19680\\888\\-497
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\2\\-432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-1\\948
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-1\\394
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-5\\1020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\-198
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-24\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\24\\0\\-120\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-779280, -35160, 19680; -35160, -1560, 888; 19680, 888, -497],[120, 24, 1],[-11, 24, 10, 26, -5; 2, -1, -1, -5, 0; -432, 948, 394, 1020, -198],126,26]
%
% END OF LATTICE L127.26
%
\hbox{}\par\smallskip%
%
% LATTICE L127.27
%
\leavevmode%
\llap{\spacer}%
${L_{127.27}}=\pdualwitharg{2.3}({L_{127.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{}{4}{24}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2017680\\19080\\50760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19080\\-168\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50760\\-480\\-1277
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-1\\1272
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-1\\358
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-5\\240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\-159
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-24\\-24\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\24\\0\\-120\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\4\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2017680, 19080, 50760; 19080, -168, -480; 50760, -480, -1277],[120, 24, 1],[-3, 32, 9, 6, -4; 2, -1, -1, -5, 0; -120, 1272, 358, 240, -159],126,27]
%
% END OF LATTICE L127.27
%
\hbox{}\par\smallskip%
%
% LATTICE L127.28
%
\leavevmode%
\llap{\spacer}%
${L_{127.28}}=\pdualwitharg{2.3.5}(\main({L_{127.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{}{4}{120}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1500\\-8940\\2400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8940\\53400\\-14340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2400\\-14340\\3851
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-18\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-21\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-22\\-75
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-60\\-60\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\120\\0\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\20\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-120\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-30\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1500, -8940, 2400; -8940, 53400, -14340; 2400, -14340, 3851],[60, 60, 1],[-11, -6, -2, -10, -11; -18, -1, -3, -21, -22; -60, 0, -10, -72, -75],126,28]
%
% END OF LATTICE L127.28
%
\hbox{}\par\smallskip%
%
% LATTICE L127.29
%
\leavevmode%
\llap{\spacer}%
${L_{127.29}}=\pdualwitharg{2.5}({L_{127.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{*}{4}{40}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-343920\\30120\\-14160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30120\\-2600\\1240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14160\\1240\\-583
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\2\\320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-1\\-780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\-3\\-930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-1\\-148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\170
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-40\\-120\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\40\\0\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\60\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-60\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-343920, 30120, -14160; 30120, -2600, 1240; -14160, 1240, -583],[120, 40, 1],[-13, 32, 38, 6, -7; 2, -1, -3, -1, 0; 320, -780, -930, -148, 170],126,29]
%
% END OF LATTICE L127.29
%
\hbox{}\par\smallskip%
%
% LATTICE L127.30
%
\leavevmode%
\llap{\spacer}%
${L_{127.30}}=\pdualwitharg{2.5}({L_{127.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{}{4}{40}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1006320\\1023720\\41640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1023720\\-1041400\\-42360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41640\\-42360\\-1723
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\2\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-1\\-1160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-3\\-990
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\145
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-40\\-120\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\40\\0\\-40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\60\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-30\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1006320, 1023720, 41640; 1023720, -1041400, -42360; 41640, -42360, -1723],[120, 40, 1],[7, -49, -44, -3, 6; 2, -1, -3, -1, 0; 120, -1160, -990, -48, 145],126,30]
%
% END OF LATTICE L127.30
%
\hbox{}\par\smallskip%
%
% LATTICE L127.31
%
\leavevmode%
\llap{\spacer}%
${L_{127.31}}=\pdualwitharg{2.3.5}({L_{127.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{*}{4}{120}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7800\\-36240\\9840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36240\\-138360\\37680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9840\\37680\\-10261
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\257\\960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\-948\\-3540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-324\\-1210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-106\\-396
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\233\\870
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\-120\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\120\\0\\-120\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\20\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-240\\-60\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7800, -36240, 9840; -36240, -138360, 37680; 9840, 37680, -10261],[120, 120, 1],[17, -61, -21, -7, 15; 257, -948, -324, -106, 233; 960, -3540, -1210, -396, 870],126,31]
%
% END OF LATTICE L127.31
%
\hbox{}\par\smallskip%
%
% LATTICE L127.32
%
\leavevmode%
\llap{\spacer}%
${L_{127.32}}=\pdualwitharg{2.3.5}({L_{127.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{}{4}{120}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-30600\\-1920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30600\\-1036560\\-65040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1920\\-65040\\-4081
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\15\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-128\\2040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-37\\590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\-255
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\-120\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\120\\0\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\20\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\0\\24\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-30\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-840, -30600, -1920; -30600, -1036560, -65040; -1920, -65040, -4081],[120, 120, 1],[2, -1, -1, -1, 0; 15, -128, -37, -6, 16; -240, 2040, 590, 96, -255],126,32]
%
% END OF LATTICE L127.32
%

\medskip%
%------------------------------WEYL-GROUP-W128--------------------------------
%
\leavevmode\llap{}%
$W_{128}$%
\qquad\llap{64} lattices, $\chi=12$%
\hfill%
$222222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L128.1
%
\leavevmode%
\llap{\spacer}%
${L_{128.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{3m,3,2}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{360}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1187640\\-8640\\6120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8640\\-51\\42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6120\\42\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\56\\272
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\465\\2205
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\59\\277
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\660\\3060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-58\\-276
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-60\\-279
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-8\\-360\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\90\\0\\-720\\-90\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\2\\0\\-6\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-720\\0\\360\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-90\\-6\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-9\\-180\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1187640, -8640, 6120; -8640, -51, 42; 6120, 42, -31],[360, 3, 1],[1, 8, 1, 11, -1, -1; 56, 465, 59, 660, -58, -60; 272, 2205, 277, 3060, -276, -279],127,1]
%
% END OF LATTICE L128.1
%
\hbox{}\par\smallskip%
%
% LATTICE L128.2
%
\leavevmode%
\llap{\spacer}%
${L_{128.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{32\rightarrow N'_{14},3,m}%
\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1281240\\-428040\\4680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-428040\\-142998\\1563
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4680\\1563\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-60\\-288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-35\\-190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\200\\1040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\35\\189
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\16\\94
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\0\\-72\\-360\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-80\\-15\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\18\\0\\-9\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-80\\0\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-15\\-9\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-18\\-40\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1281240, -428040, 4680; -428040, -142998, 1563; 4680, 1563, -17],[360, 3, 1],[19, 11, -1, -63, -11, -5; -60, -35, 3, 200, 35, 16; -288, -190, 0, 1040, 189, 94],127,2]
%
% END OF LATTICE L128.2
%
\hbox{}\par\smallskip%
%
% LATTICE L128.3
%
\leavevmode%
\llap{\spacer}%
${L_{128.3}}=\pfillwitharg{3.2}({L_{128.2}})$%
{${}={}$}Nikulin $14'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
390\\180\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\83\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-15\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-60\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\26\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-237\\510\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74\\159\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\45\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-4\\-30\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-40\\-15\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\2\\0\\-3\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-40\\0\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-15\\-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-3\\-10\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[390, 180, -30; 180, 83, -15; -30, -15, -14],[30, 1, 1],[-1, 28, -12, -237, -74, -21; 2, -60, 26, 510, 159, 45; 0, 5, -3, -50, -15, -4],127,3]
%
% END OF LATTICE L128.3
%
\hbox{}\par\smallskip%
%
% LATTICE L128.4
%
\leavevmode%
\llap{\spacer}%
${L_{128.4}}=\main(\pfillwitharg{3}({L_{128.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\120\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\34\\-19
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-19\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-16\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\10\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\13\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\100\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-13\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-4\\-60\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-40\\-15\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\1\\0\\-3\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-40\\0\\20\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-15\\-3\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-3\\-20\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[420, 120, -60; 120, 34, -19; -60, -19, -3],[60, 1, 1],[5, -3, -4, -31, -1, 4; -16, 10, 13, 100, 3, -13; 0, -5, -3, -20, 0, 2],127,4]
%
% END OF LATTICE L128.4
%
\hbox{}\par\smallskip%
%
% LATTICE L128.5
%
\leavevmode%
\llap{\spacer}%
${L_{128.5}}=\pfillwitharg{3}({L_{128.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2040\\120\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-7\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-20\\-39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\115\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\59\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\340\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-8\\-120\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-80\\-30\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\2\\0\\-6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-80\\0\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-30\\-6\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-3\\-20\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-2040, 120, 360; 120, -7, -20; 360, -20, -39],[120, 1, 1],[1, 6, 3, 17, -1, -1; 16, 115, 59, 340, -18, -20; 0, -5, -3, -20, 0, 1],127,5]
%
% END OF LATTICE L128.5
%
\hbox{}\par\smallskip%
%
% LATTICE L128.6
%
\leavevmode%
\llap{\spacer}%
${L_{128.6}}=\pfillwitharg{3}({L_{128.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4314840\\110640\\-6240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110640\\-2837\\160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6240\\160\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\116\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\235\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\79\\17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\280\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-39\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-40\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-8\\-120\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-80\\-15\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\2\\0\\-3\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-80\\0\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-15\\-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-6\\-40\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-4314840, 110640, -6240; 110640, -2837, 160; -6240, 160, -9],[120, 1, 1],[3, 6, 2, 7, -1, -1; 116, 235, 79, 280, -39, -40; -20, 15, 17, 120, 0, -18],127,6]
%
% END OF LATTICE L128.6
%
\hbox{}\par\smallskip%
%
% LATTICE L128.7
%
\leavevmode%
\llap{\spacer}%
${L_{128.7}}=\pfillwitharg{2}({L_{128.1}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{90}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16290\\-1170\\-540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1170\\84\\39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\39\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\15\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\13\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\150\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-24\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-4\\-90\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\90\\0\\-360\\-45\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\2\\0\\-3\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-360\\0\\90\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-45\\-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-9\\-90\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[16290, -1170, -540; -1170, 84, 39; -540, 39, 17],[90, 3, 1],[-1, 1, 1, 11, 0, -2; -12, 15, 13, 150, 1, -24; -4, 0, 2, 0, -3, -9],127,7]
%
% END OF LATTICE L128.7
%
\hbox{}\par\smallskip%
%
% LATTICE L128.8
%
\leavevmode%
\llap{\spacer}%
${L_{128.8}}=\main({L_{128.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2426580\\115200\\-2340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115200\\5469\\-111
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2340\\-111\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-276\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-85\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\21\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\360\\100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\106\\33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-36\\-180\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-40\\-15\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\9\\0\\-9\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-40\\0\\20\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-15\\-9\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-9\\-20\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[2426580, 115200, -2340; 115200, 5469, -111; -2340, -111, 2],[180, 3, 1],[13, 4, -1, -17, -5, 0; -276, -85, 21, 360, 106, 0; -108, -35, 0, 100, 33, -1],127,8]
%
% END OF LATTICE L128.8
%
\hbox{}\par\smallskip%
%
% LATTICE L128.9
%
\leavevmode%
\llap{\spacer}%
${L_{128.9}}=\pdualwitharg{2}(\pfillwitharg{3.2}({L_{128.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-930\\-30\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\8\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-16\\-189
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\4\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\-65\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
94\\-67\\-206
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
253\\-185\\-555
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\-54\\-156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\7\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\-15\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-40\\-30\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\4\\0\\-6\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-40\\0\\5\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-30\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-6\\-10\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-930, -30, -420; -30, 8, -16; -420, -16, -189],[30, 2, 1],[-5, 96, 94, 253, 71, -11; 4, -65, -67, -185, -54, 7; 11, -210, -206, -555, -156, 24],127,9]
%
% END OF LATTICE L128.9
%
\hbox{}\par\smallskip%
%
% LATTICE L128.10
%
\leavevmode%
\llap{\spacer}%
${L_{128.10}}=\pdualwitharg{3}(\pfillwitharg{3.2}({L_{128.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{30}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\12\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-21\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\8\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\-65\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-67\\-174
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
261\\-370\\-930
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-18\\-43
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\7\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-12\\-90\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-120\\-15\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\6\\0\\-3\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-120\\0\\30\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-15\\-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-9\\-30\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-270, -60, -60; -60, 12, -21; -60, -21, -11],[30, 3, 1],[-5, 51, 49, 261, 12, -6; 8, -65, -67, -370, -18, 7; 18, -180, -174, -930, -43, 21],127,10]
%
% END OF LATTICE L128.10
%
\hbox{}\par\smallskip%
%
% LATTICE L128.11
%
\leavevmode%
\llap{\spacer}%
${L_{128.11}}=\pdualwitharg{5}(\pfillwitharg{3.2}({L_{128.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
510\\-120\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\20\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-25\\27
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\8\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-13\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-67\\-110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-74\\-114
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-54\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-20\\-30\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-8\\-15\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\10\\0\\-15\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-8\\0\\2\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-15\\-15\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-15\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[510, -120, 120; -120, 20, -25; 120, -25, 27],[30, 5, 1],[-1, 3, 13, 13, 8, -2; 8, -13, -67, -74, -54, 7; 10, -24, -110, -114, -75, 15],127,11]
%
% END OF LATTICE L128.11
%
\hbox{}\par\smallskip%
%
% LATTICE L128.12
%
\leavevmode%
\llap{\spacer}%
${L_{128.12}}=\pdualwitharg{2.3}(\pfillwitharg{3.2}({L_{128.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
85260\\-3270\\40830
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3270\\126\\-1566
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40830\\-1566\\19553
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82\\3\\-171
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-719\\-20\\1500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-955\\-30\\1992
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2927\\-95\\6105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-327\\-11\\682
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\1\\-96
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-45\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-120\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-6\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-120\\0\\15\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-30\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-18\\-30\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[85260, -3270, 40830; -3270, 126, -1566; 40830, -1566, 19553],[30, 6, 1],[82, -719, -955, -2927, -327, 46; 3, -20, -30, -95, -11, 1; -171, 1500, 1992, 6105, 682, -96],127,12]
%
% END OF LATTICE L128.12
%
\hbox{}\par\smallskip%
%
% LATTICE L128.13
%
\leavevmode%
\llap{\spacer}%
${L_{128.13}}=\pdualwitharg{3}(\main(\pfillwitharg{3}({L_{128.1}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1260\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\3\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-40\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-25\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-180\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-120\\-15\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\0\\-3\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-120\\0\\60\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-15\\-3\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-9\\-60\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1260, -60, -60; -60, 3, 3; -60, 3, 2],[60, 3, 1],[-3, -2, 0, 1, 0, -1; -40, -25, 1, 20, 0, -14; -24, -15, 0, 0, -1, -9],127,13]
%
% END OF LATTICE L128.13
%
\hbox{}\par\smallskip%
%
% LATTICE L128.14
%
\leavevmode%
\llap{\spacer}%
${L_{128.14}}=\pdualwitharg{2}(\main(\pfillwitharg{3}({L_{128.1}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84120\\-5220\\-20820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5220\\324\\1292
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20820\\1292\\5153
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\14\\49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69\\75\\260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\19\\68
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\15\\65
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-18\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\-15\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-40\\-60\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\4\\0\\-12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-40\\0\\5\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-60\\-12\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-12\\-20\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[84120, -5220, -20820; -5220, 324, 1292; -20820, 1292, 5153],[60, 4, 1],[13, 69, 18, 17, -13, 1; 14, 75, 19, 15, -18, 0; 49, 260, 68, 65, -48, 4],127,14]
%
% END OF LATTICE L128.14
%
\hbox{}\par\smallskip%
%
% LATTICE L128.15
%
\leavevmode%
\llap{\spacer}%
${L_{128.15}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{128.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17190\\-1980\\180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1980\\228\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\-21\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\42\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\25\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-60\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-17\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-36\\-90\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-40\\-15\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\18\\0\\-9\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-40\\0\\10\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-15\\-9\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-9\\-10\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17190, -1980, 180; -1980, 228, -21; 180, -21, 1],[90, 3, 1],[5, 3, -1, -7, -2, 0; 42, 25, -9, -60, -17, 0; -18, -10, 0, 10, 3, -1],127,15]
%
% END OF LATTICE L128.15
%
\hbox{}\par\smallskip%
%
% LATTICE L128.16
%
\leavevmode%
\llap{\spacer}%
${L_{128.16}}=\pdualwitharg{5}(\main(\pfillwitharg{3}({L_{128.1}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1380\\-540\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\210\\25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\25\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-16\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\7\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\12\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-9\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-60\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-8\\-15\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\5\\0\\-15\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-8\\0\\4\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-15\\-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-15\\-20\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1380, -540, -60; -540, 210, 25; -60, 25, 1],[60, 5, 1],[-7, 0, 3, 5, 1, -4; -16, 0, 7, 12, 3, -9; -20, -1, 5, 8, 0, -10],127,16]
%
% END OF LATTICE L128.16
%
\hbox{}\par\smallskip%
%
% LATTICE L128.17
%
\leavevmode%
\llap{\spacer}%
${L_{128.17}}=\pdualwitharg{2.5}(\pfillwitharg{3.2}({L_{128.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20820\\-2130\\9870
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2130\\210\\-1010
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9870\\-1010\\4679
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\3\\-185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-4\\284
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-951\\-30\\2000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-593\\-19\\1247
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1013\\-33\\2130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\1\\-80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-20\\-15\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-30\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-8\\0\\1\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-30\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-30\\-10\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20820, -2130, 9870; -2130, 210, -1010; 9870, -1010, 4679],[30, 10, 1],[88, -135, -951, -593, -1013, 38; 3, -4, -30, -19, -33, 1; -185, 284, 2000, 1247, 2130, -80],127,17]
%
% END OF LATTICE L128.17
%
\hbox{}\par\smallskip%
%
% LATTICE L128.18
%
\leavevmode%
\llap{\spacer}%
${L_{128.18}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9720\\-240\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\6\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\3\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-100\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-65\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\40\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-34\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-360\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-240\\-15\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\6\\0\\-3\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-240\\0\\120\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-15\\-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-18\\-120\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9720, -240, -120; -240, 6, 3; -120, 3, 1],[120, 3, 1],[-3, -2, 0, 1, 0, -1; -100, -65, 1, 40, 0, -34; -48, -30, 0, 0, -1, -18],127,18]
%
% END OF LATTICE L128.18
%
\hbox{}\par\smallskip%
%
% LATTICE L128.19
%
\leavevmode%
\llap{\spacer}%
${L_{128.19}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{128.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\0\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\51\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-54\\-41
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\128\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\15\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-35\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\-180\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\12\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\39\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-360\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-240\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\6\\0\\-6\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-240\\0\\120\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-30\\-6\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-9\\-60\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1320, 0, 360; 0, 51, -54; 360, -54, -41],[120, 3, 1],[33, 4, -9, -47, 3, 10; 128, 15, -35, -180, 12, 39; 120, 15, -33, -180, 10, 36],127,19]
%
% END OF LATTICE L128.19
%
\hbox{}\par\smallskip%
%
% LATTICE L128.20
%
\leavevmode%
\llap{\spacer}%
${L_{128.20}}=\pdualwitharg{3.5}(\pfillwitharg{3.2}({L_{128.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27870\\3000\\-6780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3000\\315\\-735
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6780\\-735\\1646
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-131\\268\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\-204\\321
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
706\\-1442\\2265
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
881\\-1800\\2826
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
251\\-513\\805
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\57\\-90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\0\\-60\\-90\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-24\\-15\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\30\\0\\-15\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-24\\0\\6\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-15\\-15\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-45\\-30\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[27870, 3000, -6780; 3000, 315, -735; -6780, -735, 1646],[30, 15, 1],[-131, 100, 706, 881, 251, -28; 268, -204, -1442, -1800, -513, 57; -420, 321, 2265, 2826, 805, -90],127,20]
%
% END OF LATTICE L128.20
%
\hbox{}\par\smallskip%
%
% LATTICE L128.21
%
\leavevmode%
\llap{\spacer}%
${L_{128.21}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{128.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61380\\8190\\30330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8190\\1014\\4044
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30330\\4044\\14987
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110\\9\\-225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-137\\-10\\280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-317\\-24\\648
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71\\-5\\145
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\7\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89\\7\\-182
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-36\\-45\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-40\\-30\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\36\\0\\-18\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-40\\0\\5\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-30\\-18\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-18\\-10\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[61380, 8190, 30330; 8190, 1014, 4044; 30330, 4044, 14987],[90, 6, 1],[110, -137, -317, -71, 88, 89; 9, -10, -24, -5, 7, 7; -225, 280, 648, 145, -180, -182],127,21]
%
% END OF LATTICE L128.21
%
\hbox{}\par\smallskip%
%
% LATTICE L128.22
%
\leavevmode%
\llap{\spacer}%
${L_{128.22}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{128.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{180}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-1170\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1170\\978\\-546
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-546\\-629
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\1\\-41
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
571\\30\\-1170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-857\\-45\\1755
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-11\\426
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-167\\-9\\342
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\-45\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\180\\0\\-360\\-90\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\4\\0\\-6\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-360\\0\\45\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-90\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-18\\-90\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2520, -1170, -1260; -1170, 978, -546; -1260, -546, -629],[90, 6, 1],[20, 571, -1, -857, -208, -167; 1, 30, 0, -45, -11, -9; -41, -1170, 2, 1755, 426, 342],127,22]
%
% END OF LATTICE L128.22
%
\hbox{}\par\smallskip%
%
% LATTICE L128.23
%
\leavevmode%
\llap{\spacer}%
${L_{128.23}}=\pdualwitharg{3}(\main({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{180}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8820\\540\\540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
540\\33\\33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
540\\33\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-12\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-15\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-120\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-14\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-24\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-4\\-180\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\45\\0\\-360\\-45\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\1\\0\\-3\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-360\\0\\180\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-45\\-3\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-9\\-180\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8820, 540, 540; 540, 33, 33; 540, 33, 34],[180, 3, 1],[1, 1, 0, 7, 1, 2; -12, -15, -1, -120, -14, -24; -4, 0, 1, 0, -3, -9],127,23]
%
% END OF LATTICE L128.23
%
\hbox{}\par\smallskip%
%
% LATTICE L128.24
%
\leavevmode%
\llap{\spacer}%
${L_{128.24}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{128.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7080\\-360\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\10\\5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-64\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\19\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-40\\-120\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-16\\-30\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\10\\0\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-16\\0\\8\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-30\\-30\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-15\\-20\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7080, -360, 0; -360, 10, 5; 0, 5, -3],[120, 5, 1],[-3, 0, 1, 1, -1, -1; -64, -1, 19, 20, -18, -20; -120, -4, 30, 32, -30, -35],127,24]
%
% END OF LATTICE L128.24
%
\hbox{}\par\smallskip%
%
% LATTICE L128.25
%
\leavevmode%
\llap{\spacer}%
${L_{128.25}}=\pdualwitharg{5}(\pfillwitharg{3}({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20280\\2040\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2040\\-205\\60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-600\\60\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\36\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\11\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-24\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-40\\-120\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-16\\-15\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\10\\0\\-15\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-16\\0\\8\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-15\\-15\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-30\\-40\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-20280, 2040, -600; 2040, -205, 60; -600, 60, -17],[120, 5, 1],[3, 1, 1, -1, -2, -1; 36, 11, 9, -16, -24, -10; 20, 3, -5, -24, -15, 0],127,25]
%
% END OF LATTICE L128.25
%
\hbox{}\par\smallskip%
%
% LATTICE L128.26
%
\leavevmode%
\llap{\spacer}%
${L_{128.26}}=\pdualwitharg{2.3}(\main(\pfillwitharg{3}({L_{128.1}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-660\\60\\180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\216\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\-12\\-49
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-1\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\1\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-1\\52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-3\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-45\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-120\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-12\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-120\\0\\15\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-60\\-12\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-36\\-60\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-660, 60, 180; 60, 216, -12; 180, -12, -49],[60, 12, 1],[7, -17, -13, 4, 14, 32; -1, 0, 1, 0, -1, -3; 27, -60, -48, 15, 52, 120],127,26]
%
% END OF LATTICE L128.26
%
\hbox{}\par\smallskip%
%
% LATTICE L128.27
%
\leavevmode%
\llap{\spacer}%
${L_{128.27}}=\pdualwitharg{2.3.5}(\pfillwitharg{3.2}({L_{128.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
94380\\53850\\22110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53850\\30690\\12600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22110\\12600\\5173
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-87\\195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\338\\-768
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67\\1644\\-3720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\881\\-1989
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\409\\-920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-197\\450
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-60\\-45\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-30\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\60\\0\\-30\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-24\\0\\3\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-30\\-30\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-90\\-30\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[94380, 53850, 22110; 53850, 30690, 12600; 22110, 12600, 5173],[30, 30, 1],[4, -13, -67, -37, -18, 7; -87, 338, 1644, 881, 409, -197; 195, -768, -3720, -1989, -920, 450],127,27]
%
% END OF LATTICE L128.27
%
\hbox{}\par\smallskip%
%
% LATTICE L128.28
%
\leavevmode%
\llap{\spacer}%
${L_{128.28}}=\pdualwitharg{3.5}(\main(\pfillwitharg{3}({L_{128.1}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1980\\-240\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\30\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\15\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-60\\-180\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-24\\-15\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\15\\0\\-15\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-24\\0\\12\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-15\\-15\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-45\\-60\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1980, -240, -120; -240, 30, 15; -120, 15, 7],[60, 15, 1],[1, 0, -1, -3, -1, 0; 8, 1, -2, -8, -3, 1; 0, -3, -15, -36, -10, 0],127,28]
%
% END OF LATTICE L128.28
%
\hbox{}\par\smallskip%
%
% LATTICE L128.29
%
\leavevmode%
\llap{\spacer}%
${L_{128.29}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{128.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12720\\3720\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3720\\-984\\152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-600\\152\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\53
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-36\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-40\\-190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\21\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\35\\152
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-8\\-30\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-160\\-120\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\16\\0\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-160\\0\\20\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-120\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-24\\-40\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12720, 3720, -600; 3720, -984, 152; -600, 152, -23],[120, 8, 1],[1, -1, -3, -3, 2, 3; 12, -10, -36, -40, 21, 35; 53, -40, -160, -190, 84, 152],127,29]
%
% END OF LATTICE L128.29
%
\hbox{}\par\smallskip%
%
% LATTICE L128.30
%
\leavevmode%
\llap{\spacer}%
${L_{128.30}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10680\\2400\\1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2400\\536\\368
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\368\\237
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-41\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\89\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\125\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\66\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-35\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-16\\-30\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-80\\-120\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\16\\0\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-80\\0\\5\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-120\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-24\\-20\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10680, 2400, 1680; 2400, 536, 368; 1680, 368, 237],[120, 8, 1],[7, 1, -15, -21, -11, 6; -41, -5, 89, 125, 66, -35; 14, 0, -32, -45, -24, 12],127,30]
%
% END OF LATTICE L128.30
%
\hbox{}\par\smallskip%
%
% LATTICE L128.31
%
\leavevmode%
\llap{\spacer}%
${L_{128.31}}=\pdualwitharg{3}({L_{128.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
122760\\-24480\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24480\\4881\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\72\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\24\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-60\\-340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-14\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\0\\-72\\-360\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-80\\-30\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\18\\0\\-18\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-80\\0\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-30\\-18\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-9\\-20\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[122760, -24480, -360; -24480, 4881, 72; -360, 72, 1],[360, 3, 1],[5, 1, -2, -13, -3, 0; 24, 5, -9, -60, -14, 0; 72, 5, -63, -340, -72, -1],127,31]
%
% END OF LATTICE L128.31
%
\hbox{}\par\smallskip%
%
% LATTICE L128.32
%
\leavevmode%
\llap{\spacer}%
${L_{128.32}}=\pdualwitharg{3}({L_{128.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{360}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-15\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-8\\-360\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\90\\0\\-720\\-45\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\2\\0\\-3\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-720\\0\\360\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-45\\-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-18\\-360\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[360, 0, 0; 0, 3, 0; 0, 0, -1],[360, 3, 1],[-1, -2, 0, 1, 0, -1; -4, -15, -1, 0, 1, 0; -20, -45, -1, 0, 0, -18],127,32]
%
% END OF LATTICE L128.32
%
\hbox{}\par\smallskip%
%
% LATTICE L128.33
%
\leavevmode%
\llap{\spacer}%
${L_{128.33}}=\pdualwitharg{2.5}(\main(\pfillwitharg{3}({L_{128.1}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\480\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
480\\740\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-400\\189
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\32\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\40\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\43\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-129\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-29\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-20\\-15\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-60\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-8\\0\\1\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\-60\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-60\\-20\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-60, 480, -360; 480, 740, -400; -360, -400, 189],[60, 20, 1],[-12, -15, -16, 2, 49, 11; 32, 40, 43, -5, -129, -29; 45, 56, 60, -7, -180, -40],127,33]
%
% END OF LATTICE L128.33
%
\hbox{}\par\smallskip%
%
% LATTICE L128.34
%
\leavevmode%
\llap{\spacer}%
${L_{128.34}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{128.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${90}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\-90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\60\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-2\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-5\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-2\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\0\\-180\\-90\\-90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-8\\-15\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\90\\0\\-45\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-8\\0\\2\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-15\\-45\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-45\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[90, -90, 0; -90, 60, 15; 0, 15, -7],[90, 15, 1],[1, 1, 1, -3, -6, -2; 0, 1, 3, -2, -5, -2; 0, 2, 0, -8, -15, -5],127,34]
%
% END OF LATTICE L128.34
%
\hbox{}\par\smallskip%
%
% LATTICE L128.35
%
\leavevmode%
\llap{\spacer}%
${L_{128.35}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{128.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35010\\15390\\-990
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15390\\6765\\-435
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-990\\-435\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-12\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-12\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-9\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-20\\-90\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-72\\-45\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\10\\0\\-15\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-72\\0\\18\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-45\\-15\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-45\\-90\\0\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[35010, 15390, -990; 15390, 6765, -435; -990, -435, 28],[90, 15, 1],[5, 5, 1, 1, 0, 4; -12, -12, -2, 0, 1, -9; -10, -9, 5, 36, 15, 0],127,35]
%
% END OF LATTICE L128.35
%
\hbox{}\par\smallskip%
%
% LATTICE L128.36
%
\leavevmode%
\llap{\spacer}%
${L_{128.36}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46440\\-600\\1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-600\\30\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\15\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\8\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\2\\155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\2\\270
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\-120\\-360\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-48\\-15\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\30\\0\\-15\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-48\\0\\24\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-15\\-15\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-90\\-120\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-46440, -600, 1200; -600, 30, 15; 1200, 15, -31],[120, 15, 1],[-3, -2, 0, 13, 4, 7; -4, -1, 1, 8, 2, 2; -120, -78, 0, 504, 155, 270],127,36]
%
% END OF LATTICE L128.36
%
\hbox{}\par\smallskip%
%
% LATTICE L128.37
%
\leavevmode%
\llap{\spacer}%
${L_{128.37}}=\pdualwitharg{3.5}(\pfillwitharg{3}({L_{128.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6840\\-1320\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1320\\30\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\-1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-16\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\28\\384
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\16\\230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\75
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\-120\\-360\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-48\\-30\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\30\\0\\-30\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-48\\0\\24\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-30\\-30\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-45\\-60\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6840, -1320, 0; -1320, 30, 15; 0, 15, -1],[120, 15, 1],[-3, -1, 0, 5, 3, 1; -16, -5, 1, 28, 16, 5; -240, -78, 0, 384, 230, 75],127,37]
%
% END OF LATTICE L128.37
%
\hbox{}\par\smallskip%
%
% LATTICE L128.38
%
\leavevmode%
\llap{\spacer}%
${L_{128.38}}=\pdualwitharg{2.3}(\main({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{180}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\2700\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2700\\-552\\672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\672\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\1\\53
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-221\\15\\900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
232\\-15\\-945
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\-7\\-456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62\\-3\\-252
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\-45\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\180\\0\\-360\\-180\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\4\\0\\-12\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-360\\0\\45\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-180\\-12\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-36\\-180\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[180, 2700, 0; 2700, -552, 672; 0, 672, -11],[180, 12, 1],[-13, -221, -1, 232, 112, 62; 1, 15, 0, -15, -7, -3; 53, 900, 4, -945, -456, -252],127,38]
%
% END OF LATTICE L128.38
%
\hbox{}\par\smallskip%
%
% LATTICE L128.39
%
\leavevmode%
\llap{\spacer}%
${L_{128.39}}=\pdualwitharg{2}(\main({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2672820\\-334980\\658620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-334980\\41496\\-82536
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
658620\\-82536\\162293
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\12\\765
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
919\\-60\\-3760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1795\\-117\\-7344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
842\\-55\\-3445
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\17\\1080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-570\\37\\2332
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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9\\0\\-36\\-45\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-40\\-60\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\36\\0\\-36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-40\\0\\5\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-60\\-36\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-36\\-20\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2672820, -334980, 658620; -334980, 41496, -82536; 658620, -82536, 162293],[180, 12, 1],[-187, 919, 1795, 842, -264, -570; 12, -60, -117, -55, 17, 37; 765, -3760, -7344, -3445, 1080, 2332],127,39]
%
% END OF LATTICE L128.39
%
\hbox{}\par\smallskip%
%
% LATTICE L128.40
%
\leavevmode%
\llap{\spacer}%
${L_{128.40}}=\pdualwitharg{3.5}(\main({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
128340\\55620\\-3600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55620\\24105\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3600\\-1560\\101
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-32\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-15\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-27\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-180\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-72\\-45\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\5\\0\\-15\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-72\\0\\36\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-45\\-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-45\\-180\\0\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[128340, 55620, -3600; 55620, 24105, -1560; -3600, -1560, 101],[180, 15, 1],[15, 7, 1, 1, 0, 13; -32, -15, -2, 0, 1, -27; 40, 18, 5, 36, 15, 45],127,40]
%
% END OF LATTICE L128.40
%
\hbox{}\par\smallskip%
%
% LATTICE L128.41
%
\leavevmode%
\llap{\spacer}%
${L_{128.41}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{128.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{180}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-90\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\210\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-30\\-89
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-3\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\6\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\1\\-19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\1\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-1\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-180\\-45\\-90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-30\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\180\\0\\-90\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-8\\0\\1\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-30\\-90\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-90\\-10\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-360, -90, -180; -90, 210, -30; -180, -30, -89],[90, 30, 1],[-22, -1, 43, 9, 14, -5; -3, 0, 6, 1, 1, -1; 45, 2, -90, -19, -30, 10],127,41]
%
% END OF LATTICE L128.41
%
\hbox{}\par\smallskip%
%
% LATTICE L128.42
%
\leavevmode%
\llap{\spacer}%
${L_{128.42}}=\pdualwitharg{5}(\main({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30420\\180\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\-15\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\180
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-172
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\0\\-180\\-180\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-8\\-15\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\45\\0\\-45\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-8\\0\\4\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-15\\-45\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-45\\-20\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30420, 180, -180; 180, -15, 0; -180, 0, 1],[180, 15, 1],[1, 0, -1, -1, -1, 0; 12, 0, -9, -8, -7, 1; 180, -1, -180, -172, -165, 5],127,42]
%
% END OF LATTICE L128.42
%
\hbox{}\par\smallskip%
%
% LATTICE L128.43
%
\leavevmode%
\llap{\spacer}%
${L_{128.43}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{128.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{90}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58410\\-6210\\28710
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6210\\-60\\-3030
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28710\\-3030\\14111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\4\\125
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
334\\-21\\-684
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
332\\-21\\-680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
145\\-9\\-297
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-249\\16\\510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-703\\45\\1440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-20\\-45\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-72\\-90\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0\\-30\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-72\\0\\9\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-90\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-90\\-90\\0\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[58410, -6210, 28710; -6210, -60, -3030; 28710, -3030, 14111],[90, 30, 1],[-61, 334, 332, 145, -249, -703; 4, -21, -21, -9, 16, 45; 125, -684, -680, -297, 510, 1440],127,43]
%
% END OF LATTICE L128.43
%
\hbox{}\par\smallskip%
%
% LATTICE L128.44
%
\leavevmode%
\llap{\spacer}%
${L_{128.44}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-90\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\240\\0\\-240\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\0\\-24\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-240\\0\\15\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-120\\-24\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-72\\-60\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[840, -240, 0; -240, 72, 0; 0, 0, -1],[120, 24, 1],[-1, -1, 1, 1, 0, -1; -3, -5, 3, 5, 1, -2; -6, 0, 0, -15, -8, -12],127,44]
%
% END OF LATTICE L128.44
%
\hbox{}\par\smallskip%
%
% LATTICE L128.45
%
\leavevmode%
\llap{\spacer}%
${L_{128.45}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{128.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1560\\-480\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\216\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\48\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\35\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\8\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-24\\-90\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\240\\0\\-480\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\48\\0\\-24\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-480\\0\\60\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-120\\-24\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-72\\-120\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1560, -480, -120; -480, 216, 48; -120, 48, 11],[120, 24, 1],[-1, -7, -1, -1, 0, -1; 6, 35, 3, 5, 1, 8; -39, -240, -24, -30, -4, -48],127,45]
%
% END OF LATTICE L128.45
%
\hbox{}\par\smallskip%
%
% LATTICE L128.46
%
\leavevmode%
\llap{\spacer}%
${L_{128.46}}=\pdualwitharg{2.3.5}(\main(\pfillwitharg{3}({L_{128.1}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
540\\-420\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-1560\\420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\420\\-113
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\26\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\65\\240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\17\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-49\\-180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-60\\-45\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\0\\60\\0\\-60\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-24\\0\\3\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\-60\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-180\\-60\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[540, -420, 120; -420, -1560, 420; 120, 420, -113],[60, 60, 1],[0, -1, -3, -1, 0, 2; 4, 26, 65, 17, -11, -49; 15, 96, 240, 63, -40, -180],127,46]
%
% END OF LATTICE L128.46
%
\hbox{}\par\smallskip%
%
% LATTICE L128.47
%
\leavevmode%
\llap{\spacer}%
${L_{128.47}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{128.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68520\\25200\\-1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25200\\9160\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1200\\-440\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\5\\505
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\5\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\7\\720
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-40\\-30\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-32\\-120\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\80\\0\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-32\\0\\4\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-120\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-120\\-40\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[68520, 25200, -1200; 25200, 9160, -440; -1200, -440, 21],[120, 40, 1],[7, 7, -1, -1, 1, 10; 5, 5, -1, -1, 0, 7; 505, 504, -80, -78, 60, 720],127,47]
%
% END OF LATTICE L128.47
%
\hbox{}\par\smallskip%
%
% LATTICE L128.48
%
\leavevmode%
\llap{\spacer}%
${L_{128.48}}=\pdualwitharg{2.5}(\pfillwitharg{3}({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-80\\-30\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-16\\-120\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\80\\0\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-16\\0\\1\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-120\\-20\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120, 0, 0; 0, -40, 0; 0, 0, 1],[120, 40, 1],[-1, -1, -1, 0, 1, 0; 5, 3, 1, 0, 0, 3; -30, -16, 0, 1, 0, -20],127,48]
%
% END OF LATTICE L128.48
%
\hbox{}\par\smallskip%
%
% LATTICE L128.49
%
\leavevmode%
\llap{\spacer}%
${L_{128.49}}=\pdualwitharg{5}({L_{128.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{72}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\1440\\-720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1440\\-795\\375
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\375\\-158
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-96\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-30\\-27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\8\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\48\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-57\\-45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-40\\-360\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-144\\-90\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\10\\0\\-30\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-144\\0\\72\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-90\\-30\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-45\\-180\\0\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2520, 1440, -720; 1440, -795, 375; -720, 375, -158],[360, 15, 1],[-33, -10, 3, 17, -1, -20; -96, -30, 8, 48, -2, -57; -80, -27, 5, 36, 0, -45],127,49]
%
% END OF LATTICE L128.49
%
\hbox{}\par\smallskip%
%
% LATTICE L128.50
%
\leavevmode%
\llap{\spacer}%
${L_{128.50}}=\pdualwitharg{5}({L_{128.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${360}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3960\\-360\\-720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-30\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-75\\-97
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-12\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\7\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-48\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-49\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-36\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\-360\\-360\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-16\\-15\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\90\\0\\-45\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-16\\0\\8\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-15\\-45\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-90\\-40\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3960, -360, -720; -360, -30, -75; -720, -75, -97],[360, 15, 1],[1, -1, -1, 7, 7, 5; -12, 7, 9, -48, -49, -36; 0, 2, 0, -16, -15, -10],127,50]
%
% END OF LATTICE L128.50
%
\hbox{}\par\smallskip%
%
% LATTICE L128.51
%
\leavevmode%
\llap{\spacer}%
${L_{128.51}}=\pdualwitharg{3.5}({L_{128.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{180}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50040\\2880\\-1080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2880\\-165\\60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\60\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\60\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\39\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-144\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-40\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-40\\-360\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-144\\-45\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\10\\0\\-15\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-144\\0\\72\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-45\\-15\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-90\\-360\\0\\180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-50040, 2880, -1080; 2880, -165, 60; -1080, 60, -17],[360, 15, 1],[3, 2, 0, -7, -2, -1; 60, 39, -1, -144, -40, -18; 20, 9, -5, -72, -15, 0],127,51]
%
% END OF LATTICE L128.51
%
\hbox{}\par\smallskip%
%
% LATTICE L128.52
%
\leavevmode%
\llap{\spacer}%
${L_{128.52}}=\pdualwitharg{3.5}({L_{128.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${360}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-105\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-24\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-7\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\52\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\94\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\14\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\-360\\-360\\-360\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-16\\-30\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\90\\0\\-90\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-16\\0\\8\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-30\\-90\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-45\\-20\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[360, 0, -360; 0, -30, -105; -360, -105, -7],[360, 15, 1],[7, 2, -1, -15, -27, -4; -24, -7, 3, 52, 94, 14; 0, 2, 0, -16, -30, -5],127,52]
%
% END OF LATTICE L128.52
%
\hbox{}\par\smallskip%
%
% LATTICE L128.53
%
\leavevmode%
\llap{\spacer}%
${L_{128.53}}=\pdualwitharg{2}({L_{128.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{s}{2}{144}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
218520\\-74880\\-2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74880\\25656\\864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\864\\29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-15\\-162
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-15\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-144\\-90\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-80\\-120\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\0\\144\\0\\-72\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-80\\0\\5\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-120\\-72\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-72\\-20\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[218520, -74880, -2520; -74880, 25656, 864; -2520, 864, 29],[360, 24, 1],[-7, -7, 1, 2, 2, -1; -15, -15, 3, 5, 5, -2; -162, -160, 0, 25, 24, -28],127,53]
%
% END OF LATTICE L128.53
%
\hbox{}\par\smallskip%
%
% LATTICE L128.54
%
\leavevmode%
\llap{\spacer}%
${L_{128.54}}=\pdualwitharg{2.3}({L_{128.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34920\\-15120\\-1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15120\\6216\\744
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\744\\89
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\18\\-171
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\55\\-520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\15\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-25\\230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-22\\204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-72\\-90\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-160\\-120\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\144\\0\\-72\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-160\\0\\20\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-120\\-72\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-72\\-40\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[34920, -15120, -1800; -15120, 6216, 744; -1800, 744, 89],[360, 24, 1],[-1, -3, -1, 1, 1, 0; 18, 55, 15, -25, -22, -1; -171, -520, -144, 230, 204, 8],127,54]
%
% END OF LATTICE L128.54
%
\hbox{}\par\smallskip%
%
% LATTICE L128.55
%
\leavevmode%
\llap{\spacer}%
${L_{128.55}}=\pdualwitharg{2}({L_{128.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{720}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{180}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-360\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-4728\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\336\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-30\\-450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\15\\216
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-8\\-90\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\720\\0\\-1440\\-360\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\16\\0\\-24\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-1440\\0\\180\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-360\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-72\\-360\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[720, -360, 0; -360, -4728, 336; 0, 336, -23],[360, 24, 1],[1, -1, -3, -13, 1, 8; 2, 0, -6, -30, 1, 15; 29, 0, -88, -450, 12, 216],127,55]
%
% END OF LATTICE L128.55
%
\hbox{}\par\smallskip%
%
% LATTICE L128.56
%
\leavevmode%
\llap{\spacer}%
${L_{128.56}}=\pdualwitharg{2.3}({L_{128.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{720}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-3240\\-1440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3240\\14136\\6288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1440\\6288\\2797
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\14\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\60\\-135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\11\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-15\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-16\\-90\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\720\\0\\-720\\-360\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\16\\0\\-24\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-720\\0\\45\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-360\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-72\\-180\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[720, -3240, -1440; -3240, 14136, 6288; -1440, 6288, 2797],[360, 24, 1],[1, 1, -1, -1, 1, 4; -6, 0, 14, 60, 11, -15; 14, 0, -32, -135, -24, 36],127,56]
%
% END OF LATTICE L128.56
%
\hbox{}\par\smallskip%
%
% LATTICE L128.57
%
\leavevmode%
\llap{\spacer}%
${L_{128.57}}=\pdualwitharg{2.3.5}(\main({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{360}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140760\\12780\\-25380
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12780\\-780\\-3660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25380\\-3660\\3629
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\-192\\275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-158\\453\\-648
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\559\\-800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\195\\-279
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
351\\-1006\\1440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1009\\-2892\\4140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-20\\-45\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-72\\-180\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0\\-60\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-72\\0\\9\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-180\\-60\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-180\\-180\\0\\360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[140760, 12780, -25380; 12780, -780, -3660; -25380, -3660, 3629],[180, 60, 1],[67, -158, -195, -68, 351, 1009; -192, 453, 559, 195, -1006, -2892; 275, -648, -800, -279, 1440, 4140],127,57]
%
% END OF LATTICE L128.57
%
\hbox{}\par\smallskip%
%
% LATTICE L128.58
%
\leavevmode%
\llap{\spacer}%
${L_{128.58}}=\pdualwitharg{2.5}(\main({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{180}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1598040\\33660\\-365760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33660\\660\\-7740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-365760\\-7740\\83689
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-33\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\-129\\176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\-267\\360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\158\\-217
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-469\\1402\\-1920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171\\512\\-700
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-180\\-45\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-60\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\180\\0\\-180\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-8\\0\\1\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-60\\-180\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-180\\-20\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1598040, 33660, -365760; 33660, 660, -7740; -365760, -7740, 83689],[180, 60, 1],[11, 43, 88, -53, -469, -171; -33, -129, -267, 158, 1402, 512; 45, 176, 360, -217, -1920, -700],127,58]
%
% END OF LATTICE L128.58
%
\hbox{}\par\smallskip%
%
% LATTICE L128.59
%
\leavevmode%
\llap{\spacer}%
${L_{128.59}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{128.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{s}{2}{240}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-3000\\-1440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3000\\33480\\16080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1440\\16080\\7723
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\72\\-150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\46\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-10\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-19\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\29\\-60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-240\\-90\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-48\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\0\\240\\0\\-120\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-48\\0\\3\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-120\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-360\\-60\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240, -3000, -1440; -3000, 33480, 16080; -1440, 16080, 7723],[120, 120, 1],[-1, -1, 1, 1, 2, 1; 72, 46, 0, -10, -19, 29; -150, -96, 0, 21, 40, -60],127,59]
%
% END OF LATTICE L128.59
%
\hbox{}\par\smallskip%
%
% LATTICE L128.60
%
\leavevmode%
\llap{\spacer}%
${L_{128.60}}=\pdualwitharg{2.3.5}(\pfillwitharg{3}({L_{128.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
159480\\152880\\75000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
152880\\146520\\71880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75000\\71880\\35263
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\82\\-165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\107\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-49\\102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-48\\100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\61\\-120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-120\\-90\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-96\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\240\\0\\-120\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-96\\0\\12\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-120\\-120\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-360\\-120\\0\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[159480, 152880, 75000; 152880, 146520, 71880; 75000, 71880, 35263],[120, 120, 1],[-1, -1, -1, -1, -1, -2; 82, 107, 1, -49, -48, 61; -165, -216, 0, 102, 100, -120],127,60]
%
% END OF LATTICE L128.60
%
\hbox{}\par\smallskip%
%
% LATTICE L128.61
%
\leavevmode%
\llap{\spacer}%
${L_{128.61}}=\pdualwitharg{2.5}({L_{128.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{720}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-37800\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37800\\1866360\\-17760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-17760\\169
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\103
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\7\\720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\0\\-720\\-90\\-360\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-16\\-120\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\0\\720\\0\\-360\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-16\\0\\1\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-120\\-360\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-360\\-20\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[720, -37800, 360; -37800, 1866360, -17760; 360, -17760, 169],[360, 120, 1],[-1, -1, 1, 1, 7, 2; -6, -2, 0, 1, 7, 1; -630, -208, 0, 103, 720, 100],127,61]
%
% END OF LATTICE L128.61
%
\hbox{}\par\smallskip%
%
% LATTICE L128.62
%
\leavevmode%
\llap{\spacer}%
${L_{128.62}}=\pdualwitharg{2.3.5}({L_{128.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{720}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\4680\\2160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4680\\19560\\9000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2160\\9000\\4141
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\21\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\26\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-38\\82
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-139\\300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-37\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-360\\-90\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-32\\-120\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\720\\0\\-360\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-32\\0\\4\\0\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-120\\-360\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-360\\-40\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[720, 4680, 2160; 4680, 19560, 9000; 2160, 9000, 4141],[360, 120, 1],[-1, -1, -1, 1, 4, 1; 21, 26, 0, -38, -139, -37; -45, -56, 0, 82, 300, 80],127,62]
%
% END OF LATTICE L128.62
%
\hbox{}\par\smallskip%
%
% LATTICE L128.63
%
\leavevmode%
\llap{\spacer}%
${L_{128.63}}=\pdualwitharg{2.5}({L_{128.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{l}{2}{360}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17640\\41760\\-720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41760\\97320\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-1680\\29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\115
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\720
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-40\\-90\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-288\\-360\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\80\\0\\-120\\-360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-288\\0\\36\\0\\-360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-360\\-120\\0\\120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-360\\-360\\0\\360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17640, 41760, -720; 41760, 97320, -1680; -720, -1680, 29],[360, 120, 1],[0, -1, -1, -1, 0, 1; 2, 3, -3, -3, 1, 12; 115, 144, -200, -198, 60, 720],127,63]
%
% END OF LATTICE L128.63
%
\hbox{}\par\smallskip%
%
% LATTICE L128.64
%
\leavevmode%
\llap{\spacer}%
${L_{128.64}}=\pdualwitharg{2.3.5}({L_{128.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}{360}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204840\\-204840\\2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204840\\-204720\\2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2520\\2520\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-1\\-410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-3\\-576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\-900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-80\\-90\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-144\\-360\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\80\\0\\-120\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-144\\0\\9\\0\\-180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-360\\-120\\0\\120\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-360\\-180\\0\\360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-204840, -204840, 2520; -204840, -204720, 2520; 2520, 2520, -31],[360, 120, 1],[-4, -4, 2, 1, -1, -11; -1, -3, -1, 0, 1, 0; -410, -576, 80, 81, 0, -900],127,64]
%
% END OF LATTICE L128.64
%

\medskip%
%------------------------------WEYL-GROUP-W129--------------------------------
%
\leavevmode\llap{}%
$W_{129}$%
\qquad\llap{92} lattices, $\chi=24$%
\hfill%
$22222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L129.1
%
\leavevmode%
\llap{\spacer}%
${L_{129.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N'_{17}}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-700728\\2016\\4032
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2016\\-5\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4032\\-12\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-68\\-140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-72\\-138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\66\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\546\\1113
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\208\\416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\141\\276
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\72\\137
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\378\\672
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-16\\-84\\-24\\-12\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-42\\-24\\-24\\-18\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\4\\0\\-8\\-18\\-20\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-42\\0\\21\\0\\-42\\-63\\-924
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\-8\\0\\8\\0\\-8\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\-18\\-42\\0\\3\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-18\\-20\\-63\\-8\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-252\\-924\\-168\\-42\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-700728, 2016, 4032; 2016, -5, -12; 4032, -12, -23],[168, 1, 1],[-1, -1, 1, 8, 3, 2, 1, 5; -68, -72, 66, 546, 208, 141, 72, 378; -140, -138, 140, 1113, 416, 276, 137, 672],128,1]
%
% END OF LATTICE L129.1
%
\hbox{}\par\smallskip%
%
% LATTICE L129.2
%
\leavevmode%
\llap{\spacer}%
${L_{129.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1848\\-504\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\137\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-18\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-11\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-63\\-42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-24\\-24\\-4\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\-18\\-84
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[1848, -504, 0; -504, 137, 1; 0, 1, -2],[168, 1, 1],[1, -5, -3, -17, -1, 5, 3, 17; 4, -18, -11, -63, -4, 18, 11, 63; 0, -12, -7, -42, -4, 6, 4, 21],128,2]
%
% END OF LATTICE L129.2
%
\hbox{}\par\smallskip%
%
% LATTICE L129.3
%
\leavevmode%
\llap{\spacer}%
${L_{129.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
241752\\336\\-2184
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-1\\-2
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2184\\-2\\19
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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-1\\3528\\1176
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0\\14\\5
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0\\-39\\-14
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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1\\76\\120
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1\\86\\124
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1\\84\\126
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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\kern3pt%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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0\\3\\0\\-42
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-16\\0\\4\\0
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-168\\-42\\0\\84
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\-8\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\-18\\-84
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-18\\-40\\-252
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0\\-84\\-252\\-1848
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[241752, 336, -2184; 336, -1, -2; -2184, -2, 19],[168, 1, 1],[1, 1, 1, 1, -1, -1, -1, -1; 76, 84, 86, 84, -88, -93, -104, -210; 120, 123, 124, 126, -124, -126, -130, -168],128,3]
%
% END OF LATTICE L129.3
%
\hbox{}\par\smallskip%
%
% LATTICE L129.4
%
\leavevmode%
\llap{\spacer}%
${L_{129.4}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-380688\\2016\\2352
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2016\\-10\\-14
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2352\\-14\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\125\\54
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\1368\\600
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5\\620\\274
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55\\6804\\3024
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1\\122\\56
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-1\\-126\\-54
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-1\\-124\\-56
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\252\\105
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2\\0\\-8\\-168\\-12\\-12\\-8\\0
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\kern3pt%
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\kern3pt%
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-12\\-48\\-8\\0\\8\\0\\-32\\-84
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\kern3pt%
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-126\\-1848\\-84\\-42\\0\\21
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-380688, 2016, 2352; 2016, -10, -14; 2352, -14, -11],[336, 2, 1],[1, 11, 5, 55, 1, -1, -1, 2; 125, 1368, 620, 6804, 122, -126, -124, 252; 54, 600, 274, 3024, 56, -54, -56, 105],128,4]
%
% END OF LATTICE L129.4
%
\hbox{}\par\smallskip%
%
% LATTICE L129.5
%
\leavevmode%
\llap{\spacer}%
${L_{129.5}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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-1594320\\4368\\9408
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4368\\-10\\-28
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9408\\-28\\-53
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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2\\251\\222
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\2376\\2112
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\499\\445
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\10332\\9240
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\122\\112
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-1\\-126\\-111
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\HardButStrongLineBreak\kern3pt%
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-1\\-124\\-112
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11\\1386\\1218
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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2\\0\\-4\\-168\\-12\\-6\\-8\\0
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-48\\-4\\0\\8\\0\\-32\\-168
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-144\\-40\\-1008\\-32\\0\\16\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-126\\-3696\\-168\\-42\\0\\84
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1594320, 4368, 9408; 4368, -10, -28; 9408, -28, -53],[336, 2, 1],[2, 19, 4, 83, 1, -1, -1, 11; 251, 2376, 499, 10332, 122, -126, -124, 1386; 222, 2112, 445, 9240, 112, -111, -112, 1218],128,5]
%
% END OF LATTICE L129.5
%
\hbox{}\par\smallskip%
%
% LATTICE L129.6
%
\leavevmode%
\llap{\spacer}%
${L_{129.6}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{336}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-423024\\2352\\2352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2352\\-2\\-14
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-423024, 2352, 2352; 2352, -2, -14; 2352, -14, -13],[336, 2, 1],[3, -1, -1, 1, 1, 4, 17, 26; 42, -12, -14, 0, 13, 54, 232, 357; 496, -168, -166, 168, 166, 663, 2816, 4305],128,6]
%
% END OF LATTICE L129.6
%
\hbox{}\par\smallskip%
%
% LATTICE L129.7
%
\leavevmode%
\llap{\spacer}%
${L_{129.7}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-72240, 0, 1344; 0, 2, 0; 1344, 0, -25],[336, 2, 1],[1, 5, 1, 19, 0, -1, -1, 1; -6, -12, -1, 0, 1, 0, -8, -42; 52, 264, 53, 1008, 0, -54, -56, 42],128,7]
%
% END OF LATTICE L129.7
%
\hbox{}\par\smallskip%
%
% LATTICE L129.8
%
\leavevmode%
\llap{\spacer}%
${L_{129.8}}=\pfillwitharg{2}({L_{129.1}})$%
{${}={}$}Nikulin $17'$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[1218, -420, 84; -420, 145, -28; 84, -28, 11],[42, 1, 1],[-5, 1, 25, 278, 43, 56, 32, 121; -14, 3, 70, 777, 120, 156, 89, 336; 2, 0, -11, -126, -20, -27, -16, -63],128,8]
%
% END OF LATTICE L129.8
%
\hbox{}\par\smallskip%
%
% LATTICE L129.9
%
\leavevmode%
\llap{\spacer}%
${L_{129.9}}=\main({L_{129.2}})$%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[924, 420, 0; 420, 190, 1; 0, 1, -1],[84, 1, 1],[-1, 4, 5, 29, 1, -4, -5, -29; 2, -9, -11, -63, -2, 9, 11, 63; 0, -12, -14, -84, -4, 6, 8, 42],128,9]
%
% END OF LATTICE L129.9
%
\hbox{}\par\smallskip%
%
% LATTICE L129.10
%
\leavevmode%
\llap{\spacer}%
${L_{129.10}}=\pfillwitharg{2}({L_{129.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3139836\\156996\\-5880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156996\\-7850\\294
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5880\\294\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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2\\41\\26
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11\\222\\48
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55\\1092\\-252
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4\\84\\105
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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-4\\-36\\-20\\-252\\-8\\0\\4\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-63\\-924\\-42\\-21\\0\\21
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3139836, 156996, -5880; 156996, -7850, 294; -5880, 294, -11],[84, 2, 1],[2, 11, 5, 55, 1, -1, -1, 4; 41, 222, 100, 1092, 19, -21, -20, 84; 26, 48, -3, -252, -28, -27, 0, 105],128,10]
%
% END OF LATTICE L129.10
%
\hbox{}\par\smallskip%
%
% LATTICE L129.11
%
\leavevmode%
\llap{\spacer}%
${L_{129.11}}=\main({L_{129.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{42}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\-14\\-5
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\28\\122
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-27\\-126
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-28\\-124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\105\\504
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1\\27\\125
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5\\137\\620
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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\kern3pt%
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0\\-168\\-504\\-1848\\-84\\-168\\0\\168
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-190344, 2352, 1008; 2352, -22, -14; 1008, -14, -5],[168, 2, 1],[1, -1, -1, 4, 1, 11, 5, 55; 28, -27, -28, 105, 27, 300, 137, 1512; 122, -126, -124, 504, 125, 1368, 620, 6804],128,11]
%
% END OF LATTICE L129.11
%
\hbox{}\par\smallskip%
%
% LATTICE L129.12
%
\leavevmode%
\llap{\spacer}%
${L_{129.12}}=\main({L_{129.6}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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168\\0\\0
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0\\2\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
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-1\\-4\\-14
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-1\\0\\-12
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0\\1\\0
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1\\0\\0
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0\\-1\\-1
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-1\\-9\\-18
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[168, 0, 0; 0, 2, 0; 0, 0, -1],[168, 2, 1],[-1, -1, 0, 1, 0, -1, -3, -11; -4, 0, 1, 0, -1, -9, -20, -63; -14, -12, 0, 0, -1, -18, -48, -168],128,12]
%
% END OF LATTICE L129.12
%
\hbox{}\par\smallskip%
%
% LATTICE L129.13
%
\leavevmode%
\llap{\spacer}%
${L_{129.13}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{129.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-50484\\3234\\-25578
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3234\\-206\\1638
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25578\\1638\\-12959
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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2623949\\-2379048\\-5477808
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1331925\\-1207612\\-2780553
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\0\\42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-24\\-18\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-18\\-20\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-126\\-924
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-50484, 3234, -25578; 3234, -206, 1638; -25578, 1638, -12959],[42, 2, 1],[78, 1250, 1619, 11146, 447, 325, -44, -121; -72, -1137, -1469, -10101, -404, -291, 41, 105; -163, -2610, -3380, -23268, -933, -678, 92, 252],128,13]
%
% END OF LATTICE L129.13
%
\hbox{}\par\smallskip%
%
% LATTICE L129.14
%
\leavevmode%
\llap{\spacer}%
${L_{129.14}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{129.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90426\\5292\\29778
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5292\\-309\\-1743
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29778\\-1743\\-9806
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-2267301\\-2632280\\-6420330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
134890\\156603\\381969
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
745380\\865368\\2110697
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\-144\\-342
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-325\\-379\\-920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1264\\-1469\\-3579
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2902\\-3367\\-8218
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6\\0\\-12\\-42\\-18\\-6\\-6\\0
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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6\\0\\-12\\-42
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-7
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-7\\0\\7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-6\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-4\\-9\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-9\\-30\\-63
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-63\\-154
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-90426, 5292, 29778; 5292, -309, -1743; 29778, -1743, -9806],[42, 3, 1],[-121, -325, -1264, -2902, -699, -85, 34, 32; -144, -379, -1469, -3367, -808, -97, 41, 35; -342, -920, -3579, -8218, -1980, -241, 96, 91],128,14]
%
% END OF LATTICE L129.14
%
\hbox{}\par\smallskip%
%
% LATTICE L129.15
%
\leavevmode%
\llap{\spacer}%
${L_{129.15}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{129.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37758\\-7938\\15792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7938\\1704\\-3318
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15792\\-3318\\6605
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-5838603\\-785323\\13562934
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1164296\\156603\\-2704632
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2446000\\-329000\\5681999
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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337\\45\\-783
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1913\\257\\-4444
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7511\\1010\\-17448
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17327\\2331\\-40250
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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3\\0\\-12\\-42\\-9\\-6\\-6\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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3\\0\\-12\\-42
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0\\2\\0\\-14
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-12\\0\\6\\0
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-42\\-14\\0\\14
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-6\\-6\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-8\\-18\\-28
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-18\\-60\\-126
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0\\-28\\-126\\-308
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[37758, -7938, 15792; -7938, 1704, -3318; 15792, -3318, 6605],[42, 6, 1],[337, 1913, 7511, 17327, 2109, 533, -173, -205; 45, 257, 1010, 2331, 284, 72, -23, -28; -783, -4444, -17448, -40250, -4899, -1238, 402, 476],128,15]
%
% END OF LATTICE L129.15
%
\hbox{}\par\smallskip%
%
% LATTICE L129.16
%
\leavevmode%
\llap{\spacer}%
${L_{129.16}}=\pdualwitharg{3}(\main({L_{129.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\168\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\66\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-39\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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223\\-504\\168
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16\\-36\\11
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-12\\27\\-8
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-29\\65\\-18
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-47\\105\\-28
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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12\\0\\-24\\-84
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-14
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-24\\0\\6\\0
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-84\\-14\\0\\14
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-12\\-12\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-8\\-18\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-18\\-60\\-126
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-126\\-308
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[420, 168, -84; 168, 66, -39; -84, -39, -7],[84, 3, 1],[-15, -12, -29, -47, -1, 4, 5, -9; 34, 27, 65, 105, 2, -9, -11, 21; -12, -8, -18, -28, 0, 2, 0, -14],128,16]
%
% END OF LATTICE L129.16
%
\hbox{}\par\smallskip%
%
% LATTICE L129.17
%
\leavevmode%
\llap{\spacer}%
${L_{129.17}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{129.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-486654, 3822, -72198; 3822, -28, 567; -72198, 567, -10711],[42, 7, 1],[193, 1522, 1966, 1931, 1081, 389, -55, -20; -16, -150, -199, -198, -114, -45, 4, 3; -1302, -10269, -13265, -13029, -7294, -2625, 371, 135],128,17]
%
% END OF LATTICE L129.17
%
\hbox{}\par\smallskip%
%
% LATTICE L129.18
%
\leavevmode%
\llap{\spacer}%
${L_{129.18}}=\pdualwitharg{2}(\main({L_{129.2}}))$%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[152376, -2604, 37464; -2604, 44, -640; 37464, -640, 9211],[84, 4, 1],[9, 17, -3, -101, -11, -23, -3, 59; -20, -42, -4, 126, 18, 36, -2, -168; -38, -72, 12, 420, 46, 96, 12, -252],128,18]
%
% END OF LATTICE L129.18
%
\hbox{}\par\smallskip%
%
% LATTICE L129.19
%
\leavevmode%
\llap{\spacer}%
${L_{129.19}}=\pdualwitharg{3}({L_{129.1}})$%
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{$1\above{1pt}{1pt}{-2}{4}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-615720, 3192, 6888; 3192, -15, -36; 6888, -36, -77],[168, 3, 1],[-1, -1, 1, 5, 7, 2, 4, 11; -12, -16, 10, 70, 104, 31, 64, 182; -84, -82, 84, 413, 576, 164, 327, 896],128,19]
%
% END OF LATTICE L129.19
%
\hbox{}\par\smallskip%
%
% LATTICE L129.20
%
\leavevmode%
\llap{\spacer}%
${L_{129.20}}=\pdualwitharg{3}({L_{129.2}})$%
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{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-69720, 17136, 1008; 17136, -4209, -249; 1008, -249, -14],[168, 3, 1],[1, -5, -3, 6, 19, 15, 18, 29; 4, -18, -11, 21, 68, 54, 65, 105; 0, -40, -21, 56, 156, 118, 138, 217],128,20]
%
% END OF LATTICE L129.20
%
\hbox{}\par\smallskip%
%
% LATTICE L129.21
%
\leavevmode%
\llap{\spacer}%
${L_{129.21}}=\pdualwitharg{3}({L_{129.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-7224\\168\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-3\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-6\\1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-169\\-5208\\-2520
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5\\154\\75
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1\\31\\14
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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0\\0\\-1
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1\\30\\12
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1\\28\\14
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\1\\0\\-14
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-72\\-12\\-24\\0
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-12\\-4\\-18\\-28
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7224, 168, 168; 168, -3, -6; 168, -6, 1],[168, 3, 1],[-3, 0, 1, 1, -1, -1, -7, -15; -92, 0, 30, 28, -32, -31, -216, -462; -48, -1, 12, 14, -12, -14, -102, -224],128,21]
%
% END OF LATTICE L129.21
%
\hbox{}\par\smallskip%
%
% LATTICE L129.22
%
\leavevmode%
\llap{\spacer}%
${L_{129.22}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{129.4}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-924\\168\\252
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-30\\-42
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252\\-42\\-41
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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6\\41\\-6
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9\\60\\-8
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11\\72\\-9
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\HardButStrongLineBreak\kern3pt%
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37\\238\\-28
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1\\5\\0
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-1\\-7\\1
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-1\\-6\\0
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6\\42\\-7
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6\\0\\-12\\-84\\-18\\-6\\-12\\0
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\kern3pt%
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0\\4\\0\\-28\\-12\\-8\\-36\\-28
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\kern3pt%
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-12\\0\\3\\0\\-6\\-9\\-60\\-63
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-6\\-8\\-9\\-28\\0\\1\\0\\-7
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-12\\-36\\-60\\-252\\-24\\0\\12\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-63\\-308\\-42\\-7\\0\\7
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-924, 168, 252; 168, -30, -42; 252, -42, -41],[84, 6, 1],[6, 9, 11, 37, 1, -1, -1, 6; 41, 60, 72, 238, 5, -7, -6, 42; -6, -8, -9, -28, 0, 1, 0, -7],128,22]
%
% END OF LATTICE L129.22
%
\hbox{}\par\smallskip%
%
% LATTICE L129.23
%
\leavevmode%
\llap{\spacer}%
${L_{129.23}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{129.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4949448\\-10290\\2383836
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10290\\70\\-4956
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2383836\\-4956\\1148143
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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626131211\\1270416\\-1300003320
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-2434705\\-4941\\5055050
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-5128\\-10\\10647
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-114994\\-233\\238756
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-21\\-42\\-14\\0
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-42\\-168\\-126\\-84
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-14\\-126\\-140\\-126
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4949448, -10290, 2383836; -10290, 70, -4956; 2383836, -4956, 1148143],[42, 14, 1],[-5128, -87793, -114994, -113735, -32329, -24578, 2623, 1364; -10, -177, -233, -231, -66, -51, 5, 3; 10647, 182280, 238756, 236142, 67123, 51030, -5446, -2832],128,23]
%
% END OF LATTICE L129.23
%
\hbox{}\par\smallskip%
%
% LATTICE L129.24
%
\leavevmode%
\llap{\spacer}%
${L_{129.24}}=\pdualwitharg{7}(\main({L_{129.2}}))$%
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{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[34692, 16800, -420; 16800, 8134, -203; -420, -203, 5],[84, 7, 1],[-1, 11, 16, 16, 9, 1, -4, -4; 2, -27, -39, -39, -22, -3, 9, 9; 0, -168, -238, -240, -140, -42, 28, 30],128,24]
%
% END OF LATTICE L129.24
%
\hbox{}\par\smallskip%
%
% LATTICE L129.25
%
\leavevmode%
\llap{\spacer}%
${L_{129.25}}=\pdualwitharg{2}(\main({L_{129.6}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-8102472\\402192\\-13776
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
402192\\-19964\\684
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13776\\684\\-23
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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-128\\-2541\\1092
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13\\258\\-114
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8102472, 402192, -13776; 402192, -19964, 684; -13776, 684, -23],[168, 4, 1],[-11, -31, -13, -128, -1, 13, 8, 1; -218, -615, -258, -2541, -20, 258, 159, 21; 104, 276, 113, 1092, 4, -114, -64, 21],128,25]
%
% END OF LATTICE L129.25
%
\hbox{}\par\smallskip%
%
% LATTICE L129.26
%
\leavevmode%
\llap{\spacer}%
${L_{129.26}}=\pdualwitharg{2}(\main({L_{129.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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291144\\504\\-2352
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2352\\-4\\19
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1\\-2\\-124
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-252
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\HardButStrongLineBreak\kern3pt%
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-1\\-10\\-130
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-63\\-168
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\HardButStrongLineBreak\kern3pt%
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1\\-4\\120
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\HardButStrongLineBreak\kern3pt%
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1\\0\\123
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1\\1\\124
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1\\0\\126
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\-18\\-84\\0\\3\\0\\-42
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[291144, 504, -2352; 504, -4, -4; -2352, -4, 19],[168, 4, 1],[-1, -2, -1, -1, 1, 1, 1, 1; -2, -9, -10, -63, -4, 0, 1, 0; -124, -252, -130, -168, 120, 123, 124, 126],128,26]
%
% END OF LATTICE L129.26
%
\hbox{}\par\smallskip%
%
% LATTICE L129.27
%
\leavevmode%
\llap{\spacer}%
${L_{129.27}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{129.1}}))$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4143846\\11676\\1441398
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\kern3pt%
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11676\\105\\4011
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1441398\\4011\\501412
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-7\\0\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-42\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-63\\-210\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-63\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4143846, 11676, 1441398; 11676, 105, 4011; 1441398, 4011, 501412],[42, 21, 1],[-8301, -23686, -93074, -30685, -52333, -6631, 2123, 368; 16582, 47313, 185915, 61293, 104534, 13245, -4241, -735; 23730, 67711, 266070, 87719, 149604, 18956, -6069, -1052],128,27]
%
% END OF LATTICE L129.27
%
\hbox{}\par\smallskip%
%
% LATTICE L129.28
%
\leavevmode%
\llap{\spacer}%
${L_{129.28}}=\pdualwitharg{3}(\main({L_{129.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-1848\\672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-1302\\486
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
672\\486\\-181
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\91\\266
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\44\\129
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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9\\128\\376
\end{tabular}\endgroup%
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11\\153\\450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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37\\504\\1484
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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12\\0\\-48\\-84\\-18\\-24\\-12\\0
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0\\2\\0\\-14\\-6\\-16\\-18\\-56
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-18\\-6\\-12\\0\\3\\0\\-12\\-84
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\kern3pt%
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-24\\-16\\-72\\-56\\0\\8\\0\\-56
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\kern3pt%
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-12\\-18\\-120\\-126\\-12\\0\\6\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-504\\-616\\-84\\-56\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1848, -1848, 672; -1848, -1302, 486; 672, 486, -181],[168, 6, 1],[1, -1, -1, 6, 3, 9, 11, 37; 10, -15, -12, 91, 44, 128, 153, 504; 30, -44, -36, 266, 129, 376, 450, 1484],128,28]
%
% END OF LATTICE L129.28
%
\hbox{}\par\smallskip%
%
% LATTICE L129.29
%
\leavevmode%
\llap{\spacer}%
${L_{129.29}}=\pdualwitharg{3}(\main({L_{129.6}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-291144\\0\\2352
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2352\\0\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-1\\-4\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\616
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\123
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\HardButStrongLineBreak\kern3pt%
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1\\-3\\122
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-20\\360
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-21\\112
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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12\\0\\-24\\-168\\-18\\-12\\-24\\0
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\kern3pt%
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0\\8\\0\\-56\\-12\\-16\\-72\\-56
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-18\\-12\\-6\\0\\3\\0\\-24\\-42
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\kern3pt%
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-12\\-16\\-18\\-56\\0\\2\\0\\-14
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-126\\-616\\-42\\-14\\0\\14
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-291144, 0, 2352; 0, 6, 0; 2352, 0, -19],[168, 6, 1],[-1, -1, 0, 5, 1, 1, 3, 1; -4, 0, 1, 0, -1, -3, -20, -21; -126, -124, 0, 616, 123, 122, 360, 112],128,29]
%
% END OF LATTICE L129.29
%
\hbox{}\par\smallskip%
%
% LATTICE L129.30
%
\leavevmode%
\llap{\spacer}%
${L_{129.30}}=\pdualwitharg{2.3}(\main({L_{129.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
169176\\-15540\\42588
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15540\\1428\\-3912
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42588\\-3912\\10721
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4733\\2366\\19656
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
468\\-235\\-1944
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-1196\\598\\4967
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-59\\32\\246
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-51\\26\\212
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-87\\40\\360
\end{tabular}\endgroup%
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-109\\42\\448
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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12\\0\\-48\\-168
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0\\8\\0\\-56
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-168\\-56\\0\\56
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-24\\-24\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-32\\-72\\-112
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-72\\-240\\-504
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0\\-112\\-504\\-1232
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[169176, -15540, 42588; -15540, 1428, -3912; 42588, -3912, 10721],[84, 12, 1],[-59, -51, -87, -109, 7, -1, -69, -255; 32, 26, 40, 42, -6, 0, 38, 140; 246, 212, 360, 448, -30, 4, 288, 1064],128,30]
%
% END OF LATTICE L129.30
%
\hbox{}\par\smallskip%
%
% LATTICE L129.31
%
\leavevmode%
\llap{\spacer}%
${L_{129.31}}=\pdualwitharg{7}({L_{129.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69384\\16800\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16800\\4067\\-203
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-203\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\240\\840
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\49\\175
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-54\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-39\\-119
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-39\\-120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-56\\-84\\-168\\-168\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-42\\-168\\-336\\-126\\-84
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-84\\-42\\0\\3\\0\\-84\\-63\\-66
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-56\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-42
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-42\\0\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-336\\-126\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-126\\-70\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-63\\-66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[69384, 16800, -840; 16800, 4067, -203; -840, -203, 10],[168, 7, 1],[-1, 11, 8, 8, 9, 1, -2, -2; 4, -54, -39, -39, -44, -6, 9, 9; 0, -168, -119, -120, -140, -42, 14, 15],128,31]
%
% END OF LATTICE L129.31
%
\hbox{}\par\smallskip%
%
% LATTICE L129.32
%
\leavevmode%
\llap{\spacer}%
${L_{129.32}}=\pdualwitharg{7}({L_{129.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10248\\1176\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1176\\-119\\14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\14\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-72\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-74\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-30\\-93
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\15\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\18\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-112\\-84\\-168\\-84\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-42\\-168\\-168\\-126\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\28\\0\\-56\\-126\\-140\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-42\\0\\3\\0\\-42\\-63\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-56\\0\\56\\0\\-56\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-168\\-126\\-42\\0\\21\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-126\\-140\\-63\\-56\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-252\\-132\\-168\\-42\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10248, 1176, -168; 1176, -119, 14; -168, 14, -1],[168, 7, 1],[-1, -5, -5, -2, -1, 1, 1, 1; -12, -72, -74, -30, -16, 15, 16, 18; -28, -210, -224, -93, -56, 42, 49, 60],128,32]
%
% END OF LATTICE L129.32
%
\hbox{}\par\smallskip%
%
% LATTICE L129.33
%
\leavevmode%
\llap{\spacer}%
${L_{129.33}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{129.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9996\\5040\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\-2534\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\168\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-17\\-147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-42\\-324
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-17\\-112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-12\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-28\\-84\\-42\\-42\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-84\\-84\\-168\\-252\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\7\\0\\-14\\-63\\-140\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\0\\12\\0\\-84\\-252\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-84\\-14\\0\\14\\0\\-56\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-168\\-63\\-84\\0\\21\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-252\\-140\\-252\\-56\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-63\\-132\\-42\\-21\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9996, 5040, -336; 5040, -2534, 168; -336, 168, -11],[84, 14, 1],[2, -1, -4, -11, -5, -4, -1, 1; 5, -12, -17, -42, -17, -12, -2, 3; 14, -168, -147, -324, -112, -63, 0, 15],128,33]
%
% END OF LATTICE L129.33
%
\hbox{}\par\smallskip%
%
% LATTICE L129.34
%
\leavevmode%
\llap{\spacer}%
${L_{129.34}}=\pdualwitharg{7}({L_{129.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2856\\840\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\245\\-49
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-49\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143\\-432\\504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\-115\\133
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\24\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\3\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\22\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\30\\-54
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-112\\-168\\-168\\-84\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-42\\-84\\-84\\-126\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\28\\0\\-56\\-126\\-280\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-42\\0\\12\\0\\-84\\-252\\-264
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-112\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-42\\0\\12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-84\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\-126\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-126\\-280\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-252\\-264
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2856, 840, -168; 840, 245, -49; -168, -49, 10],[168, 7, 1],[1, -2, -9, -11, -9, -4, -3, -1; -4, 3, 22, 30, 28, 15, 14, 6; 0, -21, -56, -54, -28, 0, 14, 12],128,34]
%
% END OF LATTICE L129.34
%
\hbox{}\par\smallskip%
%
% LATTICE L129.35
%
\leavevmode%
\llap{\spacer}%
${L_{129.35}}=\pdualwitharg{2}({L_{129.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24864\\168\\-168
\end{tabular}\endgroup%
\kern3pt%
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-168\\0\\1
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-162
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\21\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-33\\-336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-49\\-496
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-357\\-3612
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\0\\-16\\-168\\-6\\-24\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\24\\0\\-168\\-12\\-96\\-72\\-336
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\8\\0\\-4\\-72\\-80\\-504
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\0\\168\\0\\-336\\-504\\-3696
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-12\\-4\\0\\1\\0\\-8\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-96\\-72\\-336\\0\\24\\0\\-168
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-504\\-3696\\-84\\-168\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24864, 168, -168; 168, -8, 0; -168, 0, 1],[168, 8, 1],[-1, -1, 0, 1, 0, -2, -3, -22; -16, -15, 1, 21, 0, -33, -49, -357; -162, -156, 4, 168, -1, -336, -496, -3612],128,35]
%
% END OF LATTICE L129.35
%
\hbox{}\par\smallskip%
%
% LATTICE L129.36
%
\leavevmode%
\llap{\spacer}%
${L_{129.36}}=\pdualwitharg{2}({L_{129.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-22\\-63\\-336
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-48
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\6\\31
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\-2\\-14
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-12
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-12\\-24\\-8\\0
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-24\\-96\\-72\\-336
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[168, 0, 0; 0, 8, 0; 0, 0, -1],[168, 8, 1],[-1, -1, 0, 1, 0, -2, -3, -22; -2, 0, 1, 0, -1, -9, -10, -63; -14, -12, 0, 0, -2, -36, -48, -336],128,36]
%
% END OF LATTICE L129.36
%
\hbox{}\par\smallskip%
%
% LATTICE L129.37
%
\leavevmode%
\llap{\spacer}%
${L_{129.37}}=\pdualwitharg{2}({L_{129.3}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6888\\840\\504
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\112\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\48\\19
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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377\\-17199\\33264
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4\\-182\\351
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\-45\\86
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-1\\47\\-92
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1\\-42\\84
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\24\\0\\-168
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-12\\-24\\-8\\0
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-24\\-96\\-72\\-336
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-8\\-72\\-80\\-504
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6888, 840, 504; 840, 112, 48; 504, 48, 19],[168, 8, 1],[1, -1, -1, 1, 1, 7, 7, 41; -45, 48, 47, -42, -46, -321, -320, -1869; 86, -96, -92, 84, 90, 624, 620, 3612],128,37]
%
% END OF LATTICE L129.37
%
\hbox{}\par\smallskip%
%
% LATTICE L129.38
%
\leavevmode%
\llap{\spacer}%
${L_{129.38}}=\pdualwitharg{2.3.7}(\pfillwitharg{2}({L_{129.1}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145916148\\354875010\\172466154
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
354875010\\-863072826\\-419445852
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
172466154\\-419445852\\-203847019
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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3936269\\-13123160832\\27006203448
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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21\\0\\-84\\-42
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0\\14\\0\\-14
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-84\\0\\42\\0
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-42\\-14\\0\\2
\end{tabular}\endgroup%
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-63\\-42\\-42\\0
\end{tabular}\endgroup%
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-42\\-56\\-126\\-28
\end{tabular}\endgroup%
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-42\\-126\\-420\\-126
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-145916148, 354875010, 172466154; 354875010, -863072826, -419445852; 172466154, -419445852, -203847019],[42, 42, 1],[-8, -50, -199, -66, -57, -15, 4, 1; 32788, 172553, 668787, 218979, 183916, 44151, -18675, -2275; -67473, -355096, -1376298, -450638, -378483, -90860, 38430, 4682],128,38]
%
% END OF LATTICE L129.38
%
\hbox{}\par\smallskip%
%
% LATTICE L129.39
%
\leavevmode%
\llap{\spacer}%
${L_{129.39}}=\pdualwitharg{3.7}(\main({L_{129.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31164\\14532\\-588
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14532\\6762\\-273
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-588\\-273\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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7\\-24\\-168
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2\\-7\\-42
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0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\2\\0
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1\\-5\\-70
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2\\-11\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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0\\-1\\-26
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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84\\0\\-168\\-84\\-252\\-84\\-84\\0
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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84\\0\\-168\\-84
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-14
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\42\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-14\\0\\2
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-84\\-84\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-56\\-126\\-28
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-84\\-126\\-420\\-126
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0\\-28\\-126\\-44
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31164, 14532, -588; 14532, 6762, -273; -588, -273, 11],[84, 21, 1],[-1, 1, 2, 0, -3, -3, -8, -2; 2, -5, -11, -1, 10, 11, 29, 7; 0, -70, -168, -26, 84, 112, 294, 68],128,39]
%
% END OF LATTICE L129.39
%
\hbox{}\par\smallskip%
%
% LATTICE L129.40
%
\leavevmode%
\llap{\spacer}%
${L_{129.40}}=\pdualwitharg{3}({L_{129.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1330224\\7392\\7392
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7392\\-30\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7392\\-42\\-41
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\41\\498
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\120\\1496
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\144\\1830
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\476\\6160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\168
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\-166
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\42\\497
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-24\\-168\\-36\\-12\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-112\\-48\\-32\\-144\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\12\\0\\-24\\-36\\-240\\-126
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-48\\-24\\0\\24\\0\\-96\\-84
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-32\\-36\\-112\\0\\4\\0\\-14
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\kern3pt%
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-24\\-144\\-240\\-1008\\-96\\0\\48\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-126\\-616\\-84\\-14\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1330224, 7392, 7392; 7392, -30, -42; 7392, -42, -41],[336, 6, 1],[3, 9, 11, 37, 1, -1, -1, 3; 41, 120, 144, 476, 10, -14, -12, 42; 498, 1496, 1830, 6160, 168, -166, -168, 497],128,40]
%
% END OF LATTICE L129.40
%
\hbox{}\par\smallskip%
%
% LATTICE L129.41
%
\leavevmode%
\llap{\spacer}%
${L_{129.41}}=\pdualwitharg{2.3}(\main({L_{129.6}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-474600\\66864\\-6048
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66864\\-9420\\852
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6048\\852\\-77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\52\\24
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\31\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\32\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\91\\140
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-10
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\-24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\35\\-7
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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24\\0\\-24\\-168\\-72\\-24\\-24\\0
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\kern3pt%
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0\\4\\0\\-28\\-24\\-16\\-36\\-28
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-24\\0\\3\\0\\-12\\-18\\-60\\-63
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\kern3pt%
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-168\\-28\\0\\28\\0\\-56\\-252\\-308
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-24\\-12\\0\\24\\0\\-48\\-84
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-16\\-18\\-56\\0\\4\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-36\\-60\\-252\\-48\\0\\12\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-63\\-308\\-84\\-14\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-474600, 66864, -6048; 66864, -9420, 852; -6048, 852, -77],[168, 12, 1],[7, 4, 4, 11, -1, -1, 1, 5; 52, 31, 32, 91, -6, -8, 5, 35; 24, 28, 39, 140, 12, -10, -24, -7],128,41]
%
% END OF LATTICE L129.41
%
\hbox{}\par\smallskip%
%
% LATTICE L129.42
%
\leavevmode%
\llap{\spacer}%
${L_{129.42}}=\pdualwitharg{3}({L_{129.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3696\\336\\1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-30\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\-84\\-167
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\83\\-6
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\232\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\135\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\868\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\182\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-12\\-168\\-36\\-6\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-112\\-48\\-16\\-144\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\0\\-12\\-9\\-120\\-126
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-112\\0\\112\\0\\-56\\-1008\\-1232
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-48\\-12\\0\\24\\0\\-96\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-16\\-9\\-56\\0\\1\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-144\\-120\\-1008\\-96\\0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-126\\-1232\\-168\\-14\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3696, 336, 1008; 336, -30, -84; 1008, -84, -167],[336, 6, 1],[6, 17, 10, 65, 1, -1, -1, 13; 83, 232, 135, 868, 10, -14, -12, 182; -6, -16, -9, -56, 0, 1, 0, -14],128,42]
%
% END OF LATTICE L129.42
%
\hbox{}\par\smallskip%
%
% LATTICE L129.43
%
\leavevmode%
\llap{\spacer}%
${L_{129.43}}=\pdualwitharg{2.3}(\main({L_{129.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-10\\-102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-21\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-48\\-168\\-72\\-12\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-28\\-24\\-8\\-36\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\12\\0\\-24\\-18\\-120\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-28\\0\\28\\0\\-28\\-252\\-616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-24\\-24\\0\\24\\0\\-48\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-8\\-18\\-28\\0\\1\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-36\\-120\\-252\\-48\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-252\\-616\\-168\\-14\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-168, 168, 0; 168, -12, -12; 0, -12, 1],[168, 12, 1],[-1, -2, -7, -15, -3, 0, 1, 1; -2, -3, -10, -21, -4, 0, 1, 0; -12, -28, -102, -224, -48, -1, 12, 14],128,43]
%
% END OF LATTICE L129.43
%
\hbox{}\par\smallskip%
%
% LATTICE L129.44
%
\leavevmode%
\llap{\spacer}%
${L_{129.44}}=\pdualwitharg{3}({L_{129.6}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1287888\\-35952\\14448
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35952\\-1002\\402
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14448\\402\\-161
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-218\\-276
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\68\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\72\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-336\\-392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-139\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-141\\-173
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-1632\\-2016
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-784\\-973
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-48\\-336\\-36\\-12\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-112\\-24\\-16\\-144\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\12\\0\\-12\\-18\\-240\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-112\\0\\112\\0\\-56\\-1008\\-616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-24\\-12\\0\\6\\0\\-48\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-16\\-18\\-56\\0\\1\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-144\\-240\\-1008\\-48\\0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-126\\-616\\-42\\-7\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1287888, -35952, 14448; -35952, -1002, 402; 14448, 402, -161],[336, 6, 1],[3, -1, -1, 5, 2, 2, 23, 11; -218, 68, 72, -336, -139, -141, -1632, -784; -276, 80, 90, -392, -168, -173, -2016, -973],128,44]
%
% END OF LATTICE L129.44
%
\hbox{}\par\smallskip%
%
% LATTICE L129.45
%
\leavevmode%
\llap{\spacer}%
${L_{129.45}}=\pdualwitharg{3}({L_{129.7}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{112}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2286480\\0\\13776
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13776\\0\\-83
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-6\\828
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-4\\1160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\663
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\4144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-166
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-14\\826
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-336\\-36\\-24\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-112\\-24\\-32\\-144\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\3\\0\\-6\\-18\\-120\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-112\\0\\112\\0\\-112\\-1008\\-1232
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-24\\-6\\0\\6\\0\\-48\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-32\\-18\\-112\\0\\4\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-144\\-120\\-1008\\-48\\0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-126\\-1232\\-84\\-28\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2286480, 0, 13776; 0, 6, 0; 13776, 0, -83],[336, 6, 1],[5, 7, 4, 25, 0, -1, -1, 5; -6, -4, -1, 0, 1, 0, -8, -14; 828, 1160, 663, 4144, 0, -166, -168, 826],128,45]
%
% END OF LATTICE L129.45
%
\hbox{}\par\smallskip%
%
% LATTICE L129.46
%
\leavevmode%
\llap{\spacer}%
${L_{129.46}}=\pdualwitharg{7}(\main({L_{129.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{l}{2}{42}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34440\\5880\\-672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5880\\938\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672\\-112\\13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-20\\-434
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-15\\-336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-19\\-378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-48\\-996
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-112\\-84\\-42\\-168\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-42\\-42\\-336\\-126\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\56\\0\\-28\\-504\\-280\\-504
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\kern3pt%
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-84\\-42\\0\\6\\0\\-168\\-126\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-28\\0\\7\\0\\-28\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-336\\-504\\-168\\0\\168\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-126\\-280\\-126\\-28\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-504\\-264\\-84\\-168\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[34440, 5880, -672; 5880, 938, -112; -672, -112, 13],[168, 14, 1],[-5, -4, -1, 1, 1, -1, -4, -11; -20, -15, -4, 3, 3, -12, -19, -48; -434, -336, -84, 78, 77, -168, -378, -996],128,46]
%
% END OF LATTICE L129.46
%
\hbox{}\par\smallskip%
%
% LATTICE L129.47
%
\leavevmode%
\llap{\spacer}%
${L_{129.47}}=\pdualwitharg{7}(\main({L_{129.6}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41496\\0\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\0\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-4\\-350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-9\\-546
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-20\\-1344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-9\\-648
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-56\\-168\\-42\\-84\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-168\\-84\\-336\\-504\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\14\\0\\-14\\-126\\-280\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\0\\24\\0\\-168\\-504\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-84\\-14\\0\\7\\0\\-56\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-336\\-126\\-168\\0\\42\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-504\\-280\\-504\\-56\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-126\\-264\\-42\\-42\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[41496, 0, 840; 0, 14, 0; 840, 0, 17],[168, 14, 1],[7, 5, 0, 1, 1, 11, 27, 13; -4, 0, 1, 0, -1, -9, -20, -9; -350, -252, 0, -48, -49, -546, -1344, -648],128,47]
%
% END OF LATTICE L129.47
%
\hbox{}\par\smallskip%
%
% LATTICE L129.48
%
\leavevmode%
\llap{\spacer}%
${L_{129.48}}=\pdualwitharg{2.7}(\main({L_{129.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
453012\\21504\\-93828
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21504\\980\\-4480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93828\\-4480\\19417
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5759\\-15840\\24192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\-551\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1240\\3410\\-5209
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-123\\331\\-518
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-319\\855\\-1344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\83\\-140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
157\\-429\\660
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-112\\-168\\-84\\-168\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-168\\-168\\-672\\-504\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\56\\0\\-56\\-504\\-560\\-504
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\0\\24\\0\\-336\\-504\\-528
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-112\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\0\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-168\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-672\\-504\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-504\\-560\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-504\\-528
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[453012, 21504, -93828; 21504, 980, -4480; -93828, -4480, 19417],[84, 28, 1],[-123, -319, -33, 157, 163, 439, 153, -37; 331, 855, 83, -429, -441, -1185, -413, 99; -518, -1344, -140, 660, 686, 1848, 644, -156],128,48]
%
% END OF LATTICE L129.48
%
\hbox{}\par\smallskip%
%
% LATTICE L129.49
%
\leavevmode%
\llap{\spacer}%
${L_{129.49}}=\pdualwitharg{2}({L_{129.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{336}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1102416\\-21168\\5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21168\\-392\\96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\96\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\13\\272
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\72\\1392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\15\\281
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\294\\5376
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-296
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-8\\-336\\-24\\-24\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-336\\-48\\-96\\-144\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\1\\0\\-4\\-18\\-40\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-336\\0\\336\\0\\-336\\-1008\\-3696
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\-4\\0\\8\\0\\-32\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-96\\-18\\-336\\0\\12\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-144\\-40\\-1008\\-32\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-126\\-3696\\-168\\-84\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1102416, -21168, 5040; -21168, -392, 96; 5040, 96, -23],[336, 8, 1],[1, 5, 1, 19, 0, -1, -1, 1; 13, 72, 15, 294, 1, -15, -18, 0; 272, 1392, 281, 5376, 4, -282, -296, 210],128,49]
%
% END OF LATTICE L129.49
%
\hbox{}\par\smallskip%
%
% LATTICE L129.50
%
\leavevmode%
\llap{\spacer}%
${L_{129.50}}=\pdualwitharg{2}({L_{129.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[133695408, -6663552, 50400; -6663552, 332120, -2512; 50400, -2512, 19],[336, 8, 1],[3, 1, 1, 1, 0, 5, 4, 61; 61, 21, 22, 21, -1, 96, 79, 1218; 104, 123, 256, 126, -132, -576, -170, -840],128,50]
%
% END OF LATTICE L129.50
%
\hbox{}\par\smallskip%
%
% LATTICE L129.51
%
\leavevmode%
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${L_{129.51}}=\pdualwitharg{2}({L_{129.4}})$%
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{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{l}{2}{336}\above{1pt}{1pt}{}{2}$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-48149808, 2427264, 48720; 2427264, -122360, -2456; 48720, -2456, -49],[336, 8, 1],[8, 13, -1, -73, -19, -59, -16, -85; 159, 258, -20, -1449, -377, -1170, -317, -1680; -16, -6, 8, 42, 4, -24, -22, -336],128,51]
%
% END OF LATTICE L129.51
%
\hbox{}\par\smallskip%
%
% LATTICE L129.52
%
\leavevmode%
\llap{\spacer}%
${L_{129.52}}=\pdualwitharg{2}({L_{129.7}})$%
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{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-83119344, 4155984, -30240; 4155984, -207800, 1512; -30240, 1512, -11],[336, 8, 1],[2, 11, 5, 55, 1, -1, -1, 2; 41, 222, 100, 1092, 19, -21, -20, 42; 136, 264, -6, -1176, -140, -138, 0, 273],128,52]
%
% END OF LATTICE L129.52
%
\hbox{}\par\smallskip%
%
% LATTICE L129.53
%
\leavevmode%
\llap{\spacer}%
${L_{129.53}}=\pdualwitharg{3.7}({L_{129.2}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-112\\-126\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-126\\-210\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-63\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[247128, 54936, -3024; 54936, 12201, -672; -3024, -672, 37],[168, 21, 1],[3, 3, 4, 1, 1, -1, -1, 0; -4, 2, 4, 1, 0, -4, -7, -2; 168, 280, 399, 100, 84, -154, -210, -37],128,53]
%
% END OF LATTICE L129.53
%
\hbox{}\par\smallskip%
%
% LATTICE L129.54
%
\leavevmode%
\llap{\spacer}%
${L_{129.54}}=\pdualwitharg{3.7}({L_{129.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12264\\11424\\-672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11424\\10731\\-630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672\\-630\\37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-24\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-4\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-154
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-34\\-672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-121
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-168\\-168\\-504\\-168\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-14\\-84\\-56\\-126\\-14
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\21\\0\\-84\\-126\\-420\\-63
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-14\\0\\4\\0\\-56\\-252\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-84\\-84\\0\\168\\0\\-336\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-56\\-126\\-56\\0\\28\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-126\\-420\\-252\\-336\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-63\\-44\\-84\\-14\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12264, 11424, -672; 11424, 10731, -630; -672, -630, 37],[168, 21, 1],[-5, -1, -2, -1, -1, -1, -5, -1; -24, -3, -4, -2, -4, -8, -34, -6; -504, -70, -105, -52, -84, -154, -672, -121],128,54]
%
% END OF LATTICE L129.54
%
\hbox{}\par\smallskip%
%
% LATTICE L129.55
%
\leavevmode%
\llap{\spacer}%
${L_{129.55}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{129.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5082\\294
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\294\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-3\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-76
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-168\\-42\\-126\\-84\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-7\\-42\\-56\\-63\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\84\\0\\-84\\-252\\-420\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-7\\0\\1\\0\\-28\\-63\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-42\\-84\\0\\42\\0\\-84\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-56\\-252\\-28\\0\\28\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-63\\-420\\-63\\-84\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-252\\-44\\-84\\-28\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[84, 0, 0; 0, -5082, 294; 0, 294, -17],[84, 42, 1],[-2, 0, 1, 0, -1, -3, -5, -3; -7, -2, 0, 1, 5, 2, -3, -4; -126, -35, 0, 17, 84, 28, -63, -76],128,55]
%
% END OF LATTICE L129.55
%
\hbox{}\par\smallskip%
%
% LATTICE L129.56
%
\leavevmode%
\llap{\spacer}%
${L_{129.56}}=\pdualwitharg{3.7}({L_{129.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21000\\5376\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5376\\-1365\\84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\84\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\80\\1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-399
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\28\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-336\\-168\\-504\\-84\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-14\\-84\\-28\\-126\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-14\\0\\4\\0\\-28\\-252\\-88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-336\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-14\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-84\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-28\\-126\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-126\\-840\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-252\\-88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21000, 5376, -336; 5376, -1365, 84; -336, 84, -5],[168, 21, 1],[1, 1, 5, 1, -1, -1, -5, -1; 0, 5, 28, 6, -4, -6, -34, -8; -84, 14, 126, 32, 0, -35, -252, -74],128,56]
%
% END OF LATTICE L129.56
%
\hbox{}\par\smallskip%
%
% LATTICE L129.57
%
\leavevmode%
\llap{\spacer}%
${L_{129.57}}=\pdualwitharg{2.3}({L_{129.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19488\\-6552\\504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6552\\-2136\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\168\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\7\\280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\51
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-7\\-28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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12\\0\\-48\\-168\\-18\\-24\\-24\\0
\end{tabular}\endgroup%
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0\\8\\0\\-56\\-12\\-32\\-72\\-112
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\24\\0\\-12\\-72\\-240\\-504
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-32\\-72\\-112\\0\\8\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-504\\-1232\\-84\\-56\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19488, -6552, 504; -6552, -2136, 168; 504, 168, -13],[168, 24, 1],[-1, -1, 0, 5, 1, 2, 3, 2; -2, -1, 1, 7, 1, 1, -1, -7; -66, -52, 12, 280, 51, 88, 96, -28],128,57]
%
% END OF LATTICE L129.57
%
\hbox{}\par\smallskip%
%
% LATTICE L129.58
%
\leavevmode%
\llap{\spacer}%
${L_{129.58}}=\pdualwitharg{2.3}({L_{129.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-291144\\0\\2352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2352\\0\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
370\\-1113\\44520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-10\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\9\\-361
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\616
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-168\\-36\\-24\\-24\\0
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\kern3pt%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-168
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-56
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\24\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-56\\0\\56
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-32\\-72\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-72\\-240\\-504
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0\\-112\\-504\\-1232
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-291144, 0, 2352; 0, 24, 0; 2352, 0, -19],[168, 24, 1],[-1, -1, 0, 5, 2, 2, 3, 2; -2, 0, 1, 0, -1, -3, -10, -21; -126, -124, 0, 616, 246, 244, 360, 224],128,58]
%
% END OF LATTICE L129.58
%
\hbox{}\par\smallskip%
%
% LATTICE L129.59
%
\leavevmode%
\llap{\spacer}%
${L_{129.59}}=\pdualwitharg{2.3}({L_{129.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30072\\-2856\\-672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2856\\-240\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672\\-48\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
475\\-8925\\17136
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\-1051\\2016
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-300\\575
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-17\\30
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-1\\20\\-40
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\19\\-36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-98\\196
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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12\\0\\-48\\-168\\-36\\-24\\-24\\0
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\kern3pt%
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0\\8\\0\\-56\\-24\\-32\\-72\\-112
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-168
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-56
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-48\\0\\24\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-56\\0\\56
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-32\\-72\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-72\\-240\\-504
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-504\\-1232
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30072, -2856, -672; -2856, -240, -48; -672, -48, -7],[168, 24, 1],[1, -1, -1, 5, 3, 5, 13, 23; -17, 20, 19, -98, -58, -95, -244, -427; 30, -40, -36, 196, 114, 184, 468, 812],128,59]
%
% END OF LATTICE L129.59
%
\hbox{}\par\smallskip%
%
% LATTICE L129.60
%
\leavevmode%
\llap{\spacer}%
${L_{129.60}}=\pdualwitharg{7}({L_{129.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{r}{2}{336}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-995568\\-52080\\5376
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52080\\-2674\\280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5376\\280\\-29
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-34\\-1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-30\\-1218
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\243
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\11\\476
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\24\\1512
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-60\\-1896
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-224\\-84\\-84\\-336\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-42\\-84\\-672\\-252\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224\\0\\112\\0\\-56\\-1008\\-560\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-42\\0\\3\\0\\-168\\-126\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\-56\\0\\14\\0\\-56\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-672\\-1008\\-168\\0\\336\\0\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-252\\-560\\-126\\-56\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-1008\\-264\\-168\\-336\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-995568, -52080, 5376; -52080, -2674, 280; 5376, 280, -29],[336, 14, 1],[-5, -5, -1, 1, 2, 7, -1, -7; -34, -30, -4, 6, 11, 24, -16, -60; -1260, -1218, -224, 243, 476, 1512, -350, -1896],128,60]
%
% END OF LATTICE L129.60
%
\hbox{}\par\smallskip%
%
% LATTICE L129.61
%
\leavevmode%
\llap{\spacer}%
${L_{129.61}}=\pdualwitharg{2.7}(\main({L_{129.6}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27384\\-336\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\-364
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-177
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\161
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3\\312
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-112\\-84\\-168\\-168\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-42\\-168\\-336\\-126\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\28\\0\\-56\\-252\\-140\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-42\\0\\3\\0\\-84\\-63\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-56\\0\\56\\0\\-56\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-336\\-252\\-84\\0\\84\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-126\\-140\\-63\\-56\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-252\\-132\\-168\\-84\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-27384, -336, 168; -336, 140, 0; 168, 0, -1],[168, 28, 1],[1, -1, -2, -1, -1, 1, 1, 2; 0, -6, -7, -3, -2, 3, 2, 3; 140, -210, -364, -177, -168, 168, 161, 312],128,61]
%
% END OF LATTICE L129.61
%
\hbox{}\par\smallskip%
%
% LATTICE L129.62
%
\leavevmode%
\llap{\spacer}%
${L_{129.62}}=\pdualwitharg{7}({L_{129.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{}{2}{336}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-830256\\-4032\\3024
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4032\\14\\14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3024\\14\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-34\\-1428
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-12\\-567
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\-336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-17\\-581
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-84\\-3168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-224\\-168\\-84\\-336\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-42\\-42\\-336\\-63\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224\\0\\112\\0\\-56\\-1008\\-280\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-42\\0\\12\\0\\-336\\-126\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-42\\-56\\0\\14\\0\\-28\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-336\\-1008\\-336\\0\\336\\0\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-63\\-280\\-126\\-28\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-1008\\-528\\-168\\-336\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-830256, -4032, 3024; -4032, 14, 14; 3024, 14, -11],[336, 14, 1],[-5, -2, -1, 1, 1, -1, -2, -11; -34, -12, -4, 6, 5, -24, -17, -84; -1428, -567, -280, 282, 280, -336, -581, -3168],128,62]
%
% END OF LATTICE L129.62
%
\hbox{}\par\smallskip%
%
% LATTICE L129.63
%
\leavevmode%
\llap{\spacer}%
${L_{129.63}}=\pdualwitharg{7}({L_{129.6}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{112}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22512\\1008\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\-14\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-14\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-12\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-112\\-42\\-84\\-168\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-21\\-84\\-336\\-126\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\112\\0\\-112\\-1008\\-560\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-21\\0\\3\\0\\-168\\-126\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\-112\\0\\56\\0\\-112\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-336\\-1008\\-168\\0\\336\\0\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-126\\-560\\-126\\-112\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-1008\\-264\\-336\\-336\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22512, 1008, -336; 1008, -14, -14; -336, -14, 5],[336, 14, 1],[1, 1, -1, -1, -3, -7, -1, 1; 1, 0, -8, -3, -6, -12, -2, 0; 70, 63, -112, -81, -224, -504, -70, 72],128,63]
%
% END OF LATTICE L129.63
%
\hbox{}\par\smallskip%
%
% LATTICE L129.64
%
\leavevmode%
\llap{\spacer}%
${L_{129.64}}=\pdualwitharg{7}({L_{129.7}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{s}{2}{336}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84290640\\0\\759696
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
759696\\0\\-6847
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\0\\7434
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
161\\-8\\17864
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\-6\\7878
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-6\\2996
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\-12\\-5880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-1\\-2219
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\0\\-6768
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-112\\-84\\-84\\-168\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-84\\-168\\-672\\-126\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\112\\0\\-112\\-1008\\-280\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\0\\12\\0\\-336\\-126\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-168\\-112\\0\\56\\0\\-56\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-672\\-1008\\-336\\0\\336\\0\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-126\\-280\\-126\\-56\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-1008\\-528\\-336\\-336\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-84290640, 0, 759696; 0, 14, 0; 759696, 0, -6847],[336, 14, 1],[0, 67, 161, 71, 27, -53, -20, -61; 1, 0, -8, -6, -6, -12, -1, 0; 0, 7434, 17864, 7878, 2996, -5880, -2219, -6768],128,64]
%
% END OF LATTICE L129.64
%
\hbox{}\par\smallskip%
%
% LATTICE L129.65
%
\leavevmode%
\llap{\spacer}%
${L_{129.65}}=\pdualwitharg{2.7}(\main({L_{129.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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\kern3pt%
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0\\-84\\-252\\-264\\-168\\-84\\0\\12
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1176, -504, -168; -504, 28, 28; -168, 28, 17],[168, 28, 1],[-1, 1, 2, 1, -1, -5, -5, -4; 22, -9, -23, -12, 12, 69, 74, 63; -56, 21, 56, 30, -28, -168, -182, -156],128,65]
%
% END OF LATTICE L129.65
%
\hbox{}\par\smallskip%
%
% LATTICE L129.66
%
\leavevmode%
\llap{\spacer}%
${L_{129.66}}=\pdualwitharg{3.7}(\main({L_{129.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[35448, 59136, 3864; 59136, 91434, 5964; 3864, 5964, 389],[168, 42, 1],[-5, -2, -1, 1, 3, 5, 2, -1; 154, 57, 28, -27, -80, -120, -23, 48; -2310, -854, -420, 404, 1197, 1792, 336, -724],128,66]
%
% END OF LATTICE L129.66
%
\hbox{}\par\smallskip%
%
% LATTICE L129.67
%
\leavevmode%
\llap{\spacer}%
${L_{129.67}}=\pdualwitharg{3.7}(\main({L_{129.6}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{l}{2}{42}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-54264, 0, 840; 0, 42, 0; 840, 0, -13],[168, 42, 1],[-7, -3, 0, 1, 1, -1, -15, -3; -4, 0, 1, 0, -1, -3, -20, -3; -462, -196, 0, 64, 63, -70, -1008, -200],128,67]
%
% END OF LATTICE L129.67
%
\hbox{}\par\smallskip%
%
% LATTICE L129.68
%
\leavevmode%
\llap{\spacer}%
${L_{129.68}}=\pdualwitharg{2.3.7}(\main({L_{129.2}}))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2940, -39732, -9492; -39732, 520968, 124488; -9492, 124488, 29747],[84, 84, 1],[-24, -8, 26, 16, 26, 10, -20, -14; 229, 73, -279, -167, -269, -113, 159, 127; -966, -308, 1176, 704, 1134, 476, -672, -536],128,68]
%
% END OF LATTICE L129.68
%
\hbox{}\par\smallskip%
%
% LATTICE L129.69
%
\leavevmode%
\llap{\spacer}%
${L_{129.69}}=\pdualwitharg{2.3}({L_{129.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{112}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2117136\\-155232\\12768
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-155232\\-11352\\936
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12768\\936\\-77
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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5\\0\\826
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7\\6\\1232
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2117136, -155232, 12768; -155232, -11352, 936; 12768, 936, -77],[336, 24, 1],[0, -1, -1, 5, 5, 7, 4, 25; 1, -1, -4, 0, 3, 6, 4, 28; 12, -178, -216, 826, 864, 1232, 711, 4480],128,69]
%
% END OF LATTICE L129.69
%
\hbox{}\par\smallskip%
%
% LATTICE L129.70
%
\leavevmode%
\llap{\spacer}%
${L_{129.70}}=\pdualwitharg{2.3}({L_{129.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{b}{2}{112}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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104496\\-15120\\336
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\kern3pt%
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-15120\\2184\\-48
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-48\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\-1\\-36
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-1\\-10\\-128
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-2\\-19\\-222
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-9\\-84\\-952
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0\\0\\-1
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1\\7\\14
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[104496, -15120, 336; -15120, 2184, -48; 336, -48, 1],[336, 24, 1],[0, -1, -2, -9, -1, 0, 1, 1; -1, -10, -19, -84, -9, 0, 8, 7; -36, -128, -222, -952, -96, -1, 48, 14],128,70]
%
% END OF LATTICE L129.70
%
\hbox{}\par\smallskip%
%
% LATTICE L129.71
%
\leavevmode%
\llap{\spacer}%
${L_{129.71}}=\pdualwitharg{2.3}({L_{129.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{l}{2}{112}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-4096176\\581280\\-25200
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\kern3pt%
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581280\\-82488\\3576
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25200\\3576\\-155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6\\47\\108
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9\\68\\104
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37\\266\\112
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1\\5\\-48
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6\\49\\154
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4096176, 581280, -25200; 581280, -82488, 3576; -25200, 3576, -155],[336, 24, 1],[6, 9, 11, 37, 1, -1, -1, 6; 47, 68, 81, 266, 5, -8, -6, 49; 108, 104, 78, 112, -48, -22, 24, 154],128,71]
%
% END OF LATTICE L129.71
%
\hbox{}\par\smallskip%
%
% LATTICE L129.72
%
\leavevmode%
\llap{\spacer}%
${L_{129.72}}=\pdualwitharg{2.3}({L_{129.7}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9910320\\1651776\\-20160
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1651776\\-275304\\3360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20160\\3360\\-41
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\5\\-84
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3\\21\\245
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9\\60\\488
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11\\72\\486
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9910320, 1651776, -20160; 1651776, -275304, 3360; -20160, 3360, -41],[336, 24, 1],[1, -1, -1, 3, 6, 9, 11, 37; 5, -7, -6, 21, 41, 60, 72, 238; -84, -82, 0, 245, 408, 488, 486, 1288],128,72]
%
% END OF LATTICE L129.72
%
\hbox{}\par\smallskip%
%
% LATTICE L129.73
%
\leavevmode%
\llap{\spacer}%
${L_{129.73}}=\pdualwitharg{2.7}({L_{129.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3864\\840\\-336
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840\\224\\-112
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-336\\-112\\55
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-39\\-84
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\38\\84
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1\\33\\72
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0\\-3\\-7
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-5\\-228\\-504
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{$\left.\llap{\phantom{%
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0\\0\\0
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\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
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28\\0\\-112\\-168\\-42\\-168\\-56\\0
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3864, 840, -336; 840, 224, -112; -336, -112, 55],[168, 56, 1],[-2, -1, 1, 1, 0, -5, -7, -7; -83, -39, 38, 33, -3, -228, -305, -300; -182, -84, 84, 72, -7, -504, -672, -660],128,73]
%
% END OF LATTICE L129.73
%
\hbox{}\par\smallskip%
%
% LATTICE L129.74
%
\leavevmode%
\llap{\spacer}%
${L_{129.74}}=\pdualwitharg{2.7}({L_{129.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41496\\-41496\\-840
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41496\\41552\\840
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-840\\840\\17
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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611\\153\\22848
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-648\\-163\\-24192
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-12\\-3\\-449
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-5\\0\\-252
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1\\1\\0
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-1\\0\\-48
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[41496, -41496, -840; -41496, 41552, 840; -840, 840, 17],[168, 56, 1],[-9, -5, 1, -1, -3, -31, -37, -35; -2, 0, 1, 0, -1, -9, -10, -9; -350, -252, 0, -48, -98, -1092, -1344, -1296],128,74]
%
% END OF LATTICE L129.74
%
\hbox{}\par\smallskip%
%
% LATTICE L129.75
%
\leavevmode%
\llap{\spacer}%
${L_{129.75}}=\pdualwitharg{2.7}({L_{129.3}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-560\\13
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-1\\-1\\-28
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1\\0\\-12
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2184, -7224, 168; -7224, 24080, -560; 168, -560, 13],[168, 56, 1],[-5, -7, -1, 1, 1, -5, -11, -13; -5, -6, -1, 0, 0, -9, -14, -15; -154, -168, -28, -12, -14, -336, -476, -492],128,75]
%
% END OF LATTICE L129.75
%
\hbox{}\par\smallskip%
%
% LATTICE L129.76
%
\leavevmode%
\llap{\spacer}%
${L_{129.76}}=\pdualwitharg{3.7}({L_{129.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-691152\\-66864\\5712
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66864\\-6258\\546
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5712\\546\\-47
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-18\\-574
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\191
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\35\\1134
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\72\\2408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\56\\1974
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\12\\496
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-672\\-84\\-252\\-336\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-14\\-84\\-224\\-252\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672\\0\\336\\0\\-168\\-1008\\-1680\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-14\\0\\1\\0\\-56\\-126\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-84\\-168\\0\\42\\0\\-168\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-224\\-1008\\-56\\0\\112\\0\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-252\\-1680\\-126\\-168\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-1008\\-88\\-168\\-112\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-691152, -66864, 5712; -66864, -6258, 546; 5712, 546, -47],[336, 42, 1],[-5, -3, -1, 1, 6, 13, 11, 3; -34, -18, -4, 6, 35, 72, 56, 12; -1008, -574, -168, 191, 1134, 2408, 1974, 496],128,76]
%
% END OF LATTICE L129.76
%
\hbox{}\par\smallskip%
%
% LATTICE L129.77
%
\leavevmode%
\llap{\spacer}%
${L_{129.77}}=\pdualwitharg{2.3.7}(\main({L_{129.6}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
247128\\109872\\-3024
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
109872\\48804\\-1344
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3024\\-1344\\37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-154
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-37
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\1\\280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\2\\399
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-336\\-84\\-504\\-168\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-14\\-168\\-112\\-126\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\0\\84\\0\\-168\\-252\\-420\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-14\\0\\1\\0\\-28\\-63\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-168\\-168\\0\\168\\0\\-168\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-112\\-252\\-28\\0\\28\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-126\\-420\\-63\\-168\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-252\\-44\\-168\\-28\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[247128, 109872, -3024; 109872, 48804, -1344; -3024, -1344, 37],[168, 84, 1],[1, -1, -2, 0, 3, 3, 4, 2; 0, -2, -7, -1, -2, 1, 2, 1; 84, -154, -420, -37, 168, 280, 399, 200],128,77]
%
% END OF LATTICE L129.77
%
\hbox{}\par\smallskip%
%
% LATTICE L129.78
%
\leavevmode%
\llap{\spacer}%
${L_{129.78}}=\pdualwitharg{3.7}({L_{129.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{336}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{42}\above{1pt}{1pt}{}{2}{112}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-162960\\-29568\\1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29568\\-4998\\294
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\294\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-34\\-1092
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-203
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\202
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\17\\588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\24\\896
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-672\\-168\\-252\\-336\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-14\\-42\\-112\\-63\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672\\0\\336\\0\\-168\\-1008\\-840\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-14\\0\\4\\0\\-112\\-126\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-42\\-168\\0\\42\\0\\-84\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-112\\-1008\\-112\\0\\112\\0\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-63\\-840\\-126\\-84\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-1008\\-176\\-168\\-112\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-162960, -29568, 1680; -29568, -4998, 294; 1680, 294, -17],[336, 42, 1],[-5, -1, -1, 1, 3, 5, 1, -1; -34, -6, -4, 6, 17, 24, 1, -12; -1092, -203, -168, 202, 588, 896, 105, -320],128,78]
%
% END OF LATTICE L129.78
%
\hbox{}\par\smallskip%
%
% LATTICE L129.79
%
\leavevmode%
\llap{\spacer}%
${L_{129.79}}=\pdualwitharg{3.7}({L_{129.6}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{336}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-523824\\-6384\\-1344
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6384\\42\\42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1344\\42\\25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-287\\588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-141\\287
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-672\\1344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\446\\-924
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\436\\-896
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\144\\-294
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-144\\296
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-336\\-42\\-252\\-168\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-7\\-84\\-112\\-126\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\0\\336\\0\\-336\\-1008\\-1680\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-7\\0\\1\\0\\-56\\-126\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-84\\-336\\0\\168\\0\\-336\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-112\\-1008\\-56\\0\\112\\0\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-126\\-1680\\-126\\-336\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-1008\\-88\\-336\\-112\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-523824, -6384, -1344; -6384, 42, 42; -1344, 42, 25],[336, 42, 1],[2, 1, 5, 0, -3, -3, -1, 1; -287, -141, -672, 4, 446, 436, 144, -144; 588, 287, 1344, -11, -924, -896, -294, 296],128,79]
%
% END OF LATTICE L129.79
%
\hbox{}\par\smallskip%
%
% LATTICE L129.80
%
\leavevmode%
\llap{\spacer}%
${L_{129.80}}=\pdualwitharg{3.7}({L_{129.7}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}{336}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22512\\0\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\0\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-8\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-74
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\64
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-336\\-84\\-252\\-168\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-28\\-168\\-224\\-126\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\0\\336\\0\\-336\\-1008\\-840\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-28\\0\\4\\0\\-112\\-126\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-168\\-336\\0\\168\\0\\-168\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-224\\-1008\\-112\\0\\112\\0\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-126\\-840\\-126\\-168\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-1008\\-176\\-336\\-112\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-22512, 0, 336; 0, 42, 0; 336, 0, -5],[336, 42, 1],[0, -1, -7, -1, -1, 1, 1, 1; 1, 0, -8, -2, -6, -4, -1, 0; 0, -70, -504, -74, -84, 56, 63, 64],128,80]
%
% END OF LATTICE L129.80
%
\hbox{}\par\smallskip%
%
% LATTICE L129.81
%
\leavevmode%
\llap{\spacer}%
${L_{129.81}}=\pdualwitharg{2.3.7}(\main({L_{129.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-3192\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3192\\-2436\\672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\672\\-185
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\24\\91
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\133\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\28\\106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-59\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-166\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-39\\-148
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-336\\-168\\-504\\-168\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-14\\-84\\-56\\-126\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\0\\84\\0\\-168\\-252\\-840\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-14\\0\\4\\0\\-56\\-252\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-84\\-168\\0\\168\\0\\-336\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-56\\-252\\-56\\0\\28\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-126\\-840\\-252\\-336\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-252\\-88\\-168\\-28\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[168, -3192, 840; -3192, -2436, 672; 840, 672, -185],[168, 84, 1],[1, 1, 5, 1, -1, -2, -5, -1; 0, 24, 133, 28, -22, -59, -166, -39; 0, 91, 504, 106, -84, -224, -630, -148],128,81]
%
% END OF LATTICE L129.81
%
\hbox{}\par\smallskip%
%
% LATTICE L129.82
%
\leavevmode%
\llap{\spacer}%
${L_{129.82}}=\pdualwitharg{2.7}({L_{129.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{b}{2}{112}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{336}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112560\\56448\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56448\\-28168\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\168\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\2\\-728
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-354
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\161
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\624
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-224\\-168\\-168\\-336\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-84\\-168\\-672\\-126\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224\\0\\112\\0\\-112\\-1008\\-280\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-84\\0\\12\\0\\-336\\-126\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-112\\0\\56\\0\\-56\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-672\\-1008\\-336\\0\\336\\0\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-126\\-280\\-126\\-56\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-1008\\-528\\-336\\-336\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-112560, 56448, -336; 56448, -28168, 168; -336, 168, -1],[336, 56, 1],[1, 2, 3, 1, 0, -1, 0, 1; 3, 3, 2, 0, -1, 0, 1, 6; 140, -210, -728, -354, -168, 336, 161, 624],128,82]
%
% END OF LATTICE L129.82
%
\hbox{}\par\smallskip%
%
% LATTICE L129.83
%
\leavevmode%
\llap{\spacer}%
${L_{129.83}}=\pdualwitharg{2.7}({L_{129.7}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{l}{2}{112}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{336}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3713808\\-1584576\\8400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1584576\\676088\\-3584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8400\\-3584\\19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\10\\560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\123
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\12\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\4\\-574
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\6\\-1080
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-224\\-84\\-168\\-336\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-42\\-168\\-672\\-252\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224\\0\\112\\0\\-112\\-1008\\-560\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-42\\0\\3\\0\\-168\\-126\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-112\\0\\56\\0\\-112\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-672\\-1008\\-168\\0\\336\\0\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-252\\-560\\-126\\-112\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-1008\\-264\\-336\\-336\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3713808, -1584576, 8400; -1584576, 676088, -3584; 8400, -3584, 19],[336, 56, 1],[1, 1, 3, 1, 2, 7, 3, 5; 1, 3, 10, 3, 5, 12, 4, 6; -252, 126, 560, 123, 56, -840, -574, -1080],128,83]
%
% END OF LATTICE L129.83
%
\hbox{}\par\smallskip%
%
% LATTICE L129.84
%
\leavevmode%
\llap{\spacer}%
${L_{129.84}}=\pdualwitharg{2.7}({L_{129.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{b}{2}{336}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
955920\\584304\\-4032
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4032\\-2464\\17
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\-6\\-630
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-1\\2\\56
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-1\\3\\198
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\HardButStrongLineBreak\kern3pt%
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2\\-13\\-1414
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-30\\-3168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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56\\0\\-224\\-168\\-168\\-336\\-56\\0
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\kern3pt%
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0\\-336\\-1008\\-528\\-336\\-336\\0\\48
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[955920, 584304, -4032; 584304, 357112, -2464; -4032, -2464, 17],[336, 56, 1],[2, 1, -1, -1, -1, 1, 2, 5; -11, -6, 2, 3, 3, -12, -13, -30; -1120, -630, 56, 198, 196, -1512, -1414, -3168],128,84]
%
% END OF LATTICE L129.84
%
\hbox{}\par\smallskip%
%
% LATTICE L129.85
%
\leavevmode%
\llap{\spacer}%
${L_{129.85}}=\pdualwitharg{2.7}({L_{129.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{112}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{l}{2}{336}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39984\\20160\\-672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20160\\-10136\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672\\336\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-17\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-6\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\28
\end{tabular}\endgroup%
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-1\\-12\\-336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-4\\-17\\-294
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-42\\-648
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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56\\0\\-224\\-168\\-168\\-336\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\21\\0\\-42\\-84\\-336\\-126\\-168
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-168\\-84\\-112\\0\\56\\0\\-112\\-336
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\kern3pt%
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-56\\-126\\-560\\-252\\-112\\0\\28\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-1008\\-528\\-336\\-336\\0\\48
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-39984, 20160, -672; 20160, -10136, 336; -672, 336, -11],[336, 56, 1],[-5, -2, -1, 1, 2, -1, -4, -11; -17, -6, -2, 3, 5, -12, -17, -42; -224, -63, 0, 30, 28, -336, -294, -648],128,85]
%
% END OF LATTICE L129.85
%
\hbox{}\par\smallskip%
%
% LATTICE L129.86
%
\leavevmode%
\llap{\spacer}%
${L_{129.86}}=\pdualwitharg{2.3.7}({L_{129.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
657888\\354312\\-4368
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
354312\\190680\\-2352
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4368\\-2352\\29
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-2\\-1218
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\-532
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\231
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\HardButStrongLineBreak\kern3pt%
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-2\\1\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-1\\-2352
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\HardButStrongLineBreak\kern3pt%
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-6\\-1\\-988
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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84\\0\\-336\\-168\\-126\\-168\\-168\\0
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\kern3pt%
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0\\56\\0\\-56\\-84\\-224\\-504\\-112
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-224\\-504\\-112\\0\\56\\0\\-56
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-504\\-1680\\-504\\-168\\0\\168\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-504\\-176\\-84\\-56\\0\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[657888, 354312, -4368; 354312, 190680, -2352; -4368, -2352, 29],[168, 168, 1],[-7, -3, 0, 1, 1, -2, -15, -6; -2, -1, 1, 1, 1, 1, -1, -1; -1218, -532, 84, 232, 231, -224, -2352, -988],128,86]
%
% END OF LATTICE L129.86
%
\hbox{}\par\smallskip%
%
% LATTICE L129.87
%
\leavevmode%
\llap{\spacer}%
${L_{129.87}}=\pdualwitharg{2.3.7}({L_{129.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54264\\0\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\0\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-326\\-195\\-21840
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-10\\-1008
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\3\\335
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-2\\-462
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\64
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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84\\0\\-336\\-168\\-252\\-168\\-168\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-336\\-168
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-56
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\0\\168\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-56\\0\\8
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-168\\-168\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-224\\-504\\-112
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-504\\-1680\\-504
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-504\\-176
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-54264, 0, 840; 0, 168, 0; 840, 0, -13],[168, 168, 1],[-7, -3, 0, 1, 2, -2, -15, -6; -2, 0, 1, 0, -1, -3, -10, -3; -462, -196, 0, 64, 126, -140, -1008, -400],128,87]
%
% END OF LATTICE L129.87
%
\hbox{}\par\smallskip%
%
% LATTICE L129.88
%
\leavevmode%
\llap{\spacer}%
${L_{129.88}}=\pdualwitharg{2.3.7}({L_{129.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-287112\\-124824\\-60480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-124824\\-54096\\-26208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60480\\-26208\\-12697
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-299\\37101\\-75096
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-136\\16931\\-34272
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-66\\8217\\-16633
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3\\224\\-448
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-1\\43\\-84
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-74\\148
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
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0\\56\\0\\-56
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-252\\-168\\-168\\0
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-168\\-224\\-504\\-112
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-287112, -124824, -60480; -124824, -54096, -26208; -60480, -26208, -12697],[168, 168, 1],[-5, -3, -1, 1, 3, 1, -5, -3; 439, 224, 43, -74, -190, 25, 704, 323; -882, -448, -84, 148, 378, -56, -1428, -652],128,88]
%
% END OF LATTICE L129.88
%
\hbox{}\par\smallskip%
%
% LATTICE L129.89
%
\leavevmode%
\llap{\spacer}%
${L_{129.89}}=\pdualwitharg{2.3.7}({L_{129.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{b}{2}{336}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{112}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
195216\\219744\\-2688
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
219744\\247128\\-3024
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2688\\-3024\\37
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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0\\1\\84
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-1\\-1\\-154
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-7\\-4\\-840
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-1\\0\\-74
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-1\\3\\168
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1\\6\\560
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1\\4\\399
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1\\4\\400
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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0\\-112\\-1008\\-176\\-336\\-112\\0\\16
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[195216, 219744, -2688; 219744, 247128, -3024; -2688, -3024, 37],[336, 168, 1],[0, -1, -7, -1, -1, 1, 1, 1; 1, -1, -4, 0, 3, 6, 4, 4; 84, -154, -840, -74, 168, 560, 399, 400],128,89]
%
% END OF LATTICE L129.89
%
\hbox{}\par\smallskip%
%
% LATTICE L129.90
%
\leavevmode%
\llap{\spacer}%
${L_{129.90}}=\pdualwitharg{2.3.7}({L_{129.7}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{336}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110544\\672\\-336
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672\\-168\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\0\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2\\-1\\-588
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\14
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\336
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0\\0\\-1
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-1\\-3\\-336
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-3\\-6\\-952
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-5\\-8\\-1554
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-3\\-4\\-920
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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168\\0\\-672\\-84\\-504\\-336\\-168\\0
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\kern3pt%
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-1008\\-88\\-336\\-112\\0\\16
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[110544, 672, -336; 672, -168, 0; -336, 0, 1],[336, 168, 1],[-2, 0, 1, 0, -1, -3, -5, -3; -1, 1, 4, 0, -3, -6, -8, -4; -588, 14, 336, -1, -336, -952, -1554, -920],128,90]
%
% END OF LATTICE L129.90
%
\hbox{}\par\smallskip%
%
% LATTICE L129.91
%
\leavevmode%
\llap{\spacer}%
${L_{129.91}}=\pdualwitharg{2.3.7}({L_{129.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{b}{2}{112}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}{336}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1834896\\1334592\\-11088
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1334592\\970536\\-8064
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11088\\-8064\\67
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\5\\1428
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\2\\1064
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\294
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\-1176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-2\\-406
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\286
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-336\\-336\\-504\\-168\\-336\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\112\\0\\-112\\-336\\-224\\-1008\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\0\\84\\0\\-168\\-252\\-1680\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-112\\0\\16\\0\\-112\\-1008\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-336\\-168\\0\\168\\0\\-672\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-224\\-252\\-112\\0\\28\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-1008\\-1680\\-1008\\-672\\0\\336\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-252\\-176\\-168\\-28\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1834896, 1334592, -11088; 1334592, 970536, -8064; -11088, -8064, 67],[336, 168, 1],[5, 5, 4, 1, -2, -1, 1, 1; 5, 2, -3, -4, -7, -2, 0, 1; 1428, 1064, 294, -320, -1176, -406, 168, 286],128,91]
%
% END OF LATTICE L129.91
%
\hbox{}\par\smallskip%
%
% LATTICE L129.92
%
\leavevmode%
\llap{\spacer}%
${L_{129.92}}=\pdualwitharg{2.3.7}({L_{129.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{l}{2}{112}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{336}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-239064\\2016
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2016\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-3\\-378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-488
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-119
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\118
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-336\\-336\\-504\\-84\\-336\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\112\\0\\-112\\-336\\-112\\-1008\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\0\\84\\0\\-168\\-126\\-1680\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-112\\0\\16\\0\\-56\\-1008\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-336\\-168\\0\\168\\0\\-672\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-112\\-126\\-56\\0\\7\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-1008\\-1680\\-1008\\-672\\0\\336\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-252\\-176\\-168\\-14\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[336, 0, 0; 0, -239064, 2016; 0, 2016, -17],[336, 168, 1],[-1, -3, -5, -3, -2, 0, 1, 0; 5, 2, -3, -4, -7, -1, 0, 1; 588, 224, -378, -488, -840, -119, 0, 118],128,92]
%
% END OF LATTICE L129.92
%

\medskip%
%------------------------------WEYL-GROUP-W130--------------------------------
%
\leavevmode\llap{}%
$W_{130}$%
\qquad\llap{64} lattices, $\chi=18$%
\hfill%
$2222222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L130.1
%
\leavevmode%
\llap{\spacer}%
${L_{130.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3,2}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{63}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-94248\\-16128\\1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16128\\-2757\\171
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\171\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\20\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\14\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-32\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-105\\-189
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-20\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\112\\196
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-24\\-63\\-12\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-63\\-28\\-126\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24\\-216\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-63\\0\\63\\0\\-252\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-28\\-24\\0\\8\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-126\\-216\\-252\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-224\\-504\\-756\\-56\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-94248, -16128, 1008; -16128, -2757, 171; 1008, 171, -10],[504, 3, 1],[-3, -2, 5, 16, 3, -1, -17; 20, 14, -32, -105, -20, 6, 112; 39, 35, -48, -189, -40, 0, 196],129,1]
%
% END OF LATTICE L130.1
%
\hbox{}\par\smallskip%
%
% LATTICE L130.2
%
\leavevmode%
\llap{\spacer}%
${L_{130.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{32\rightarrow N'_{18},3m,3,m}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{*}{2}{252}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
915768\\504\\-7560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\-3\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7560\\-3\\62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\40\\123
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\126\\371
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\128\\372
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\42\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-44\\-124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-51\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-24\\-126\\-12\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-126\\-28\\-63\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24\\-108\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-126\\0\\252\\0\\-252\\-1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-28\\-24\\0\\8\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-63\\-108\\-252\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-224\\-504\\-1512\\-56\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[915768, 504, -7560; 504, -3, -3; -7560, -3, 62],[504, 3, 1],[1, 3, 3, 1, -1, -1, 1; 40, 126, 128, 42, -44, -51, 0; 123, 371, 372, 126, -124, -126, 112],129,2]
%
% END OF LATTICE L130.2
%
\hbox{}\par\smallskip%
%
% LATTICE L130.3
%
\leavevmode%
\llap{\spacer}%
${L_{130.3}}=\pfillwitharg{3.2}({L_{130.2}})$%
{${}={}$}Nikulin $18'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-462\\-210\\126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-95\\63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
126\\63\\38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-42\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\-84\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\78\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-123\\259\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\120\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\139\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-299\\630\\-56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-12\\-21\\-6\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-21\\-14\\-21\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\6\\0\\-6\\-18\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-21\\0\\7\\0\\-14\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-14\\-6\\0\\2\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-21\\-18\\-14\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-126\\-126\\-14\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-462, -210, 126; -210, -95, 63; 126, 63, 38],[42, 1, 1],[20, 40, -37, -123, -57, -66, -299; -42, -84, 78, 259, 120, 139, 630; 3, 7, -6, -21, -10, -12, -56],129,3]
%
% END OF LATTICE L130.3
%
\hbox{}\par\smallskip%
%
% LATTICE L130.4
%
\leavevmode%
\llap{\spacer}%
${L_{130.4}}=\main(\pfillwitharg{3}({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23772\\-672\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672\\19\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-36\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-35\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\36\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\105\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\34\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-196\\-84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-21\\-6\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-21\\-14\\-21\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-12\\-36\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-21\\0\\14\\0\\-28\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-14\\-12\\0\\4\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-21\\-36\\-28\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-252\\-252\\-28\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[23772, -672, 0; -672, 19, 0; 0, 0, -1],[84, 1, 1],[-1, -1, 1, 3, 1, 0, -5; -36, -35, 36, 105, 34, -3, -196; -3, 0, 0, -7, -6, -13, -84],129,4]
%
% END OF LATTICE L130.4
%
\hbox{}\par\smallskip%
%
% LATTICE L130.5
%
\leavevmode%
\llap{\spacer}%
${L_{130.5}}=\pfillwitharg{3}({L_{130.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8904\\-168\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\5\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-3\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-72\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-70\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\72\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\210\\182
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\68\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-392\\-336
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-24\\-42\\-12\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-42\\-28\\-21\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24\\-36\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\0\\28\\0\\-28\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-28\\-24\\0\\8\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-21\\-36\\-28\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-224\\-504\\-504\\-56\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-8904, -168, 336; -168, 5, -3; 336, -3, -2],[168, 1, 1],[-1, -1, 1, 3, 1, 0, -5; -72, -70, 72, 210, 68, -3, -392; -63, -63, 60, 182, 60, -2, -336],129,5]
%
% END OF LATTICE L130.5
%
\hbox{}\par\smallskip%
%
% LATTICE L130.6
%
\leavevmode%
\llap{\spacer}%
${L_{130.6}}=\pfillwitharg{3}({L_{130.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73752\\1512\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1512\\31\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-99\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-147\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-48\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\49\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\48\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-52\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-756\\-140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-24\\-21\\-12\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-21\\-28\\-42\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24\\-72\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-21\\0\\7\\0\\-28\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-28\\-24\\0\\8\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-42\\-72\\-28\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-224\\-504\\-252\\-56\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[73752, 1512, 0; 1512, 31, 0; 0, 0, -1],[168, 1, 1],[2, 3, 1, -1, -1, 1, 15; -99, -147, -48, 49, 48, -52, -756; -9, -7, 0, 0, -4, -18, -140],129,6]
%
% END OF LATTICE L130.6
%
\hbox{}\par\smallskip%
%
% LATTICE L130.7
%
\leavevmode%
\llap{\spacer}%
${L_{130.7}}=\pfillwitharg{2}({L_{130.1}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{63}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-626094\\-28728\\10836
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28728\\-1317\\498
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10836\\498\\-187
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-127\\183
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-329\\455
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-160\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-84\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\42\\-62
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\15\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-266\\392
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-12\\-63\\-6\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-63\\-14\\-63\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\6\\0\\-6\\-54\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-63\\0\\63\\0\\-126\\-378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-14\\-6\\0\\2\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-63\\-54\\-126\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-126\\-378\\-14\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-626094, -28728, 10836; -28728, -1317, 498; 10836, 498, -187],[126, 3, 1],[9, 23, 11, 5, -3, -1, 19; -127, -329, -160, -84, 42, 15, -266; 183, 455, 210, 63, -62, -18, 392],129,7]
%
% END OF LATTICE L130.7
%
\hbox{}\par\smallskip%
%
% LATTICE L130.8
%
\leavevmode%
\llap{\spacer}%
${L_{130.8}}=\main({L_{130.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{126}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47124\\-16128\\504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16128\\-5514\\171
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\171\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\10\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-16\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-105\\-378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-10\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\56\\196
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-63\\-6\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-63\\-14\\-63\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-12\\-108\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-63\\0\\126\\0\\-252\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-14\\-12\\0\\4\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-63\\-108\\-252\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-252\\-756\\-28\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-47124, -16128, 504; -16128, -5514, 171; 504, 171, -5],[252, 3, 1],[-3, -2, 5, 32, 3, -1, -17; 10, 7, -16, -105, -10, 3, 56; 39, 35, -48, -378, -40, 0, 196],129,8]
%
% END OF LATTICE L130.8
%
\hbox{}\par\smallskip%
%
% LATTICE L130.9
%
\leavevmode%
\llap{\spacer}%
${L_{130.9}}=\pdualwitharg{2}(\pfillwitharg{3.2}({L_{130.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210\\-168\\126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\26\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
126\\-84\\73
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-9\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\-21\\-140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-3\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\7\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\3\\19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\4\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-42\\-6\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-42\\-14\\-42\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\0\\-3\\-18\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\0\\14\\0\\-28\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-14\\-3\\0\\1\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-42\\-18\\-28\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-63\\-126\\-7\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[210, -168, 126; -168, 26, -84; 126, -84, 73],[42, 2, 1],[29, 67, 10, -20, -9, -11, 5; -9, -21, -3, 7, 3, 4, 0; -60, -140, -21, 42, 19, 24, -7],129,9]
%
% END OF LATTICE L130.9
%
\hbox{}\par\smallskip%
%
% LATTICE L130.10
%
\leavevmode%
\llap{\spacer}%
${L_{130.10}}=\pdualwitharg{3}(\pfillwitharg{3.2}({L_{130.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1050\\-252\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\39\\84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\84\\107
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-3\\-23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-21\\-147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-2\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\7\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\6\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\4\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\0\\-84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-4\\-21\\-6\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-63\\-42\\-63\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\2\\0\\-6\\-18\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-63\\0\\21\\0\\-42\\-378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-42\\-6\\0\\6\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-63\\-18\\-42\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-126\\-378\\-42\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1050, -252, -336; -252, 39, 84; -336, 84, 107],[42, 3, 1],[-8, -52, -5, 15, 13, 6, -25; -3, -21, -2, 7, 6, 4, 0; -23, -147, -14, 42, 36, 15, -84],129,10]
%
% END OF LATTICE L130.10
%
\hbox{}\par\smallskip%
%
% LATTICE L130.11
%
\leavevmode%
\llap{\spacer}%
${L_{130.11}}=\pdualwitharg{3}(\main(\pfillwitharg{3}({L_{130.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4788\\-168\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\6\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\3\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-91\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-68\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-203\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\28\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\-21\\-6\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-63\\-42\\-63\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\4\\0\\-12\\-36\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-63\\0\\42\\0\\-84\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-42\\-12\\0\\12\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-63\\-36\\-84\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-252\\-756\\-84\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4788, -168, -84; -168, 6, 3; -84, 3, 1],[84, 3, 1],[0, -4, -3, -9, -1, 0, 1; 0, -91, -68, -203, -22, 1, 28; -1, -63, -44, -126, -12, 0, 0],129,11]
%
% END OF LATTICE L130.11
%
\hbox{}\par\smallskip%
%
% LATTICE L130.12
%
\leavevmode%
\llap{\spacer}%
${L_{130.12}}=\pdualwitharg{2.3}(\pfillwitharg{3.2}({L_{130.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12810\\-1008\\5334
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1008\\78\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5334\\-420\\2221
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-11\\34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-21\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\1\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\7\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-3\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-32\\102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-126\\399
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-4\\-42\\-6\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-126\\-42\\-126\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\1\\0\\-3\\-18\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-126\\0\\42\\0\\-84\\-378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-42\\-3\\0\\3\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-126\\-18\\-84\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-63\\-378\\-21\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12810, -1008, 5334; -1008, 78, -420; 5334, -420, 2221],[42, 6, 1],[-15, -19, 3, 18, -4, -45, -176; -11, -21, 1, 7, -3, -32, -126; 34, 42, -7, -42, 9, 102, 399],129,12]
%
% END OF LATTICE L130.12
%
\hbox{}\par\smallskip%
%
% LATTICE L130.13
%
\leavevmode%
\llap{\spacer}%
${L_{130.13}}=\pdualwitharg{7}(\pfillwitharg{3.2}({L_{130.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{42}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14154\\0\\2184
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2184\\0\\337
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-9\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-1\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\0\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-2\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-9\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-84\\-21\\-42\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-3\\-14\\-21\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\42\\0\\-42\\-126\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-3\\0\\1\\0\\-14\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-14\\-42\\0\\14\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-21\\-126\\-14\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-16\\-126\\-18\\-14\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14154, 0, 2184; 0, -7, 0; 2184, 0, 337],[42, 7, 1],[16, 6, 13, -2, -13, -11, -3; -9, -1, 0, 0, -2, -9, -10; -105, -39, -84, 13, 84, 70, 18],129,13]
%
% END OF LATTICE L130.13
%
\hbox{}\par\smallskip%
%
% LATTICE L130.14
%
\leavevmode%
\llap{\spacer}%
${L_{130.14}}=\pdualwitharg{2}(\main(\pfillwitharg{3}({L_{130.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4620\\2688\\1176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2688\\1132\\672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1176\\672\\299
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103\\-12\\432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\7\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\3\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\-21\\784
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-113\\-13\\474
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-493\\-57\\2068
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-786\\-91\\3297
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-84\\-12\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-84\\-28\\-84\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\0\\-6\\-36\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\0\\56\\0\\-112\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-28\\-6\\0\\4\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-84\\-36\\-112\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-63\\-252\\-14\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4620, 2688, 1176; 2688, 1132, 672; 1176, 672, 299],[84, 4, 1],[-103, 60, 25, -187, -113, -493, -786; -12, 7, 3, -21, -13, -57, -91; 432, -252, -105, 784, 474, 2068, 3297],129,14]
%
% END OF LATTICE L130.14
%
\hbox{}\par\smallskip%
%
% LATTICE L130.15
%
\leavevmode%
\llap{\spacer}%
${L_{130.15}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{130.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{126}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{126}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63630\\-2898\\1386
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2898\\132\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1386\\-63\\31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\19\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\21\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\22\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\63\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\102\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\60\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\1092\\-252
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-12\\-21\\-18\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\126\\0\\-63\\-126\\-63\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\6\\0\\-18\\-18\\-378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-63\\0\\7\\0\\-14\\-378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-126\\-18\\0\\18\\0\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-63\\-18\\-14\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1008\\-378\\-378\\-126\\0\\126
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[63630, -2898, 1386; -2898, 132, -63; 1386, -63, 31],[126, 3, 1],[1, 1, 1, 3, 5, 3, 55; 19, 21, 22, 63, 102, 60, 1092; -6, 0, 0, -7, -18, -13, -252],129,15]
%
% END OF LATTICE L130.15
%
\hbox{}\par\smallskip%
%
% LATTICE L130.16
%
\leavevmode%
\llap{\spacer}%
${L_{130.16}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{130.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51912\\25536\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25536\\12561\\-165
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-165\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
87\\-182\\-399
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\-136\\-296
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
97\\-203\\-441
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-44\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\56\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-8\\-21\\-12\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-63\\-84\\-126\\-672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\8\\0\\-24\\-72\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-63\\0\\21\\0\\-84\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-84\\-24\\0\\24\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-126\\-72\\-84\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-672\\-504\\-756\\-168\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[51912, 25536, -336; 25536, 12561, -165; -336, -165, 2],[168, 3, 1],[0, 87, 65, 97, 21, -1, -27; 0, -182, -136, -203, -44, 2, 56; -1, -399, -296, -441, -96, 0, 84],129,16]
%
% END OF LATTICE L130.16
%
\hbox{}\par\smallskip%
%
% LATTICE L130.17
%
\leavevmode%
\llap{\spacer}%
${L_{130.17}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-21\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-12\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-28\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-8\\-42\\-12\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-126\\-84\\-63\\-672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\8\\0\\-24\\-36\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-126\\0\\84\\0\\-84\\-1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-84\\-24\\0\\24\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-63\\-36\\-84\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-672\\-504\\-1512\\-168\\0\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[168, 0, 0; 0, 3, 0; 0, 0, -1],[168, 3, 1],[0, -4, -3, -9, -1, 0, 1; -1, -21, -12, -28, 0, 1, 0; -1, -63, -44, -126, -12, 0, 0],129,17]
%
% END OF LATTICE L130.17
%
\hbox{}\par\smallskip%
%
% LATTICE L130.18
%
\leavevmode%
\llap{\spacer}%
${L_{130.18}}=\pdualwitharg{2.7}(\pfillwitharg{3.2}({L_{130.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
508998\\0\\245238
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
245238\\0\\118157
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1801\\9\\-3738
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
159\\1\\-330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-172\\0\\357
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\0\\110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
172\\1\\-357
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1774\\9\\-3682
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
993\\5\\-2061
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-84\\-42\\-42\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-6\\-14\\-42\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\21\\0\\-21\\-126\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-6\\0\\2\\0\\-28\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-14\\-21\\0\\7\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-126\\-28\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-16\\-63\\-18\\-7\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[508998, 0, 245238; 0, -14, 0; 245238, 0, 118157],[42, 14, 1],[1801, 159, -172, -53, 172, 1774, 993; 9, 1, 0, 0, 1, 9, 5; -3738, -330, 357, 110, -357, -3682, -2061],129,18]
%
% END OF LATTICE L130.18
%
\hbox{}\par\smallskip%
%
% LATTICE L130.19
%
\leavevmode%
\llap{\spacer}%
${L_{130.19}}=\pdualwitharg{7}(\main(\pfillwitharg{3}({L_{130.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\8\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\4\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-84\\-21\\-42\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-3\\-14\\-21\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\84\\0\\-84\\-252\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-3\\0\\2\\0\\-28\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-14\\-84\\0\\28\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-21\\-252\\-28\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-16\\-252\\-36\\-28\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[84, 0, 0; 0, -7, -7; 0, -7, -6],[84, 7, 1],[-1, 0, 1, 0, -1, -3, -3; -3, -1, 0, 2, 8, 8, 4; 0, 1, 0, -3, -14, -21, -16],129,19]
%
% END OF LATTICE L130.19
%
\hbox{}\par\smallskip%
%
% LATTICE L130.20
%
\leavevmode%
\llap{\spacer}%
${L_{130.20}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{130.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{252}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{63}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57960\\-12978\\-28728
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12978\\-2454\\-6426
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28728\\-6426\\-14239
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-127\\-4\\258
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1427\\-42\\2898
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\1\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
710\\21\\-1442
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
616\\18\\-1251
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
521\\15\\-1058
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3692\\105\\-7497
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-42\\-18\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\252\\0\\-126\\-126\\-126\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\0\\-9\\-18\\-189
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-126\\0\\14\\0\\-28\\-378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-126\\-9\\0\\9\\0\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-126\\-18\\-28\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1008\\-189\\-378\\-63\\0\\63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-57960, -12978, -28728; -12978, -2454, -6426; -28728, -6426, -14239],[126, 6, 1],[-127, -1427, 31, 710, 616, 521, 3692; -4, -42, 1, 21, 18, 15, 105; 258, 2898, -63, -1442, -1251, -1058, -7497],129,20]
%
% END OF LATTICE L130.20
%
\hbox{}\par\smallskip%
%
% LATTICE L130.21
%
\leavevmode%
\llap{\spacer}%
${L_{130.21}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{130.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{126}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12536118\\449820\\6157872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
449820\\16158\\220956
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6157872\\220956\\3024811
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1617\\-53\\-3288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8689\\-287\\-17668
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3566\\-118\\-7251
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16483\\-546\\-33516
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-1\\-61
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62\\3\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1525\\-49\\-3101
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-126\\-6\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-126\\-14\\-126\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\0\\-3\\-54\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-126\\0\\126\\0\\-252\\-378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-14\\-3\\0\\1\\0\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-126\\-54\\-252\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-63\\-378\\-7\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12536118, 449820, 6157872; 449820, 16158, 220956; 6157872, 220956, 3024811],[126, 6, 1],[1617, 8689, 3566, 16483, 30, -62, 1525; -53, -287, -118, -546, -1, 3, -49; -3288, -17668, -7251, -33516, -61, 126, -3101],129,21]
%
% END OF LATTICE L130.21
%
\hbox{}\par\smallskip%
%
% LATTICE L130.22
%
\leavevmode%
\llap{\spacer}%
${L_{130.22}}=\pdualwitharg{3}(\main({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{63}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{252}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30492\\-1512\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1512\\75\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\21\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-21\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\18\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\39\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\924\\-252
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-21\\-18\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\63\\0\\-63\\-126\\-63\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-36\\-36\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-63\\0\\14\\0\\-28\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-126\\-36\\0\\36\\0\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-63\\-36\\-28\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1008\\-756\\-756\\-252\\0\\252
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30492, -1512, 0; -1512, 75, 0; 0, 0, -1],[252, 3, 1],[1, 1, -1, -1, 1, 2, 47; 20, 21, -20, -21, 18, 39, 924; -3, 0, 0, -7, -18, -13, -252],129,22]
%
% END OF LATTICE L130.22
%
\hbox{}\par\smallskip%
%
% LATTICE L130.23
%
\leavevmode%
\llap{\spacer}%
${L_{130.23}}=\pdualwitharg{3.7}(\pfillwitharg{3.2}({L_{130.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
423990\\-42\\148260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
148260\\0\\51833
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
487\\-971\\-1393
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
129\\-257\\-369
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93\\186\\266
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\86\\123
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
279\\-556\\-798
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1439\\-2869\\-4116
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1611\\-3212\\-4608
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-28\\-21\\-42\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-9\\-42\\-63\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\14\\0\\-42\\-126\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-9\\0\\3\\0\\-42\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-42\\0\\42\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-63\\-126\\-42\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-126\\-54\\-42\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[423990, -42, 148260; -42, -21, 0; 148260, 0, 51833],[42, 21, 1],[487, 129, -93, -43, 279, 1439, 1611; -971, -257, 186, 86, -556, -2869, -3212; -1393, -369, 266, 123, -798, -4116, -4608],129,23]
%
% END OF LATTICE L130.23
%
\hbox{}\par\smallskip%
%
% LATTICE L130.24
%
\leavevmode%
\llap{\spacer}%
${L_{130.24}}=\pdualwitharg{2.3}(\main(\pfillwitharg{3}({L_{130.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3192\\84\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\324\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-756\\24\\-179
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-1\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-340\\-21\\1428
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\-3\\193
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-401\\-28\\1680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\2\\-156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-4\\-84\\-12\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-252\\-84\\-252\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\1\\0\\-6\\-36\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-252\\0\\168\\0\\-336\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-84\\-6\\0\\12\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-252\\-36\\-336\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-336\\-63\\-756\\-42\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3192, 84, -756; 84, 324, 24; -756, 24, -179],[84, 12, 1],[-19, -340, -46, -401, 7, 37, -5; -1, -21, -3, -28, 0, 2, 0; 80, 1428, 193, 1680, -30, -156, 21],129,24]
%
% END OF LATTICE L130.24
%
\hbox{}\par\smallskip%
%
% LATTICE L130.25
%
\leavevmode%
\llap{\spacer}%
${L_{130.25}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\168\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-14\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-7\\39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-13\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\38\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\164\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\112\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\160\\-32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-168\\-42\\-84\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-6\\-28\\-21\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\168\\0\\-168\\-252\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-6\\0\\4\\0\\-28\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-28\\-168\\0\\56\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-21\\-252\\-28\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-32\\-504\\-72\\-56\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1848, 168, 168; 168, -14, -7; 168, -7, 39],[168, 7, 1],[-1, -1, 1, 3, 13, 9, 13; -15, -13, 12, 38, 164, 112, 160; 0, 2, 0, -6, -28, -21, -32],129,25]
%
% END OF LATTICE L130.25
%
\hbox{}\par\smallskip%
%
% LATTICE L130.26
%
\leavevmode%
\llap{\spacer}%
${L_{130.26}}=\pdualwitharg{7}(\pfillwitharg{3}({L_{130.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\8\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\4\\-32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-168\\-21\\-84\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-3\\-28\\-42\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\168\\0\\-168\\-504\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-3\\0\\1\\0\\-28\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-28\\-168\\0\\56\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-504\\-28\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-32\\-504\\-36\\-56\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[168, 0, 0; 0, -14, -7; 0, -7, -3],[168, 7, 1],[-1, 0, 1, 0, -1, -3, -3; -3, -1, 0, 1, 8, 8, 4; 0, 2, 0, -3, -28, -42, -32],129,26]
%
% END OF LATTICE L130.26
%
\hbox{}\par\smallskip%
%
% LATTICE L130.27
%
\leavevmode%
\llap{\spacer}%
${L_{130.27}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27384\\1008\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\72\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-8\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-7\\-448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\146
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\9\\584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\14\\875
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-168\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\112\\0\\-168\\-56\\-168\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\12\\0\\-12\\-72\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\0\\56\\0\\-112\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-56\\-12\\0\\4\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-168\\-72\\-112\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-224\\-126\\-252\\-14\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-27384, 1008, 168; 1008, 72, -8; 168, -8, -1],[168, 8, 1],[1, -3, -1, 1, 1, 4, 6; 3, -7, -3, 0, 2, 9, 14; 144, -448, -150, 140, 146, 584, 875],129,27]
%
% END OF LATTICE L130.27
%
\hbox{}\par\smallskip%
%
% LATTICE L130.28
%
\leavevmode%
\llap{\spacer}%
${L_{130.28}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{130.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140952\\-21840\\672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21840\\3384\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
672\\-104\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-33\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-35\\-112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\70\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\13\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\13\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-24\\-168\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\112\\0\\-168\\-56\\-168\\-448
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\3\\0\\-6\\-36\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\0\\56\\0\\-112\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-56\\-6\\0\\4\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-168\\-36\\-112\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-448\\-126\\-504\\-28\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[140952, -21840, 672; -21840, 3384, -104; 672, -104, 3],[168, 8, 1],[-5, -5, 1, 11, 2, 2, -1; -33, -35, 6, 70, 13, 13, -7; -24, -112, -21, -56, 2, 4, -14],129,28]
%
% END OF LATTICE L130.28
%
\hbox{}\par\smallskip%
%
% LATTICE L130.29
%
\leavevmode%
\llap{\spacer}%
${L_{130.29}}=\pdualwitharg{3}({L_{130.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{b}{2}{126}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{504}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
986328\\235368\\5544
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
235368\\56166\\1323
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5544\\1323\\31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-25\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\21\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\88\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\63\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\72\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-12\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143\\-588\\-504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-24\\-21\\-36\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\126\\0\\-63\\-252\\-126\\-2016
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-72\\-72\\-1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-63\\0\\7\\0\\-28\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-252\\-72\\0\\72\\0\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-126\\-72\\-28\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2016\\-1512\\-756\\-504\\0\\504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[986328, 235368, 5544; 235368, 56166, 1323; 5544, 1323, 31],[504, 3, 1],[6, -5, -21, -15, -17, 3, 143; -25, 21, 88, 63, 72, -12, -588; -6, 0, 0, -7, -36, -26, -504],129,29]
%
% END OF LATTICE L130.29
%
\hbox{}\par\smallskip%
%
% LATTICE L130.30
%
\leavevmode%
\llap{\spacer}%
${L_{130.30}}=\pdualwitharg{3}({L_{130.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{126}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{504}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-447048\\3024\\1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3024\\-3\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\-9\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\40\\321
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\42\\315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-40\\-324
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-42\\-322
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\36\\324
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\39\\322
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\1848\\15120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-24\\-42\\-36\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\126\\0\\-126\\-252\\-63\\-2016
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-72\\-36\\-1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-126\\0\\28\\0\\-28\\-1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-252\\-72\\0\\72\\0\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-63\\-36\\-28\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2016\\-1512\\-1512\\-504\\0\\504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-447048, 3024, 1008; 3024, -3, -9; 1008, -9, -2],[504, 3, 1],[1, 1, -1, -1, 1, 1, 47; 40, 42, -40, -42, 36, 39, 1848; 321, 315, -324, -322, 324, 322, 15120],129,30]
%
% END OF LATTICE L130.30
%
\hbox{}\par\smallskip%
%
% LATTICE L130.31
%
\leavevmode%
\llap{\spacer}%
${L_{130.31}}=\pdualwitharg{3.7}(\main(\pfillwitharg{3}({L_{130.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252\\-168\\84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\126\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-63\\31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-32\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-11\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-28\\-21\\-42\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-9\\-42\\-63\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\28\\0\\-84\\-252\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-9\\0\\6\\0\\-84\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-84\\0\\84\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-63\\-252\\-84\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-252\\-108\\-84\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[252, -168, 84; -168, 126, -63; 84, -63, 31],[84, 21, 1],[0, -1, -5, -2, -1, 1, 1; -3, -8, -32, -11, -2, 1, -4; -7, -15, -56, -18, 0, 0, -12],129,31]
%
% END OF LATTICE L130.31
%
\hbox{}\par\smallskip%
%
% LATTICE L130.32
%
\leavevmode%
\llap{\spacer}%
${L_{130.32}}=\pdualwitharg{2.3.7}(\pfillwitharg{3.2}({L_{130.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4353930\\-10580892\\-5142186
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10580892\\25713366\\12496386
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5142186\\12496386\\6073093
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\1829\\-3766
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1997\\-4110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\711\\-1463
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-659\\1356
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2113\\4347
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-3392\\6972
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-368\\753
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-28\\-42\\-42\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-18\\-42\\-126\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\7\\0\\-21\\-126\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-18\\0\\6\\0\\-84\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-42\\-21\\0\\21\\0\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-126\\-126\\-84\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-63\\-54\\-21\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4353930, -10580892, -5142186; -10580892, 25713366, 12496386; -5142186, 12496386, 6073093],[42, 42, 1],[-3, -1, 0, 0, -1, -9, -5; 1829, 1997, 711, -659, -2113, -3392, -368; -3766, -4110, -1463, 1356, 4347, 6972, 753],129,32]
%
% END OF LATTICE L130.32
%
\hbox{}\par\smallskip%
%
% LATTICE L130.33
%
\leavevmode%
\llap{\spacer}%
${L_{130.33}}=\pdualwitharg{2.7}(\main(\pfillwitharg{3}({L_{130.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\-84\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\168\\140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\140\\121
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-15\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-11\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-9\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-17\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-84\\-84\\-84\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12\\-28\\-84\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\21\\0\\-42\\-252\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-12\\0\\8\\0\\-112\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-28\\-42\\0\\28\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\-252\\-112\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-16\\-63\\-36\\-14\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[84, -84, -84; -84, 168, 140; -84, 140, 121],[84, 28, 1],[1, 1, 5, 4, 4, 10, 2; 0, -3, -15, -11, -9, -17, -3; 0, 4, 21, 16, 14, 28, 5],129,33]
%
% END OF LATTICE L130.33
%
\hbox{}\par\smallskip%
%
% LATTICE L130.34
%
\leavevmode%
\llap{\spacer}%
${L_{130.34}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{130.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{l}{2}{42}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{126}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15246\\-5166\\126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5166\\1743\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
126\\-42\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\-324
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-84\\-21\\-126\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-9\\-126\\-63\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\42\\0\\-126\\-126\\-378
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-9\\0\\1\\0\\-14\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-126\\-126\\0\\126\\0\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-63\\-126\\-14\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-378\\-54\\-126\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15246, -5166, 126; -5166, 1743, -42; 126, -42, 1],[126, 21, 1],[1, 1, 1, 0, -1, 0, 1; 1, 3, 4, 0, -6, -3, -6; -63, 9, 42, -1, -126, -112, -324],129,34]
%
% END OF LATTICE L130.34
%
\hbox{}\par\smallskip%
%
% LATTICE L130.35
%
\leavevmode%
\llap{\spacer}%
${L_{130.35}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{130.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{42}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{63}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23310\\-7812\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7812\\2625\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\168\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-11\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-11\\-61
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-68\\-378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-24\\-135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-84\\-63\\-42\\-63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-9\\-14\\-63\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\42\\0\\-42\\-378\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-9\\0\\9\\0\\-126\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-14\\-42\\0\\14\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-63\\-378\\-126\\0\\63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-16\\-126\\-54\\-14\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[23310, -7812, -504; -7812, 2625, 168; -504, 168, 11],[126, 21, 1],[-5, -5, -31, -11, -1, 1, -1; -11, -11, -68, -24, -2, 3, -2; -63, -61, -378, -135, -14, 0, -16],129,35]
%
% END OF LATTICE L130.35
%
\hbox{}\par\smallskip%
%
% LATTICE L130.36
%
\leavevmode%
\llap{\spacer}%
${L_{130.36}}=\pdualwitharg{2}(\main({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{504}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9801288\\-92484\\-2479932
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92484\\876\\23400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2479932\\23400\\627475
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\-29\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-242\\-119\\-952
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-106\\-52\\-417
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1217\\-588\\-4788
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-14\\-122
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-311\\-138\\-1224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-201\\-91\\-791
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-252\\-12\\-36\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-252\\-28\\-252\\-112
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-12\\0\\3\\0\\-6\\-108\\-63
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-252\\-252\\0\\504\\0\\-1008\\-756
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-28\\-6\\0\\4\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-252\\-108\\-1008\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-112\\-63\\-756\\-14\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9801288, -92484, -2479932; -92484, 876, 23400; -2479932, 23400, 627475],[252, 12, 1],[-61, -242, -106, -1217, -31, -311, -201; -29, -119, -52, -588, -14, -138, -91; -240, -952, -417, -4788, -122, -1224, -791],129,36]
%
% END OF LATTICE L130.36
%
\hbox{}\par\smallskip%
%
% LATTICE L130.37
%
\leavevmode%
\llap{\spacer}%
${L_{130.37}}=\pdualwitharg{2.3}(\main({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{252}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{63}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9324\\6048\\-2268
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6048\\1452\\-1512
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2268\\-1512\\551
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\-4\\264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-320\\21\\-1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\1\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
327\\-21\\1288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
425\\-27\\1674
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
525\\-33\\2068
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2351\\-147\\9261
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-84\\-36\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\252\\0\\-252\\-252\\-252\\-1008
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\0\\-18\\-36\\-189
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-252\\0\\56\\0\\-112\\-756
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-252\\-18\\0\\36\\0\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-252\\-36\\-112\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1008\\-189\\-756\\-126\\0\\63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9324, 6048, -2268; 6048, 1452, -1512; -2268, -1512, 551],[252, 12, 1],[67, -320, -16, 327, 425, 525, 2351; -4, 21, 1, -21, -27, -33, -147; 264, -1260, -63, 1288, 1674, 2068, 9261],129,37]
%
% END OF LATTICE L130.37
%
\hbox{}\par\smallskip%
%
% LATTICE L130.38
%
\leavevmode%
\llap{\spacer}%
${L_{130.38}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{130.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-8\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-56\\-21\\-84\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-9\\-84\\-126\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\56\\0\\-168\\-504\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-9\\0\\3\\0\\-84\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\-168\\0\\168\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-126\\-504\\-84\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-96\\-504\\-108\\-168\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[504, -168, 0; -168, 63, 0; 0, 0, -1],[168, 21, 1],[0, -1, -5, -1, -1, 1, 1; 1, -1, -8, -2, -4, 2, 4; -7, -15, -56, -9, 0, 0, -12],129,38]
%
% END OF LATTICE L130.38
%
\hbox{}\par\smallskip%
%
% LATTICE L130.39
%
\leavevmode%
\llap{\spacer}%
${L_{130.39}}=\pdualwitharg{3.7}(\pfillwitharg{3}({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41496\\0\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\0\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-3\\-249
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-12\\-1148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-4\\-450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-56\\-42\\-84\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-18\\-84\\-63\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\56\\0\\-168\\-252\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-18\\0\\12\\0\\-84\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\-168\\0\\168\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-63\\-252\\-84\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-96\\-504\\-216\\-168\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[41496, 0, 840; 0, 21, 0; 840, 0, 17],[168, 21, 1],[1, 5, 23, 9, 5, 0, 1; -1, -3, -12, -4, 0, 1, 0; -49, -249, -1148, -450, -252, 0, -48],129,39]
%
% END OF LATTICE L130.39
%
\hbox{}\par\smallskip%
%
% LATTICE L130.40
%
\leavevmode%
\llap{\spacer}%
${L_{130.40}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{130.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-577080\\-15960\\5208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15960\\-432\\144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5208\\144\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-7\\2184
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\107
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-14\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-114
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\7\\462
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-8\\-168\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\336\\0\\-504\\-168\\-504\\-1344
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\1\\0\\-6\\-36\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-504\\0\\168\\0\\-336\\-1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-168\\-6\\0\\12\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-504\\-36\\-336\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1344\\-126\\-1512\\-84\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-577080, -15960, 5208; -15960, -432, 144; 5208, 144, -47],[168, 24, 1],[2, 20, 1, 5, -1, -1, 4; 1, -7, -1, -14, -1, 1, 7; 224, 2184, 107, 504, -114, -108, 462],129,40]
%
% END OF LATTICE L130.40
%
\hbox{}\par\smallskip%
%
% LATTICE L130.41
%
\leavevmode%
\llap{\spacer}%
${L_{130.41}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{336}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-207816\\-26376\\1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26376\\-3312\\192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1512\\192\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-21\\2352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\218
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-14\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\273
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\-168\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\336\\0\\-504\\-168\\-504\\-672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\4\\0\\-12\\-72\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-504\\0\\168\\0\\-336\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-168\\-12\\0\\12\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-504\\-72\\-336\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-672\\-126\\-756\\-42\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-207816, -26376, 1512; -26376, -3312, 192; 1512, 192, -11],[168, 24, 1],[2, 20, 2, 5, -1, -1, 2; -1, -21, -3, -14, 0, 1, 0; 256, 2352, 218, 420, -138, -120, 273],129,41]
%
% END OF LATTICE L130.41
%
\hbox{}\par\smallskip%
%
% LATTICE L130.42
%
\leavevmode%
\llap{\spacer}%
${L_{130.42}}=\pdualwitharg{7}(\main({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{b}{2}{126}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1015812\\334404\\-5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
334404\\110082\\-1659
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5040\\-1659\\25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-67
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-28\\-1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-15\\-594
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\-196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1015812, 334404, -5040; 334404, 110082, -1659; -5040, -1659, 25],[252, 21, 1],[-1, 0, 3, 2, 1, -1, -1; 4, -1, -28, -15, -6, 3, 4; 63, -67, -1260, -594, -196, 0, 64],129,42]
%
% END OF LATTICE L130.42
%
\hbox{}\par\smallskip%
%
% LATTICE L130.43
%
\leavevmode%
\llap{\spacer}%
${L_{130.43}}=\pdualwitharg{2.3.7}(\pfillwitharg{2}({L_{130.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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309834\\-10332\\152334
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\kern3pt%
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152334\\-5082\\74897
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\HardButStrongLineBreak\kern3pt%
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31\\1\\-63
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63\\2\\-128
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31\\0\\-63
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-696\\-25\\1414
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-1360\\-48\\2763
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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84\\0\\-84\\-42\\-126\\-42\\0
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0\\36\\0\\-18\\-126\\-126\\-144
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-84\\0\\21\\0\\-63\\-126\\-189
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-42\\-18\\0\\2\\0\\-28\\-54
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-126\\-126\\-63\\0\\63\\0\\-63
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-42\\-126\\-126\\-28\\0\\14\\0
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0\\-144\\-189\\-54\\-63\\0\\9
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[309834, -10332, 152334; -10332, 210, -5082; 152334, -5082, 74897],[126, 42, 1],[-1075, -443, 31, 63, 31, -696, -1360; -37, -15, 1, 2, 0, -25, -48; 2184, 900, -63, -128, -63, 1414, 2763],129,43]
%
% END OF LATTICE L130.43
%
\hbox{}\par\smallskip%
%
% LATTICE L130.44
%
\leavevmode%
\llap{\spacer}%
${L_{130.44}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{130.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-365022\\18648\\-179172
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18648\\-798\\9156
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-179172\\9156\\-87947
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-537\\-19\\1092
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-63\\-3\\128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
196\\4\\-399
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407\\12\\-828
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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148\\5\\-301
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62\\3\\-126
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89\\-3\\181
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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84\\0\\-84\\-126\\-42\\-126\\0
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0\\4\\0\\-18\\-14\\-126\\-16
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-84\\0\\21\\0\\-21\\-378\\-63
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-126\\-18\\0\\18\\0\\-252\\-54
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-42\\-14\\-21\\0\\7\\0\\-7
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0\\-16\\-63\\-54\\-7\\0\\1
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-365022, 18648, -179172; 18648, -798, 9156; -179172, 9156, -87947],[126, 42, 1],[-537, -63, 196, 407, 148, 62, -89; -19, -3, 4, 12, 5, 3, -3; 1092, 128, -399, -828, -301, -126, 181],129,44]
%
% END OF LATTICE L130.44
%
\hbox{}\par\smallskip%
%
% LATTICE L130.45
%
\leavevmode%
\llap{\spacer}%
${L_{130.45}}=\pdualwitharg{3.7}(\main({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{252}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\252\\252
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\kern3pt%
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252\\-42\\-63
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252\\-63\\-31
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-31\\-21
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-4\\-12\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-45\\-28
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\HardButStrongLineBreak\kern3pt%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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21\\0\\-84\\-21\\-126\\-21\\0
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0\\9\\0\\-9\\-126\\-63\\-144
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-84\\0\\84\\0\\-252\\-252\\-756
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-21\\-9\\0\\2\\0\\-28\\-108
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-126\\-126\\-252\\0\\252\\0\\-252
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-21\\-63\\-252\\-28\\0\\14\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1260, 252, 252; 252, -42, -63; 252, -63, -31],[252, 21, 1],[-10, -4, 1, 1, -1, -14, -53; -31, -12, 4, 3, -6, -45, -168; -21, -9, 0, 2, 0, -28, -108],129,45]
%
% END OF LATTICE L130.45
%
\hbox{}\par\smallskip%
%
% LATTICE L130.46
%
\leavevmode%
\llap{\spacer}%
${L_{130.46}}=\pdualwitharg{2.3.7}(\main(\pfillwitharg{3}({L_{130.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3\\48\\-196
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5\\85\\-348
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\70\\-287
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\73\\-300
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-31\\126
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\HardButStrongLineBreak\kern3pt%
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-2\\-41\\168
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\HardButStrongLineBreak\kern3pt%
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1\\14\\-57
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-28\\-84\\-84\\-84\\0
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\kern3pt%
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0\\12\\0\\-36\\-84\\-252\\-48
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\kern3pt%
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-28\\0\\7\\0\\-42\\-252\\-63
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-84\\-36\\0\\24\\0\\-336\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-84\\-42\\0\\84\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-252\\-252\\-336\\0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-63\\-108\\-42\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-756, 2100, 504; 2100, -2520, -588; 504, -588, -137],[84, 84, 1],[3, 5, 4, 4, -2, -2, 1; 48, 85, 70, 73, -31, -41, 14; -196, -348, -287, -300, 126, 168, -57],129,46]
%
% END OF LATTICE L130.46
%
\hbox{}\par\smallskip%
%
% LATTICE L130.47
%
\leavevmode%
\llap{\spacer}%
${L_{130.47}}=\pdualwitharg{2.7}(\pfillwitharg{3}({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-1176\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1176\\280\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-56\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\13\\44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\37\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\115\\392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\22\\75
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\0\\-336\\-168\\-168\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-24\\-56\\-168\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\0\\84\\0\\-84\\-504\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-24\\0\\8\\0\\-112\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-56\\-84\\0\\28\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-504\\-112\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-32\\-126\\-36\\-14\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[336, -1176, 336; -1176, 280, -56; 336, -56, 9],[168, 56, 1],[1, -1, -1, 1, 3, 10, 2; 0, -14, -12, 13, 37, 115, 22; 0, -48, -42, 44, 126, 392, 75],129,47]
%
% END OF LATTICE L130.47
%
\hbox{}\par\smallskip%
%
% LATTICE L130.48
%
\leavevmode%
\llap{\spacer}%
${L_{130.48}}=\pdualwitharg{2.7}(\pfillwitharg{3}({L_{130.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7728\\-3192\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3192\\-728\\56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\56\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\48\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-9\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-19\\-136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-29\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-49\\-364
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-10\\-78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\0\\-168\\-168\\-168\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-24\\-56\\-168\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\21\\0\\-42\\-252\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-24\\0\\8\\0\\-112\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-56\\-42\\0\\28\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-252\\-112\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-64\\-126\\-72\\-28\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7728, -3192, 336; -3192, -728, 56; 336, 56, -3],[168, 56, 1],[-5, -1, 1, 2, 3, 5, 1; 48, 10, -9, -19, -29, -49, -10; 336, 72, -63, -136, -210, -364, -78],129,48]
%
% END OF LATTICE L130.48
%
\hbox{}\par\smallskip%
%
% LATTICE L130.49
%
\leavevmode%
\llap{\spacer}%
${L_{130.49}}=\pdualwitharg{7}({L_{130.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{s}{2}{252}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
126504\\30744\\-2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30744\\7413\\-609
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-609\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-56\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\148
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-168\\-63\\-84\\-126\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-9\\-28\\-126\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\168\\0\\-168\\-1512\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-9\\0\\9\\0\\-252\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-28\\-168\\0\\56\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-126\\-1512\\-252\\0\\252\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-32\\-504\\-108\\-56\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[126504, 30744, -2520; 30744, 7413, -609; -2520, -609, 50],[504, 21, 1],[1, 0, -3, -1, -1, 1, 1; 8, -2, -56, -15, -12, 6, 8; 147, -25, -840, -234, -196, 126, 148],129,49]
%
% END OF LATTICE L130.49
%
\hbox{}\par\smallskip%
%
% LATTICE L130.50
%
\leavevmode%
\llap{\spacer}%
${L_{130.50}}=\pdualwitharg{7}({L_{130.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{l}{2}{63}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85176\\4536\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4536\\-21\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-38\\-399
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\56\\588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\18\\198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\-196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-135\\-1386
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-96\\-992
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-168\\-126\\-84\\-63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-18\\-28\\-63\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\168\\0\\-168\\-756\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-126\\-18\\0\\36\\0\\-252\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-28\\-168\\0\\56\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-63\\-756\\-252\\0\\63\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-32\\-504\\-216\\-56\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-85176, 4536, 0; 4536, -21, -21; 0, -21, 2],[504, 21, 1],[-2, 0, 3, 1, -1, -7, -5; -38, 0, 56, 18, -20, -135, -96; -399, -1, 588, 198, -196, -1386, -992],129,50]
%
% END OF LATTICE L130.50
%
\hbox{}\par\smallskip%
%
% LATTICE L130.51
%
\leavevmode%
\llap{\spacer}%
${L_{130.51}}=\pdualwitharg{3.7}({L_{130.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{504}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
239400\\80136\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80136\\26817\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-168\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-7\\-147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\16\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-36\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-30\\-476
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-84\\-1404
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-168\\-21\\-252\\-42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-9\\-252\\-126\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\168\\0\\-504\\-504\\-1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-9\\0\\1\\0\\-28\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-252\\-504\\0\\504\\0\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-126\\-504\\-28\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-288\\-1512\\-108\\-504\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[239400, 80136, -504; 80136, 26817, -168; -504, -168, 1],[504, 21, 1],[2, -1, -5, 0, 11, 9, 25; -7, 3, 16, 0, -36, -30, -84; -147, 9, 168, -1, -504, -476, -1404],129,51]
%
% END OF LATTICE L130.51
%
\hbox{}\par\smallskip%
%
% LATTICE L130.52
%
\leavevmode%
\llap{\spacer}%
${L_{130.52}}=\pdualwitharg{3.7}({L_{130.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{b}{2}{168}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{504}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4447800\\16128\\16632
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16128\\-21\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16632\\-63\\-62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\178\\2499
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\72\\999
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\123\\1750
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\936\\13248
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-168\\-42\\-252\\-21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-18\\-252\\-63\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\168\\0\\-504\\-252\\-1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-18\\0\\4\\0\\-28\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\-252\\-504\\0\\504\\0\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-63\\-252\\-28\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-288\\-1512\\-216\\-504\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4447800, 16128, 16632; 16128, -21, -63; 16632, -63, -62],[504, 21, 1],[10, 4, -1, -1, 1, 7, 53; 178, 72, -16, -18, 12, 123, 936; 2499, 999, -252, -250, 252, 1750, 13248],129,52]
%
% END OF LATTICE L130.52
%
\hbox{}\par\smallskip%
%
% LATTICE L130.53
%
\leavevmode%
\llap{\spacer}%
${L_{130.53}}=\pdualwitharg{2.3}({L_{130.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{1008}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{252}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\-504\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-3336\\216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\216\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\7\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-11\\-188
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59\\-126\\-2142
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-24\\-168\\-72\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1008\\0\\-504\\-504\\-504\\-4032
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\3\\0\\-18\\-36\\-378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-504\\0\\56\\0\\-112\\-1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-504\\-18\\0\\36\\0\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-504\\-36\\-112\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4032\\-378\\-1512\\-252\\0\\252
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1008, -504, 0; -504, -3336, 216; 0, 216, -13],[504, 24, 1],[-5, -1, 1, 4, -1, -5, -59; -10, 0, 2, 7, -3, -11, -126; -168, 0, 33, 112, -54, -188, -2142],129,53]
%
% END OF LATTICE L130.53
%
\hbox{}\par\smallskip%
%
% LATTICE L130.54
%
\leavevmode%
\llap{\spacer}%
${L_{130.54}}=\pdualwitharg{2.3}({L_{130.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{s}{2}{1008}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{63}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\-504\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\264\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\7\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\-63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-168\\-72\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1008\\0\\-504\\-504\\-504\\-2016
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\12\\0\\-36\\-72\\-378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-504\\0\\56\\0\\-112\\-756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-504\\-36\\0\\36\\0\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-504\\-72\\-112\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2016\\-378\\-756\\-126\\0\\63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1008, -504, 0; -504, 264, 0; 0, 0, -1],[504, 24, 1],[-1, 1, 1, 3, 1, 0, -2; -2, 0, 2, 7, 3, 1, 0; 0, 0, -6, -28, -18, -16, -63],129,54]
%
% END OF LATTICE L130.54
%
\hbox{}\par\smallskip%
%
% LATTICE L130.55
%
\leavevmode%
\llap{\spacer}%
${L_{130.55}}=\pdualwitharg{2}({L_{130.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{504}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9845640\\-425880\\38304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-425880\\-18384\\1656
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38304\\1656\\-149
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6\\47\\2064
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2\\14\\669
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5\\21\\1512
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\HardButStrongLineBreak\kern3pt%
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6\\49\\2086
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9845640, -425880, 38304; -425880, -18384, 1656; 38304, 1656, -149],[504, 24, 1],[6, 16, 2, 5, -1, -1, 6; 47, 119, 14, 21, -8, -6, 49; 2064, 5432, 669, 1512, -346, -324, 2086],129,55]
%
% END OF LATTICE L130.55
%
\hbox{}\par\smallskip%
%
% LATTICE L130.56
%
\leavevmode%
\llap{\spacer}%
${L_{130.56}}=\pdualwitharg{2}({L_{130.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2520\\-5544\\504
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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6\\41\\480
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16\\105\\1232
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4\\25\\294
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5\\21\\252
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\HardButStrongLineBreak\kern3pt%
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-1\\-6\\-72
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\HardButStrongLineBreak\kern3pt%
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3\\21\\245
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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48\\0\\-48\\-504\\-24\\-72\\0
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0\\112\\0\\-504\\-56\\-504\\-224
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-48\\0\\12\\0\\-12\\-216\\-126
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-504\\-504\\0\\504\\0\\-1008\\-756
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-24\\-56\\-12\\0\\4\\0\\-14
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-72\\-504\\-216\\-1008\\0\\72\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2520, -5544, 504; -5544, -3984, 408; 504, 408, -41],[504, 24, 1],[6, 16, 4, 5, -1, -1, 3; 41, 105, 25, 21, -7, -6, 21; 480, 1232, 294, 252, -82, -72, 245],129,56]
%
% END OF LATTICE L130.56
%
\hbox{}\par\smallskip%
%
% LATTICE L130.57
%
\leavevmode%
\llap{\spacer}%
${L_{130.57}}=\pdualwitharg{2.7}(\main({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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12189492\\175392\\2960244
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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2960244\\42924\\718645
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1067\\3268\\4200
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-816\\2499\\3212
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2150\\6584\\8463
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2323\\7113\\9144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
345\\-1057\\-1358
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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3137\\-9609\\-12348
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\HardButStrongLineBreak\kern3pt%
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47\\-144\\-185
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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84\\0\\-84\\-252\\-84\\-252\\0
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-84\\0\\21\\0\\-42\\-756\\-63
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-84\\-28\\-42\\0\\28\\0\\-14
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0\\-16\\-63\\-108\\-14\\0\\1
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12189492, 175392, 2960244; 175392, 2100, 42924; 2960244, 42924, 718645],[252, 84, 1],[-1067, -816, -2150, -2323, 345, 3137, 47; 3268, 2499, 6584, 7113, -1057, -9609, -144; 4200, 3212, 8463, 9144, -1358, -12348, -185],129,57]
%
% END OF LATTICE L130.57
%
\hbox{}\par\smallskip%
%
% LATTICE L130.58
%
\leavevmode%
\llap{\spacer}%
${L_{130.58}}=\pdualwitharg{2.3.7}(\main({L_{130.2}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7812\\8064\\-4032
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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37\\-108\\-144
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16\\-47\\-63
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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-84\\0\\21\\0\\-126\\-252\\-189
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-84\\-36\\0\\8\\0\\-112\\-108
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-252\\-252\\-126\\0\\252\\0\\-126
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-84\\-252\\-252\\-112\\0\\56\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7812, 8064, -4032; 8064, 8148, -4032; -4032, -4032, 1985],[252, 84, 1],[44, 37, 16, -1, -31, 17, 32; -127, -108, -47, 3, 93, -45, -90; -168, -144, -63, 4, 126, -56, -117],129,58]
%
% END OF LATTICE L130.58
%
\hbox{}\par\smallskip%
%
% LATTICE L130.59
%
\leavevmode%
\llap{\spacer}%
${L_{130.59}}=\pdualwitharg{2.3.7}(\pfillwitharg{3}({L_{130.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-164304\\-164472\\2184
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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2184\\2184\\-29
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-3\\-2\\-384
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\0\\-112\\-168\\-168\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-72\\-168\\-504\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\28\\0\\-84\\-504\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-72\\0\\24\\0\\-336\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-84\\0\\84\\0\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-504\\-504\\-336\\0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-96\\-126\\-108\\-42\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-164304, -164472, 2184; -164472, -164472, 2184; 2184, 2184, -29],[168, 168, 1],[-1, -3, -3, -2, 0, 1, 0; 4, -2, -6, -7, -5, -1, 1; 224, -384, -686, -684, -378, 0, 75],129,59]
%
% END OF LATTICE L130.59
%
\hbox{}\par\smallskip%
%
% LATTICE L130.60
%
\leavevmode%
\llap{\spacer}%
${L_{130.60}}=\pdualwitharg{2.3.7}(\pfillwitharg{3}({L_{130.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${112}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3024\\2184\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2184\\504\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\168\\55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-42\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\44\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\44\\-119
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\89\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\47\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-31\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-38\\102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\0\\-56\\-168\\-168\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-72\\-168\\-504\\-192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\7\\0\\-42\\-252\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-72\\0\\24\\0\\-336\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-168\\-42\\0\\84\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-504\\-252\\-336\\0\\168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-192\\-126\\-216\\-84\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3024, 2184, 840; 2184, 504, 168; 840, 168, 55],[168, 168, 1],[1, -1, -1, -2, -1, 1, 1; -42, 44, 44, 89, 47, -31, -38; 112, -120, -119, -240, -126, 84, 102],129,60]
%
% END OF LATTICE L130.60
%
\hbox{}\par\smallskip%
%
% LATTICE L130.61
%
\leavevmode%
\llap{\spacer}%
${L_{130.61}}=\pdualwitharg{2.3.7}({L_{130.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{252}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17136\\-34776\\15120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34776\\-35448\\15456
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15120\\15456\\-6739
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-874\\-2016
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-156\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\173\\399
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\163\\376
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\273\\630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-425\\-980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-960\\-2214
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\0\\-168\\-168\\-504\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-72\\-504\\-504\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\21\\0\\-126\\-252\\-378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-72\\0\\8\\0\\-112\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-504\\-126\\0\\252\\0\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-504\\-252\\-112\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-576\\-378\\-216\\-252\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17136, -34776, 15120; -34776, -35448, 15456; 15120, 15456, -6739],[504, 168, 1],[-5, -1, 1, 1, 2, -2, -5; -874, -156, 173, 163, 273, -425, -960; -2016, -360, 399, 376, 630, -980, -2214],129,61]
%
% END OF LATTICE L130.61
%
\hbox{}\par\smallskip%
%
% LATTICE L130.62
%
\leavevmode%
\llap{\spacer}%
${L_{130.62}}=\pdualwitharg{2.3.7}({L_{130.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{s}{2}{144}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{252}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31248\\11592\\-3024
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11592\\1848\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3024\\-504\\137
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-86\\-336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\72\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\74\\294
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-159\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-241\\-952
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-162\\-639
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\0\\-336\\-168\\-504\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-72\\-504\\-504\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\0\\84\\0\\-252\\-504\\-378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-72\\0\\8\\0\\-112\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-504\\-252\\0\\252\\0\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-504\\-504\\-112\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-288\\-378\\-108\\-126\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31248, 11592, -3024; 11592, 1848, -504; -3024, -504, 137],[504, 168, 1],[-1, 1, 1, 0, -2, -3, -2; -86, 72, 74, -1, -159, -241, -162; -336, 288, 294, -4, -630, -952, -639],129,62]
%
% END OF LATTICE L130.62
%
\hbox{}\par\smallskip%
%
% LATTICE L130.63
%
\leavevmode%
\llap{\spacer}%
${L_{130.63}}=\pdualwitharg{2.7}({L_{130.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{b}{2}{72}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{504}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-540456\\5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5040\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\11\\1176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5\\-546
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-972
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-322
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\107
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\0\\-336\\-504\\-168\\-504\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-72\\-56\\-504\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\0\\84\\0\\-84\\-1512\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-72\\0\\72\\0\\-1008\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-56\\-84\\0\\28\\0\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-504\\-1512\\-1008\\0\\504\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-32\\-126\\-108\\-14\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[504, 0, 0; 0, -540456, 5040; 0, 5040, -47],[504, 168, 1],[-1, -1, -3, -2, 0, 1, 0; 11, 1, -5, -9, -3, 0, 1; 1176, 104, -546, -972, -322, 0, 107],129,63]
%
% END OF LATTICE L130.63
%
\hbox{}\par\smallskip%
%
% LATTICE L130.64
%
\leavevmode%
\llap{\spacer}%
${L_{130.64}}=\pdualwitharg{2.7}({L_{130.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${336}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}{504}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2223144\\1318464\\-9072
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1318464\\781032\\-5376
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9072\\-5376\\37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\17\\3696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-1800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-826
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\682
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\0\\-168\\-504\\-168\\-504\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-72\\-56\\-504\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\21\\0\\-42\\-756\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-72\\0\\72\\0\\-1008\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-56\\-42\\0\\28\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-504\\-756\\-1008\\0\\504\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-64\\-126\\-216\\-28\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2223144, 1318464, -9072; 1318464, 781032, -5376; -9072, -5376, 37],[504, 168, 1],[5, 1, 0, -2, -1, 1, 1; 17, 3, -1, -9, -4, 0, 3; 3696, 680, -147, -1800, -826, 252, 682],129,64]
%
% END OF LATTICE L130.64
%

\medskip%
%------------------------------WEYL-GROUP-W131--------------------------------
%
\leavevmode\llap{}%
$W_{131}$%
\qquad\llap{64} lattices, $\chi=36$%
\hfill%
$2222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L131.1
%
\leavevmode%
\llap{\spacer}%
${L_{131.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{23\rightarrow N'_{20},3m,3,2,m}%
\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{b}{2}{198}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{792}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{88}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84744\\0\\792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
792\\-9\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-64\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\66\\99
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\64\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79\\5016\\8316
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\190\\316
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\568\\948
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\440\\737
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\62\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\176\\308
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-66\\-108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-2376\\-96\\-312\\-264\\-48\\-264\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\198\\0\\-3960\\-198\\-792\\-792\\-198\\-1584\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\6\\0\\-6\\-48\\-66\\-24\\-264\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2376\\-3960\\0\\792\\0\\-792\\-1584\\-792\\-11880\\-3168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-198\\-6\\0\\4\\0\\-22\\-18\\-352\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-792\\-48\\-792\\0\\24\\0\\-24\\-792\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-792\\-66\\-1584\\-22\\0\\22\\0\\-440\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-198\\-24\\-792\\-18\\-24\\0\\6\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-1584\\-264\\-11880\\-352\\-792\\-440\\0\\88\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-198\\-54\\-3168\\-108\\-288\\-198\\-18\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-84744, 0, 792; 0, 15, -9; 792, -9, -2],[792, 3, 1],[-1, 1, 1, 79, 3, 9, 7, 1, 3, -1; -64, 66, 64, 5016, 190, 568, 440, 62, 176, -66; -108, 99, 105, 8316, 316, 948, 737, 105, 308, -108],130,1]
%
% END OF LATTICE L131.1
%
\hbox{}\par\smallskip%
%
% LATTICE L131.2
%
\leavevmode%
\llap{\spacer}%
${L_{131.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3m,3,m}%
\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{}{2}{99}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{1584}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{176}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2051280\\-1018512\\1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1018512\\-505662\\783
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1584\\783\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-64\\-1008
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\33\\495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\64\\1002
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2429\\5016\\79200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\95\\1504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-275\\568\\9024
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-213\\440\\7018
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\31\\501
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85\\176\\2992
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-33\\-504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-4752\\-96\\-624\\-528\\-48\\-528\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\99\\0\\-3960\\-99\\-792\\-792\\-99\\-1584\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\12\\0\\-6\\-96\\-132\\-24\\-528\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4752\\-3960\\0\\1584\\0\\-1584\\-3168\\-792\\-23760\\-3168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-99\\-6\\0\\2\\0\\-22\\-9\\-352\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-792\\-96\\-1584\\0\\48\\0\\-24\\-1584\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-792\\-132\\-3168\\-22\\0\\44\\0\\-880\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-99\\-24\\-792\\-9\\-24\\0\\3\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-1584\\-528\\-23760\\-352\\-1584\\-880\\0\\176\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-99\\-54\\-3168\\-54\\-288\\-198\\-9\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2051280, -1018512, 1584; -1018512, -505662, 783; 1584, 783, -1],[1584, 3, 1],[31, -16, -31, -2429, -46, -275, -213, -15, -85, 16; -64, 33, 64, 5016, 95, 568, 440, 31, 176, -33; -1008, 495, 1002, 79200, 1504, 9024, 7018, 501, 2992, -504],130,2]
%
% END OF LATTICE L131.2
%
\hbox{}\par\smallskip%
%
% LATTICE L131.3
%
\leavevmode%
\llap{\spacer}%
${L_{131.3}}=\pfillwitharg{2.3}({L_{131.1}})$%
{${}={}$}Nikulin $20'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{22}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1254\\198\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
198\\-31\\11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\11\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-529\\-2640\\4224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\449\\-720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-50\\79
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-22\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-12\\-3
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7\\22\\-176
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2\\9\\-25
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1254, 198, -66; 198, -31, 11; -66, 11, -2],[66, 1, 1],[1, -4, -2, 7, 2, 11, 26, 8, 37, 2; 6, -22, -12, 22, 9, 54, 132, 42, 198, 11; 0, 11, -3, -176, -25, -96, -187, -45, -176, -7],130,3]
%
% END OF LATTICE L131.3
%
\hbox{}\par\smallskip%
%
% LATTICE L131.4
%
\leavevmode%
\llap{\spacer}%
${L_{131.4}}=\main(\pfillwitharg{3}({L_{131.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1716\\132\\396
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\-10\\-33
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-793\\-8844\\-528
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186\\2077\\123
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\12\\0
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-15\\-165\\-11
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-13\\-144\\-9
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-261\\-2904\\-176
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-27\\-301\\-18
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-396\\-220\\0\\44\\0
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-132\\-264\\-132\\-1980\\-176
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1716, 132, 396; 132, -10, -33; 396, -33, -49],[132, 1, 1],[1, -15, -13, -261, -27, -73, -51, -5, -3, 3; 12, -165, -144, -2904, -301, -816, -572, -57, -44, 33; 0, -11, -9, -176, -18, -48, -33, -3, 0, 2],130,4]
%
% END OF LATTICE L131.4
%
\hbox{}\par\smallskip%
%
% LATTICE L131.5
%
\leavevmode%
\llap{\spacer}%
${L_{131.5}}=\main(\pfillwitharg{3}({L_{131.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{88}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-503976\\-125400\\2376
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-125400\\-31202\\591
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-1189\\4752\\-2376
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-295\\1179\\-590
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5\\-20\\9
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{$\left.\llap{\phantom{%
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0\\0\\0
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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29\\-120\\-192
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-503976, -125400, 2376; -125400, -31202, 591; 2376, 591, -11],[264, 1, 1],[29, -3, -14, -237, -11, -41, -8, 11, 193, 9; -120, 11, 57, 968, 45, 168, 33, -45, -792, -37; -192, -66, 36, 792, 41, 168, 44, -42, -880, -45],130,5]
%
% END OF LATTICE L131.5
%
\hbox{}\par\smallskip%
%
% LATTICE L131.6
%
\leavevmode%
\llap{\spacer}%
${L_{131.6}}=\pfillwitharg{3}({L_{131.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{88}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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5\\1206\\657
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3\\704\\396
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-726\\-394
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-24\\0\\6\\0\\-6
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-312\\-264\\-48\\-264\\0
\end{tabular}\endgroup%
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-48\\-66\\-24\\-264\\-18
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-528\\-264\\-3960\\-352
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-4080648, 5544, 20856; 5544, -5, -33; 20856, -33, -98],[264, 1, 1],[73, 51, 5, 3, -3, -1, 15, 13, 261, 27; 17640, 12320, 1206, 704, -726, -240, 3630, 3144, 63096, 6526; 9588, 6699, 657, 396, -394, -132, 1969, 1707, 34276, 3546],130,6]
%
% END OF LATTICE L131.6
%
\hbox{}\par\smallskip%
%
% LATTICE L131.7
%
\leavevmode%
\llap{\spacer}%
${L_{131.7}}=\pfillwitharg{3}({L_{131.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{176}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{176}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-475728\\-155760\\6864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-155760\\-50998\\2247
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6864\\2247\\-37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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55\\-168\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1145\\-3498\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
436\\-1332\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16073\\-49104\\-352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
799\\-2441\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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4085\\-12480\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1325\\-4048\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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163\\-498\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-893\\2728\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109\\333\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-96\\-66\\-3\\0\\2\\0\\-11\\-18\\-352\\-18
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-528\\-48\\-528\\0\\48\\0\\-48\\-1584\\-96
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-132\\-24\\-528\\-18\\-48\\0\\12\\0\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-1056\\-264\\-7920\\-352\\-1584\\-440\\0\\176\\0
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0\\-22\\-9\\-352\\-18\\-96\\-33\\-6\\0\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-475728, -155760, 6864; -155760, -50998, 2247; 6864, 2247, -37],[528, 1, 1],[55, 1145, 436, 16073, 799, 4085, 1325, 163, -893, -109; -168, -3498, -1332, -49104, -2441, -12480, -4048, -498, 2728, 333; 0, -22, -9, -352, -18, -96, -33, -6, 0, 2],130,7]
%
% END OF LATTICE L131.7
%
\hbox{}\par\smallskip%
%
% LATTICE L131.8
%
\leavevmode%
\llap{\spacer}%
${L_{131.8}}=\pfillwitharg{2}({L_{131.1}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{r}{2}{198}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{198}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{22}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-173250\\-14256\\1386
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14256\\-1173\\114
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1386\\114\\-11
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-3\\37\\3
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-47\\594\\198
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-1\\13\\8
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-1\\14\\18
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1\\-11\\11
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1\\-13\\-9
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\HardButStrongLineBreak\kern3pt%
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1\\-22\\-110
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\-36
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-24\\-99\\-3\\0\\1\\0\\-11\\-9\\-88\\-27
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-78\\-396\\-24\\-198\\0\\6\\0\\-12\\-198\\-72
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-132\\-792\\-66\\-792\\-11\\0\\22\\0\\-220\\-99
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-792\\-132\\-2970\\-88\\-198\\-220\\0\\22\\0
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0\\-99\\-27\\-792\\-27\\-72\\-99\\-9\\0\\9
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-173250, -14256, 1386; -14256, -1173, 114; 1386, 114, -11],[198, 3, 1],[-3, -25, -3, -47, -1, -1, 1, 1, 1, -1; 34, 297, 37, 594, 13, 14, -11, -13, -22, 9; -30, -99, 3, 198, 8, 18, 11, -9, -110, -36],130,8]
%
% END OF LATTICE L131.8
%
\hbox{}\par\smallskip%
%
% LATTICE L131.9
%
\leavevmode%
\llap{\spacer}%
${L_{131.9}}=\main({L_{131.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{99}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{396}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{44}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42372\\0\\396
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\-9
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
396\\-9\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1\\-32\\-108
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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1\\32\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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79\\2508\\8316
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\HardButStrongLineBreak\kern3pt%
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3\\95\\316
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\HardButStrongLineBreak\kern3pt%
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9\\284\\948
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\HardButStrongLineBreak\kern3pt%
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7\\220\\737
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\31\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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3\\88\\308
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-33\\-108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-1188\\-48\\-156\\-132\\-24\\-132\\0
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-42372, 0, 396; 0, 30, -9; 396, -9, -1],[396, 3, 1],[-1, 1, 1, 79, 3, 9, 7, 1, 3, -1; -32, 33, 32, 2508, 95, 284, 220, 31, 88, -33; -108, 99, 105, 8316, 316, 948, 737, 105, 308, -108],130,9]
%
% END OF LATTICE L131.9
%
\hbox{}\par\smallskip%
%
% LATTICE L131.10
%
\leavevmode%
\llap{\spacer}%
${L_{131.10}}=\main({L_{131.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47520\\-4575\\228
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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-3\\35\\75
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\HardButStrongLineBreak\kern3pt%
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-137\\1584\\3168
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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0\\-99\\-27\\-1584\\-27\\-144\\-99\\-9\\0\\9
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-493416, -47520, 2376; -47520, -4575, 228; 2376, 228, -11],[792, 3, 1],[3, -8, -3, -137, -2, -7, -1, 2, 31, 4; -32, 99, 35, 1584, 23, 80, 11, -23, -352, -45; -24, 297, 75, 3168, 44, 144, 11, -45, -616, -72],130,10]
%
% END OF LATTICE L131.10
%
\hbox{}\par\smallskip%
%
% LATTICE L131.11
%
\leavevmode%
\llap{\spacer}%
${L_{131.11}}=\pdualwitharg{2}(\pfillwitharg{2.3}({L_{131.1}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-2394546\\82500\\-1200936
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82500\\-2842\\41376
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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103\\-111\\-213
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2394546, 82500, -1200936; 82500, -2842, 41376; -1200936, 41376, -602305],[66, 2, 1],[103, -3301, -3409, -19861, -4364, -6421, -19865, -2909, -3305, 152; -111, 3465, 3591, 20955, 4607, 6783, 20999, 3081, 3509, -159; -213, 6820, 7044, 41041, 9018, 13269, 41052, 6012, 6831, -314],130,11]
%
% END OF LATTICE L131.11
%
\hbox{}\par\smallskip%
%
% LATTICE L131.12
%
\leavevmode%
\llap{\spacer}%
${L_{131.12}}=\pdualwitharg{3}(\pfillwitharg{2.3}({L_{131.1}}))$%
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{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3837570, 127908, -1282974; 127908, -4263, 42762; -1282974, 42762, -428923],[66, 3, 1],[43, -2102, -722, -25213, -2769, -2715, -12594, -614, -4179, 97; -74, 3465, 1197, 41910, 4607, 4522, 20999, 1027, 7018, -159; -136, 6633, 2279, 79596, 8742, 8572, 39765, 1939, 13200, -306],130,12]
%
% END OF LATTICE L131.12
%
\hbox{}\par\smallskip%
%
% LATTICE L131.13
%
\leavevmode%
\llap{\spacer}%
${L_{131.13}}=\pdualwitharg{3}(\main(\pfillwitharg{3}({L_{131.1}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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17\\-16\\548
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40\\-33\\1287
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25\\0\\792
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-63228, 1056, 1980; 1056, -15, -33; 1980, -33, -62],[132, 3, 1],[17, 40, 2, 25, -1, -1, 4, 2, 151, 17; -16, -33, -1, 0, 2, 0, -11, -3, -176, -18; 548, 1287, 64, 792, -33, -32, 132, 65, 4884, 549],130,13]
%
% END OF LATTICE L131.13
%
\hbox{}\par\smallskip%
%
% LATTICE L131.14
%
\leavevmode%
\llap{\spacer}%
${L_{131.14}}=\pdualwitharg{2.3}(\pfillwitharg{2.3}({L_{131.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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-767844\\22506\\-376800
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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1880092792\\25513712\\-3829734625
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{$\left[\!\llap{\phantom{%
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158245\\2145\\-322344
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54335\\737\\-110680
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209070\\2837\\-425874
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26290770, -767844, 12901548; -767844, 22506, -376800; 12901548, -376800, 6331117],[66, 6, 1],[-1541, 158245, 54335, 950977, 209070, 102643, 953877, 46767, 161145, -6866; -21, 2145, 737, 12903, 2837, 1393, 12947, 635, 2189, -93; 3139, -322344, -110680, -1937133, -425874, -209083, -1943040, -95264, -328251, 13986],130,14]
%
% END OF LATTICE L131.14
%
\hbox{}\par\smallskip%
%
% LATTICE L131.15
%
\leavevmode%
\llap{\spacer}%
${L_{131.15}}=\pdualwitharg{2}(\main(\pfillwitharg{3}({L_{131.1}})))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[220440, 22044, 55440; 22044, 2204, 5544; 55440, 5544, 13943],[132, 4, 1],[8, 52, 19, 37, 5, -5, -41, -16, -26, -1; 3, 33, 21, 99, 40, 27, 77, 9, 11, 0; -33, -220, -84, -187, -36, 9, 132, 60, 99, 4],130,15]
%
% END OF LATTICE L131.15
%
\hbox{}\par\smallskip%
%
% LATTICE L131.16
%
\leavevmode%
\llap{\spacer}%
${L_{131.16}}=\pdualwitharg{11}(\pfillwitharg{2.3}({L_{131.1}}))$%
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{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-22517682, 52602, 4040190; 52602, -121, -9438; 4040190, -9438, -724903],[66, 11, 1],[-521, 739, 8425, 8939, 10809, 31829, 4479, 7229, 1497, -373; 18, -17, -207, -226, -276, -822, -117, -195, -42, 8; -2904, 4119, 46959, 49824, 60247, 177408, 24965, 40293, 8344, -2079],130,16]
%
% END OF LATTICE L131.16
%
\hbox{}\par\smallskip%
%
% LATTICE L131.17
%
\leavevmode%
\llap{\spacer}%
${L_{131.17}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{131.1}}))$%
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{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-370392, 3432, 3432; 3432, -30, -33; 3432, -33, -31],[264, 3, 1],[-1, 4, 2, 151, 17, 17, 40, 2, 25, -1; -44, 165, 85, 6468, 730, 732, 1727, 87, 1100, -42; -64, 264, 130, 9768, 1098, 1096, 2574, 128, 1584, -66],130,17]
%
% END OF LATTICE L131.17
%
\hbox{}\par\smallskip%
%
% LATTICE L131.18
%
\leavevmode%
\llap{\spacer}%
${L_{131.18}}=\pdualwitharg{3}(\main(\pfillwitharg{3}({L_{131.2}})))$%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1848, 528, 264; 528, -150, -69; 264, -69, 11],[264, 3, 1],[13, 74, 11, 401, 15, 15, 3, -4, -93, -1; 48, 275, 41, 1496, 56, 56, 11, -15, -352, -4; -16, -66, -8, -264, -9, -8, 0, 2, 0, -3],130,18]
%
% END OF LATTICE L131.18
%
\hbox{}\par\smallskip%
%
% LATTICE L131.19
%
\leavevmode%
\llap{\spacer}%
${L_{131.19}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{131.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
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229482\\-5304\\113711
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1593\\-89\\-3219
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19499\\-1089\\-39402
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1737\\-97\\-3510
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\HardButStrongLineBreak\kern3pt%
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12395\\-693\\-25047
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482\\-27\\-974
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577\\-33\\-1166
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193\\-11\\-390
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2945\\-165\\-5951
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2414\\-135\\-4878
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{$\left.\llap{\phantom{%
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0\\0\\0
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\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[463122, -10692, 229482; -10692, 462, -5304; 229482, -5304, 113711],[198, 6, 1],[1593, 19499, 1737, 12395, 482, 337, 577, 193, 2945, 2414; -89, -1089, -97, -693, -27, -19, -33, -11, -165, -135; -3219, -39402, -3510, -25047, -974, -681, -1166, -390, -5951, -4878],130,19]
%
% END OF LATTICE L131.19
%
\hbox{}\par\smallskip%
%
% LATTICE L131.20
%
\leavevmode%
\llap{\spacer}%
${L_{131.20}}=\pdualwitharg{11}(\main(\pfillwitharg{3}({L_{131.1}})))$%
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{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[132, 0, 0; 0, -3322, 341; 0, 341, -35],[132, 11, 1],[-13, -1, -2, -1, 0, 1, 0, -1, -3, -4; 84, 4, -3, -8, -9, 0, 3, 24, 36, 37; 792, 37, -33, -80, -88, 0, 29, 231, 344, 352],130,20]
%
% END OF LATTICE L131.20
%
\hbox{}\par\smallskip%
%
% LATTICE L131.21
%
\leavevmode%
\llap{\spacer}%
${L_{131.21}}=\pdualwitharg{2.11}(\pfillwitharg{2.3}({L_{131.1}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[52207386, 122892, 24869988; 122892, 418, 58542; 24869988, 58542, 11847295],[66, 22, 1],[8033, -24855, -281831, -149523, -361637, -532709, -150037, -242845, -25369, 11853; 6, -17, -195, -104, -252, -372, -105, -171, -18, 8; -16863, 52176, 591624, 313881, 759154, 1118271, 314960, 509784, 53255, -24882],130,21]
%
% END OF LATTICE L131.21
%
\hbox{}\par\smallskip%
%
% LATTICE L131.22
%
\leavevmode%
\llap{\spacer}%
${L_{131.22}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{131.2}}))$%
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{$1\above{1pt}{1pt}{-2}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-50160, 1056, 1584; 1056, -15, -33; 1584, -33, -50],[528, 3, 1],[-1, 4, 3, 203, 11, 21, 47, 1, 17, -1; 0, -11, -6, -352, -18, -32, -66, -1, 0, 2; -32, 132, 98, 6600, 357, 680, 1518, 32, 528, -33],130,22]
%
% END OF LATTICE L131.22
%
\hbox{}\par\smallskip%
%
% LATTICE L131.23
%
\leavevmode%
\llap{\spacer}%
${L_{131.23}}=\pdualwitharg{2.3}(\main(\pfillwitharg{3}({L_{131.1}})))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1408836, 3828, 340032; 3828, 24, 924; 340032, 924, 82069],[132, 12, 1],[355, 3313, 167, 661, 0, 7, 669, 195, 3321, 1448; 9, 88, 5, 22, 1, 0, 11, 4, 77, 35; -1471, -13728, -692, -2739, 0, -29, -2772, -808, -13761, -6000],130,23]
%
% END OF LATTICE L131.23
%
\hbox{}\par\smallskip%
%
% LATTICE L131.24
%
\leavevmode%
\llap{\spacer}%
${L_{131.24}}=\pdualwitharg{2}(\main(\pfillwitharg{3}({L_{131.2}})))$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38544, -16632, 4488; -16632, -6872, 1592; 4488, 1592, -133],[264, 8, 1],[-61, -1011, -463, -690, -202, -68, 65, 205, 217, 30; 186, 3080, 1410, 2101, 615, 207, -198, -624, -660, -91; 165, 2728, 1248, 1859, 544, 183, -176, -552, -583, -80],130,24]
%
% END OF LATTICE L131.24
%
\hbox{}\par\smallskip%
%
% LATTICE L131.25
%
\leavevmode%
\llap{\spacer}%
${L_{131.25}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{131.1}}))$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[31152, 2904, 1056; 2904, 248, 88; 1056, 88, 31],[264, 8, 1],[-1, -1, 1, 5, 1, 1, 1, -1, -5, -1; 66, 110, -36, -220, -47, -48, -44, 54, 286, 59; -150, -264, 72, 462, 100, 102, 88, -120, -638, -132],130,25]
%
% END OF LATTICE L131.25
%
\hbox{}\par\smallskip%
%
% LATTICE L131.26
%
\leavevmode%
\llap{\spacer}%
${L_{131.26}}=\pdualwitharg{3.11}(\pfillwitharg{2.3}({L_{131.1}}))$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17700474, -87252, -5371806; -87252, 627, 26598; -5371806, 26598, 1630327],[66, 33, 1],[-2433, 11292, 42680, 135861, 164297, 161345, 68164, 36776, 23051, -5385; 4870, -22601, -85425, -271930, -328846, -322938, -136433, -73609, -46138, 10778; -8096, 37575, 142021, 452088, 546711, 536888, 226821, 122375, 76704, -17919],130,26]
%
% END OF LATTICE L131.26
%
\hbox{}\par\smallskip%
%
% LATTICE L131.27
%
\leavevmode%
\llap{\spacer}%
${L_{131.27}}=\pdualwitharg{11}(\pfillwitharg{3}({L_{131.1}}))$%
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{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1219416, -45144, 9240; -45144, -1595, 341; 9240, 341, -70],[264, 11, 1],[-1, 3, 25, 39, 41, 97, 5, -1, -7, -9; 0, 6, 48, 72, 74, 168, 8, -6, -16, -18; -132, 425, 3531, 5492, 5764, 13596, 697, -165, -1004, -1276],130,27]
%
% END OF LATTICE L131.27
%
\hbox{}\par\smallskip%
%
% LATTICE L131.28
%
\leavevmode%
\llap{\spacer}%
${L_{131.28}}=\pdualwitharg{11}(\main(\pfillwitharg{3}({L_{131.2}})))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[264, 0, 0; 0, -77, 44; 0, 44, -25],[264, 11, 1],[1, 0, -1, -3, -2, -13, -1, -2, -1, 0; 0, 3, 21, 24, 10, 24, -1, -15, -16, -6; 0, 5, 33, 32, 11, 0, -5, -33, -32, -11],130,28]
%
% END OF LATTICE L131.28
%
\hbox{}\par\smallskip%
%
% LATTICE L131.29
%
\leavevmode%
\llap{\spacer}%
${L_{131.29}}=\pdualwitharg{3.11}(\main(\pfillwitharg{3}({L_{131.1}})))$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10428, 3564, 1056; 3564, 1023, 297; 1056, 297, 86],[132, 33, 1],[-3, -1, -1, -1, 1, 3, 1, 1, 1, -1; 100, 35, 39, 68, 8, -60, -25, -29, -28, 32; -308, -108, -121, -216, -33, 176, 75, 88, 84, -99],130,29]
%
% END OF LATTICE L131.29
%
\hbox{}\par\smallskip%
%
% LATTICE L131.30
%
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\llap{\spacer}%
${L_{131.30}}=\pdualwitharg{2.3.11}(\pfillwitharg{2.3}({L_{131.1}}))$%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-44\\-99\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-60\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\0\\44\\0\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-60\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-198\\-66\\0\\66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-143\\-132\\-88\\-33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-144\\-132\\-72\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\-132\\-176\\-132\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-72\\-132\\-135\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-66\\-198\\-264\\-594
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1260973362, -836381172, -386427096; -836381172, -554756730, -256310208; -386427096, -256310208, -118421137],[66, 66, 1],[3, -17, -69, -113, -276, -137, -117, -65, -21, 8; 2623, -22595, -85725, -136331, -329615, -161719, -136501, -73371, -22765, 11423; -5687, 48960, 185768, 295443, 714318, 350471, 295824, 159016, 49341, -24750],130,30]
%
% END OF LATTICE L131.30
%
\hbox{}\par\smallskip%
%
% LATTICE L131.31
%
\leavevmode%
\llap{\spacer}%
${L_{131.31}}=\pdualwitharg{2}(\main({L_{131.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{396}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{99}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{44}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38870568\\-367092\\-9819612
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-367092\\3468\\92736
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9819612\\92736\\2480663
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\1\\-99
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800\\33\\-3168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-197\\5\\-780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3326\\33\\-13167
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-483\\2\\-1912
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-1\\-1425
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1145\\-11\\-4532
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-188\\-5\\-744
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-303\\-11\\-1199
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-91\\-6\\-360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-297\\-48\\-39\\-132\\-24\\-33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\396\\0\\-1980\\-396\\-396\\-1584\\-396\\-792\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-12\\-24\\-132\\-48\\-132\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-297\\-1980\\0\\99\\0\\-99\\-792\\-396\\-1485\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-396\\-12\\0\\8\\0\\-44\\-36\\-176\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-396\\-24\\-99\\0\\3\\0\\-12\\-99\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-1584\\-132\\-792\\-44\\0\\44\\0\\-220\\-396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-396\\-48\\-396\\-36\\-12\\0\\12\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-792\\-132\\-1485\\-176\\-99\\-220\\0\\11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-396\\-108\\-1584\\-216\\-144\\-396\\-36\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[38870568, -367092, -9819612; -367092, 3468, 92736; -9819612, 92736, 2480663],[396, 12, 1],[-25, -800, -197, -3326, -483, -360, -1145, -188, -303, -91; 1, 33, 5, 33, 2, -1, -11, -5, -11, -6; -99, -3168, -780, -13167, -1912, -1425, -4532, -744, -1199, -360],130,31]
%
% END OF LATTICE L131.31
%
\hbox{}\par\smallskip%
%
% LATTICE L131.32
%
\leavevmode%
\llap{\spacer}%
${L_{131.32}}=\pdualwitharg{3}({L_{131.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{396}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{1584}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{176}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
shares genus with 3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3914064\\30096\\727056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30096\\-150\\-7173
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
727056\\-7173\\-104293
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-177\\-10264\\-528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3385\\-196284\\-10098
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-357\\-20701\\-1065
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32389\\-1878096\\-96624
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-478\\-27717\\-1426
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1979\\-114752\\-5904
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-424\\-24585\\-1265
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
175\\10148\\522
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2891\\167640\\8624
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
356\\20643\\1062
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-24\\-4752\\-96\\-624\\-264\\-96\\-528\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\396\\0\\-7920\\-198\\-1584\\-792\\-396\\-3168\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\3\\0\\-3\\-48\\-33\\-24\\-264\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4752\\-7920\\0\\1584\\0\\-1584\\-1584\\-1584\\-23760\\-3168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-198\\-3\\0\\2\\0\\-11\\-18\\-352\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-1584\\-48\\-1584\\0\\48\\0\\-48\\-1584\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-792\\-33\\-1584\\-11\\0\\11\\0\\-440\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-396\\-24\\-1584\\-18\\-48\\0\\12\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-3168\\-264\\-23760\\-352\\-1584\\-440\\0\\176\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-198\\-27\\-3168\\-54\\-288\\-99\\-18\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3914064, 30096, 727056; 30096, -150, -7173; 727056, -7173, -104293],[1584, 3, 1],[-177, -3385, -357, -32389, -478, -1979, -424, 175, 2891, 356; -10264, -196284, -20701, -1878096, -27717, -114752, -24585, 10148, 167640, 20643; -528, -10098, -1065, -96624, -1426, -5904, -1265, 522, 8624, 1062],130,32]
%
% END OF LATTICE L131.32
%
\hbox{}\par\smallskip%
%
% LATTICE L131.33
%
\leavevmode%
\llap{\spacer}%
${L_{131.33}}=\pdualwitharg{11}(\pfillwitharg{3}({L_{131.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${528}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{132}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{b}{2}{528}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6602640\\1922976\\25344
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1922976\\560054\\7381
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25344\\7381\\97
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
581\\-2016\\1584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-52\\37
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\18\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-39\\136\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\87\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-24\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-66\\58
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\-264\\231
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
235\\-816\\688
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
123\\-427\\352
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\0\\-528\\-1584\\-1056\\-6864\\-528\\-528\\-528\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-40\\-33\\-264\\-24\\-33\\-48\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\0\\132\\0\\-66\\-1056\\-132\\-264\\-528\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1584\\-40\\0\\16\\0\\-528\\-96\\-264\\-720\\-352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1056\\-33\\-66\\0\\22\\0\\-22\\-99\\-352\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6864\\-264\\-1056\\-528\\0\\528\\0\\-264\\-1584\\-1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-24\\-132\\-96\\-22\\0\\4\\0\\-80\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-33\\-264\\-264\\-99\\-264\\0\\33\\0\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-48\\-528\\-720\\-352\\-1584\\-80\\0\\16\\0
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6602640, 1922976, 25344; 1922976, 560054, 7381; 25344, 7381, 97],[528, 11, 1],[581, 15, -5, -39, -25, 7, 19, 76, 235, 123; -2016, -52, 18, 136, 87, -24, -66, -264, -816, -427; 1584, 37, -66, -160, -88, 0, 58, 231, 688, 352],130,33]
%
% END OF LATTICE L131.33
%
\hbox{}\par\smallskip%
%
% LATTICE L131.34
%
\leavevmode%
\llap{\spacer}%
${L_{131.34}}=\pdualwitharg{2.11}(\main(\pfillwitharg{3}({L_{131.1}})))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1599972, 49632, -351780; 49632, 1540, -10912; -351780, -10912, 77345],[132, 44, 1],[127, 37, 68, 10, 11, 8, 8, 67, 35, 169; -585, -172, -321, -46, -43, -24, -31, -288, -157, -769; 495, 144, 264, 39, 44, 33, 32, 264, 137, 660],130,34]
%
% END OF LATTICE L131.34
%
\hbox{}\par\smallskip%
%
% LATTICE L131.35
%
\leavevmode%
\llap{\spacer}%
${L_{131.35}}=\pdualwitharg{2.3}(\main(\pfillwitharg{3}({L_{131.2}})))$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L131.35
%
\hbox{}\par\smallskip%
%
% LATTICE L131.36
%
\leavevmode%
\llap{\spacer}%
${L_{131.36}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{131.1}}))$%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-528\\-64\\-528\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-3168\\-528\\-5940\\-1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-264\\-72\\-1056\\-216
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[528, -2376, -1056; -2376, 10248, 4560; -1056, 4560, 2029],[264, 24, 1],[1, 1, -1, -5, 0, 1, 21, 3, 27, 4; -6, 0, 14, 264, 59, 28, 242, 8, -22, -15; 14, 0, -32, -594, -132, -62, -528, -16, 66, 36],130,36]
%
% END OF LATTICE L131.36
%
\hbox{}\par\smallskip%
%
% LATTICE L131.37
%
\leavevmode%
\llap{\spacer}%
${L_{131.37}}=\pdualwitharg{11}(\pfillwitharg{2}({L_{131.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${66}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{99}\above{1pt}{1pt}{}{2}{66}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
990\\-396\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\165\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\2\\-132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\5\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
66\\0\\-132\\-198\\-792\\-858\\-396\\-264\\-198\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-20\\-99\\-132\\-72\\-66\\-72\\-11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\0\\66\\0\\-99\\-264\\-198\\-264\\-396\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-20\\0\\2\\0\\-66\\-72\\-132\\-270\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-792\\-99\\-99\\0\\99\\0\\-99\\-297\\-792\\-297
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-858\\-132\\-264\\-66\\0\\66\\0\\-132\\-594\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-72\\-198\\-72\\-99\\0\\18\\0\\-180\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-66\\-264\\-132\\-297\\-132\\0\\66\\0\\-33
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-72\\-396\\-270\\-792\\-594\\-180\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\-99\\-88\\-297\\-264\\-99\\-33\\0\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[990, -396, 0; -396, 165, 0; 0, 0, -1],[198, 33, 1],[5, 1, 3, 1, 1, -1, -1, -1, 1, 1; 2, 1, 5, 2, 3, -2, -3, -5, -6, -1; -132, -19, -33, -8, 0, 0, -9, -33, -108, -44],130,37]
%
% END OF LATTICE L131.37
%
\hbox{}\par\smallskip%
%
% LATTICE L131.38
%
\leavevmode%
\llap{\spacer}%
${L_{131.38}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{131.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{176}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{32}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{176}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}{32}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
292512\\25872\\-5280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25872\\2288\\-464
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5280\\-464\\67
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-81\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\11\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\231\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\704\\66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\196\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-12\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\-649\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\-405\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\-671\\-77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-428\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-24\\-198\\-96\\-78\\-264\\-48\\-33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\176\\0\\-880\\-528\\-528\\-2112\\-528\\-528\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\48\\0\\-48\\-96\\-528\\-192\\-264\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-880\\0\\44\\0\\-132\\-1056\\-528\\-990\\-704
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-528\\-48\\0\\32\\0\\-176\\-144\\-352\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-528\\-96\\-132\\0\\12\\0\\-48\\-198\\-192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-2112\\-528\\-1056\\-176\\0\\176\\0\\-440\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-528\\-192\\-528\\-144\\-48\\0\\48\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-528\\-264\\-990\\-352\\-198\\-440\\0\\11\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-176\\-144\\-704\\-288\\-192\\-528\\-48\\0\\32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[292512, 25872, -5280; 25872, 2288, -464; -5280, -464, 67],[528, 16, 1],[7, -1, -20, -61, -17, 1, 56, 35, 58, 37; -81, 11, 231, 704, 196, -12, -649, -405, -671, -428; -9, 0, 24, 66, 16, -6, -88, -48, -77, -48],130,38]
%
% END OF LATTICE L131.38
%
\hbox{}\par\smallskip%
%
% LATTICE L131.39
%
\leavevmode%
\llap{\spacer}%
${L_{131.39}}=\pdualwitharg{3.11}(\pfillwitharg{3}({L_{131.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${88}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{22}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20856\\22968\\1848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22968\\23430\\1881
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1848\\1881\\151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-384\\4752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-593\\7326
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\593
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-52\\616
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-25\\300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\-31\\374
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-28\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\38\\-462
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\0\\-88\\-792\\-1056\\-1144\\-264\\-176\\-264\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-120\\-198\\-264\\-72\\-66\\-144\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\0\\22\\0\\-66\\-176\\-66\\-88\\-264\\-198
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-792\\-120\\0\\24\\0\\-264\\-144\\-264\\-1080\\-1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1056\\-198\\-66\\0\\132\\0\\-66\\-198\\-1056\\-1188
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\0\\-88\\-792\\-1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-120\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\0\\22\\0\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-792\\-120\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1056\\-198\\-66\\0\\132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1144\\-264\\-176\\-264\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-72\\-66\\-144\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-66\\-88\\-264\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-144\\-264\\-1080\\-1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-66\\-198\\-1056\\-1188
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20856, 22968, 1848; 22968, 23430, 1881; 1848, 1881, 151],[264, 33, 1],[3, 1, 1, 1, -1, -3, -1, -1, -1, 1; -52, -25, -31, -28, 38, 116, 41, 47, 92, 26; 616, 300, 374, 336, -462, -1408, -498, -572, -1128, -330],130,39]
%
% END OF LATTICE L131.39
%
\hbox{}\par\smallskip%
%
% LATTICE L131.40
%
\leavevmode%
\llap{\spacer}%
${L_{131.40}}=\pdualwitharg{3.11}(\main(\pfillwitharg{3}({L_{131.2}})))$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-264\\-72\\-66\\-144\\-33
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-176\\-66\\-88\\-264\\-99
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\kern3pt%
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-264\\-144\\-264\\-1080\\-528
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[642840, 0, -7392; 0, 33, 0; -7392, 0, 85],[264, 33, 1],[75, 16, 9, 13, 0, 1, 3, 10, 63, 38; -16, -3, -1, 0, 1, 0, -1, -3, -16, -9; 6512, 1389, 781, 1128, 0, 88, 261, 869, 5472, 3300],130,40]
%
% END OF LATTICE L131.40
%
\hbox{}\par\smallskip%
%
% LATTICE L131.41
%
\leavevmode%
\llap{\spacer}%
${L_{131.41}}=\pdualwitharg{11}(\main({L_{131.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{r}{2}{22}\above{1pt}{1pt}{l}{2}{132}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{198}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1277892\\-143748\\7920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-143748\\16170\\-891
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7920\\-891\\49
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-80\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-36\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-55\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-180\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-49\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-7\\-76\\-264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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0\\-1\\-19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2\\16\\-33
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\HardButStrongLineBreak\kern3pt%
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1\\8\\-16
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-132\\-1188\\-528\\-1716\\-132\\-264\\-132\\0
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-528\\-99\\-33\\0\\22\\0\\-11\\-99\\-176\\-594
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\kern3pt%
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-1716\\-396\\-264\\-396\\0\\132\\0\\-132\\-396\\-1584
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\kern3pt%
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-132\\-36\\-33\\-72\\-11\\0\\1\\0\\-20\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-72\\-132\\-540\\-176\\-396\\-20\\0\\4\\0
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0\\-99\\-297\\-1584\\-594\\-1584\\-99\\-99\\0\\198
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1277892, -143748, 7920; -143748, 16170, -891; 7920, -891, 49],[396, 33, 1],[-9, -4, -6, -19, -5, -7, 0, 2, 1, -1; -80, -36, -55, -180, -49, -76, -1, 16, 8, -9; 0, -9, -33, -216, -88, -264, -19, -33, -16, 0],130,41]
%
% END OF LATTICE L131.41
%
\hbox{}\par\smallskip%
%
% LATTICE L131.42
%
\leavevmode%
\llap{\spacer}%
${L_{131.42}}=\pdualwitharg{2.11}(\pfillwitharg{2}({L_{131.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{132}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{198}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{s}{2}{132}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{22}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17622\\792\\-8712
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8712\\396\\-4307
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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-53\\-1\\107
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98\\3\\-198
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49\\1\\-99
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-107\\-3\\216
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-523\\-13\\1056
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\HardButStrongLineBreak\kern3pt%
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-887\\-21\\1791
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-730\\-17\\1474
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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0\\4\\0\\-20\\-198\\-132\\-144\\-132\\-72\\-22
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-132\\0\\132\\0\\-198\\-264\\-396\\-528\\-396\\-198
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-20\\0\\1\\0\\-33\\-72\\-132\\-135\\-88
\end{tabular}\endgroup%
\kern3pt%
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-429\\-132\\-264\\-33\\0\\33\\0\\-132\\-297\\-264
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-144\\-396\\-72\\-198\\0\\36\\0\\-180\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-132\\-528\\-132\\-594\\-132\\0\\132\\0\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-72\\-396\\-135\\-792\\-297\\-180\\0\\9\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-22\\-198\\-88\\-594\\-264\\-198\\-66\\0\\22
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17622, 792, -8712; 792, 330, 396; -8712, 396, -4307],[198, 66, 1],[-1095, -313, -523, -53, 98, 49, -107, -523, -887, -730; -25, -7, -11, -1, 3, 1, -3, -13, -21, -17; 2211, 632, 1056, 107, -198, -99, 216, 1056, 1791, 1474],130,42]
%
% END OF LATTICE L131.42
%
\hbox{}\par\smallskip%
%
% LATTICE L131.43
%
\leavevmode%
\llap{\spacer}%
${L_{131.43}}=\pdualwitharg{3.11}(\pfillwitharg{3}({L_{131.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${176}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}{66}\above{1pt}{1pt}{l}{2}{176}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{44}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
297264\\167376\\-13200
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
167376\\94083\\-7326
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13200\\-7326\\515
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[297264, 167376, -13200; 167376, 94083, -7326; -13200, -7326, 515],[528, 33, 1],[-377, -137, -73, -143, 77, 469, 80, 169, 235, -31; 760, 276, 147, 288, -155, -944, -161, -340, -472, 63; 1144, 414, 220, 432, -231, -1408, -240, -506, -696, 99],130,43]
%
% END OF LATTICE L131.43
%
\hbox{}\par\smallskip%
%
% LATTICE L131.44
%
\leavevmode%
\llap{\spacer}%
${L_{131.44}}=\pdualwitharg{2.3.11}(\main(\pfillwitharg{3}({L_{131.1}})))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[218328, 637956, 154440; 637956, 1863708, 451176; 154440, 451176, 109223],[132, 132, 1],[1, 5, 12, 14, 59, 12, 8, 1, -1, -7; -3, -115, -313, -373, -1586, -321, -209, -11, 49, 290; 11, 468, 1276, 1521, 6468, 1309, 852, 44, -201, -1188],130,44]
%
% END OF LATTICE L131.44
%
\hbox{}\par\smallskip%
%
% LATTICE L131.45
%
\leavevmode%
\llap{\spacer}%
${L_{131.45}}=\pdualwitharg{2}(\main({L_{131.2}}))$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[286704, -21384, 7920; -21384, -600, 288; 7920, 288, -133],[792, 24, 1],[1, -1, -3, -3, -1, 1, 47, 7, 31, 3; 179, -176, -538, -550, -201, 172, 8250, 1240, 5511, 535; 447, -440, -1344, -1375, -504, 429, 20592, 3096, 13761, 1336],130,45]
%
% END OF LATTICE L131.45
%
\hbox{}\par\smallskip%
%
% LATTICE L131.46
%
\leavevmode%
\llap{\spacer}%
${L_{131.46}}=\pdualwitharg{2}({L_{131.1}})$%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7055928, 598752, 58608; 598752, 50808, 4968; 58608, 4968, 455],[792, 24, 1],[51, 45, -59, -313, -194, -73, -377, 37, 697, 50; -611, -539, 707, 3751, 2325, 875, 4521, -443, -8349, -599; 102, 88, -120, -638, -396, -150, -792, 72, 1386, 100],130,46]
%
% END OF LATTICE L131.46
%
\hbox{}\par\smallskip%
%
% LATTICE L131.47
%
\leavevmode%
\llap{\spacer}%
${L_{131.47}}=\pdualwitharg{2.11}(\pfillwitharg{3}({L_{131.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{528}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{88}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-264\\0\\5
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\1\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-1\\-16\\-56
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-5\\-78\\-264
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5808, 1320, -264; 1320, -88, 0; -264, 0, 5],[264, 88, 1],[-1, -1, -5, -1, -1, 1, 1, 5, 1, 1; -18, -16, -78, -16, -17, 12, 14, 72, 14, 13; -66, -56, -264, -50, -44, 66, 56, 264, 50, 44],130,47]
%
% END OF LATTICE L131.47
%
\hbox{}\par\smallskip%
%
% LATTICE L131.48
%
\leavevmode%
\llap{\spacer}%
${L_{131.48}}=\pdualwitharg{2.11}(\main(\pfillwitharg{3}({L_{131.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25080, -25080, 792; -25080, 25168, -792; 792, -792, 25],[264, 88, 1],[-1, -4, -34, -8, -37, -26, -14, -20, -1, 1; 0, -1, -9, -2, -9, -6, -3, -3, 0, 1; 33, 96, 792, 189, 880, 627, 344, 528, 31, 0],130,48]
%
% END OF LATTICE L131.48
%
\hbox{}\par\smallskip%
%
% LATTICE L131.49
%
\leavevmode%
\llap{\spacer}%
${L_{131.49}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{131.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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139\\-33\\-1056
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8\\-1\\-40
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-16\\-198\\-192\\-26\\-528\\-32\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\528\\0\\-1320\\-1584\\-264\\-6336\\-528\\-3168\\-528
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\16\\0\\-48\\-16\\-528\\-64\\-528\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\-1584\\-48\\0\\96\\0\\-528\\-144\\-2112\\-864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-264\\-16\\-33\\0\\1\\0\\-8\\-198\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-6336\\-528\\-1584\\-528\\0\\528\\0\\-2640\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-528\\-64\\-264\\-144\\-8\\0\\16\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-3168\\-528\\-2970\\-2112\\-198\\-2640\\0\\132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-528\\-144\\-1056\\-864\\-96\\-1584\\-48\\0\\96
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1056, 12144, -528; 12144, -139056, 6048; -528, 6048, -263],[528, 48, 1],[-1, 4, 5, 56, 53, 7, 139, 8, 31, -1; 0, -11, -3, -22, -18, -2, -33, -1, 0, 2; 2, -264, -80, -627, -528, -61, -1056, -40, -66, 48],130,49]
%
% END OF LATTICE L131.49
%
\hbox{}\par\smallskip%
%
% LATTICE L131.50
%
\leavevmode%
\llap{\spacer}%
${L_{131.50}}=\pdualwitharg{11}({L_{131.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${264}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{b}{2}{72}\above{1pt}{1pt}{*}{2}{44}\above{1pt}{1pt}{*}{2}{264}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{66}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{*}{2}{396}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5378472\\209088\\-22968
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
209088\\8085\\-891
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22968\\-891\\98
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-160\\-3564
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-72\\-1593
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-110\\-2409
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-360\\-7740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-98\\-2068
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-152\\-3036
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\32\\759
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-396
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
264\\0\\-264\\-2376\\-1056\\-3432\\-264\\-528\\-264\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-360\\-198\\-792\\-72\\-198\\-144\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\66\\0\\-66\\-528\\-66\\-264\\-264\\-594
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2376\\-360\\0\\72\\0\\-792\\-144\\-792\\-1080\\-3168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1056\\-198\\-66\\0\\44\\0\\-22\\-198\\-352\\-1188
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3432\\-792\\-528\\-792\\0\\264\\0\\-264\\-792\\-3168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-72\\-66\\-144\\-22\\0\\2\\0\\-40\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-198\\-264\\-792\\-198\\-264\\0\\66\\0\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-144\\-264\\-1080\\-352\\-792\\-40\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-198\\-594\\-3168\\-1188\\-3168\\-198\\-198\\0\\396
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5378472, 209088, -22968; 209088, 8085, -891; -22968, -891, 98],[792, 33, 1],[-9, -4, -6, -19, -5, -7, 0, 2, 1, -1; -160, -72, -110, -360, -98, -152, -2, 32, 16, -18; -3564, -1593, -2409, -7740, -2068, -3036, -19, 759, 380, -396],130,50]
%
% END OF LATTICE L131.50
%
\hbox{}\par\smallskip%
%
% LATTICE L131.51
%
\leavevmode%
\llap{\spacer}%
${L_{131.51}}=\pdualwitharg{11}(\main({L_{131.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${264}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{}{2}{264}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{66}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{99}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27720\\3168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3168\\363\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-8\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\-9\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-19\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-72\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-11\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-40\\-264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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0\\-1\\-19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\7\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\8\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
264\\0\\-264\\-2376\\-528\\-3432\\-264\\-528\\-264\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\18\\0\\-360\\-99\\-792\\-72\\-198\\-144\\-99
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2376\\-360\\0\\72\\0\\-792\\-144\\-792\\-1080\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-99\\-33\\0\\11\\0\\-11\\-99\\-176\\-297
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\kern3pt%
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-3432\\-792\\-528\\-792\\0\\264\\0\\-264\\-792\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-72\\-66\\-144\\-11\\0\\2\\0\\-40\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-198\\-264\\-792\\-99\\-264\\0\\66\\0\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-144\\-264\\-1080\\-176\\-792\\-40\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-99\\-297\\-1584\\-297\\-1584\\-99\\-99\\0\\99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[27720, 3168, 0; 3168, 363, 0; 0, 0, -1],[792, 33, 1],[1, 1, 2, 7, 1, 3, 0, -1, -1, -1; -8, -9, -19, -72, -11, -40, -1, 7, 8, 9; 0, -9, -33, -216, -44, -264, -19, -33, -16, 0],130,51]
%
% END OF LATTICE L131.51
%
\hbox{}\par\smallskip%
%
% LATTICE L131.52
%
\leavevmode%
\llap{\spacer}%
${L_{131.52}}=\pdualwitharg{11}({L_{131.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${528}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{132}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{b}{2}{22}\above{1pt}{1pt}{b}{2}{528}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{198}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9977616\\31680\\-22176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31680\\33\\-66
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22176\\-66\\49
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-4\\-132\\-1989
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-13\\-430\\-6468
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-47\\-1560\\-23400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-7\\-233\\-3487
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\HardButStrongLineBreak\kern3pt%
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-27\\-904\\-13464
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\HardButStrongLineBreak\kern3pt%
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-1\\-34\\-500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\32\\495
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\32\\496
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-33\\-495
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\0\\-528\\-4752\\-1056\\-6864\\-528\\-528\\-528\\0
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\kern3pt%
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0\\9\\0\\-360\\-99\\-792\\-72\\-99\\-144\\-99
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\kern3pt%
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-528\\0\\132\\0\\-66\\-1056\\-132\\-264\\-528\\-594
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4752\\-360\\0\\144\\0\\-1584\\-288\\-792\\-2160\\-3168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1056\\-99\\-66\\0\\22\\0\\-22\\-99\\-352\\-594
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6864\\-792\\-1056\\-1584\\0\\528\\0\\-264\\-1584\\-3168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-72\\-132\\-288\\-22\\0\\4\\0\\-80\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-99\\-264\\-792\\-99\\-264\\0\\33\\0\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-144\\-528\\-2160\\-352\\-1584\\-80\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-99\\-594\\-3168\\-594\\-3168\\-198\\-99\\0\\198
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9977616, 31680, -22176; 31680, 33, -66; -22176, -66, 49],[1584, 33, 1],[-17, -4, -13, -47, -7, -27, -1, 1, 1, -1; -560, -132, -430, -1560, -233, -904, -34, 32, 32, -33; -8448, -1989, -6468, -23400, -3487, -13464, -500, 495, 496, -495],130,52]
%
% END OF LATTICE L131.52
%
\hbox{}\par\smallskip%
%
% LATTICE L131.53
%
\leavevmode%
\llap{\spacer}%
${L_{131.53}}=\pdualwitharg{2.11}(\main({L_{131.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}{132}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{88}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{132}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{792}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-380556\\-60192\\-50688
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\kern3pt%
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-60192\\36564\\-42768
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50688\\-42768\\19453
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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75\\-224\\-297
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2045\\-6108\\-8100
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5832\\-17419\\-23100
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7496\\-22389\\-29691
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11053\\-33013\\-43780
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7190\\-21475\\-28479
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1740\\-5197\\-6892
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1433\\-4280\\-5676
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-77\\230\\305
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-4099\\12243\\16236
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
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0\\-396\\-1188\\-1584\\-2376\\-1584\\-396\\-396\\0\\792
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-380556, -60192, -50688; -60192, 36564, -42768; -50688, -42768, 19453],[396, 132, 1],[75, 2045, 5832, 7496, 11053, 7190, 1740, 1433, -77, -4099; -224, -6108, -17419, -22389, -33013, -21475, -5197, -4280, 230, 12243; -297, -8100, -23100, -29691, -43780, -28479, -6892, -5676, 305, 16236],130,53]
%
% END OF LATTICE L131.53
%
\hbox{}\par\smallskip%
%
% LATTICE L131.54
%
\leavevmode%
\llap{\spacer}%
${L_{131.54}}=\pdualwitharg{3.11}({L_{131.2}})$%
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{$1\above{1pt}{1pt}{-2}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
shares genus with 3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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26508688272\\7184680272\\50200128
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7184680272\\1947272118\\13605801
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50200128\\13605801\\95065
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-3791\\12624\\195120
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-354\\1179\\18194
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1029\\-3424\\-53328
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155\\-516\\-7999
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1673\\-5570\\-86262
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1521\\-5064\\-78416
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1448\\-4821\\-74646
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26508688272, 7184680272, 50200128; 7184680272, 1947272118, 13605801; 50200128, 13605801, 95065],[1584, 33, 1],[-257, -737, -740, -3791, -354, 1029, 155, 1673, 1521, 1448; 856, 2454, 2464, 12624, 1179, -3424, -516, -5570, -5064, -4821; 13200, 37962, 38115, 195120, 18194, -53328, -7999, -86262, -78416, -74646],130,54]
%
% END OF LATTICE L131.54
%
\hbox{}\par\smallskip%
%
% LATTICE L131.55
%
\leavevmode%
\llap{\spacer}%
${L_{131.55}}=\pdualwitharg{2.3.11}(\pfillwitharg{3}({L_{131.1}}))$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{s}{2}{176}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{264}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-2640\\4488\\-1848
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\kern3pt%
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4488\\8184\\-3432
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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1\\36\\88
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15\\821\\1980
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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-1056\\-792\\-264\\0\\264\\0\\-264\\-792\\-1056\\-2376
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\right.$}%
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44\\0\\-176\\-396\\-1056
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-240\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\0\\176\\0\\-264
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2640, 4488, -1848; 4488, 8184, -3432; -1848, -3432, 1439],[264, 264, 1],[-1, -1, 1, 5, 15, 9, 9, 7, 3, 1; -64, -70, 36, 266, 821, 502, 508, 402, 172, 55; -154, -168, 88, 642, 1980, 1210, 1224, 968, 414, 132],130,55]
%
% END OF LATTICE L131.55
%
\hbox{}\par\smallskip%
%
% LATTICE L131.56
%
\leavevmode%
\llap{\spacer}%
${L_{131.56}}=\pdualwitharg{2.3.11}(\main(\pfillwitharg{3}({L_{131.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${11}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{176}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{264}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-433488\\-433752\\6072
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-433752\\-433752\\6072
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6072\\6072\\-85
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-437\\3270\\201432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-428\\3209\\197736
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-45\\-2773
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-143
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2\\11\\639
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\61\\3696
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\0\\-88\\-99\\-528\\-143\\-264\\-176\\-33\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-792\\-264\\0\\264\\0\\-264\\-792\\-528\\-2376
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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11\\0\\-88\\-99\\-528
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-120\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\0\\176\\0\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-120\\0\\3\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-792\\-264\\0\\264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-143\\-264\\-176\\-33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-576\\-528\\-144\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\-528\\-704\\-264\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-144\\-264\\-135\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-264\\-792\\-528\\-2376
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-433488, -433752, 6072; -433752, -433752, 6072; 6072, 6072, -85],[264, 264, 1],[0, -1, -3, -2, -9, -2, -3, -1, 0, 1; -2, -2, 8, 11, 61, 17, 32, 22, 4, -1; -143, -216, 352, 639, 3696, 1067, 2064, 1496, 285, 0],130,56]
%
% END OF LATTICE L131.56
%
\hbox{}\par\smallskip%
%
% LATTICE L131.57
%
\leavevmode%
\llap{\spacer}%
${L_{131.57}}=\pdualwitharg{2.3}({L_{131.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{176}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{b}{2}{44}\above{1pt}{1pt}{*}{2}{288}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{1584}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{99}\above{1pt}{1pt}{r}{2}{32}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54570384\\-2654784\\68112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2654784\\-128976\\3312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68112\\3312\\-85
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\64\\7299
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\429\\48752
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\76\\8568
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\99\\11066
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\-1152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-11\\-1230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\792
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\41\\4800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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47\\495\\56925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\159\\18208
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-24\\-198\\-288\\-78\\-792\\-48\\-99\\0
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-198\\-880\\0\\44\\0\\-132\\-3168\\-528\\-2970\\-704
\end{tabular}\endgroup%
\kern3pt%
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-288\\-1584\\-144\\0\\288\\0\\-1584\\-432\\-3168\\-864
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\kern3pt%
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-78\\-528\\-96\\-132\\0\\12\\0\\-48\\-594\\-192
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\kern3pt%
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-792\\-6336\\-1584\\-3168\\-1584\\0\\1584\\0\\-3960\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-528\\-192\\-528\\-432\\-48\\0\\48\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-1584\\-792\\-2970\\-3168\\-594\\-3960\\0\\99\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\-176\\-144\\-704\\-864\\-192\\-1584\\-48\\0\\32
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-54570384, -2654784, 68112; -2654784, -128976, 3312; 68112, 3312, -85],[1584, 48, 1],[6, 40, 7, 9, -1, -1, 1, 4, 47, 15; 64, 429, 76, 99, -9, -11, 0, 41, 495, 159; 7299, 48752, 8568, 11066, -1152, -1230, 792, 4800, 56925, 18208],130,57]
%
% END OF LATTICE L131.57
%
\hbox{}\par\smallskip%
%
% LATTICE L131.58
%
\leavevmode%
\llap{\spacer}%
${L_{131.58}}=\pdualwitharg{2}({L_{131.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}11\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{176}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{11}\above{1pt}{1pt}{r}{2}{288}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{1584}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{b}{2}{396}\above{1pt}{1pt}{*}{2}{32}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
shares genus with 3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1584\\-50688\\1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50688\\-346128\\10992
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1584\\10992\\-349
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\271\\8646
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\913\\29128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\164\\5232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\110\\3509
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\-288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-11\\-351
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\85\\2712
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
193\\2079\\66330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\335\\10688
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-198\\-576\\-78\\-1584\\-96\\-396\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\176\\0\\-440\\-1584\\-264\\-6336\\-528\\-3168\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\0\\-144\\-48\\-1584\\-192\\-1584\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-440\\0\\11\\0\\-33\\-1584\\-264\\-2970\\-352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-576\\-1584\\-144\\0\\288\\0\\-1584\\-432\\-6336\\-864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-264\\-48\\-33\\0\\3\\0\\-24\\-594\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1584\\-6336\\-1584\\-1584\\-1584\\0\\1584\\0\\-7920\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-528\\-192\\-264\\-432\\-24\\0\\48\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-3168\\-1584\\-2970\\-6336\\-594\\-7920\\0\\396\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-176\\-144\\-352\\-864\\-96\\-1584\\-48\\0\\32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1584, -50688, 1584; -50688, -346128, 10992; 1584, 10992, -349],[1584, 48, 1],[25, 84, 15, 10, -1, -1, 1, 8, 193, 31; 271, 913, 164, 110, -9, -11, 0, 85, 2079, 335; 8646, 29128, 5232, 3509, -288, -351, 0, 2712, 66330, 10688],130,58]
%
% END OF LATTICE L131.58
%
\hbox{}\par\smallskip%
%
% LATTICE L131.59
%
\leavevmode%
\llap{\spacer}%
${L_{131.59}}=\pdualwitharg{2.11}(\pfillwitharg{3}({L_{131.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{528}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{352}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{528}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{352}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
279312\\-289344\\-34320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-289344\\299728\\35552
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34320\\35552\\4217
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-79\\6\\-693
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\4\\-416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88\\9\\-792
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\1\\-98
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-3\\66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-1\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\0\\-528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\1\\-171
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-203\\13\\-1760
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\0\\-264\\-198\\-1056\\-858\\-264\\-528\\-33\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-528\\-528\\-192\\-528\\-48\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\528\\0\\-528\\-1056\\-528\\-2112\\-264\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-80\\0\\4\\0\\-132\\-96\\-528\\-90\\-704
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1056\\-528\\-528\\0\\352\\0\\-176\\-1584\\-352\\-3168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-858\\-528\\-1056\\-132\\0\\132\\0\\-528\\-198\\-2112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\-192\\-528\\-96\\-176\\0\\16\\0\\-40\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-528\\-2112\\-528\\-1584\\-528\\0\\528\\0\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-48\\-264\\-90\\-352\\-198\\-40\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-176\\-1584\\-704\\-3168\\-2112\\-528\\-528\\0\\352
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[279312, -289344, -34320; -289344, 299728, 35552; -34320, 35552, 4217],[528, 176, 1],[-79, -47, -88, -11, -1, 5, -4, -65, -20, -203; 6, 4, 9, 1, -1, -3, -1, 0, 1, 13; -693, -416, -792, -98, 0, 66, -24, -528, -171, -1760],130,59]
%
% END OF LATTICE L131.59
%
\hbox{}\par\smallskip%
%
% LATTICE L131.60
%
\leavevmode%
\llap{\spacer}%
${L_{131.60}}=\pdualwitharg{2.11}({L_{131.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{528}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{792}\above{1pt}{1pt}{b}{2}{132}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{s}{2}{528}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{b}{2}{88}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77616\\-24552\\6336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24552\\-5016\\1320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6336\\1320\\-347
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\604\\2442
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\210\\848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\524\\2112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\78\\314
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\99\\396
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-82\\-330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-108\\-432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\288\\1170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\315\\1276
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-528\\-396\\-3168\\-1716\\-1584\\-1056\\-396\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-792\\-528\\-576\\-528\\-288\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\0\\528\\0\\-792\\-1056\\-1584\\-2112\\-1584\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-80\\0\\4\\0\\-132\\-288\\-528\\-540\\-352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3168\\-792\\-792\\0\\792\\0\\-792\\-2376\\-3168\\-2376
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1716\\-528\\-1056\\-132\\0\\132\\0\\-528\\-1188\\-1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1584\\-576\\-1584\\-288\\-792\\0\\144\\0\\-720\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1056\\-528\\-2112\\-528\\-2376\\-528\\0\\528\\0\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-288\\-1584\\-540\\-3168\\-1188\\-720\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-88\\-792\\-352\\-2376\\-1056\\-792\\-264\\0\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-77616, -24552, 6336; -24552, -5016, 1320; 6336, 1320, -347],[792, 264, 1],[9, 3, 7, 1, 1, -1, -1, 1, 5, 5; 604, 210, 524, 78, 99, -82, -108, -2, 288, 315; 2442, 848, 2112, 314, 396, -330, -432, 0, 1170, 1276],130,60]
%
% END OF LATTICE L131.60
%
\hbox{}\par\smallskip%
%
% LATTICE L131.61
%
\leavevmode%
\llap{\spacer}%
${L_{131.61}}=\pdualwitharg{2.11}(\main({L_{131.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{528}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{792}\above{1pt}{1pt}{}{2}{33}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{*}{2}{528}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{88}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
792\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\25080\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-792\\25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\20\\627
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\11\\344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\17\\528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\31
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\25\\792
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\18\\567
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\28\\880
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\0\\-264\\-99\\-1584\\-429\\-792\\-528\\-99\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-40\\-792\\-264\\-576\\-528\\-144\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\528\\0\\-792\\-528\\-1584\\-2112\\-792\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-40\\0\\1\\0\\-33\\-144\\-264\\-135\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1584\\-792\\-792\\0\\792\\0\\-792\\-2376\\-1584\\-2376
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-429\\-264\\-528\\-33\\0\\33\\0\\-264\\-297\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-792\\-576\\-1584\\-144\\-792\\0\\144\\0\\-360\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-528\\-2112\\-264\\-2376\\-264\\0\\528\\0\\-264
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-144\\-792\\-135\\-1584\\-297\\-360\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-88\\-792\\-176\\-2376\\-528\\-792\\-264\\0\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[792, 0, 0; 0, 25080, -792; 0, -792, 25],[792, 264, 1],[-2, -1, -1, 0, 1, 0, -1, -3, -2, -3; 20, 11, 17, 1, 0, 1, 9, 25, 18, 28; 627, 344, 528, 31, 0, 33, 288, 792, 567, 880],130,61]
%
% END OF LATTICE L131.61
%
\hbox{}\par\smallskip%
%
% LATTICE L131.62
%
\leavevmode%
\llap{\spacer}%
${L_{131.62}}=\pdualwitharg{2.3.11}(\pfillwitharg{3}({L_{131.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${44}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{l}{2}{176}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{1056}\above{1pt}{1pt}{l}{2}{11}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}{176}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{1056}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1050720\\-1501104\\3696
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1501104\\-2144208\\5280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3696\\5280\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-286
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-1232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-537
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-18\\-3696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-253
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\528
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\0\\-176\\-198\\-2112\\-286\\-528\\-352\\-132\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-120\\-1584\\-264\\-576\\-528\\-288\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-176\\0\\176\\0\\-528\\-176\\-528\\-704\\-528\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-120\\0\\3\\0\\-33\\-144\\-264\\-270\\-1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2112\\-1584\\-528\\0\\1056\\0\\-528\\-1584\\-2112\\-9504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-286\\-264\\-176\\-33\\0\\11\\0\\-88\\-198\\-1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-576\\-528\\-144\\-528\\0\\48\\0\\-240\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-352\\-528\\-704\\-264\\-1584\\-88\\0\\176\\0\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132\\-288\\-528\\-270\\-2112\\-198\\-240\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-528\\-1584\\-1056\\-9504\\-1056\\-1584\\-528\\0\\1056
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1050720, -1501104, 3696; -1501104, -2144208, 5280; 3696, 5280, -13],[528, 528, 1],[-1, -1, 0, 1, 13, 2, 4, 3, 1, -1; 0, -1, -3, -2, -18, -2, -3, -1, 0, 2; -286, -696, -1232, -537, -3696, -253, -96, 440, 282, 528],130,62]
%
% END OF LATTICE L131.62
%
\hbox{}\par\smallskip%
%
% LATTICE L131.63
%
\leavevmode%
\llap{\spacer}%
${L_{131.63}}=\pdualwitharg{2.3.11}({L_{131.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${33}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{528}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{3168}\above{1pt}{1pt}{*}{2}{132}\above{1pt}{1pt}{b}{2}{144}\above{1pt}{1pt}{l}{2}{528}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{352}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3168\\90288\\11088
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90288\\2566608\\315216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11088\\315216\\38713
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\85\\-693
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\51\\-416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\97\\-792
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\-98
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\9\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\65\\-528
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\63\\-513
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\216\\-1760
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\0\\-264\\-198\\-3168\\-858\\-792\\-528\\-99\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-1584\\-528\\-576\\-528\\-144\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\0\\528\\0\\-1584\\-1056\\-1584\\-2112\\-792\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-80\\0\\4\\0\\-132\\-288\\-528\\-270\\-704
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3168\\-1584\\-1584\\0\\3168\\0\\-1584\\-4752\\-3168\\-9504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-858\\-528\\-1056\\-132\\0\\132\\0\\-528\\-594\\-2112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-792\\-576\\-1584\\-288\\-1584\\0\\144\\0\\-360\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\-528\\-2112\\-528\\-4752\\-528\\0\\528\\0\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-144\\-792\\-270\\-3168\\-594\\-360\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-176\\-1584\\-704\\-9504\\-2112\\-1584\\-528\\0\\352
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3168, 90288, 11088; 90288, 2566608, 315216; 11088, 315216, 38713],[1584, 528, 1],[4, 3, 8, 1, -1, -3, -4, -3, 1, 7; 85, 51, 97, 12, 0, -8, 9, 65, 63, 216; -693, -416, -792, -98, 0, 66, -72, -528, -513, -1760],130,63]
%
% END OF LATTICE L131.63
%
\hbox{}\par\smallskip%
%
% LATTICE L131.64
%
\leavevmode%
\llap{\spacer}%
${L_{131.64}}=\pdualwitharg{2.11}({L_{131.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}11\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${132}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{528}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{3168}\above{1pt}{1pt}{l}{2}{33}\above{1pt}{1pt}{}{2}{144}\above{1pt}{1pt}{r}{2}{528}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}{352}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
shares genus with 3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3168\\49104\\-3168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49104\\760848\\-49104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3168\\-49104\\3169
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-14\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-3\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-2\\-33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-9\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-17\\-264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-18\\-270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-24\\-352
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\0\\-528\\-198\\-6336\\-858\\-1584\\-1056\\-396\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-40\\-1584\\-264\\-576\\-528\\-288\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\0\\528\\0\\-1584\\-528\\-1584\\-2112\\-1584\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-40\\0\\1\\0\\-33\\-144\\-264\\-270\\-352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6336\\-1584\\-1584\\0\\3168\\0\\-1584\\-4752\\-6336\\-9504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-858\\-264\\-528\\-33\\0\\33\\0\\-264\\-594\\-1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1584\\-576\\-1584\\-144\\-1584\\0\\144\\0\\-720\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1056\\-528\\-2112\\-264\\-4752\\-264\\0\\528\\0\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-396\\-288\\-1584\\-270\\-6336\\-594\\-720\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-176\\-1584\\-352\\-9504\\-1056\\-1584\\-528\\0\\352
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3168, 49104, -3168; 49104, 760848, -49104; -3168, -49104, 3169],[1584, 528, 1],[21, 7, 16, 1, -1, -2, -4, 1, 11, 23; -14, -3, -1, 0, 0, -2, -9, -17, -18, -24; -198, -40, 0, 1, 0, -33, -144, -264, -270, -352],130,64]
%
% END OF LATTICE L131.64
%

\medskip%
%------------------------------WEYL-GROUP-W132--------------------------------
%
\leavevmode\llap{}%
$W_{132}$%
\qquad\llap{32} lattices, $\chi=48$%
\hfill%
$222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L132.1
%
\leavevmode%
\llap{\spacer}%
${L_{132.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N'_{23}}%
\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9764440\\82320\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82320\\-694\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\-7\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-949971\\-112791560\\8248520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7995\\949259\\-69420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82\\-9736\\711
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\475\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\1781\\-128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
347\\41200\\-3000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82\\9736\\-711
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5763\\684250\\-50050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1069\\126924\\-9288
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-120\\-35\\-2730\\-520\\-110\\-20\\-80\\-5\\-70\\0
\end{tabular}\endgroup%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-9764440, 82320, 840; 82320, -694, -7; 840, -7, 1],[280, 1, 1],[4, 15, 347, 82, 5763, 1069, 201, 26, 63, 0, -23, -3; 475, 1781, 41200, 9736, 684250, 126924, 23865, 3087, 7480, 0, -2730, -356; -30, -128, -3000, -711, -50050, -9288, -1750, -228, -560, -1, 210, 32],131,1]
%
% END OF LATTICE L132.1
%
\hbox{}\par\smallskip%
%
% LATTICE L132.2
%
\leavevmode%
\llap{\spacer}%
${L_{132.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[466760, -4480, 0; -4480, 43, 0; 0, 0, -1],[280, 1, 1],[-1, 1, 37, 19, 349, 131, 26, 4, 13, 1, 1, -1; -105, 101, 3780, 1944, 35735, 13416, 2665, 411, 1340, 104, 105, -104; -5, -21, -500, -238, -4200, -1560, -295, -39, -100, -2, 0, 0],131,2]
%
% END OF LATTICE L132.2
%
\hbox{}\par\smallskip%
%
% LATTICE L132.3
%
\leavevmode%
\llap{\spacer}%
${L_{132.3}}=\pfillwitharg{2}({L_{132.1}})$%
{${}={}$}Nikulin $23'$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[2590, 1260, 0; 1260, 613, 0; 0, 0, -1],[70, 1, 1],[-907, -135, -213, -16, -102, -1, 17, -43, -677, -340, -12358, -2313; 1875, 279, 440, 33, 210, 2, -35, 89, 1400, 703, 25550, 4782; -405, -59, -90, -6, -35, 0, 5, -21, -310, -154, -5565, -1040],131,3]
%
% END OF LATTICE L132.3
%
\hbox{}\par\smallskip%
%
% LATTICE L132.4
%
\leavevmode%
\llap{\spacer}%
${L_{132.4}}=\main({L_{132.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-4620, -2240, 140; -2240, -1086, 69; 140, 69, 17],[140, 1, 1],[99, 26, 127, 18, -17, -35, -34, -13, 3, 34, 1837, 373; -200, -53, -260, -37, 35, 72, 70, 27, 0, -67, -3675, -748; -35, -3, 0, 2, 0, -4, -5, -5, -80, -34, -1120, -204],131,4]
%
% END OF LATTICE L132.4
%
\hbox{}\par\smallskip%
%
% LATTICE L132.5
%
\leavevmode%
\llap{\spacer}%
${L_{132.5}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{132.1}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[83370, -2520, 41720; -2520, 74, -1260; 41720, -1260, 20877],[70, 2, 1],[39, -149, -1082, -1071, -38606, -3605, -2799, -403, -298, -33, -34, 17; -35, 167, 1190, 1174, 42245, 3943, 3055, 437, 320, 34, 35, -17; -80, 308, 2235, 2212, 79730, 7445, 5780, 832, 615, 68, 70, -35],131,5]
%
% END OF LATTICE L132.5
%
\hbox{}\par\smallskip%
%
% LATTICE L132.6
%
\leavevmode%
\llap{\spacer}%
${L_{132.6}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{132.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[213570, -6300, -42700; -6300, 185, 1260; -42700, 1260, 8537],[70, 5, 1],[-4, 54, 163, 406, 2937, 2745, 214, 156, 47, 14, 3, -15; -7, 167, 476, 1174, 8449, 7886, 611, 437, 128, 34, 7, -34; -19, 245, 744, 1855, 13426, 12550, 979, 715, 216, 65, 14, -70],131,6]
%
% END OF LATTICE L132.6
%
\hbox{}\par\smallskip%
%
% LATTICE L132.7
%
\leavevmode%
\llap{\spacer}%
${L_{132.7}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{132.1}}))$%
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{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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299670\\-8820\\-85610
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[299670, -8820, -85610; -8820, 259, 2520; -85610, 2520, 24457],[70, 7, 1],[-31, 63, 991, 497, 2577, 3375, 1321, 195, 299, 19, 3, -21; -35, 167, 2380, 1174, 6035, 7886, 3055, 437, 640, 34, 5, -34; -105, 203, 3220, 1617, 8390, 10990, 4305, 637, 980, 63, 10, -70],131,7]
%
% END OF LATTICE L132.7
%
\hbox{}\par\smallskip%
%
% LATTICE L132.8
%
\leavevmode%
\llap{\spacer}%
${L_{132.8}}=\pdualwitharg{2}(\main({L_{132.2}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12309080, -185780, -3074820; -185780, 2804, 46408; -3074820, 46408, 768093],[140, 4, 1],[-4207, -536, -319, 1, 71, -9, -123, -330, -1846, -3465, -121311, -5620; -6975, -889, -530, 0, 70, -17, -205, -547, -3060, -5744, -201110, -9317; -16420, -2092, -1245, 4, 280, -35, -480, -1288, -7205, -13524, -473480, -21935],131,8]
%
% END OF LATTICE L132.8
%
\hbox{}\par\smallskip%
%
% LATTICE L132.9
%
\leavevmode%
\llap{\spacer}%
${L_{132.9}}=\pdualwitharg{5}(\main({L_{132.2}}))$%
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{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7980, -3500, 0; -3500, -1510, -5; 0, -5, 1],[140, 5, 1],[61, 42, 23, 4, -3, -7, 0, 19, 99, 240, 1711, 1593; -138, -95, -52, -9, 7, 16, 0, -43, -224, -543, -3871, -3604; -679, -465, -252, -40, 42, 80, -1, -215, -1108, -2680, -19082, -17760],131,9]
%
% END OF LATTICE L132.9
%
\hbox{}\par\smallskip%
%
% LATTICE L132.10
%
\leavevmode%
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${L_{132.10}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{132.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[409990, -50400, 194740; -50400, 6130, -23940; 194740, -23940, 92499],[70, 10, 1],[-321, -5501, -6355, -30039, -211180, -97940, -14739, -9559, -1175, -81, 340, 50; -7, -121, -140, -662, -4655, -2159, -325, -211, -26, -2, 7, 1; 674, 11550, 13343, 63070, 443394, 205635, 30946, 20070, 2467, 170, -714, -105],131,10]
%
% END OF LATTICE L132.10
%
\hbox{}\par\smallskip%
%
% LATTICE L132.11
%
\leavevmode%
\llap{\spacer}%
${L_{132.11}}=\pdualwitharg{7}(\main({L_{132.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1260, -280, 0; -280, 63, 0; 0, 0, -1],[140, 7, 1],[54, 7, 17, 0, -1, -1, 1, 4, 93, 44, 221, 287; 205, 26, 60, -1, -5, -4, 5, 16, 360, 169, 845, 1096; -350, -49, -140, -7, -5, 0, 0, -21, -560, -273, -1395, -1820],131,11]
%
% END OF LATTICE L132.11
%
\hbox{}\par\smallskip%
%
% LATTICE L132.12
%
\leavevmode%
\llap{\spacer}%
${L_{132.12}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{132.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[141890, -35280, 65870; -35280, 8582, -16380; 65870, -16380, 30579],[70, 14, 1],[-1601, -5509, -31842, -30107, -151198, -98173, -73879, -9587, -5898, -85, 238, 49; -35, -121, -700, -662, -3325, -2159, -1625, -211, -130, -2, 5, 1; 3430, 11802, 68215, 64498, 323910, 210315, 158270, 20538, 12635, 182, -510, -105],131,12]
%
% END OF LATTICE L132.12
%
\hbox{}\par\smallskip%
%
% LATTICE L132.13
%
\leavevmode%
\llap{\spacer}%
${L_{132.13}}=\pdualwitharg{5}({L_{132.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[840, 280, 280; 280, 90, 75; 280, 75, -7],[280, 5, 1],[3, 10, 17, 12, 103, 79, 0, -7, -11, -6, -19, -1; -11, -37, -64, -46, -406, -316, -1, 25, 40, 22, 70, 4; 2, 0, -24, -35, -546, -520, -22, -20, -16, -5, -14, 0],131,13]
%
% END OF LATTICE L132.13
%
\hbox{}\par\smallskip%
%
% LATTICE L132.14
%
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${L_{132.14}}=\pdualwitharg{5}({L_{132.2}})$%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\10\\0\\-30\\-175\\-360
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-273\\-175\\-84\\0\\7\\0
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-22\\-20\\-16\\-10\\-7\\0
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-20\\-30\\-40\\-50\\-105\\-160
\end{tabular}\endgroup%
\kern3pt%
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-16\\-40\\-88\\-160\\-476\\-840
\end{tabular}\endgroup%
\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2520, 280, 280; 280, -30, -35; 280, -35, -17],[280, 5, 1],[1, -1, -17, -47, -179, -339, -14, -12, -9, -5, -3, 1; 7, -11, -140, -380, -1435, -2712, -111, -93, -68, -36, -21, 8; 2, 0, -24, -70, -273, -520, -22, -20, -16, -10, -7, 0],131,14]
%
% END OF LATTICE L132.14
%
\hbox{}\par\smallskip%
%
% LATTICE L132.15
%
\leavevmode%
\llap{\spacer}%
${L_{132.15}}=\pdualwitharg{7}({L_{132.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{280}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-8680\\0\\280
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\kern3pt%
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0\\7\\-7
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\-7\\-2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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-2531\\-81880\\-83720
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22\\711\\728
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55\\1780\\1819
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
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-1\\-30\\-35
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-4\\-128\\-133
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\HardButStrongLineBreak\kern3pt%
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-93\\-3000\\-3080
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\HardButStrongLineBreak\kern3pt%
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-22\\-711\\-728
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\HardButStrongLineBreak\kern3pt%
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-221\\-7150\\-7310
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-287\\-9288\\-9492
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8680, 0, 280; 0, 7, -7; 280, -7, -2],[280, 7, 1],[-1, -4, -93, -22, -221, -287, -54, -7, -17, 0, 1, 1; -30, -128, -3000, -711, -7150, -9288, -1750, -228, -560, -1, 30, 32; -35, -133, -3080, -728, -7310, -9492, -1785, -231, -560, 0, 30, 28],131,15]
%
% END OF LATTICE L132.15
%
\hbox{}\par\smallskip%
%
% LATTICE L132.16
%
\leavevmode%
\llap{\spacer}%
${L_{132.16}}=\pdualwitharg{7}({L_{132.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}{280}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-139720\\8680\\-560
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8680\\-539\\35
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\35\\-2
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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16469\\270840\\256200
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45\\740\\699
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-2\\-30\\35
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-4\\-64\\-21
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-81\\-1320\\-980
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-37\\-606\\-518
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-91\\-1495\\-1380
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-235\\-3864\\-3640
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-224\\-5880\\-2856\\-7280\\-18984
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-139720, 8680, -560; 8680, -539, 35; -560, 35, -2],[280, 7, 1],[-2, -4, -81, -37, -91, -235, -43, -5, -9, 1, 1, 1; -30, -64, -1320, -606, -1495, -3864, -710, -84, -160, 14, 15, 16; 35, -21, -980, -518, -1380, -3640, -735, -119, -420, -42, -20, 0],131,16]
%
% END OF LATTICE L132.16
%
\hbox{}\par\smallskip%
%
% LATTICE L132.17
%
\leavevmode%
\llap{\spacer}%
${L_{132.17}}=\pdualwitharg{2}({L_{132.2}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[280, 0, 0; 0, -8, 8; 0, 8, -7],[280, 8, 1],[-39, -5, -3, 0, 1, 0, -1, -3, -17, -16, -561, -26; -225, -33, -25, -3, 0, 1, 5, -11, -85, -85, -3080, -144; -520, -72, -50, -4, 0, 1, 0, -32, -210, -204, -7280, -339],131,17]
%
% END OF LATTICE L132.17
%
\hbox{}\par\smallskip%
%
% LATTICE L132.18
%
\leavevmode%
\llap{\spacer}%
${L_{132.18}}=\pdualwitharg{2}({L_{132.1}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19320, 0, -9520; 0, 8, 0; -9520, 0, -4691],[280, 8, 1],[3021, 387, 116, 0, -69, 1, 79, 233, 659, 1240, 43469, 4029; -35, -3, 0, 1, 0, -1, -5, -5, -10, -17, -560, -51; -6120, -784, -235, 0, 140, -2, -160, -472, -1335, -2512, -88060, -8162],131,18]
%
% END OF LATTICE L132.18
%
\hbox{}\par\smallskip%
%
% LATTICE L132.19
%
\leavevmode%
\llap{\spacer}%
${L_{132.19}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{132.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1190, -1190, 210; -1190, 21455, -7560; 210, -7560, 2629],[70, 35, 1],[113, 1939, 4481, 10591, 10637, 69065, 5197, 3371, 829, 29, -17, -35; -233, -3999, -9242, -21844, -21939, -142448, -10719, -6953, -1710, -60, 35, 72; -679, -11655, -26936, -63665, -63942, -415170, -31241, -20265, -4984, -175, 102, 210],131,19]
%
% END OF LATTICE L132.19
%
\hbox{}\par\smallskip%
%
% LATTICE L132.20
%
\leavevmode%
\llap{\spacer}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2553320, -53900, -651840; -53900, 1140, 13760; -651840, 13760, 166409],[140, 20, 1],[-961, -674, -97, -103, -137, -9, -1, -288, -384, -3745, -26799, -6244; -345, -245, -36, -42, -56, -3, 1, -99, -136, -1334, -9576, -2233; -3736, -2620, -377, -400, -532, -35, -4, -1120, -1493, -14560, -104188, -24275],131,20]
%
% END OF LATTICE L132.20
%
\hbox{}\par\smallskip%
%
% LATTICE L132.21
%
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${L_{132.21}}=\pdualwitharg{2.7}(\main({L_{132.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[140, 1120, 0; 1120, 252, -280; 0, -280, -9],[140, 28, 1],[3086, 445, 332, 79, 41, -9, -36, 165, 1193, 2361, 12159, 3974; -315, -46, -35, -9, -5, 1, 5, -16, -120, -239, -1235, -404; 12040, 1736, 1295, 308, 160, -35, -140, 644, 4655, 9212, 47440, 15505],131,21]
%
% END OF LATTICE L132.21
%
\hbox{}\par\smallskip%
%
% LATTICE L132.22
%
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${L_{132.22}}=\pdualwitharg{5.7}(\main({L_{132.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[140, 0, 0; 0, -805, -175; 0, -175, -38],[140, 35, 1],[0, -4, -21, -51, -52, -339, -13, -9, -5, -1, 0, 1; 3, 21, 76, 168, 162, 1040, 37, 19, 4, -8, -2, 0; -14, -105, -392, -875, -849, -5460, -196, -105, -28, 35, 9, 0],131,22]
%
% END OF LATTICE L132.22
%
\hbox{}\par\smallskip%
%
% LATTICE L132.23
%
\leavevmode%
\llap{\spacer}%
${L_{132.23}}=\pdualwitharg{2.5.7}(\pfillwitharg{2}({L_{132.1}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3172610, -12328120, -6031970; -12328120, 47904570, 23439010; -6031970, 23439010, 11468367],[70, 70, 1],[-7, 167, 238, 1174, 1207, 3943, 611, 437, 64, 34, 1, -17; -221, -505, -86, 131, 436, 1683, 445, 729, 198, 317, 12, -227; 448, 1120, 301, 350, -256, -1365, -588, -1260, -371, -630, -24, 455],131,23]
%
% END OF LATTICE L132.23
%
\hbox{}\par\smallskip%
%
% LATTICE L132.24
%
\leavevmode%
\llap{\spacer}%
${L_{132.24}}=\pdualwitharg{5.7}({L_{132.1}})$%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[97720, -22960, 2240; -22960, 5390, -525; 2240, -525, 51],[280, 35, 1],[-2, 1, 3, 2, 1, 1, -1, -4, -9, -8, -13, -79; -45, -15, 8, 12, 6, 4, -7, -37, -112, -116, -214, -1356; -392, -210, -56, 35, 18, 0, -28, -210, -784, -875, -1698, -10920],131,24]
%
% END OF LATTICE L132.24
%
\hbox{}\par\smallskip%
%
% LATTICE L132.25
%
\leavevmode%
\llap{\spacer}%
${L_{132.25}}=\pdualwitharg{5.7}({L_{132.2}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[280, 0, 0; 0, -1610, 175; 0, 175, -19],[280, 35, 1],[-13, -9, -5, -1, 0, 1, 0, -4, -21, -51, -26, -339; -37, -19, -4, 8, 1, 0, -3, -21, -76, -168, -81, -1040; -392, -210, -56, 70, 9, 0, -28, -210, -784, -1750, -849, -10920],131,25]
%
% END OF LATTICE L132.25
%
\hbox{}\par\smallskip%
%
% LATTICE L132.26
%
\leavevmode%
\llap{\spacer}%
${L_{132.26}}=\pdualwitharg{2.5}({L_{132.2}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[185080, 0, -3360; 0, -40, 0; -3360, 0, 61],[280, 40, 1],[65, 47, 7, 4, 1, -1, -1, 17, 25, 124, 895, 209; 59, 39, 5, 1, 0, 0, 1, 21, 25, 119, 840, 195; 3536, 2560, 382, 220, 56, -55, -56, 920, 1358, 6740, 48664, 11365],131,26]
%
% END OF LATTICE L132.26
%
\hbox{}\par\smallskip%
%
% LATTICE L132.27
%
\leavevmode%
\llap{\spacer}%
${L_{132.27}}=\pdualwitharg{2.5}({L_{132.1}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2520, 2520, -280; 2520, -2480, 280; -280, 280, -31],[280, 40, 1],[176, 114, 7, 1, -3, 0, 4, 66, 38, 359, 2523, 1170; -7, -3, 0, 1, 0, -1, -1, -5, -2, -17, -112, -51; -1752, -1120, -67, 0, 28, -10, -48, -680, -383, -3600, -25228, -11690],131,27]
%
% END OF LATTICE L132.27
%
\hbox{}\par\smallskip%
%
% LATTICE L132.28
%
\leavevmode%
\llap{\spacer}%
${L_{132.28}}=\pdualwitharg{2.7}({L_{132.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26040, 10640, -280; 10640, 3864, -112; -280, -112, 3],[280, 56, 1],[-13, 1, 2, 2, 1, -1, -7, -5, -7, -10, -41, -25; -35, -3, 0, 1, 0, -1, -5, -5, -10, -17, -80, -51; -2800, -56, 175, 224, 100, -126, -840, -672, -1085, -1680, -7380, -4606],131,28]
%
% END OF LATTICE L132.28
%
\hbox{}\par\smallskip%
%
% LATTICE L132.29
%
\leavevmode%
\llap{\spacer}%
${L_{132.29}}=\pdualwitharg{2.7}({L_{132.2}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4760, -1960, 280; -1960, -336, 56; 280, 56, -9],[280, 56, 1],[-22, -4, -4, -1, -1, 0, 2, 0, -6, -7, -39, -13; -215, -47, -55, -17, -20, -1, 35, 11, -35, -55, -340, -116; -2240, -448, -490, -140, -160, -7, 280, 56, -490, -644, -3760, -1267],131,29]
%
% END OF LATTICE L132.29
%
\hbox{}\par\smallskip%
%
% LATTICE L132.30
%
\leavevmode%
\llap{\spacer}%
${L_{132.30}}=\pdualwitharg{2.5.7}(\main({L_{132.2}}))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5320, 24500, -6300; 24500, 113260, -29120; -6300, -29120, 7487],[140, 140, 1],[1, 42, 53, 511, 519, 845, 129, 88, 12, 9, 1, 0; 7, 171, 210, 2014, 2040, 3319, 505, 341, 46, 34, 8, 9; 28, 700, 861, 8260, 8368, 13615, 2072, 1400, 189, 140, 32, 35],131,30]
%
% END OF LATTICE L132.30
%
\hbox{}\par\smallskip%
%
% LATTICE L132.31
%
\leavevmode%
\llap{\spacer}%
${L_{132.31}}=\pdualwitharg{2.5.7}({L_{132.1}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12880, -74760, 19040; -74760, -433160, 110320; 19040, 110320, -28097],[280, 280, 1],[-3, -5, 1, 18, 23, 79, 15, 17, 2, 6, 1, -1; -28, -498, -287, -2711, -2721, -8832, -1328, -858, -52, -1, 9, 18; -112, -1960, -1127, -10640, -10676, -34650, -5208, -3360, -203, 0, 36, 70],131,31]
%
% END OF LATTICE L132.31
%
\hbox{}\par\smallskip%
%
% LATTICE L132.32
%
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\llap{\spacer}%
${L_{132.32}}=\pdualwitharg{2.5.7}({L_{132.2}})$%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12880, -147000, 1400; -147000, -1297240, 12320; 1400, 12320, -117],[280, 280, 1],[-3, -21, -19, -84, -81, -130, -37, -19, -1, 4, 1, 0; 28, 178, 154, 671, 641, 1026, 288, 138, 4, -39, -9, 1; 2912, 18480, 15974, 69580, 66456, 106365, 29848, 14280, 406, -4060, -936, 105],131,32]
%
% END OF LATTICE L132.32
%

\medskip%
%------------------------------WEYL-GROUP-W133--------------------------------
%
\leavevmode\llap{}%
$W_{133}$%
\qquad\llap{32} lattices, $\chi=54$%
\hfill%
$422222422222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L133.1
%
\leavevmode%
\llap{\spacer}%
${L_{133.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{24}}%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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8680\\-2240\\0
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-2240\\578\\-1
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0\\-1\\-15
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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7349\\28560\\-2520
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-1925\\-7481\\660
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0\\0\\0
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1\\4\\-1
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223\\868\\-84
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126\\490\\-45
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443\\1722\\-154
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351\\1364\\-120
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{$\left.\llap{\phantom{%
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0\\0\\0
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-15\\-15\\-280\\-105\\-280\\-180
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-14\\-28\\-672\\-280\\-812\\-560
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[8680, -2240, 0; -2240, 578, -1; 0, -1, -15],[280, 1, 1],[-8, 1, 223, 126, 443, 351, 113, 34, 267, 49, 47, -1; -31, 4, 868, 490, 1722, 1364, 439, 132, 1036, 190, 182, -4; 2, -1, -84, -45, -154, -120, -38, -11, -84, -15, -14, 0],132,1]
%
% END OF LATTICE L133.1
%
\hbox{}\par\smallskip%
%
% LATTICE L133.2
%
\leavevmode%
\llap{\spacer}%
${L_{133.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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6440\\1019\\-62
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{$\left.\llap{\phantom{%
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0\\0\\0
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23309\\-31080\\1906240
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-668920, -105840, 6440; -105840, -16745, 1019; 6440, 1019, -62],[280, 1, 1],[-1, 3, 73, 73, 60, 91, 28, 15, 53, 17, 3, -1; 2, -2, -84, -90, -77, -120, -38, -22, -84, -30, -7, 0; -71, 278, 6188, 6090, 4956, 7464, 2279, 1194, 4116, 1270, 196, -104],132,2]
%
% END OF LATTICE L133.2
%
\hbox{}\par\smallskip%
%
% LATTICE L133.3
%
\leavevmode%
\llap{\spacer}%
${L_{133.3}}=\pfillwitharg{2}({L_{133.1}})$%
{${}={}$}Nikulin $24'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-70, 0, 0; 0, 1, 0; 0, 0, 1],[70, 1, 1],[0, 0, -5, -6, -11, -9, -6, -2, -9, -4, -3, -1; -1, 1, 0, -5, -14, -14, -11, -5, -28, -15, -14, -6; 1, 0, -42, -50, -91, -74, -49, -16, -70, -30, -21, -6],132,3]
%
% END OF LATTICE L133.3
%
\hbox{}\par\smallskip%
%
% LATTICE L133.4
%
\leavevmode%
\llap{\spacer}%
${L_{133.4}}=\main({L_{133.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-334460, -105840, 3220; -105840, -33490, 1019; 3220, 1019, -31],[140, 1, 1],[-1, 3, 73, 73, 120, 91, 28, 15, 53, 17, 6, -1; 1, -1, -42, -45, -77, -60, -19, -11, -42, -15, -7, 0; -71, 278, 6188, 6090, 9912, 7464, 2279, 1194, 4116, 1270, 392, -104],132,4]
%
% END OF LATTICE L133.4
%
\hbox{}\par\smallskip%
%
% LATTICE L133.5
%
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\llap{\spacer}%
${L_{133.5}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{133.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-365190, 7070, -180810; 7070, -136, 3500; -180810, 3500, -89521],[70, 2, 1],[32, -101, 3914, 9871, 18539, 7697, 10408, 3581, 8266, 7529, 5821, 1003; -35, 102, -3857, -9740, -18305, -7603, -10285, -3542, -8183, -7460, -5775, -997; -66, 208, -8057, -20320, -38164, -15845, -21426, -7372, -17017, -15500, -11984, -2065],132,5]
%
% END OF LATTICE L133.5
%
\hbox{}\par\smallskip%
%
% LATTICE L133.6
%
\leavevmode%
\llap{\spacer}%
${L_{133.6}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{133.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-575610, 14000, 116550; 14000, -340, -2835; 116550, -2835, -23599],[70, 5, 1],[-13, 44, -3477, -876, -8222, -6825, -4613, -1586, -7317, -666, -2572, -885; -35, 102, -7714, -1948, -18305, -15206, -10285, -3542, -16366, -1492, -5775, -1994; -60, 205, -16240, -4091, -38395, -31870, -21540, -7405, -34160, -3109, -12005, -4130],132,6]
%
% END OF LATTICE L133.6
%
\hbox{}\par\smallskip%
%
% LATTICE L133.7
%
\leavevmode%
\llap{\spacer}%
${L_{133.7}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{133.1}}))$%
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{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-727790, 18620, 209300; 18620, -476, -5355; 209300, -5355, -60191],[70, 7, 1],[-17, 61, -699, -6159, -1651, -9591, -6481, -2227, -1467, -4671, -515, -1239; -35, 102, -1102, -9740, -2615, -15206, -10285, -3542, -2338, -7460, -825, -1994; -56, 203, -2332, -20545, -5507, -31990, -21616, -7427, -4892, -15575, -1717, -4130],132,7]
%
% END OF LATTICE L133.7
%
\hbox{}\par\smallskip%
%
% LATTICE L133.8
%
\leavevmode%
\llap{\spacer}%
${L_{133.8}}=\pdualwitharg{2}(\main({L_{133.2}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[32817960, 9380, -8451380; 9380, 4, -2416; -8451380, -2416, 2176425],[140, 4, 1],[0, -589, -5561, -10637, -17047, -3179, -3846, -1975, -3413, -2183, -901, -26; 1, -748, -7084, -13560, -21742, -4056, -4909, -2524, -4368, -2800, -1162, -34; 0, -2288, -21602, -41320, -66220, -12349, -14940, -7672, -13258, -8480, -3500, -101],132,8]
%
% END OF LATTICE L133.8
%
\hbox{}\par\smallskip%
%
% LATTICE L133.9
%
\leavevmode%
\llap{\spacer}%
${L_{133.9}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{133.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[664020, -87850, 315350; -87850, 11730, -41720; 315350, -41720, 149763],[70, 10, 1],[19, 1723, -88807, -44211, -412434, -170520, -229651, -78267, -179097, -32333, -123374, -20840; 0, 29, -1484, -739, -6895, -2851, -3840, -1309, -2996, -541, -2065, -349; -40, -3620, 186585, 92888, 866530, 358265, 482500, 164440, 376285, 67932, 259210, 43785],132,9]
%
% END OF LATTICE L133.9
%
\hbox{}\par\smallskip%
%
% LATTICE L133.10
%
\leavevmode%
\llap{\spacer}%
${L_{133.10}}=\pdualwitharg{5}(\main({L_{133.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-355180, -243880, 72520; -243880, -167450, 49795; 72520, 49795, -14807],[140, 5, 1],[-16, 113, 2173, 415, 3312, 2459, 739, 369, 1201, 67, 62, -49; 1, -1, -42, -9, -77, -60, -19, -11, -42, -3, -7, 0; -75, 550, 10500, 2002, 15960, 11840, 3555, 1770, 5740, 318, 280, -240],132,10]
%
% END OF LATTICE L133.10
%
\hbox{}\par\smallskip%
%
% LATTICE L133.11
%
\leavevmode%
\llap{\spacer}%
${L_{133.11}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{133.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[217980, -59290, 101150; -59290, 16422, -27510; 101150, -27510, 46937],[70, 14, 1],[13, 1593, -11708, -204017, -54380, -157387, -211969, -72245, -23618, -149243, -16272, -19243; 0, 29, -212, -3695, -985, -2851, -3840, -1309, -428, -2705, -295, -349; -28, -3416, 25107, 437500, 116614, 337505, 454552, 154924, 50647, 320040, 34894, 41265],132,11]
%
% END OF LATTICE L133.11
%
\hbox{}\par\smallskip%
%
% LATTICE L133.12
%
\leavevmode%
\llap{\spacer}%
${L_{133.12}}=\pdualwitharg{7}(\main({L_{133.2}}))$%
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{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-417620, -312900, 120680; -312900, -234430, 90419; 120680, 90419, -34873],[140, 7, 1],[-23, 199, 527, 3493, 792, 4099, 1226, 603, 275, 517, 10, -89; 1, -1, -6, -45, -11, -60, -19, -11, -6, -15, -1, 0; -77, 686, 1808, 11970, 2712, 14028, 4193, 2058, 936, 1750, 32, -308],132,12]
%
% END OF LATTICE L133.12
%
\hbox{}\par\smallskip%
%
% LATTICE L133.13
%
\leavevmode%
\llap{\spacer}%
${L_{133.13}}=\pdualwitharg{5}({L_{133.1}})$%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3640, 2240, 840; 2240, -1230, -445; 840, -445, -159],[280, 5, 1],[91, 25, 181, 6, 23, -3, -4, 4, 225, 23, 383, 293; 641, 177, 1288, 43, 168, -20, -29, 27, 1568, 161, 2688, 2060; -1320, -365, -2660, -89, -350, 40, 60, -55, -3220, -331, -5530, -4240],132,13]
%
% END OF LATTICE L133.13
%
\hbox{}\par\smallskip%
%
% LATTICE L133.14
%
\leavevmode%
\llap{\spacer}%
${L_{133.14}}=\pdualwitharg{5}({L_{133.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{*}{4}{20}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-710360\\-243880\\145040
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-243880\\-83725\\49795
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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369\\-22\\1770
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1201\\-84\\5740
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67\\-6\\318
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31\\-7\\140
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-710360, -243880, 145040; -243880, -83725, 49795; 145040, 49795, -29614],[280, 5, 1],[739, 369, 1201, 67, 31, -49, -16, 113, 2173, 415, 1656, 2459; -38, -22, -84, -6, -7, 0, 2, -2, -84, -18, -77, -120; 3555, 1770, 5740, 318, 140, -240, -75, 550, 10500, 2002, 7980, 11840],132,14]
%
% END OF LATTICE L133.14
%
\hbox{}\par\smallskip%
%
% LATTICE L133.15
%
\leavevmode%
\llap{\spacer}%
${L_{133.15}}=\pdualwitharg{7}({L_{133.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6440, -5880, 1120; -5880, -5285, 994; 1120, 994, -185],[280, 7, 1],[-214, -56, -55, -59, -5, 11, 7, -13, -83, -286, -133, -701; 533, 139, 136, 145, 12, -28, -17, 33, 208, 715, 332, 1748; 1561, 406, 396, 420, 34, -84, -49, 98, 612, 2100, 974, 5124],132,15]
%
% END OF LATTICE L133.15
%
\hbox{}\par\smallskip%
%
% LATTICE L133.16
%
\leavevmode%
\llap{\spacer}%
${L_{133.16}}=\pdualwitharg{7}({L_{133.2}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1770440, -455560, 762720; -455560, -117215, 196259; 762720, 196259, -328586],[280, 7, 1],[3505, 1797, 863, 1787, 33, -193, -94, 477, 1411, 9583, 1104, 11563; -38, -22, -12, -30, -1, 0, 2, -2, -12, -90, -11, -120; 8113, 4158, 1996, 4130, 76, -448, -217, 1106, 3268, 22190, 2556, 26768],132,16]
%
% END OF LATTICE L133.16
%
\hbox{}\par\smallskip%
%
% LATTICE L133.17
%
\leavevmode%
\llap{\spacer}%
${L_{133.17}}=\pdualwitharg{2}({L_{133.1}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-560, 840, -280; 840, -1240, 416; -280, 416, -139],[280, 8, 1],[-1, 6, 55, 104, 165, 61, 73, 18, 29, 16, 3, -1; 2, -1, -21, -45, -77, -30, -38, -11, -21, -15, -7, 0; 8, -16, -182, -360, -588, -222, -272, -72, -126, -80, -28, 2],132,17]
%
% END OF LATTICE L133.17
%
\hbox{}\par\smallskip%
%
% LATTICE L133.18
%
\leavevmode%
\llap{\spacer}%
${L_{133.18}}=\pdualwitharg{2}({L_{133.2}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-43512560, 78120, 155960; 78120, -136, -280; 155960, -280, -559],[280, 8, 1],[-1, 12, 104, 194, 305, 56, 133, 32, 50, 26, 3, -1; 2, -1, -21, -45, -77, -15, -38, -11, -21, -15, -7, 0; -280, 3348, 29022, 54140, 85120, 15629, 37120, 8932, 13958, 7260, 840, -279],132,18]
%
% END OF LATTICE L133.18
%
\hbox{}\par\smallskip%
%
% LATTICE L133.19
%
\leavevmode%
\llap{\spacer}%
${L_{133.19}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{133.1}}))$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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2590\\6160\\-1680
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6160\\41055\\-13055
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2590, 6160, -1680; 6160, 41055, -13055; -1680, -13055, 4198],[70, 35, 1],[-6, -592, 8713, 15182, 20233, 117115, 78864, 26878, 17573, 11104, 6053, 14315; 12, 1213, -17850, -31103, -41451, -239932, -161568, -55065, -36002, -22749, -12401, -29328; 35, 3535, -52020, -90643, -120800, -699230, -470855, -160475, -104920, -66297, -36140, -85470],132,19]
%
% END OF LATTICE L133.19
%
\hbox{}\par\smallskip%
%
% LATTICE L133.20
%
\leavevmode%
\llap{\spacer}%
${L_{133.20}}=\pdualwitharg{2.5}(\main({L_{133.2}}))$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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536200\\1540\\-156100
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[536200, 1540, -156100; 1540, 20, -460; -156100, -460, 45453],[140, 20, 1],[0, -289, -2731, -1045, -8375, -1562, -1890, -971, -1679, -215, -445, -13; 1, -748, -7084, -2712, -21742, -4056, -4909, -2524, -4368, -560, -1162, -34; 0, -1000, -9450, -3616, -28980, -5405, -6540, -3360, -5810, -744, -1540, -45],132,20]
%
% END OF LATTICE L133.20
%
\hbox{}\par\smallskip%
%
% LATTICE L133.21
%
\leavevmode%
\llap{\spacer}%
${L_{133.21}}=\pdualwitharg{2.7}(\main({L_{133.2}}))$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[512680, -2100, -106120; -2100, 28, 448; -106120, 448, 21975],[140, 28, 1],[0, 223, 301, 4031, 923, 1205, 1458, 749, 185, 829, 49, 10; 1, -748, -1012, -13560, -3106, -4056, -4909, -2524, -624, -2800, -166, -34; 0, 1092, 1474, 19740, 4520, 5901, 7140, 3668, 906, 4060, 240, 49],132,21]
%
% END OF LATTICE L133.21
%
\hbox{}\par\smallskip%
%
% LATTICE L133.22
%
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${L_{133.22}}=\pdualwitharg{2.5.7}(\pfillwitharg{2}({L_{133.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8050420, 31318210, 15323910; 31318210, -121835910, -59613960; 15323910, -59613960, -29168939],[70, 70, 1],[-35, 102, -551, -1948, -2615, -7603, -10285, -3542, -1169, -1492, -825, -997; -146, 1225, -8012, -28059, -37487, -108663, -146566, -50129, -16440, -20845, -11417, -13597; 280, -2450, 16085, 56322, 75240, 218085, 294140, 100590, 32985, 41818, 22900, 27265],132,22]
%
% END OF LATTICE L133.22
%
\hbox{}\par\smallskip%
%
% LATTICE L133.23
%
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${L_{133.23}}=\pdualwitharg{5.7}(\main({L_{133.2}}))$%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1395940, 116760, 679140; 116760, -9730, -56805; 679140, -56805, -330409],[140, 35, 1],[6384, 2962, 1245, 388, -39, -613, -51, 1260, 2977, 3834, 4261, 21723; -19, -11, -6, -3, -1, 0, 1, -1, -6, -9, -11, -60; 13125, 6090, 2560, 798, -80, -1260, -105, 2590, 6120, 7882, 8760, 44660],132,23]
%
% END OF LATTICE L133.23
%
\hbox{}\par\smallskip%
%
% LATTICE L133.24
%
\leavevmode%
\llap{\spacer}%
${L_{133.24}}=\pdualwitharg{5.7}({L_{133.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38920, 4760, 280; 4760, -525, -35; 280, -35, -2],[280, 35, 1],[1, 1, 1, 0, -1, -9, -4, -2, -3, -1, -1, -1; 2, -1, -12, -9, -22, -120, -38, -11, -12, -3, -2, 0; 105, 140, 260, 98, 110, 140, -105, -140, -260, -98, -110, -140],132,24]
%
% END OF LATTICE L133.24
%
\hbox{}\par\smallskip%
%
% LATTICE L133.25
%
\leavevmode%
\llap{\spacer}%
${L_{133.25}}=\pdualwitharg{5.7}({L_{133.2}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-23293480, 19598600, 11537960; 19598600, -16489795, -9707775; 11537960, -9707775, -5715098],[280, 35, 1],[-85, 3639, 8361, 10685, 5905, 59953, 17530, 7991, 3269, 945, -90, -1803; 2, -2, -12, -18, -11, -120, -38, -22, -12, -6, -1, 0; -175, 7350, 16900, 21602, 11940, 121240, 35455, 16170, 6620, 1918, -180, -3640],132,25]
%
% END OF LATTICE L133.25
%
\hbox{}\par\smallskip%
%
% LATTICE L133.26
%
\leavevmode%
\llap{\spacer}%
${L_{133.26}}=\pdualwitharg{2.5}({L_{133.1}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4243120, -162680, 76160; -162680, -6200, 2920; 76160, 2920, -1367],[280, 40, 1],[637, 162, 271, 32, 43, -7, -13, 46, 457, 176, 1413, 527; -38, -11, -21, -3, -7, 0, 2, -1, -21, -9, -77, -30; 35400, 9000, 15050, 1776, 2380, -390, -720, 2560, 25410, 9784, 78540, 29290],132,26]
%
% END OF LATTICE L133.26
%
\hbox{}\par\smallskip%
%
% LATTICE L133.27
%
\leavevmode%
\llap{\spacer}%
${L_{133.27}}=\pdualwitharg{2.5}({L_{133.2}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-34436080, 155400, 13836480; 155400, -680, -62440; 13836480, -62440, -5559523],[280, 40, 1],[29685, 7144, 11166, 1162, 675, -223, -225, 2676, 23204, 8658, 68065, 12498; -38, -11, -21, -3, -7, 0, 2, -1, -21, -9, -77, -15; 73880, 17780, 27790, 2892, 1680, -555, -560, 6660, 57750, 21548, 169400, 31105],132,27]
%
% END OF LATTICE L133.27
%
\hbox{}\par\smallskip%
%
% LATTICE L133.28
%
\leavevmode%
\llap{\spacer}%
${L_{133.28}}=\pdualwitharg{2.7}({L_{133.1}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-23140880, -449400, 193200; -449400, -8680, 3752; 193200, 3752, -1613],[280, 56, 1],[783, 198, 47, 192, 7, -9, -15, 58, 81, 1088, 249, 649; -38, -11, -3, -15, -1, 0, 2, -1, -3, -45, -11, -30; 93688, 23688, 5622, 22960, 836, -1078, -1792, 6944, 9694, 130200, 29796, 77658],132,28]
%
% END OF LATTICE L133.28
%
\hbox{}\par\smallskip%
%
% LATTICE L133.29
%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24419920, 24265080, -7021280; 24265080, -24111192, 6976760; -7021280, 6976760, -2018777],[280, 56, 1],[-21211, -5107, -1141, -4161, -70, 159, 163, -1909, -2367, -30919, -6946, -8929; -38, -11, -3, -15, -1, 0, 2, -1, -3, -45, -11, -15; 73640, 17724, 3958, 14420, 240, -553, -560, 6636, 8222, 107380, 24120, 31003],132,29]
%
% END OF LATTICE L133.29
%
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%
% LATTICE L133.30
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[140, -13860, 3360; -13860, 45099880, -10934420; 3360, -10934420, 2651039],[140, 140, 1],[431, 220, 54, 48, 14, 3, 1, 68, 90, 240, 274, 357; 42870, 22029, 5443, 4881, 1445, 297, 0, 6551, 8847, 23699, 27135, 35428; 176820, 90860, 22450, 20132, 5960, 1225, 0, 27020, 36490, 97748, 111920, 146125],132,30]
%
% END OF LATTICE L133.30
%
\hbox{}\par\smallskip%
%
% LATTICE L133.31
%
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${L_{133.31}}=\pdualwitharg{2.5.7}({L_{133.1}})$%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-43400, -490560, 126560; -490560, -5164040, 1332520; 126560, 1332520, -343841],[280, 280, 1],[-2593, -641, -148, -115, -16, 35, 37, -211, -278, -737, -836, -2165; -91873, -22698, -5237, -4064, -561, 1243, 1297, -7494, -9859, -26128, -29631, -76723; -357000, -88200, -20350, -15792, -2180, 4830, 5040, -29120, -38310, -101528, -115140, -298130],132,31]
%
% END OF LATTICE L133.31
%
\hbox{}\par\smallskip%
%
% LATTICE L133.32
%
\leavevmode%
\llap{\spacer}%
${L_{133.32}}=\pdualwitharg{2.5.7}({L_{133.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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191\\851520\\-1751680
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\kern3pt%
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28740\\127461899\\-262204600
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13971\\61961385\\-127462091
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-11\\-38724\\79660
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-3\\-8648\\17790
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-3\\-6302\\12964
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\HardButStrongLineBreak\kern3pt%
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-1\\-525\\1080
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1208\\-2485
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-490\\-1260\\-1400\\-1785
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4760, -695800, -338240; -695800, -98622160, -47941880; -338240, -47941880, -23305349],[280, 280, 1],[-38, -11, -3, -3, -1, 0, 2, -1, -3, -9, -11, -15; -160885, -38724, -8648, -6302, -525, 1208, 1225, -14496, -17962, -46918, -52695, -67733; 330960, 79660, 17790, 12964, 1080, -2485, -2520, 29820, 36950, 96516, 108400, 139335],132,32]
%
% END OF LATTICE L133.32
%

\medskip%
%------------------------------WEYL-GROUP-W134--------------------------------
%
\leavevmode\llap{}%
$W_{134}$%
\qquad\llap{32} lattices, $\chi=21$%
\hfill%
$2222224$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L134.1
%
\leavevmode%
\llap{\spacer}%
${L_{134.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N'_{27}}%
\EasyButWeakLineBreak%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1962168\\6864\\9672
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\kern3pt%
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6864\\-22\\-35
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9672\\-35\\-47
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-2\\-169\\-286
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3\\248\\432
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6\\499\\862
\end{tabular}\endgroup%
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113\\9412\\16224
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1\\84\\143
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2\\0\\-8\\-12\\-208\\-18\\-1
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-18\\-78\\-24\\-6\\0\\12\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-13\\-8\\-5\\-52\\0\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1962168, 6864, 9672; 6864, -22, -35; 9672, -35, -47],[312, 1, 1],[-1, -2, 3, 6, 113, 11, 1; -83, -169, 248, 499, 9412, 918, 84; -144, -286, 432, 862, 16224, 1578, 143],133,1]
%
% END OF LATTICE L134.1
%
\hbox{}\par\smallskip%
%
% LATTICE L134.2
%
\leavevmode%
\llap{\spacer}%
${L_{134.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29016\\-101\\-52
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14664\\-52\\-25
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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3\\621\\467
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17\\3523\\2639
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\1452\\1084
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\623\\463
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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21\\4368\\3224
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\HardButStrongLineBreak\kern3pt%
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-1\\-207\\-156
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-208\\-154
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-8\\-12\\-208\\-9\\-2
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\kern3pt%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\8\\0\\-104\\-12\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-26\\0\\2\\0\\-3\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-624\\-104\\0\\104\\0\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-39\\-12\\-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-26\\-16\\-10\\-104\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-8293272, 29016, 14664; 29016, -101, -52; 14664, -52, -25],[312, 1, 1],[3, 17, 7, 3, 21, -1, -1; 621, 3523, 1452, 623, 4368, -207, -208; 467, 2639, 1084, 463, 3224, -156, -154],133,2]
%
% END OF LATTICE L134.2
%
\hbox{}\par\smallskip%
%
% LATTICE L134.3
%
\leavevmode%
\llap{\spacer}%
${L_{134.3}}=\pfillwitharg{2}({L_{134.1}})$%
{${}={}$}Nikulin $27'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{26}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{26}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2262\\390\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
390\\67\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-13\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\26\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-50\\-19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
87\\-546\\-208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-63\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-19\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-4\\-12\\-104\\-9\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\26\\0\\-26\\-312\\-39\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\2\\0\\-26\\-6\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-26\\0\\2\\0\\-3\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-312\\-26\\0\\26\\0\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-39\\-6\\-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-13\\-4\\-5\\-26\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[2262, 390, -78; 390, 67, -13; -78, -13, 2],[78, 1, 1],[0, -4, 1, 8, 87, 10, 3; 0, 26, -6, -50, -546, -63, -19; 1, 13, -2, -19, -208, -24, -7],133,3]
%
% END OF LATTICE L134.3
%
\hbox{}\par\smallskip%
%
% LATTICE L134.4
%
\leavevmode%
\llap{\spacer}%
${L_{134.4}}=\main({L_{134.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{52}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3588\\-1248\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1248\\434\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\104\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\72\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\46\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
163\\468\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-23\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\-6\\-104\\-9\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\13\\0\\-13\\-312\\-39\\-13
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\4\\0\\-52\\-12\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-13\\0\\1\\0\\-3\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-312\\-52\\0\\52\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-39\\-12\\-3\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-13\\-8\\-5\\-52\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[3588, -1248, 0; -1248, 434, -1; 0, -1, -11],[156, 1, 1],[1, 36, 25, 16, 163, -1, -8; 3, 104, 72, 46, 468, -3, -23; -1, -13, -8, -5, -52, 0, 2],133,4]
%
% END OF LATTICE L134.4
%
\hbox{}\par\smallskip%
%
% LATTICE L134.5
%
\leavevmode%
\llap{\spacer}%
${L_{134.5}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{134.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{52}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1878474\\9594\\-932568
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9594\\-44\\4762
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-932568\\4762\\-462973
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
220\\-83\\-444
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5230\\-2041\\-10556
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1033\\-406\\-2085
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3474\\-1373\\-7012
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11767\\-4667\\-23751
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1189\\-477\\-2400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-219\\84\\442
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-4\\-24\\-104\\-18\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\52\\0\\-52\\-312\\-78\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\1\\0\\-13\\-6\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-52\\0\\4\\0\\-6\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-312\\-13\\0\\13\\0\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-78\\-6\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-26\\-4\\-10\\-26\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1878474, 9594, -932568; 9594, -44, 4762; -932568, 4762, -462973],[78, 2, 1],[220, 5230, 1033, 3474, 11767, 1189, -219; -83, -2041, -406, -1373, -4667, -477, 84; -444, -10556, -2085, -7012, -23751, -2400, 442],133,5]
%
% END OF LATTICE L134.5
%
\hbox{}\par\smallskip%
%
% LATTICE L134.6
%
\leavevmode%
\llap{\spacer}%
${L_{134.6}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{134.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{78}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2321358\\13026\\779844
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13026\\-66\\-4377
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
779844\\-4377\\-261983
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-83\\-588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4702\\-2041\\-13962
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1857\\-812\\-5514
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3122\\-1373\\-9270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21147\\-9334\\-62790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-356\\-159\\-1057
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
197\\84\\585
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-12\\-36\\-312\\-9\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\78\\0\\-78\\-936\\-39\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\6\\0\\-78\\-6\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-78\\0\\6\\0\\-3\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-936\\-78\\0\\78\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-39\\-6\\-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-39\\-12\\-15\\-78\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2321358, 13026, 779844; 13026, -66, -4377; 779844, -4377, -261983],[78, 3, 1],[-198, -4702, -1857, -3122, -21147, -356, 197; -83, -2041, -812, -1373, -9334, -159, 84; -588, -13962, -5514, -9270, -62790, -1057, 585],133,6]
%
% END OF LATTICE L134.6
%
\hbox{}\par\smallskip%
%
% LATTICE L134.7
%
\leavevmode%
\llap{\spacer}%
${L_{134.7}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{134.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{156}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22823580\\-762762\\10515648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-762762\\25494\\-351432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10515648\\-351432\\4844939
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\2\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38977\\-624\\84552
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9458\\-152\\20517
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36521\\-588\\79224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-133921\\-2158\\290511
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5636\\-91\\12226
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
697\\11\\-1512
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-72\\-312\\-18\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\156\\0\\-156\\-936\\-78\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\0\\-39\\-6\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-156\\0\\12\\0\\-6\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-936\\-39\\0\\39\\0\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-78\\-6\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-78\\-12\\-30\\-78\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22823580, -762762, 10515648; -762762, 25494, -351432; 10515648, -351432, 4844939],[78, 6, 1],[83, -38977, -9458, -36521, -133921, -5636, 697; 2, -624, -152, -588, -2158, -91, 11; -180, 84552, 20517, 79224, 290511, 12226, -1512],133,7]
%
% END OF LATTICE L134.7
%
\hbox{}\par\smallskip%
%
% LATTICE L134.8
%
\leavevmode%
\llap{\spacer}%
${L_{134.8}}=\pdualwitharg{3}(\main({L_{134.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{156}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93444\\3744\\1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3744\\-150\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1872\\-75\\-37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\117\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\80\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\51\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\520\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-25\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-18\\-312\\-9\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\39\\0\\-39\\-936\\-39\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-156\\-12\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-39\\0\\3\\0\\-3\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-936\\-156\\0\\156\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-39\\-12\\-3\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-39\\-24\\-15\\-156\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-93444, 3744, 1872; 3744, -150, -75; 1872, -75, -37],[156, 3, 1],[0, 4, 3, 2, 21, 0, -1; 4, 117, 80, 51, 520, -1, -25; -9, -39, -12, -3, 0, 2, 0],133,8]
%
% END OF LATTICE L134.8
%
\hbox{}\par\smallskip%
%
% LATTICE L134.9
%
\leavevmode%
\llap{\spacer}%
${L_{134.9}}=\pdualwitharg{2}(\main({L_{134.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{52}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3381144\\58500\\-846612
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58500\\1012\\-14648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-846612\\-14648\\211985
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\55\\-124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\221\\-468
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\14\\-23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\1\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\-65\\299
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\-54\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-4\\-24\\-104\\-36\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\52\\0\\-52\\-312\\-156\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\1\\0\\-13\\-12\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-52\\0\\4\\0\\-12\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-312\\-13\\0\\13\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-156\\-12\\-12\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-52\\-8\\-20\\-52\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3381144, 58500, -846612; 58500, 1012, -14648; -846612, -14648, 211985],[156, 4, 1],[-32, -121, -6, 9, 76, 43, 1; 55, 221, 14, 1, -65, -54, 0; -124, -468, -23, 36, 299, 168, 4],133,9]
%
% END OF LATTICE L134.9
%
\hbox{}\par\smallskip%
%
% LATTICE L134.10
%
\leavevmode%
\llap{\spacer}%
${L_{134.10}}=\pdualwitharg{3}({L_{134.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{312}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1376232\\9672\\-183144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9672\\-66\\1341
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-183144\\1341\\-22901
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\3277\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\6383\\-234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-9832\\360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-232\\-19493\\714
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4359\\-366236\\13416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-141\\-11846\\434
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\-3192\\117
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-24\\-36\\-624\\-18\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\78\\0\\-78\\-1872\\-78\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-312\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-78\\0\\6\\0\\-6\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-1872\\-312\\0\\312\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-78\\-24\\-6\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-39\\-24\\-15\\-156\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1376232, 9672, -183144; 9672, -66, 1341; -183144, 1341, -22901],[312, 3, 1],[39, 76, -117, -232, -4359, -141, -38; 3277, 6383, -9832, -19493, -366236, -11846, -3192; -120, -234, 360, 714, 13416, 434, 117],133,10]
%
% END OF LATTICE L134.10
%
\hbox{}\par\smallskip%
%
% LATTICE L134.11
%
\leavevmode%
\llap{\spacer}%
${L_{134.11}}=\pdualwitharg{3}({L_{134.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{78}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{312}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35328072\\52416\\104832
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52416\\-75\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
104832\\-156\\-311
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\467\\2799
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\2223\\13377
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\772\\4668
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\255\\1557
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\1040\\6552
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-52\\-311
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-50\\-312
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-24\\-36\\-624\\-9\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\78\\0\\-78\\-1872\\-39\\-78
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-312\\-12\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-78\\0\\6\\0\\-3\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-1872\\-312\\0\\312\\0\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-39\\-12\\-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-78\\-48\\-30\\-312\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35328072, 52416, 104832; 52416, -75, -156; 104832, -156, -311],[312, 3, 1],[9, 43, 15, 5, 21, -1, -1; 467, 2223, 772, 255, 1040, -52, -50; 2799, 13377, 4668, 1557, 6552, -311, -312],133,11]
%
% END OF LATTICE L134.11
%
\hbox{}\par\smallskip%
%
% LATTICE L134.12
%
\leavevmode%
\llap{\spacer}%
${L_{134.12}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{134.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{26}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6922734\\-65598\\-1068600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65598\\-611\\-10127
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1068600\\-10127\\-164950
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
192\\-144\\-1235
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
362\\-262\\-2329
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1871\\-1344\\-12038
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3162\\-2258\\-20345
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1653\\-1176\\-10636
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1097\\-771\\-7059
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-194\\143\\1248
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\0\\-52\\-156\\-104\\-117\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-26\\-24\\-39\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\0\\26\\0\\-26\\-78\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-26\\0\\26\\0\\-39\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-24\\-26\\0\\2\\0\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-39\\-78\\-39\\0\\39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-13\\-52\\-65\\-26\\0\\13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6922734, -65598, -1068600; -65598, -611, -10127; -1068600, -10127, -164950],[78, 13, 1],[192, 362, 1871, 3162, 1653, 1097, -194; -144, -262, -1344, -2258, -1176, -771, 143; -1235, -2329, -12038, -20345, -10636, -7059, 1248],133,12]
%
% END OF LATTICE L134.12
%
\hbox{}\par\smallskip%
%
% LATTICE L134.13
%
\leavevmode%
\llap{\spacer}%
${L_{134.13}}=\pdualwitharg{2.3}(\main({L_{134.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{156}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27768\\7956\\-7020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7956\\3612\\-1992
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7020\\-1992\\1775
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-447\\25\\-1740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2084\\117\\-8112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-195\\11\\-759
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-370\\21\\-1440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-912\\52\\-3549
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\2\\-144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-72\\-312\\-36\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\156\\0\\-156\\-936\\-156\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\0\\-39\\-12\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-156\\0\\12\\0\\-12\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-936\\-39\\0\\39\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-156\\-12\\-12\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-156\\-24\\-60\\-156\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[27768, 7956, -7020; 7956, 3612, -1992; -7020, -1992, 1775],[156, 12, 1],[-447, -2084, -195, -370, -912, 1, -37; 25, 117, 11, 21, 52, 0, 2; -1740, -8112, -759, -1440, -3549, 4, -144],133,13]
%
% END OF LATTICE L134.13
%
\hbox{}\par\smallskip%
%
% LATTICE L134.14
%
\leavevmode%
\llap{\spacer}%
${L_{134.14}}=\pdualwitharg{13}(\main({L_{134.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${13}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{52}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5772\\936\\-936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
936\\-143\\169
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\169\\-118
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-79\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-17\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\47\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\68\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\78\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-24\\13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\0\\-52\\-78\\-104\\-117\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-13\\-24\\-39\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\0\\52\\0\\-52\\-156\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-13\\0\\13\\0\\-39\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-24\\-52\\0\\4\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-39\\-156\\-39\\0\\78\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-13\\-104\\-65\\-52\\0\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5772, 936, -936; 936, -143, 169; -936, 169, -118],[156, 13, 1],[-19, -4, -1, 12, 17, 19, -6; -79, -17, -8, 47, 68, 78, -24; 39, 8, 0, -26, -36, -39, 13],133,14]
%
% END OF LATTICE L134.14
%
\hbox{}\par\smallskip%
%
% LATTICE L134.15
%
\leavevmode%
\llap{\spacer}%
${L_{134.15}}=\pdualwitharg{2.13}(\pfillwitharg{2}({L_{134.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{78}\above{1pt}{1pt}{}{2}{26}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
377937690\\-11148618\\185272698
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11148618\\328900\\-5465278
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
185272698\\-5465278\\90824423
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1402\\1\\2860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47248\\-47\\-96384
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
149127\\-148\\-304213
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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575988\\-571\\-1174992
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
162491\\-161\\-331475
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
266773\\-264\\-544206
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10961\\11\\22360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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52\\0\\-52\\-312\\-104\\-234\\-26
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0\\4\\0\\-52\\-24\\-78\\-26
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-52\\0\\13\\0\\-13\\-78\\-52
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-312\\-52\\0\\52\\0\\-78\\-130
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-104\\-24\\-13\\0\\1\\0\\-26
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-78\\-78\\-78\\0\\78\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-26\\-52\\-130\\-26\\0\\26
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[377937690, -11148618, 185272698; -11148618, 328900, -5465278; 185272698, -5465278, 90824423],[78, 26, 1],[-1402, 47248, 149127, 575988, 162491, 266773, -10961; 1, -47, -148, -571, -161, -264, 11; 2860, -96384, -304213, -1174992, -331475, -544206, 22360],133,15]
%
% END OF LATTICE L134.15
%
\hbox{}\par\smallskip%
%
% LATTICE L134.16
%
\leavevmode%
\llap{\spacer}%
${L_{134.16}}=\pdualwitharg{2}({L_{134.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{208}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-456456\\9672\\-103584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9672\\-176\\2216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103584\\2216\\-23491
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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64\\193\\-264
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\HardButStrongLineBreak\kern3pt%
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126\\377\\-520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-48\\-145\\198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-382\\-1151\\1576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-898\\-2704\\3705
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-349\\-1050\\1440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\-189\\260
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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16\\0\\-16\\-96\\-208\\-72\\-8
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0\\208\\0\\-208\\-624\\-312\\-104
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-16\\0\\4\\0\\-26\\-24\\-16
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-208\\0\\16\\0\\-24\\-40
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-624\\-26\\0\\13\\0\\-52
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-312\\-24\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-104\\-16\\-40\\-52\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-456456, 9672, -103584; 9672, -176, 2216; -103584, 2216, -23491],[312, 8, 1],[64, 126, -48, -382, -898, -349, -63; 193, 377, -145, -1151, -2704, -1050, -189; -264, -520, 198, 1576, 3705, 1440, 260],133,16]
%
% END OF LATTICE L134.16
%
\hbox{}\par\smallskip%
%
% LATTICE L134.17
%
\leavevmode%
\llap{\spacer}%
${L_{134.17}}=\pdualwitharg{2}({L_{134.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{208}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8941296\\60840\\25272
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60840\\-376\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25272\\-176\\-71
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-150\\-1408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-208\\-1976
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\60\\563
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\868\\8168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
133\\3978\\37466
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\747\\7044
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\119\\1128
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-8\\-96\\-416\\-72\\-8
\end{tabular}\endgroup%
\kern3pt%
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0\\208\\0\\-208\\-1248\\-312\\-104
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-8\\0\\1\\0\\-26\\-12\\-8
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-96\\-208\\0\\16\\0\\-24\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-416\\-1248\\-26\\0\\52\\0\\-104
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-312\\-12\\-24\\0\\24\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-104\\-8\\-40\\-104\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8941296, 60840, 25272; 60840, -376, -176; 25272, -176, -71],[312, 8, 1],[-5, -7, 2, 29, 133, 25, 4; -150, -208, 60, 868, 3978, 747, 119; -1408, -1976, 563, 8168, 37466, 7044, 1128],133,17]
%
% END OF LATTICE L134.17
%
\hbox{}\par\smallskip%
%
% LATTICE L134.18
%
\leavevmode%
\llap{\spacer}%
${L_{134.18}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{134.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${78}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{78}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
367116906\\13457418\\116375064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13457418\\493350\\4265937
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
116375064\\4265937\\36890599
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1094\\2189\\3198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36864\\-73775\\-107760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
232705\\-465706\\-680238
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
449400\\-899371\\-1313676
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
253559\\-507440\\-741198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69381\\-138850\\-202813
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8552\\17115\\24999
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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78\\0\\-156\\-468\\-312\\-117\\-39
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-78\\-72\\-39\\-39
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-156\\0\\78\\0\\-78\\-78\\-156
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-468\\-78\\0\\78\\0\\-39\\-195
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-72\\-78\\0\\6\\0\\-78
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-39\\-78\\-39\\0\\13\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-39\\-156\\-195\\-78\\0\\39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[367116906, 13457418, 116375064; 13457418, 493350, 4265937; 116375064, 4265937, 36890599],[78, 39, 1],[-1094, 36864, 232705, 449400, 253559, 69381, -8552; 2189, -73775, -465706, -899371, -507440, -138850, 17115; 3198, -107760, -680238, -1313676, -741198, -202813, 24999],133,18]
%
% END OF LATTICE L134.18
%
\hbox{}\par\smallskip%
%
% LATTICE L134.19
%
\leavevmode%
\llap{\spacer}%
${L_{134.19}}=\pdualwitharg{13}({L_{134.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{104}\above{1pt}{1pt}{b}{2}{26}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{r}{2}{52}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6275256\\90792\\402480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90792\\-1313\\-5824
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
402480\\-5824\\-25813
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\609\\455
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\259\\191
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\1356\\988
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\563\\403
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\288\\200
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-207\\-156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-184\\-130
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\0\\-104\\-156\\-208\\-117\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-26\\-48\\-39\\-26
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\0\\104\\0\\-104\\-156\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-26\\0\\26\\0\\-39\\-130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-48\\-104\\0\\8\\0\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-39\\-156\\-39\\0\\39\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-26\\-208\\-130\\-104\\0\\52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6275256, 90792, 402480; 90792, -1313, -5824; 402480, -5824, -25813],[312, 13, 1],[38, 16, 83, 34, 17, -13, -11; 609, 259, 1356, 563, 288, -207, -184; 455, 191, 988, 403, 200, -156, -130],133,19]
%
% END OF LATTICE L134.19
%
\hbox{}\par\smallskip%
%
% LATTICE L134.20
%
\leavevmode%
\llap{\spacer}%
${L_{134.20}}=\pdualwitharg{13}({L_{134.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${26}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{104}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34008\\3120\\1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3120\\-286\\-143
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1560\\-143\\-71
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\-11\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\32\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\55\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\76\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\90\\-78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\0\\-104\\-156\\-208\\-234\\-13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-26\\-48\\-78\\-13
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\0\\104\\0\\-104\\-312\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-26\\0\\26\\0\\-78\\-65
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-48\\-104\\0\\8\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-78\\-312\\-78\\0\\156\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-13\\-104\\-65\\-52\\0\\13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-34008, 3120, 1560; 3120, -286, -143; 1560, -143, -71],[312, 13, 1],[-1, 0, 3, 4, 5, 5, 0; -11, -1, 32, 55, 76, 90, 6; 0, 2, 0, -26, -48, -78, -13],133,20]
%
% END OF LATTICE L134.20
%
\hbox{}\par\smallskip%
%
% LATTICE L134.21
%
\leavevmode%
\llap{\spacer}%
${L_{134.21}}=\pdualwitharg{3.13}(\main({L_{134.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${39}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{156}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{26}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1716\\23400\\-1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23400\\-313638\\25077
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\25077\\-2005
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-61\\-741
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-24\\-291
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-116\\-1404
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-42\\-507
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-16\\-192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\15\\182
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\13\\156
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\0\\-156\\-234\\-312\\-117\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-39\\-72\\-39\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\0\\156\\0\\-156\\-156\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-39\\0\\39\\0\\-39\\-195
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-72\\-156\\0\\12\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-39\\-156\\-39\\0\\26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-39\\-312\\-195\\-156\\0\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1716, 23400, -1872; 23400, -313638, 25077; -1872, 25077, -2005],[156, 39, 1],[-24, -10, -51, -20, -9, 6, 7; -61, -24, -116, -42, -16, 15, 13; -741, -291, -1404, -507, -192, 182, 156],133,21]
%
% END OF LATTICE L134.21
%
\hbox{}\par\smallskip%
%
% LATTICE L134.22
%
\leavevmode%
\llap{\spacer}%
${L_{134.22}}=\pdualwitharg{2.3.13}(\pfillwitharg{2}({L_{134.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{26}\above{1pt}{1pt}{}{2}{78}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7044362598\\17327630658\\8531912844
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17327630658\\-42622278036\\-20986687020
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8531912844\\-20986687020\\-10333587325
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\122611\\-249132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-262\\230845\\-469044
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672\\596387\\-1211769
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2258\\2015333\\-4094844
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-588\\526702\\-1070175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-257\\232919\\-473252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143\\-123802\\251550
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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156\\0\\-156\\-936\\-312\\-234\\-78
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-156\\-72\\-78\\-78
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\kern3pt%
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-156\\0\\39\\0\\-39\\-78\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-156\\0\\156\\0\\-78\\-390
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-72\\-39\\0\\3\\0\\-78
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-78\\-78\\-78\\0\\26\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-78\\-156\\-390\\-78\\0\\78
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7044362598, 17327630658, 8531912844; 17327630658, -42622278036, -20986687020; 8531912844, -20986687020, -10333587325],[78, 78, 1],[-144, -262, -672, -2258, -588, -257, 143; 122611, 230845, 596387, 2015333, 526702, 232919, -123802; -249132, -469044, -1211769, -4094844, -1070175, -473252, 251550],133,22]
%
% END OF LATTICE L134.22
%
\hbox{}\par\smallskip%
%
% LATTICE L134.23
%
\leavevmode%
\llap{\spacer}%
${L_{134.23}}=\pdualwitharg{2.3}({L_{134.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{s}{2}{624}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-87672\\7176\\28392
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7176\\-528\\-2400
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28392\\-2400\\-9097
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-215\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-134\\-403\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\161\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
422\\1261\\984
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
987\\2951\\2301
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127\\380\\296
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\201\\156
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-288\\-624\\-72\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\624\\0\\-624\\-1872\\-312\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\12\\0\\-78\\-24\\-48
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-624\\0\\48\\0\\-24\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-1872\\-78\\0\\39\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-312\\-24\\-24\\0\\8\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-312\\-48\\-120\\-156\\0\\24
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-87672, 7176, 28392; 7176, -528, -2400; 28392, -2400, -9097],[312, 24, 1],[-72, -134, 54, 422, 987, 127, 67; -215, -403, 161, 1261, 2951, 380, 201; -168, -312, 126, 984, 2301, 296, 156],133,23]
%
% END OF LATTICE L134.23
%
\hbox{}\par\smallskip%
%
% LATTICE L134.24
%
\leavevmode%
\llap{\spacer}%
${L_{134.24}}=\pdualwitharg{2.3}({L_{134.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{624}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3291600\\64584\\22776
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64584\\-1128\\-456
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22776\\-456\\-157
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-13\\-104\\-1584
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\HardButStrongLineBreak\kern3pt%
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-23\\-182\\-2808
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\40\\609
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\614\\9384
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
359\\2860\\43758
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\183\\2804
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\95\\1464
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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48\\0\\-24\\-288\\-1248\\-72\\-24
\end{tabular}\endgroup%
\kern3pt%
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0\\624\\0\\-624\\-3744\\-312\\-312
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-24\\0\\3\\0\\-78\\-12\\-24
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-624\\0\\48\\0\\-24\\-120
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1248\\-3744\\-78\\0\\156\\0\\-312
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-72\\-312\\-12\\-24\\0\\8\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-312\\-24\\-120\\-312\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3291600, 64584, 22776; 64584, -1128, -456; 22776, -456, -157],[312, 24, 1],[-13, -23, 5, 77, 359, 23, 12; -104, -182, 40, 614, 2860, 183, 95; -1584, -2808, 609, 9384, 43758, 2804, 1464],133,24]
%
% END OF LATTICE L134.24
%
\hbox{}\par\smallskip%
%
% LATTICE L134.25
%
\leavevmode%
\llap{\spacer}%
${L_{134.25}}=\pdualwitharg{2.13}(\main({L_{134.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${52}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{}{2}{52}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{312}\above{1pt}{1pt}{*}{2}{104}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140244\\-95316\\107484
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95316\\65000\\-73216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
107484\\-73216\\82501
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
369\\-1099\\-1456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
153\\-456\\-604
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
214\\-638\\-845
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
461\\-1374\\-1820
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93\\-277\\-367
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\-117\\-156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\79\\104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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52\\0\\-52\\-312\\-104\\-468\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-52\\-24\\-156\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\0\\13\\0\\-13\\-156\\-104
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-52\\0\\52\\0\\-156\\-260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-24\\-13\\0\\1\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-156\\-156\\-156\\0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-52\\-104\\-260\\-52\\0\\104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[140244, -95316, 107484; -95316, 65000, -73216; 107484, -73216, 82501],[156, 52, 1],[369, 153, 214, 461, 93, 40, -26; -1099, -456, -638, -1374, -277, -117, 79; -1456, -604, -845, -1820, -367, -156, 104],133,25]
%
% END OF LATTICE L134.25
%
\hbox{}\par\smallskip%
%
% LATTICE L134.26
%
\leavevmode%
\llap{\spacer}%
${L_{134.26}}=\pdualwitharg{3.13}({L_{134.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${78}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{312}\above{1pt}{1pt}{r}{2}{78}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6252168\\2371824\\-62712
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2371824\\-899769\\23790
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62712\\23790\\-629
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-11\\-117
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-141
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\32\\312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\159\\2223
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\268\\3852
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\134\\1976
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\58\\897
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\0\\-312\\-468\\-624\\-234\\-39
\end{tabular}\endgroup%
\kern3pt%
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0\\6\\0\\-78\\-144\\-78\\-39
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\0\\312\\0\\-312\\-312\\-312
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\kern3pt%
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-468\\-78\\0\\78\\0\\-78\\-195
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-144\\-312\\0\\24\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-78\\-312\\-78\\0\\52\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\-39\\-312\\-195\\-156\\0\\39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6252168, 2371824, -62712; 2371824, -899769, 23790; -62712, 23790, -629],[312, 39, 1],[-3, -2, 9, 38, 63, 31, 13; -11, -9, 32, 159, 268, 134, 58; -117, -141, 312, 2223, 3852, 1976, 897],133,26]
%
% END OF LATTICE L134.26
%
\hbox{}\par\smallskip%
%
% LATTICE L134.27
%
\leavevmode%
\llap{\spacer}%
${L_{134.27}}=\pdualwitharg{3.13}({L_{134.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${78}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{312}\above{1pt}{1pt}{b}{2}{78}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{13}\above{1pt}{1pt}{r}{2}{156}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9707880\\495144\\4480320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
495144\\-25194\\-228423
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4480320\\-228423\\-2067587
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2631\\-10525\\6864
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
959\\-3837\\2502
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4305\\-17228\\11232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1405\\-5625\\3666
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
423\\-1696\\1104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-294\\1176\\-767
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-269\\1078\\-702
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\0\\-312\\-468\\-624\\-117\\-78
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\kern3pt%
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0\\6\\0\\-78\\-144\\-39\\-78
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-312\\0\\312\\0\\-312\\-156\\-624
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-468\\-78\\0\\78\\0\\-39\\-390
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-144\\-312\\0\\24\\0\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-117\\-39\\-156\\-39\\0\\13\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-78\\-624\\-390\\-312\\0\\156
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9707880, 495144, 4480320; 495144, -25194, -228423; 4480320, -228423, -2067587],[312, 39, 1],[2631, 959, 4305, 1405, 423, -294, -269; -10525, -3837, -17228, -5625, -1696, 1176, 1078; 6864, 2502, 11232, 3666, 1104, -767, -702],133,27]
%
% END OF LATTICE L134.27
%
\hbox{}\par\smallskip%
%
% LATTICE L134.28
%
\leavevmode%
\llap{\spacer}%
${L_{134.28}}=\pdualwitharg{2.3.13}(\main({L_{134.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${156}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{39}\above{1pt}{1pt}{}{2}{156}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{*}{2}{312}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63804\\-664092\\170976
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-664092\\6916728\\-1780740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
170976\\-1780740\\458459
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-301\\1176\\4680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\407\\1620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-125\\480\\1911
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-237\\901\\3588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\174\\693
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\13\\52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38\\157\\624
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156\\0\\-156\\-936\\-312\\-468\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-156\\-72\\-156\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\0\\39\\0\\-39\\-156\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-156\\0\\156\\0\\-156\\-780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-72\\-39\\0\\3\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-468\\-156\\-156\\-156\\0\\104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-156\\-156\\-312\\-780\\-156\\0\\312
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[63804, -664092, 170976; -664092, 6916728, -1780740; 170976, -1780740, 458459],[156, 156, 1],[-301, -105, -125, -237, -46, -4, -38; 1176, 407, 480, 901, 174, 13, 157; 4680, 1620, 1911, 3588, 693, 52, 624],133,28]
%
% END OF LATTICE L134.28
%
\hbox{}\par\smallskip%
%
% LATTICE L134.29
%
\leavevmode%
\llap{\spacer}%
${L_{134.29}}=\pdualwitharg{2.13}({L_{134.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{52}\above{1pt}{1pt}{*}{2}{208}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{312}\above{1pt}{1pt}{r}{2}{104}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-815880\\772512\\-209352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
772512\\-731432\\198224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-209352\\198224\\-53719
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\1\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\-1\\234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-287\\7\\1144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\2\\171
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-146\\15\\624
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\6\\-52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\0\\-208\\-1248\\-208\\-936\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-208\\-48\\-312\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\0\\52\\0\\-26\\-312\\-208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1248\\-208\\0\\208\\0\\-312\\-520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-208\\-48\\-26\\0\\1\\0\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-312\\-312\\-312\\0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-104\\-208\\-520\\-52\\0\\104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-815880, 772512, -209352; 772512, -731432, 198224; -209352, 198224, -53719],[312, 104, 1],[81, -3, -61, -287, -42, -146, 19; 1, -1, -1, 7, 2, 15, 6; -312, 8, 234, 1144, 171, 624, -52],133,29]
%
% END OF LATTICE L134.29
%
\hbox{}\par\smallskip%
%
% LATTICE L134.30
%
\leavevmode%
\llap{\spacer}%
${L_{134.30}}=\pdualwitharg{2.13}({L_{134.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${208}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{13}\above{1pt}{1pt}{r}{2}{208}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{312}\above{1pt}{1pt}{l}{2}{104}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-285168\\-223080\\-101712
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-223080\\-173368\\-79040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-101712\\-79040\\-36035
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\528\\-1144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\118\\-256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-210\\455
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\-3838\\8320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-1440\\3122
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\-3813\\8268
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-815\\1768
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\0\\-104\\-1248\\-416\\-936\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-208\\-96\\-312\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\0\\13\\0\\-26\\-156\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1248\\-208\\0\\208\\0\\-312\\-520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-416\\-96\\-26\\0\\4\\0\\-104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-312\\-156\\-312\\0\\312\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-104\\-104\\-520\\-104\\0\\104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-285168, -223080, -101712; -223080, -173368, -79040; -101712, -79040, -36035],[312, 104, 1],[-5, -1, 2, 35, 13, 34, 7; 528, 118, -210, -3838, -1440, -3813, -815; -1144, -256, 455, 8320, 3122, 8268, 1768],133,30]
%
% END OF LATTICE L134.30
%
\hbox{}\par\smallskip%
%
% LATTICE L134.31
%
\leavevmode%
\llap{\spacer}%
${L_{134.31}}=\pdualwitharg{2.3.13}({L_{134.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${624}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{*}{2}{156}\above{1pt}{1pt}{*}{2}{624}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{104}\above{1pt}{1pt}{r}{2}{312}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6864\\40872\\3120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40872\\-241176\\-18408
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3120\\-18408\\-1405
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\24\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-70\\936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-9\\123
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-7\\104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\13\\-156
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\0\\-624\\-3744\\-624\\-936\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-624\\-144\\-312\\-312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\0\\156\\0\\-78\\-312\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3744\\-624\\0\\624\\0\\-312\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-144\\-78\\0\\3\\0\\-156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-312\\-312\\-312\\0\\104\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-312\\-312\\-624\\-1560\\-156\\0\\312
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6864, 40872, 3120; 40872, -241176, -18408; 3120, -18408, -1405],[312, 312, 1],[1, -1, -1, 7, 2, 5, 6; 24, -2, -18, -70, -9, -7, 13; -312, 24, 234, 936, 123, 104, -156],133,31]
%
% END OF LATTICE L134.31
%
\hbox{}\par\smallskip%
%
% LATTICE L134.32
%
\leavevmode%
\llap{\spacer}%
${L_{134.32}}=\pdualwitharg{2.3.13}({L_{134.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}13\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${624}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{39}\above{1pt}{1pt}{r}{2}{624}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{104}\above{1pt}{1pt}{l}{2}{312}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6864\\-1029912\\7176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1029912\\-154263720\\1074840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7176\\1074840\\-7489
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\24\\3432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-9\\-1287
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6864, -1029912, 7176; -1029912, -154263720, 1074840; 7176, 1074840, -7489],[312, 312, 1],[-13, 1, 5, 41, 11, 5, -6; 24, 2, -9, -122, -42, -33, -13; 3432, 288, -1287, -17472, -6018, -4732, -1872],133,32]
%
% END OF LATTICE L134.32
%

\medskip%
%------------------------------WEYL-GROUP-W135--------------------------------
%
\leavevmode\llap{}%
$W_{135}$%
\qquad\llap{32} lattices, $\chi=72$%
\hfill%
$2222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L135.1
%
\leavevmode%
\llap{\spacer}%
${L_{135.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}23\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{30}}%
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{${6}\above{1pt}{1pt}{s}{2}{46}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{*}{2}{92}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{69}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{184}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
552\\5\\-5
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552\\-5\\-2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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0\\-11\\-5
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0\\24\\11
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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5\\230\\782
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1\\44\\156
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3\\103\\455
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-113160, 552, 552; 552, 5, -5; 552, -5, -2],[552, 1, 1],[-1, -1, 1, 5, 1, 29, 3, 67, 1, 1, -1, -5, -1, -29, -3, -67; -48, -46, 48, 230, 44, 1035, 103, 2208, 24, -46, -96, -322, -52, -1449, -149, -3312; -159, -161, 156, 782, 156, 4416, 455, 10120, 147, 115, -180, -828, -160, -4623, -478, -10672],134,1]
%
% END OF LATTICE L135.1
%
\hbox{}\par\smallskip%
%
% LATTICE L135.2
%
\leavevmode%
\llap{\spacer}%
${L_{135.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}23\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25944\\-1104\\0
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[25944, -1104, 0; -1104, 47, 0; 0, 0, -1],[552, 1, 1],[2, 5, 1, -1, -1, -83, -9, -105, -2, -5, -1, 1, 1, 83, 9, 105; 45, 115, 24, -23, -24, -2208, -242, -2852, -57, -161, -48, 0, 20, 1794, 196, 2300; -15, -23, 0, 0, -4, -1794, -212, -2668, -69, -299, -168, -161, -24, -1104, -110, -1196],134,2]
%
% END OF LATTICE L135.2
%
\hbox{}\par\smallskip%
%
% LATTICE L135.3
%
\leavevmode%
\llap{\spacer}%
${L_{135.3}}=\pfillwitharg{2}({L_{135.1}})$%
{${}={}$}Nikulin $30'$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-5934, -2898, 552; -2898, -1415, 276; 552, 276, 85],[138, 1, 1],[-3634, -14038, -3535, -5937, -267, -7036, -600, -5597, 34, 238, -65, -963, -333, -55064, -6300, -77203; 7377, 28497, 7176, 12052, 542, 14283, 1218, 11362, -69, -483, 132, 1955, 676, 111780, 12789, 156722; -351, -1357, -342, -575, -26, -690, -59, -552, 3, 23, -6, -92, -32, -5313, -608, -7452],134,3]
%
% END OF LATTICE L135.3
%
\hbox{}\par\smallskip%
%
% LATTICE L135.4
%
\leavevmode%
\llap{\spacer}%
${L_{135.4}}=\main({L_{135.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[19596, 828, 0; 828, 35, 0; 0, 0, -1],[276, 1, 1],[1, 1, -1, -5, -1, -58, -6, -67, -1, -1, 1, 5, 1, 58, 6, 67; -24, -23, 24, 115, 22, 1035, 103, 1104, 12, -23, -48, -161, -26, -1449, -149, -1656; -3, 0, 0, -23, -10, -2001, -231, -2852, -69, -276, -144, -253, -14, -483, -45, -460],134,4]
%
% END OF LATTICE L135.4
%
\hbox{}\par\smallskip%
%
% LATTICE L135.5
%
\leavevmode%
\llap{\spacer}%
${L_{135.5}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{135.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[23322, 1380, 11730; 1380, 58, 690; 11730, 690, 5899],[138, 2, 1],[1513, 3353, 164, -68, -33, 26779, 3290, 21353, 2435, 11781, 1810, 7635, 362, 41324, 4277, 24049; 549, 1219, 60, -23, -12, 9660, 1187, 7705, 879, 4255, 654, 2760, 131, 14973, 1550, 8717; -3072, -6808, -333, 138, 67, -54372, -6680, -43355, -4944, -23920, -3675, -15502, -735, -83904, -8684, -48829],134,5]
%
% END OF LATTICE L135.5
%
\hbox{}\par\smallskip%
%
% LATTICE L135.6
%
\leavevmode%
\llap{\spacer}%
${L_{135.6}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{135.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[42918, 2070, 14352; 2070, 87, 690; 14352, 690, 4799],[138, 3, 1],[328, 2180, 71, -45, -43, 5806, 2140, 27779, 528, 7664, 785, 4967, 471, 8960, 2782, 31285; 183, 1219, 40, -23, -24, 3220, 1187, 15410, 293, 4255, 436, 2760, 262, 4991, 1550, 17434; -1007, -6693, -218, 138, 132, -17825, -6570, -85284, -1621, -23529, -2410, -15249, -1446, -27508, -8541, -96048],134,6]
%
% END OF LATTICE L135.6
%
\hbox{}\par\smallskip%
%
% LATTICE L135.7
%
\leavevmode%
\llap{\spacer}%
${L_{135.7}}=\pdualwitharg{3}(\main({L_{135.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[51060, -552, -276; -552, 6, 3; -276, 3, 1],[276, 3, 1],[0, -12, -5, -55, -11, -412, -138, -1657, -12, -126, -19, -83, -1, -2, 0, 1; 0, -989, -412, -4531, -906, -33925, -11363, -136436, -988, -10373, -1564, -6831, -82, -161, 1, 92; -1, -345, -140, -1518, -300, -11132, -3726, -44712, -323, -3381, -508, -2208, -24, -46, 0, 0],134,7]
%
% END OF LATTICE L135.7
%
\hbox{}\par\smallskip%
%
% LATTICE L135.8
%
\leavevmode%
\llap{\spacer}%
${L_{135.8}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{135.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[34593978, 157320, 17145258; 157320, 714, 77970; 17145258, 77970, 8497429],[138, 6, 1],[-2833, -21887, -511, -1368, 284, -18625, -7300, -49618, -2139, -35291, -1975, -27221, -1527, -67142, -21289, -121916; 17, 161, 5, 23, -3, -230, -77, -460, -13, -115, -3, 0, 4, 299, 100, 598; 5716, 44160, 1031, 2760, -573, 37582, 14730, 100119, 4316, 71208, 3985, 54924, 3081, 135470, 42954, 245985],134,8]
%
% END OF LATTICE L135.8
%
\hbox{}\par\smallskip%
%
% LATTICE L135.9
%
\leavevmode%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19596, -3312, 4968; -3312, 284, -828; 4968, -828, 1259],[276, 4, 1],[65, -136, -17, 657, 165, 66883, 7725, 23852, 2311, 9244, 1205, 8451, 231, 15089, 1383, 3472; -12, 23, 3, -115, -29, -11799, -1363, -4209, -408, -1633, -213, -1495, -41, -2691, -247, -621; -264, 552, 69, -2668, -670, -271584, -31368, -96853, -9384, -37536, -4893, -34316, -938, -61272, -5616, -14099],134,9]
%
% END OF LATTICE L135.9
%
\hbox{}\par\smallskip%
%
% LATTICE L135.10
%
\leavevmode%
\llap{\spacer}%
${L_{135.10}}=\pdualwitharg{3}({L_{135.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-619896, 3312, 552; 3312, -3, -9; 552, -9, 2],[552, 3, 1],[16, 167, 25, 54, 1, -1, -1, -11, 0, 17, 7, 38, 15, 553, 185, 2219; 2120, 22126, 3312, 7153, 132, -138, -134, -1472, 0, 2254, 928, 5037, 1988, 73278, 24514, 294032; 5149, 53751, 8048, 17388, 324, -322, -324, -3588, -1, 5451, 2248, 12213, 4824, 177928, 59526, 714012],134,10]
%
% END OF LATTICE L135.10
%
\hbox{}\par\smallskip%
%
% LATTICE L135.11
%
\leavevmode%
\llap{\spacer}%
${L_{135.11}}=\pdualwitharg{3}({L_{135.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[102120, -552, -552; -552, 3, 3; -552, 3, 2],[552, 3, 1],[-12, -126, -19, -83, -1, -1, 0, 1, 0, -12, -5, -55, -11, -206, -69, -1657; -1976, -20746, -3128, -13662, -164, -161, 1, 184, 0, -1978, -824, -9062, -1812, -33925, -11363, -272872; -323, -3381, -508, -2208, -24, -23, 0, 0, -1, -345, -140, -1518, -300, -5566, -1863, -44712],134,11]
%
% END OF LATTICE L135.11
%
\hbox{}\par\smallskip%
%
% LATTICE L135.12
%
\leavevmode%
\llap{\spacer}%
${L_{135.12}}=\pdualwitharg{23}(\pfillwitharg{2}({L_{135.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[211278, 0, -37260; 0, -23, 0; -37260, 0, 6571],[138, 23, 1],[1667, 161, 365, -3, -73, 1273, 3598, 2031, 2665, 561, 3967, 364, 795, 1976, 4705, 2301; 51, 5, 12, 0, -2, 39, 110, 62, 81, 17, 120, 11, 24, 60, 143, 70; 9453, 913, 2070, -17, -414, 7218, 20401, 11516, 15111, 3181, 22494, 2064, 4508, 11205, 26680, 13048],134,12]
%
% END OF LATTICE L135.12
%
\hbox{}\par\smallskip%
%
% LATTICE L135.13
%
\leavevmode%
\llap{\spacer}%
${L_{135.13}}=\pdualwitharg{2.3}(\main({L_{135.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36984, 276, -9108; 276, 1092, 72; -9108, 72, -2243],[276, 12, 1],[-67, -7004, -664, -27609, -2627, -188621, -62969, -188491, -5397, -55832, -2068, -35227, -105, 113, 133, -17; -1, -115, -11, -460, -44, -3174, -1060, -3174, -91, -943, -35, -598, -2, 0, 2, 0; 272, 28428, 2695, 112056, 10662, 765532, 255564, 765003, 21904, 226596, 8393, 142968, 426, -460, -540, 69],134,13]
%
% END OF LATTICE L135.13
%
\hbox{}\par\smallskip%
%
% LATTICE L135.14
%
\leavevmode%
\llap{\spacer}%
${L_{135.14}}=\pdualwitharg{2}({L_{135.2}})$%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4200720, -45816, 11040; -45816, -488, 120; 11040, 120, -29],[552, 8, 1],[109, 385, 22, 130, 1, 13, 0, -3, -1, 29, 5, 77, 8, 1850, 207, 1245; 1506, 5336, 306, 1817, 15, 207, 1, -46, -18, 368, 66, 1035, 109, 25461, 2851, 17158; 47592, 168176, 9615, 56856, 442, 5796, 4, -1334, -456, 12512, 2169, 33488, 3486, 807300, 90340, 543398],134,14]
%
% END OF LATTICE L135.14
%
\hbox{}\par\smallskip%
%
% LATTICE L135.15
%
\leavevmode%
\llap{\spacer}%
${L_{135.15}}=\pdualwitharg{2}({L_{135.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[552, 0, 0; 0, 8, 0; 0, 0, -1],[552, 8, 1],[-35, -119, -13, -36, 0, 1, 0, -1, -1, -19, -5, -33, -3, -622, -69, -206; -273, -943, -105, -299, -1, 0, 1, 0, -3, -115, -33, -230, -22, -4761, -530, -1587; -1128, -3864, -426, -1196, -2, 0, 0, -23, -24, -552, -150, -1012, -94, -19872, -2208, -6601],134,15]
%
% END OF LATTICE L135.15
%
\hbox{}\par\smallskip%
%
% LATTICE L135.16
%
\leavevmode%
\llap{\spacer}%
${L_{135.16}}=\pdualwitharg{2.23}(\pfillwitharg{2}({L_{135.1}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6108018, 0, 2985630; 0, -46, 0; 2985630, 0, 1459391],[138, 46, 1],[5801, 829, 1855, 131, -371, -4502, -11715, -3082, -6341, -919, -2125, -176, 326, 4097, 10680, 2812; -51, -5, -6, 0, 1, -39, -110, -31, -81, -17, -60, -11, -12, -60, -143, -35; -11868, -1696, -3795, -268, 759, 9210, 23966, 6305, 12972, 1880, 4347, 360, -667, -8382, -21850, -5753],134,16]
%
% END OF LATTICE L135.16
%
\hbox{}\par\smallskip%
%
% LATTICE L135.17
%
\leavevmode%
\llap{\spacer}%
${L_{135.17}}=\pdualwitharg{23}(\main({L_{135.2}}))$%
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{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[276, 0, 0; 0, -23, -23; 0, -23, -22],[276, 23, 1],[-1, 0, 1, 0, -1, -13, -35, -19, -11, -2, -25, -2, -3, -5, -11, -5; -3, -1, 0, 4, 36, 270, 712, 380, 207, 35, 408, 30, 32, 36, 70, 28; 0, 1, 0, -5, -46, -357, -943, -504, -276, -47, -552, -41, -46, -57, -115, -48],134,17]
%
% END OF LATTICE L135.17
%
\hbox{}\par\smallskip%
%
% LATTICE L135.18
%
\leavevmode%
\llap{\spacer}%
${L_{135.18}}=\pdualwitharg{3.23}(\pfillwitharg{2}({L_{135.1}}))$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2155146, -138, -685308; -138, -69, 0; -685308, 0, 217891],[138, 69, 1],[-914, -392, -585, -62, 351, 711, 5551, 2921, 1002, 436, 673, 84, -307, -645, -5045, -2657; 1811, 779, 1166, 124, -700, -1435, -11212, -5904, -2031, -889, -1386, -179, 590, 1270, 9947, 5244; -2875, -1233, -1840, -195, 1104, 2236, 17457, 9186, 3151, 1371, 2116, 264, -966, -2029, -15870, -8358],134,18]
%
% END OF LATTICE L135.18
%
\hbox{}\par\smallskip%
%
% LATTICE L135.19
%
\leavevmode%
\llap{\spacer}%
${L_{135.19}}=\pdualwitharg{23}({L_{135.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6072, 552, 552; 552, -46, -23; 552, -23, 127],[552, 23, 1],[-1, -1, 1, 5, 45, 172, 454, 485, 265, 45, 527, 39, 43, 26, 52, 43; -15, -13, 12, 64, 576, 2199, 5804, 6200, 3387, 575, 6732, 498, 548, 330, 659, 544; 0, 2, 0, -10, -92, -357, -943, -1008, -552, -94, -1104, -82, -92, -57, -115, -96],134,19]
%
% END OF LATTICE L135.19
%
\hbox{}\par\smallskip%
%
% LATTICE L135.20
%
\leavevmode%
\llap{\spacer}%
${L_{135.20}}=\pdualwitharg{23}({L_{135.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[552, 0, 0; 0, -46, -23; 0, -23, -11],[552, 23, 1],[-1, 0, 1, 0, -1, -13, -35, -19, -11, -2, -25, -1, -3, -5, -11, -5; -3, -1, 0, 2, 36, 270, 712, 380, 207, 35, 408, 15, 32, 36, 70, 28; 0, 2, 0, -5, -92, -714, -1886, -1008, -552, -94, -1104, -41, -92, -114, -230, -96],134,20]
%
% END OF LATTICE L135.20
%
\hbox{}\par\smallskip%
%
% LATTICE L135.21
%
\leavevmode%
\llap{\spacer}%
${L_{135.21}}=\pdualwitharg{2.3}({L_{135.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{1104}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{552}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{184}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{69}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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-33120\\0\\181
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3507\\-943\\641424
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6060408, 0, -33120; 0, 24, 0; -33120, 0, 181],[552, 24, 1],[341, 3507, 129, 1088, 2, 1, 0, 20, 7, 495, 45, 913, 85, 6002, 2001, 5983; -91, -943, -35, -299, -1, 0, 1, 0, -1, -115, -11, -230, -22, -1587, -530, -1587; 62368, 641424, 23594, 198996, 366, 184, 0, 3657, 1280, 90528, 8230, 166980, 15546, 1097744, 365976, 1094271],134,21]
%
% END OF LATTICE L135.21
%
\hbox{}\par\smallskip%
%
% LATTICE L135.22
%
\leavevmode%
\llap{\spacer}%
${L_{135.22}}=\pdualwitharg{2.3}({L_{135.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}23\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{1104}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{552}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{184}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{b}{2}{276}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10488, 6624, -552; 6624, -3576, 288; -552, 288, -23],[552, 24, 1],[-13, -147, -3, -58, -1, -5, 0, 7, 1, 9, 0, -11, -2, -202, -69, -421; -173, -1909, -38, -713, -10, -46, 1, 69, 9, 23, -3, -230, -31, -2783, -944, -5727; -1912, -20976, -415, -7728, -102, -460, 12, 690, 88, 0, -41, -2760, -354, -31004, -10500, -63618],134,22]
%
% END OF LATTICE L135.22
%
\hbox{}\par\smallskip%
%
% LATTICE L135.23
%
\leavevmode%
\llap{\spacer}%
${L_{135.23}}=\pdualwitharg{3.23}(\main({L_{135.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-276, 0, -276; 0, 138, 69; -276, 69, -241],[276, 69, 1],[758, 336, 1125, 203, -1, -2, 0, 23, 22, 54, 435, 187, 781, 1172, 8970, 4657; 357, 158, 528, 95, -2, -1, 1, 12, 11, 26, 208, 89, 370, 553, 4231, 2196; -805, -357, -1196, -216, 0, 2, 0, -24, -23, -57, -460, -198, -828, -1244, -9522, -4944],134,23]
%
% END OF LATTICE L135.23
%
\hbox{}\par\smallskip%
%
% LATTICE L135.24
%
\leavevmode%
\llap{\spacer}%
${L_{135.24}}=\pdualwitharg{2.3.23}(\pfillwitharg{2}({L_{135.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[60137778, 34434588, 16531986; 34434588, 19717026, 9466110; 16531986, 9466110, 4544663],[138, 138, 1],[17, 5, 2, 0, -1, 13, 110, 31, 27, 17, 20, 11, 12, 20, 143, 35; -36955, -10713, -4056, 193, 2420, -28216, -239235, -67519, -59057, -37293, -43950, -24196, -26423, -43793, -312834, -76499; 76912, 22296, 8441, -402, -5037, 58724, 497904, 140523, 122912, 77616, 91471, 50358, 54993, 91144, 651084, 159213],134,24]
%
% END OF LATTICE L135.24
%
\hbox{}\par\smallskip%
%
% LATTICE L135.25
%
\leavevmode%
\llap{\spacer}%
${L_{135.25}}=\pdualwitharg{2.23}(\main({L_{135.2}}))$%
\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[276, -276, -276; -276, 2024, 1564; -276, 1564, 1225],[276, 92, 1],[1, 1, 17, 18, 58, 760, 1990, 264, 563, 93, 265, 76, 36, 86, 172, 18; 0, -3, -51, -53, -169, -2193, -5739, -761, -1620, -267, -759, -217, -101, -237, -471, -49; 0, 4, 69, 72, 230, 2988, 7820, 1037, 2208, 364, 1035, 296, 138, 324, 644, 67],134,25]
%
% END OF LATTICE L135.25
%
\hbox{}\par\smallskip%
%
% LATTICE L135.26
%
\leavevmode%
\llap{\spacer}%
${L_{135.26}}=\pdualwitharg{3.23}({L_{135.2}})$%
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{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}23\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36412680, 56856, 132480; 56856, -69, -207; 132480, -207, -482],[552, 69, 1],[11, 28, 227, 49, 411, 619, 4739, 2461, 401, 178, 597, 54, 1, -1, -1, 11; 22, 52, 416, 89, 740, 1106, 8462, 4392, 714, 316, 1056, 95, -4, -2, 2, 24; 3013, 7671, 62192, 13425, 112608, 169600, 1298442, 674292, 109871, 48771, 163576, 14796, 276, -274, -276, 3012],134,26]
%
% END OF LATTICE L135.26
%
\hbox{}\par\smallskip%
%
% LATTICE L135.27
%
\leavevmode%
\llap{\spacer}%
${L_{135.27}}=\pdualwitharg{3.23}({L_{135.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-552, 0, -552; 0, 69, 69; -552, 69, -482],[552, 69, 1],[22, 54, 435, 187, 781, 586, 4485, 4657, 758, 336, 1125, 203, -1, -1, 0, 23; 22, 52, 416, 178, 740, 553, 4231, 4392, 714, 316, 1056, 190, -4, -1, 1, 24; -23, -57, -460, -198, -828, -622, -4761, -4944, -805, -357, -1196, -216, 0, 1, 0, -24],134,27]
%
% END OF LATTICE L135.27
%
\hbox{}\par\smallskip%
%
% LATTICE L135.28
%
\leavevmode%
\llap{\spacer}%
${L_{135.28}}=\pdualwitharg{2.3.23}(\main({L_{135.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9660, 95220, 23736; 95220, 951096, 237084; 23736, 237084, 59099],[276, 276, 1],[-91, -41, -35, -26, -2, 0, 2, 0, -1, -5, -11, -20, -44, -138, -1060, -138; -12626, -5631, -4749, -3467, -103, -1, -69, -83, -298, -831, -1713, -2995, -6359, -19385, -148557, -19303; 50692, 22608, 19067, 13920, 414, 4, 276, 333, 1196, 3336, 6877, 12024, 25530, 77828, 596436, 77499],134,28]
%
% END OF LATTICE L135.28
%
\hbox{}\par\smallskip%
%
% LATTICE L135.29
%
\leavevmode%
\llap{\spacer}%
${L_{135.29}}=\pdualwitharg{2.23}({L_{135.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6780216, -2310672, 24840; -2310672, -786968, 8464; 24840, 8464, -91],[552, 184, 1],[-1, 43, 155, 99, 228, 2230, 5727, 1495, 2989, 455, 592, 150, 21, 11, 0, -1; -3, 43, 156, 100, 231, 2265, 5818, 1519, 3039, 463, 603, 153, 22, 12, 1, -1; -552, 15728, 56787, 36304, 83674, 818916, 2103212, 549054, 1097928, 167168, 217557, 55144, 7774, 4116, 92, -366],134,29]
%
% END OF LATTICE L135.29
%
\hbox{}\par\smallskip%
%
% LATTICE L135.30
%
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\llap{\spacer}%
${L_{135.30}}=\pdualwitharg{2.23}({L_{135.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-552, 0, -552; 0, 184, 0; -552, 0, -551],[552, 184, 1],[-1, 91, 655, 209, 481, 4702, 12075, 1576, 6301, 959, 2495, 316, 44, 23, 0, -1; -3, -5, -33, -10, -22, -207, -530, -69, -273, -41, -105, -13, -1, 0, 1, 0; 0, -96, -690, -220, -506, -4944, -12696, -1657, -6624, -1008, -2622, -332, -46, -24, 0, 1],134,30]
%
% END OF LATTICE L135.30
%
\hbox{}\par\smallskip%
%
% LATTICE L135.31
%
\leavevmode%
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${L_{135.31}}=\pdualwitharg{2.3.23}({L_{135.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7212432, -7212984, 20976; -7212984, -7212984, 20976; 20976, 20976, -61],[552, 552, 1],[-1, -5, -11, -10, -22, -69, -530, -69, -91, -41, -35, -13, -1, 0, 1, 0; 16, 38, 76, 65, 135, 403, 3083, 400, 520, 230, 192, 69, -1, -1, -1, 2; 5152, 11328, 22310, 18876, 38778, 114608, 876024, 113577, 147200, 64848, 53866, 19212, -690, -344, 0, 687],134,31]
%
% END OF LATTICE L135.31
%
\hbox{}\par\smallskip%
%
% LATTICE L135.32
%
\leavevmode%
\llap{\spacer}%
${L_{135.32}}=\pdualwitharg{2.3.23}({L_{135.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1389936, 2704248, -9936; 2704248, 5256696, -19320; -9936, -19320, 71],[552, 552, 1],[19, 45, 45, 77, 160, 478, 3657, 949, 617, 273, 114, 82, -1, -1, 0, 5; 22, 52, 52, 89, 185, 553, 4231, 1098, 714, 316, 132, 95, -1, -1, 1, 6; 8648, 20448, 20447, 34992, 72726, 217348, 1662900, 431538, 280600, 124176, 51865, 37320, -414, -412, 276, 2334],134,32]
%
% END OF LATTICE L135.32
%

\medskip%
%------------------------------WEYL-GROUP-W136--------------------------------
%
\leavevmode\llap{}%
$W_{136}$%
\qquad\llap{64} lattices, $\chi=18$%
\hfill%
$2222222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L136.1
%
\leavevmode%
\llap{\spacer}%
${L_{136.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2\rightarrow N'_{31}}%
\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{280}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
472920\\157080\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
157080\\52174\\-279
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-279\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
247\\-744\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\-250\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
251\\-756\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93\\280\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
244\\-735\\-70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-168\\-24\\-840\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-5\\-280\\-15\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\168\\0\\-840\\-84\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-5\\0\\1\\0\\-3\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-280\\-840\\0\\280\\0\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-15\\-84\\-3\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-420\\-35\\-840\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[472920, 157080, -840; 157080, 52174, -279; -840, -279, 1],[840, 1, 1],[247, 83, 251, 0, -93, -1, 244; -744, -250, -756, 0, 280, 3, -735; -72, -25, -84, -1, 0, 0, -70],135,1]
%
% END OF LATTICE L136.1
%
\hbox{}\par\smallskip%
%
% LATTICE L136.2
%
\leavevmode%
\llap{\spacer}%
${L_{136.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-867720\\288960\\3360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288960\\-96226\\-1119
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3360\\-1119\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-36\\-264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-20\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\8\\86
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\140\\1120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-35\\-350
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-168\\-48\\-840\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20\\-560\\-30\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\168\\0\\-840\\-84\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-20\\0\\4\\0\\-6\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-560\\-840\\0\\280\\0\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-30\\-84\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-420\\-70\\-840\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-867720, 288960, 3360; 288960, -96226, -1119; 3360, -1119, -13],[840, 1, 1],[-13, -7, 1, 3, 51, 1, -13; -36, -20, 0, 8, 140, 3, -35; -264, -90, 252, 86, 1120, 0, -350],135,2]
%
% END OF LATTICE L136.2
%
\hbox{}\par\smallskip%
%
% LATTICE L136.3
%
\leavevmode%
\llap{\spacer}%
${L_{136.3}}=\pfillwitharg{2}({L_{136.1}})$%
{${}={}$}Nikulin $31'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{70}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9870\\-1680\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\286\\35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\35\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\70\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-115\\-672\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-292\\-19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1091\\-6370\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92\\-537\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-174\\-1015\\-70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-42\\-12\\-210\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-5\\-140\\-15\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\42\\0\\-210\\-42\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-5\\0\\1\\0\\-3\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-140\\-210\\0\\70\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-15\\-42\\-3\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-210\\-35\\-420\\0\\70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[9870, -1680, -210; -1680, 286, 35; -210, 35, 17],[210, 1, 1],[1, 12, -115, -50, -1091, -92, -174; 6, 70, -672, -292, -6370, -537, -1015; 0, 5, -42, -19, -420, -36, -70],135,3]
%
% END OF LATTICE L136.3
%
\hbox{}\par\smallskip%
%
% LATTICE L136.4
%
\leavevmode%
\llap{\spacer}%
${L_{136.4}}=\main({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{140}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60060\\-20160\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20160\\6767\\141
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\141\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109\\-324\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74\\-220\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-113\\-336\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\140\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-106\\-315\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-84\\-24\\-420\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-10\\-280\\-15\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\84\\0\\-420\\-42\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-10\\0\\2\\0\\-3\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-280\\-420\\0\\140\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-15\\-42\\-3\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-210\\-35\\-420\\0\\35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[60060, -20160, -420; -20160, 6767, 141; -420, 141, 2],[420, 1, 1],[-109, -74, -113, 0, 47, 1, -106; -324, -220, -336, 0, 140, 3, -315; -36, -25, -42, -1, 0, 0, -35],135,4]
%
% END OF LATTICE L136.4
%
\hbox{}\par\smallskip%
%
% LATTICE L136.5
%
\leavevmode%
\llap{\spacer}%
${L_{136.5}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
170310\\34230\\84840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34230\\6488\\17046
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84840\\17046\\42263
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2738\\-81\\5529
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3719\\-110\\7510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2839\\-84\\5733
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1196\\35\\-2415
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
104\\3\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10642\\-315\\21490
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-21\\-12\\-105\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-10\\-140\\-30\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\21\\0\\-105\\-42\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-10\\0\\2\\0\\-6\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-140\\-105\\0\\35\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-30\\-42\\-6\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-210\\-70\\-420\\0\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[170310, 34230, 84840; 34230, 6488, 17046; 84840, 17046, 42263],[210, 2, 1],[-2738, -3719, -2839, 1, 1196, 104, -10642; -81, -110, -84, 0, 35, 3, -315; 5529, 7510, 5733, -2, -2415, -210, 21490],135,5]
%
% END OF LATTICE L136.5
%
\hbox{}\par\smallskip%
%
% LATTICE L136.6
%
\leavevmode%
\llap{\spacer}%
${L_{136.6}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
192990\\44100\\64050
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44100\\9732\\14631
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64050\\14631\\21257
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1207\\-54\\3674
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2459\\-110\\7485
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1251\\-56\\3808
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1587\\70\\-4830
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\1\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7037\\-315\\21420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-14\\-12\\-210\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-15\\-420\\-15\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\14\\0\\-210\\-14\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-15\\0\\3\\0\\-3\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-420\\-210\\0\\210\\0\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-15\\-14\\-3\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-210\\-105\\-1260\\0\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[192990, 44100, 64050; 44100, 9732, 14631; 64050, 14631, 21257],[210, 3, 1],[-1207, -2459, -1251, 1, 1587, 23, -7037; -54, -110, -56, 0, 70, 1, -315; 3674, 7485, 3808, -3, -4830, -70, 21420],135,6]
%
% END OF LATTICE L136.6
%
\hbox{}\par\smallskip%
%
% LATTICE L136.7
%
\leavevmode%
\llap{\spacer}%
${L_{136.7}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
247170\\63840\\-99120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63840\\16220\\-25605
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99120\\-25605\\39749
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4433\\-162\\10950
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
602\\-22\\1487
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4591\\-168\\11340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-391\\14\\-966
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85\\3\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1723\\-63\\4256
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\0\\-210\\-60\\-210\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-5\\-28\\-15\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\210\\0\\-210\\-210\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-5\\0\\5\\0\\-15\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-28\\-210\\0\\14\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-15\\-210\\-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-210\\-35\\-84\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[247170, 63840, -99120; 63840, 16220, -25605; -99120, -25605, 39749],[210, 5, 1],[4433, 602, 4591, -2, -391, -85, 1723; -162, -22, -168, 0, 14, 3, -63; 10950, 1487, 11340, -5, -966, -210, 4256],135,7]
%
% END OF LATTICE L136.7
%
\hbox{}\par\smallskip%
%
% LATTICE L136.8
%
\leavevmode%
\llap{\spacer}%
${L_{136.8}}=\pdualwitharg{3}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{420}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
118020\\-840\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\6\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\3\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-128\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-255\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-126\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\140\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-385\\-105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-28\\-24\\-420\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-30\\-840\\-15\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\28\\0\\-420\\-14\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-30\\0\\6\\0\\-3\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-840\\-420\\0\\420\\0\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-15\\-14\\-3\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-210\\-105\\-1260\\0\\105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[118020, -840, -420; -840, 6, 3; -420, 3, 1],[420, 3, 1],[-1, -2, -1, 0, 1, 0, -3; -128, -255, -126, 1, 140, 0, -385; -32, -60, -28, 0, 0, -1, -105],135,8]
%
% END OF LATTICE L136.8
%
\hbox{}\par\smallskip%
%
% LATTICE L136.9
%
\leavevmode%
\llap{\spacer}%
${L_{136.9}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{420}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6404071380\\1604610\\3183632970
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1604610\\402\\797694
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3183632970\\797694\\1582668007
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
259\\11\\-521
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-701\\205\\1410
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-957\\126\\1925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-695\\69\\1398
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-261\\35\\525
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
521\\-46\\-1048
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9187\\-420\\-18480
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-7\\-12\\-105\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-30\\-420\\-30\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\7\\0\\-105\\-14\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-30\\0\\6\\0\\-6\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-420\\-105\\0\\105\\0\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-30\\-14\\-6\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-210\\-210\\-1260\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6404071380, 1604610, 3183632970; 1604610, 402, 797694; 3183632970, 797694, 1582668007],[210, 6, 1],[259, -701, -957, -695, -261, 521, 9187; 11, 205, 126, 69, 35, -46, -420; -521, 1410, 1925, 1398, 525, -1048, -18480],135,9]
%
% END OF LATTICE L136.9
%
\hbox{}\par\smallskip%
%
% LATTICE L136.10
%
\leavevmode%
\llap{\spacer}%
${L_{136.10}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${42}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
304710\\83580\\-130830
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83580\\22708\\-35889
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-130830\\-35889\\56173
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4715\\-162\\10878
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3201\\-110\\7385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
697\\-24\\1608
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-299\\10\\-690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-91\\3\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1309\\-45\\3020
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\0\\-42\\-84\\-210\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\35\\0\\-35\\-140\\-105\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\6\\0\\-30\\-42\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-35\\0\\7\\0\\-21\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-140\\-30\\0\\10\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-105\\-42\\-21\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-30\\-35\\-60\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[304710, 83580, -130830; 83580, 22708, -35889; -130830, -35889, 56173],[210, 7, 1],[4715, 3201, 697, -3, -299, -91, 1309; -162, -110, -24, 0, 10, 3, -45; 10878, 7385, 1608, -7, -690, -210, 3020],135,10]
%
% END OF LATTICE L136.10
%
\hbox{}\par\smallskip%
%
% LATTICE L136.11
%
\leavevmode%
\llap{\spacer}%
${L_{136.11}}=\pdualwitharg{2}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{140}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10266060\\74340\\2567040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
74340\\536\\18588
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2567040\\18588\\641891
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-36\\129
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-322\\-425\\1300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-572\\-777\\2310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102\\-141\\412
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-35\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\141\\-408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
277\\420\\-1120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-42\\-24\\-105\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\40\\0\\-40\\-280\\-60\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\84\\0\\-210\\-84\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-40\\0\\8\\0\\-12\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-280\\-210\\0\\35\\0\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-60\\-84\\-12\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-420\\-140\\-420\\0\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10266060, 74340, 2567040; 74340, 536, 18588; 2567040, 18588, 641891],[420, 4, 1],[-32, -322, -572, -102, -26, 101, 277; -36, -425, -777, -141, -35, 141, 420; 129, 1300, 2310, 412, 105, -408, -1120],135,11]
%
% END OF LATTICE L136.11
%
\hbox{}\par\smallskip%
%
% LATTICE L136.12
%
\leavevmode%
\llap{\spacer}%
${L_{136.12}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50609580\\184170\\24039330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
184170\\670\\87480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24039330\\87480\\11418577
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\33\\-135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\41\\86
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-550\\378\\1155
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119\\69\\250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\7\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
257\\-138\\-540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
253\\-84\\-532
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-105\\-60\\-105\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-10\\-28\\-30\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\0\\105\\0\\-105\\-210\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-10\\0\\10\\0\\-30\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-28\\-105\\0\\7\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-30\\-210\\-30\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-210\\-70\\-84\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[50609580, 184170, 24039330; 184170, 670, 87480; 24039330, 87480, 11418577],[210, 10, 1],[64, -41, -550, -119, -10, 257, 253; 33, 41, 378, 69, 7, -138, -84; -135, 86, 1155, 250, 21, -540, -532],135,12]
%
% END OF LATTICE L136.12
%
\hbox{}\par\smallskip%
%
% LATTICE L136.13
%
\leavevmode%
\llap{\spacer}%
${L_{136.13}}=\pdualwitharg{5}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
345660\\113400\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
113400\\37195\\-275
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-275\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-24\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\84\\1050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\20\\235
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\56\\588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-7\\-119
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-420\\-120\\-420\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-10\\-56\\-15\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\420\\0\\-420\\-210\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-10\\0\\10\\0\\-15\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-56\\-420\\0\\28\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-15\\-210\\-15\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-210\\-35\\-84\\0\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[345660, 113400, -840; 113400, 37195, -275; -840, -275, 2],[420, 5, 1],[7, 0, -25, -6, -17, -1, 2; -24, 0, 84, 20, 56, 3, -7; -360, -1, 1050, 235, 588, 0, -119],135,13]
%
% END OF LATTICE L136.13
%
\hbox{}\par\smallskip%
%
% LATTICE L136.14
%
\leavevmode%
\llap{\spacer}%
${L_{136.14}}=\pdualwitharg{3}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{210}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1294440\\645120\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
645120\\321513\\-837
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-837\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127\\-256\\-452
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
253\\-510\\-900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
125\\-252\\-448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-139\\280\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
382\\-770\\-1365
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-56\\-48\\-840\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-60\\-1680\\-30\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\56\\0\\-840\\-28\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-60\\0\\12\\0\\-6\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-1680\\-840\\0\\840\\0\\-2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-30\\-28\\-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-420\\-210\\-2520\\0\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1294440, 645120, -1680; 645120, 321513, -837; -1680, -837, 2],[840, 3, 1],[127, 253, 125, -1, -139, 0, 382; -256, -510, -252, 2, 280, 0, -770; -452, -900, -448, 0, 420, -1, -1365],135,14]
%
% END OF LATTICE L136.14
%
\hbox{}\par\smallskip%
%
% LATTICE L136.15
%
\leavevmode%
\llap{\spacer}%
${L_{136.15}}=\pdualwitharg{3}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{840}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26040\\6720\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6720\\-1734\\-213
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\-213\\47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\152\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
68\\265\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
115\\448\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\35\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-143\\-560\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-39\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\35\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-56\\-24\\-840\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-15\\-840\\-15\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\56\\0\\-840\\-28\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-15\\0\\3\\0\\-3\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-840\\-840\\0\\840\\0\\-2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-15\\-28\\-3\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-420\\-105\\-2520\\0\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26040, 6720, 840; 6720, -1734, -213; 840, -213, 47],[840, 3, 1],[39, 68, 115, 9, -143, -10, 9; 152, 265, 448, 35, -560, -39, 35; -8, -15, -28, -3, 0, 2, 0],135,15]
%
% END OF LATTICE L136.15
%
\hbox{}\par\smallskip%
%
% LATTICE L136.16
%
\leavevmode%
\llap{\spacer}%
${L_{136.16}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17657220\\128730\\8197770
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
128730\\938\\59766
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8197770\\59766\\3806003
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\33\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-229\\205\\490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\54\\165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111\\69\\238
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\5\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
235\\-138\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
149\\-60\\-320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-21\\-84\\-105\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-70\\-140\\-210\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\3\\0\\-15\\-42\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-70\\0\\14\\0\\-42\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-140\\-15\\0\\5\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-210\\-42\\-42\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-30\\-70\\-60\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17657220, 128730, 8197770; 128730, 938, 59766; 8197770, 59766, 3806003],[210, 14, 1],[29, -229, -77, -111, -7, 235, 149; 33, 205, 54, 69, 5, -138, -60; -63, 490, 165, 238, 15, -504, -320],135,16]
%
% END OF LATTICE L136.16
%
\hbox{}\par\smallskip%
%
% LATTICE L136.17
%
\leavevmode%
\llap{\spacer}%
${L_{136.17}}=\pdualwitharg{7}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
92820\\18060\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18060\\3514\\-245
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-245\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\36\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-7\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\9\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\10\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-84\\-168\\-420\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\70\\0\\-70\\-280\\-105\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\12\\0\\-60\\-42\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-70\\0\\14\\0\\-21\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-280\\-60\\0\\20\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-105\\-42\\-21\\0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-30\\-35\\-60\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[92820, 18060, -1260; 18060, 3514, -245; -1260, -245, 17],[420, 7, 1],[-7, -1, 1, 1, -1, -2, -2; 36, 5, -6, -7, 0, 9, 10; 0, 0, -12, -28, -80, -21, -5],135,17]
%
% END OF LATTICE L136.17
%
\hbox{}\par\smallskip%
%
% LATTICE L136.18
%
\leavevmode%
\llap{\spacer}%
${L_{136.18}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42736470\\-207270\\15194970
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-207270\\1005\\-73695
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15194970\\-73695\\5402578
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\22\\110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\41\\-81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
275\\252\\-770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91\\69\\-255
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\14\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66\\-46\\185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-202\\-84\\567
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-70\\-60\\-210\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-15\\-84\\-15\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\70\\0\\-210\\-70\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-15\\0\\15\\0\\-15\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-84\\-210\\0\\42\\0\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-15\\-70\\-15\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-210\\-105\\-252\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[42736470, -207270, 15194970; -207270, 1005, -73695; 15194970, -73695, 5402578],[210, 15, 1],[-39, 29, 275, 91, 15, -66, -202; 22, 41, 252, 69, 14, -46, -84; 110, -81, -770, -255, -42, 185, 567],135,18]
%
% END OF LATTICE L136.18
%
\hbox{}\par\smallskip%
%
% LATTICE L136.19
%
\leavevmode%
\llap{\spacer}%
${L_{136.19}}=\pdualwitharg{5}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{30}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
691320\\226800\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
226800\\74390\\-275
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-275\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-24\\-720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\84\\2100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\10\\235
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\56\\1176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-7\\-238
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\-840\\-120\\-840\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-5\\-56\\-15\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\0\\840\\0\\-840\\-420\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-5\\0\\5\\0\\-15\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-56\\-840\\0\\56\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-15\\-420\\-15\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-420\\-35\\-168\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[691320, 226800, -840; 226800, 74390, -275; -840, -275, 1],[840, 5, 1],[7, 0, -25, -3, -17, -1, 2; -24, 0, 84, 10, 56, 3, -7; -720, -1, 2100, 235, 1176, 0, -238],135,19]
%
% END OF LATTICE L136.19
%
\hbox{}\par\smallskip%
%
% LATTICE L136.20
%
\leavevmode%
\llap{\spacer}%
${L_{136.20}}=\pdualwitharg{5}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${120}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{840}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3491880\\-1165920\\10080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1165920\\-389290\\3365
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10080\\3365\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-60\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\16\\118
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-211\\672\\4620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\124\\830
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-97\\308\\2016
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-35\\-238
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\-840\\-240\\-840\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20\\-112\\-30\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\0\\840\\0\\-840\\-420\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-20\\0\\20\\0\\-30\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-112\\-840\\0\\56\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-30\\-420\\-30\\0\\30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-420\\-70\\-168\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3491880, -1165920, 10080; -1165920, -389290, 3365; 10080, 3365, -29],[840, 5, 1],[19, -5, -211, -39, -97, -1, 11; -60, 16, 672, 124, 308, 3, -35; -360, 118, 4620, 830, 2016, 0, -238],135,20]
%
% END OF LATTICE L136.20
%
\hbox{}\par\smallskip%
%
% LATTICE L136.21
%
\leavevmode%
\llap{\spacer}%
${L_{136.21}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59663730\\-289800\\20834730
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-289800\\1407\\-101199
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20834730\\-101199\\7275542
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\22\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
331\\205\\-945
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\36\\-220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
169\\69\\-483
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\10\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-46\\343
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-236\\-60\\675
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-14\\-84\\-210\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\0\\-105\\-420\\-105\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\2\\0\\-30\\-14\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-105\\0\\21\\0\\-21\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-420\\-30\\0\\30\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-105\\-14\\-21\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-30\\-105\\-180\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[59663730, -289800, 20834730; -289800, 1407, -101199; 20834730, -101199, 7275542],[210, 21, 1],[-39, 331, 77, 169, 21, -120, -236; 22, 205, 36, 69, 10, -46, -60; 112, -945, -220, -483, -60, 343, 675],135,21]
%
% END OF LATTICE L136.21
%
\hbox{}\par\smallskip%
%
% LATTICE L136.22
%
\leavevmode%
\llap{\spacer}%
${L_{136.22}}=\pdualwitharg{2.3}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{420}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-86520\\420\\-21420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\1668\\108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21420\\108\\-5303
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-183\\-2\\739
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-847\\-10\\3420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-182
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
205\\2\\-828
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\0\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-103\\-1\\416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3328\\-35\\13440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-14\\-24\\-105\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\120\\0\\-120\\-840\\-60\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\28\\0\\-210\\-28\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-120\\0\\24\\0\\-12\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-840\\-210\\0\\105\\0\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-60\\-28\\-12\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-420\\-420\\-1260\\0\\420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-86520, 420, -21420; 420, 1668, 108; -21420, 108, -5303],[420, 12, 1],[-183, -847, 45, 205, -26, -103, -3328; -2, -10, 0, 2, 0, -1, -35; 739, 3420, -182, -828, 105, 416, 13440],135,22]
%
% END OF LATTICE L136.22
%
\hbox{}\par\smallskip%
%
% LATTICE L136.23
%
\leavevmode%
\llap{\spacer}%
${L_{136.23}}=\pdualwitharg{7}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12221160\\-315840\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-315840\\8162\\21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\21\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\504\\-336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-155\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-504\\324
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-349\\217
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-1280\\760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-39\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\155\\-110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-168\\-168\\-840\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\35\\0\\-35\\-280\\-105\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\24\\0\\-120\\-84\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-35\\0\\7\\0\\-21\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-280\\-120\\0\\40\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-105\\-84\\-21\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\-60\\-35\\-120\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12221160, -315840, -840; -315840, 8162, 21; -840, 21, -1],[840, 7, 1],[13, -4, -13, -9, -33, -1, 4; 504, -155, -504, -349, -1280, -39, 155; -336, 105, 324, 217, 760, 0, -110],135,23]
%
% END OF LATTICE L136.23
%
\hbox{}\par\smallskip%
%
% LATTICE L136.24
%
\leavevmode%
\llap{\spacer}%
${L_{136.24}}=\pdualwitharg{7}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${168}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8300040\\826560\\-16800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
826560\\82313\\-1673
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16800\\-1673\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\72\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-12\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-14\\-196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\18\\-105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\20\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\0\\-168\\-336\\-840\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\140\\0\\-140\\-560\\-210\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\0\\24\\0\\-120\\-84\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-140\\0\\28\\0\\-42\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-560\\-120\\0\\40\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-210\\-84\\-42\\0\\42\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-60\\-70\\-120\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8300040, 826560, -16800; 826560, 82313, -1673; -16800, -1673, 34],[840, 7, 1],[-7, -1, 1, 1, -1, -2, -2; 72, 10, -12, -14, 0, 18, 20; 84, 0, -96, -196, -500, -105, -5],135,24]
%
% END OF LATTICE L136.24
%
\hbox{}\par\smallskip%
%
% LATTICE L136.25
%
\leavevmode%
\llap{\spacer}%
${L_{136.25}}=\pdualwitharg{3.5}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}{84}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1912260\\333900\\22680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
333900\\58305\\3960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22680\\3960\\269
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-84\\-1040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-31\\-387
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-70\\-910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-56\\-693
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-140\\-120\\-420\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-30\\-168\\-15\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\140\\0\\-420\\-70\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-30\\0\\30\\0\\-15\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-168\\-420\\0\\84\\0\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-15\\-70\\-15\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-210\\-105\\-252\\0\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1912260, 333900, 22680; 333900, 58305, 3960; 22680, 3960, 269],[420, 15, 1],[27, 10, 23, 0, 1, 1, 18; -84, -31, -70, 1, 0, -3, -56; -1040, -387, -910, -15, -84, -40, -693],135,25]
%
% END OF LATTICE L136.25
%
\hbox{}\par\smallskip%
%
% LATTICE L136.26
%
\leavevmode%
\llap{\spacer}%
${L_{136.26}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
219942030\\4668930\\109359810
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4668930\\97320\\2321490
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
109359810\\2321490\\54376001
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69946\\-27\\140675
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56993\\-22\\114624
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72447\\-28\\145705
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
179\\0\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18492\\7\\-37191
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2680\\1\\-5390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-163118\\-63\\328062
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-35\\-60\\-105\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\6\\0\\-30\\-84\\-30\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\35\\0\\-105\\-70\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-30\\0\\30\\0\\-30\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-84\\-105\\0\\21\\0\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-30\\-70\\-30\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-210\\-210\\-252\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[219942030, 4668930, 109359810; 4668930, 97320, 2321490; 109359810, 2321490, 54376001],[210, 30, 1],[-69946, -56993, -72447, 179, 18492, 2680, -163118; -27, -22, -28, 0, 7, 1, -63; 140675, 114624, 145705, -360, -37191, -5390, 328062],135,26]
%
% END OF LATTICE L136.26
%
\hbox{}\par\smallskip%
%
% LATTICE L136.27
%
\leavevmode%
\llap{\spacer}%
${L_{136.27}}=\pdualwitharg{2}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{r}{2}{84}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{560}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-855960\\-3360\\5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3360\\8\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\16\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-36\\-195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-40\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\63\\378
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\34\\192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\70\\385
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-33\\-192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-455\\-2520
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-42\\-24\\-105\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\40\\0\\-40\\-280\\-120\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\84\\0\\-210\\-168\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-40\\0\\8\\0\\-24\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-280\\-210\\0\\35\\0\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-120\\-168\\-24\\0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-840\\-280\\-840\\0\\560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-855960, -3360, 5040; -3360, 8, 16; 5040, 16, -29],[840, 8, 1],[-1, -1, 2, 1, 2, -1, -13; -36, -40, 63, 34, 70, -33, -455; -195, -200, 378, 192, 385, -192, -2520],135,27]
%
% END OF LATTICE L136.27
%
\hbox{}\par\smallskip%
%
% LATTICE L136.28
%
\leavevmode%
\llap{\spacer}%
${L_{136.28}}=\pdualwitharg{2}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{140}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{s}{2}{560}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
328440\\-110880\\-21840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110880\\37432\\7376
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21840\\7376\\1439
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-433\\-1227\\-282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-584\\-1655\\-380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-452\\-1281\\-294
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-17\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
111\\315\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1717\\-4865\\-1120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-84\\-48\\-420\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\40\\0\\-40\\-560\\-120\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\84\\0\\-420\\-168\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-40\\0\\8\\0\\-24\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-560\\-420\\0\\140\\0\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-120\\-168\\-24\\0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-840\\-280\\-1680\\0\\560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[328440, -110880, -21840; -110880, 37432, 7376; -21840, 7376, 1439],[840, 8, 1],[-433, -584, -452, -6, 111, 1, -1717; -1227, -1655, -1281, -17, 315, 3, -4865; -282, -380, -294, -4, 70, 0, -1120],135,28]
%
% END OF LATTICE L136.28
%
\hbox{}\par\smallskip%
%
% LATTICE L136.29
%
\leavevmode%
\llap{\spacer}%
${L_{136.29}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${210}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{30}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
541247070\\-1126860\\-214289880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1126860\\2345\\446145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-214289880\\446145\\84841388
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
499\\66\\1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-302\\41\\-763
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1045\\108\\-2640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-762\\69\\-1925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\2\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1621\\-138\\4095
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
211\\-12\\533
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210\\0\\-210\\-420\\-210\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-35\\-28\\-105\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\30\\0\\-30\\-210\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-35\\0\\35\\0\\-105\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-28\\-30\\0\\2\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-105\\-210\\-105\\0\\210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-30\\-35\\-12\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[541247070, -1126860, -214289880; -1126860, 2345, 446145; -214289880, 446145, 84841388],[210, 35, 1],[499, -302, -1045, -762, -19, 1621, 211; 66, 41, 108, 69, 2, -138, -12; 1260, -763, -2640, -1925, -48, 4095, 533],135,29]
%
% END OF LATTICE L136.29
%
\hbox{}\par\smallskip%
%
% LATTICE L136.30
%
\leavevmode%
\llap{\spacer}%
${L_{136.30}}=\pdualwitharg{2.5}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67755660\\-216300\\16790340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216300\\680\\-53600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16790340\\-53600\\4160767
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\6\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-206\\55\\832
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-260\\63\\1050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
822\\-225\\-3320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1208\\-329\\-4879
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1337\\-363\\-5400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
617\\-168\\-2492
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-210\\-120\\-105\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-40\\-56\\-60\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\420\\0\\-210\\-420\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-40\\0\\40\\0\\-60\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-56\\-210\\0\\7\\0\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\-420\\-60\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-420\\-140\\-84\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[67755660, -216300, 16790340; -216300, 680, -53600; 16790340, -53600, 4160767],[420, 20, 1],[-26, -206, -260, 822, 1208, 1337, 617; 6, 55, 63, -225, -329, -363, -168; 105, 832, 1050, -3320, -4879, -5400, -2492],135,30]
%
% END OF LATTICE L136.30
%
\hbox{}\par\smallskip%
%
% LATTICE L136.31
%
\leavevmode%
\llap{\spacer}%
${L_{136.31}}=\pdualwitharg{3.7}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1719060\\-4620\\8400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\42\\21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8400\\21\\-41
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\28\\1036
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-15\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\80\\2700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\18\\623
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\35\\1245
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-28\\-168\\-420\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\210\\0\\-210\\-840\\-105\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\4\\0\\-60\\-14\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-210\\0\\42\\0\\-21\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-840\\-60\\0\\60\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-105\\-14\\-21\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-30\\-105\\-180\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1719060, -4620, 8400; -4620, 42, 21; 8400, 21, -41],[420, 21, 1],[5, -2, -1, 0, 13, 3, 6; 28, -15, -6, 1, 80, 18, 35; 1036, -420, -208, 0, 2700, 623, 1245],135,31]
%
% END OF LATTICE L136.31
%
\hbox{}\par\smallskip%
%
% LATTICE L136.32
%
\leavevmode%
\llap{\spacer}%
${L_{136.32}}=\pdualwitharg{2.3.7}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
536961390\\8597610\\266976360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8597610\\136248\\4274718
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
266976360\\4274718\\132740227
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96556\\-27\\194201
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-393317\\-110\\791070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14275\\-4\\28711
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
355\\0\\-714
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18354\\5\\-36915
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3724\\1\\-7490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160838\\-45\\323490
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-7\\-84\\-105\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\210\\0\\-210\\-420\\-210\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\1\\0\\-15\\-14\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-210\\0\\42\\0\\-42\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-420\\-15\\0\\15\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-210\\-14\\-42\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-30\\-210\\-180\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[536961390, 8597610, 266976360; 8597610, 136248, 4274718; 266976360, 4274718, 132740227],[210, 42, 1],[-96556, -393317, -14275, 355, 18354, 3724, -160838; -27, -110, -4, 0, 5, 1, -45; 194201, 791070, 28711, -714, -36915, -7490, 323490],135,32]
%
% END OF LATTICE L136.32
%
\hbox{}\par\smallskip%
%
% LATTICE L136.33
%
\leavevmode%
\llap{\spacer}%
${L_{136.33}}=\pdualwitharg{2.7}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
333480\\-21420\\83160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21420\\644\\-5320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83160\\-5320\\20737
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\3\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
589\\-70\\-2380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-6\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-395\\46\\1596
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-469\\55\\1895
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-790\\93\\3192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-297\\35\\1200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-42\\-168\\-105\\-84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\280\\0\\-280\\-280\\-420\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\12\\0\\-30\\-84\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-280\\0\\56\\0\\-84\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-280\\-30\\0\\5\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-420\\-84\\-84\\0\\84\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-60\\-140\\-60\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[333480, -21420, 83160; -21420, 644, -5320; 83160, -5320, 20737],[420, 28, 1],[-26, 589, 49, -395, -469, -790, -297; 3, -70, -6, 46, 55, 93, 35; 105, -2380, -198, 1596, 1895, 3192, 1200],135,33]
%
% END OF LATTICE L136.33
%
\hbox{}\par\smallskip%
%
% LATTICE L136.34
%
\leavevmode%
\llap{\spacer}%
${L_{136.34}}=\pdualwitharg{3.5}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}{42}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
262920\\129360\\-42000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
129360\\63645\\-20655
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42000\\-20655\\6658
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1407\\-3048\\-580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
553\\-1198\\-228
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1357\\-2940\\-560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-207\\448\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-26\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
866\\-1876\\-357
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-280\\-240\\-840\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\12\\0\\-60\\-336\\-30\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\0\\280\\0\\-840\\-140\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-60\\0\\60\\0\\-30\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-336\\-840\\0\\168\\0\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-30\\-140\\-30\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-420\\-210\\-504\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[262920, 129360, -42000; 129360, 63645, -20655; -42000, -20655, 6658],[840, 15, 1],[1407, 553, 1357, -1, -207, 12, 866; -3048, -1198, -2940, 2, 448, -26, -1876; -580, -228, -560, 0, 84, -5, -357],135,34]
%
% END OF LATTICE L136.34
%
\hbox{}\par\smallskip%
%
% LATTICE L136.35
%
\leavevmode%
\llap{\spacer}%
${L_{136.35}}=\pdualwitharg{3.5}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{280}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{42}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-654360\\-166320\\-3360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-166320\\-42270\\-855
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3360\\-855\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\40\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\11\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\56\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-7\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-112\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-11\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-280\\-120\\-840\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-15\\-168\\-15\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\0\\280\\0\\-840\\-140\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-15\\0\\15\\0\\-15\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-168\\-840\\0\\168\\0\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-15\\-140\\-15\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-420\\-105\\-504\\0\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-654360, -166320, -3360; -166320, -42270, -855; -3360, -855, -17],[840, 15, 1],[-11, -3, -15, 2, 31, 3, -2; 40, 11, 56, -7, -112, -11, 7; 160, 39, 140, -45, -504, -40, 42],135,35]
%
% END OF LATTICE L136.35
%
\hbox{}\par\smallskip%
%
% LATTICE L136.36
%
\leavevmode%
\llap{\spacer}%
${L_{136.36}}=\pdualwitharg{2.5.7}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{70}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{420}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76682130\\4193490\\36588090
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4193490\\227080\\2000880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36588090\\2000880\\17457631
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-132409\\-81\\277515
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35957\\-22\\75362
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19574\\-12\\41025
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
167\\0\\-350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1679\\1\\-3519
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5110\\3\\-10710
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14704\\-9\\30818
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\0\\-105\\-420\\-105\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-70\\-28\\-210\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\0\\15\\0\\-15\\-210\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-70\\0\\70\\0\\-210\\-70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-28\\-15\\0\\1\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-210\\-210\\-210\\0\\420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-30\\-70\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[76682130, 4193490, 36588090; 4193490, 227080, 2000880; 36588090, 2000880, 17457631],[210, 70, 1],[-132409, -35957, -19574, 167, 1679, 5110, -14704; -81, -22, -12, 0, 1, 3, -9; 277515, 75362, 41025, -350, -3519, -10710, 30818],135,36]
%
% END OF LATTICE L136.36
%
\hbox{}\par\smallskip%
%
% LATTICE L136.37
%
\leavevmode%
\llap{\spacer}%
${L_{136.37}}=\pdualwitharg{5.7}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
171780\\420\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\-35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-15\\-805
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-420\\-840\\-420\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-70\\-56\\-105\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\60\\0\\-60\\-210\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-70\\0\\70\\0\\-105\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-56\\-60\\0\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-105\\-210\\-105\\0\\105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-30\\-35\\-12\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[171780, 420, -420; 420, -35, 0; -420, 0, 1],[420, 35, 1],[1, 0, -1, -2, -1, -1, 0; 12, 1, -6, -15, -8, -9, 0; 420, 7, -390, -805, -408, -420, -1],135,37]
%
% END OF LATTICE L136.37
%
\hbox{}\par\smallskip%
%
% LATTICE L136.38
%
\leavevmode%
\llap{\spacer}%
${L_{136.38}}=\pdualwitharg{3.7}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{*}{2}{420}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{s}{2}{84}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{s}{2}{30}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3438120\\-1723680\\16800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1723680\\-864129\\8421
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16800\\8421\\-82
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\56\\1036
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-30\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-12\\-208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67\\160\\2700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\36\\623
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\70\\1245
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-56\\-336\\-840\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\420\\0\\-420\\-1680\\-210\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\8\\0\\-120\\-28\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-420\\0\\84\\0\\-42\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-1680\\-120\\0\\120\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-210\\-28\\-42\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-210\\-60\\-210\\-360\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3438120, -1723680, 16800; -1723680, -864129, 8421; 16800, 8421, -82],[840, 21, 1],[-23, 13, 5, -1, -67, -15, -29; 56, -30, -12, 2, 160, 36, 70; 1036, -420, -208, 0, 2700, 623, 1245],135,38]
%
% END OF LATTICE L136.38
%
\hbox{}\par\smallskip%
%
% LATTICE L136.39
%
\leavevmode%
\llap{\spacer}%
${L_{136.39}}=\pdualwitharg{3.7}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40616520\\0\\64680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64680\\0\\-103
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-8\\3136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-628
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\0\\8160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\1883
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-5\\3765
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-56\\-168\\-840\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\0\\-105\\-840\\-105\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\8\\0\\-120\\-28\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-105\\0\\21\\0\\-21\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-840\\-120\\0\\120\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-105\\-28\\-21\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-105\\-60\\-105\\-360\\0\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-40616520, 0, 64680; 0, 21, 0; 64680, 0, -103],[840, 21, 1],[5, -1, -1, 0, 13, 3, 6; -8, -5, 0, 1, 0, -1, -5; 3136, -630, -628, 0, 8160, 1883, 3765],135,39]
%
% END OF LATTICE L136.39
%
\hbox{}\par\smallskip%
%
% LATTICE L136.40
%
\leavevmode%
\llap{\spacer}%
${L_{136.40}}=\pdualwitharg{2.3}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{1680}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-4488\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\336\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\25\\360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\7\\98
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-35\\-525
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-14\\-24\\-105\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\120\\0\\-120\\-840\\-120\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\28\\0\\-210\\-56\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-120\\0\\24\\0\\-24\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-840\\-210\\0\\105\\0\\-2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-120\\-56\\-24\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-840\\-840\\-2520\\0\\1680
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1680, -840, 0; -840, -4488, 336; 0, 336, -23],[840, 24, 1],[1, 13, 4, -1, -16, -3, -1; 2, 25, 7, -3, -35, -6, 0; 29, 360, 98, -48, -525, -88, 0],135,40]
%
% END OF LATTICE L136.40
%
\hbox{}\par\smallskip%
%
% LATTICE L136.41
%
\leavevmode%
\llap{\spacer}%
${L_{136.41}}=\pdualwitharg{2.3}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{420}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{1680}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\-7560\\-3360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7560\\33576\\14928
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3360\\14928\\6637
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-25\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\7\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\27\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\280\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\14\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-28\\-48\\-420\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\120\\0\\-120\\-1680\\-120\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\28\\0\\-420\\-56\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-120\\0\\24\\0\\-24\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-1680\\-420\\0\\420\\0\\-5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-120\\-56\\-24\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-840\\-840\\-5040\\0\\1680
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1680, -7560, -3360; -7560, 33576, 14928; -3360, 14928, 6637],[840, 24, 1],[1, 7, 3, 1, -3, -1, 1; -6, -25, 7, 27, 280, 14, 0; 14, 60, -14, -60, -630, -32, 0],135,41]
%
% END OF LATTICE L136.41
%
\hbox{}\par\smallskip%
%
% LATTICE L136.42
%
\leavevmode%
\llap{\spacer}%
${L_{136.42}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${70}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{105}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57903510\\4456830\\-17573850
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4456830\\340620\\-1354395
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17573850\\-1354395\\5332459
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59859\\119664\\-166880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24383\\48744\\-67977
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8849\\17690\\-24670
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
113\\-226\\315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2277\\-4552\\6348
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1155\\-2309\\3220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9971\\19933\\-27798
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\0\\-70\\-420\\-210\\-140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-105\\-84\\-105\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\10\\0\\-30\\-70\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-105\\0\\105\\0\\-105\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\-84\\-30\\0\\6\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-105\\-70\\-105\\0\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-30\\-105\\-36\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[57903510, 4456830, -17573850; 4456830, 340620, -1354395; -17573850, -1354395, 5332459],[210, 105, 1],[-59859, -24383, -8849, 113, 2277, 1155, -9971; 119664, 48744, 17690, -226, -4552, -2309, 19933; -166880, -67977, -24670, 315, 6348, 3220, -27798],135,42]
%
% END OF LATTICE L136.42
%
\hbox{}\par\smallskip%
%
% LATTICE L136.43
%
\leavevmode%
\llap{\spacer}%
${L_{136.43}}=\pdualwitharg{2.3.5}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}{140}\above{1pt}{1pt}{*}{2}{120}\above{1pt}{1pt}{l}{2}{21}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{84}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42389340\\327600\\-10504620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
327600\\-2220\\81180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10504620\\81180\\-2603179
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
244\\-12\\-985
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1415\\-65\\-5712
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3867\\-175\\-15610
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1635\\-73\\-6600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-463\\21\\1869
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-872\\39\\3520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1311\\56\\5292
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-70\\-120\\-105\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-120\\-168\\-60\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\140\\0\\-210\\-140\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\0\\120\\0\\-60\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-168\\-210\\0\\21\\0\\-252
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-60\\-140\\-60\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-420\\-420\\-252\\0\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-42389340, 327600, -10504620; 327600, -2220, 81180; -10504620, 81180, -2603179],[420, 60, 1],[244, 1415, 3867, 1635, -463, -872, -1311; -12, -65, -175, -73, 21, 39, 56; -985, -5712, -15610, -6600, 1869, 3520, 5292],135,43]
%
% END OF LATTICE L136.43
%
\hbox{}\par\smallskip%
%
% LATTICE L136.44
%
\leavevmode%
\llap{\spacer}%
${L_{136.44}}=\pdualwitharg{5.7}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${840}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{120}\above{1pt}{1pt}{l}{2}{35}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{210}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\0\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-70\\-245
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-245\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-24\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\21\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\192\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\115\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\80\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-7\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\0\\-840\\-840\\-840\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-35\\-56\\-105\\-7
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\0\\120\\0\\-120\\-420\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-35\\0\\35\\0\\-105\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-56\\-120\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-105\\-420\\-105\\0\\210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-60\\-35\\-24\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[840, 0, -840; 0, -70, -245; -840, -245, -17],[840, 35, 1],[7, -6, -55, -33, -23, -1, 2; -24, 21, 192, 115, 80, 3, -7; 0, -7, -60, -35, -24, 0, 2],135,44]
%
% END OF LATTICE L136.44
%
\hbox{}\par\smallskip%
%
% LATTICE L136.45
%
\leavevmode%
\llap{\spacer}%
${L_{136.45}}=\pdualwitharg{5.7}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${840}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9240\\-4200\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4200\\-1295\\-35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-35\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\24\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-12\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-30\\-490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-16\\-268
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-18\\-315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
840\\0\\-840\\-1680\\-840\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\28\\0\\-140\\-112\\-210\\-14
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\0\\120\\0\\-120\\-420\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-140\\0\\140\\0\\-210\\-70
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-112\\-120\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-210\\-420\\-210\\0\\210\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-14\\-60\\-70\\-24\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9240, -4200, 0; -4200, -1295, -35; 0, -35, 2],[840, 35, 1],[-11, -1, 5, 13, 7, 8, 0; 24, 2, -12, -30, -16, -18, 0; 420, 42, -180, -490, -268, -315, -1],135,45]
%
% END OF LATTICE L136.45
%
\hbox{}\par\smallskip%
%
% LATTICE L136.46
%
\leavevmode%
\llap{\spacer}%
${L_{136.46}}=\pdualwitharg{2.5}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{420}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28560\\10920\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10920\\-3320\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\240\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\63\\630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\27\\280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\14\\147
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-42\\-448
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-210\\-120\\-105\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-40\\-56\\-120\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\420\\0\\-210\\-840\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-40\\0\\40\\0\\-120\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-56\\-210\\0\\7\\0\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\-840\\-120\\0\\240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-840\\-280\\-168\\0\\112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-28560, 10920, -840; 10920, -3320, 240; -840, 240, -17],[840, 40, 1],[-1, 0, 5, 2, 1, -1, -3; -15, -1, 63, 27, 14, -12, -42; -165, -16, 630, 280, 147, -120, -448],135,46]
%
% END OF LATTICE L136.46
%
\hbox{}\par\smallskip%
%
% LATTICE L136.47
%
\leavevmode%
\llap{\spacer}%
${L_{136.47}}=\pdualwitharg{2.5}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{420}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68880\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\920\\200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\200\\43
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\84\\-390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-315\\1470
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-159\\740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-238\\1106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-78\\360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\84\\-392
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-420\\-240\\-420\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-40\\-112\\-120\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\420\\0\\-420\\-840\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-40\\0\\40\\0\\-120\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-112\\-420\\0\\28\\0\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-120\\-840\\-120\\0\\240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-840\\-280\\-336\\0\\112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-68880, -840, 0; -840, 920, 200; 0, 200, 43],[840, 40, 1],[-1, 0, 4, 2, 3, 1, -1; 84, 1, -315, -159, -238, -78, 84; -390, -4, 1470, 740, 1106, 360, -392],135,47]
%
% END OF LATTICE L136.47
%
\hbox{}\par\smallskip%
%
% LATTICE L136.48
%
\leavevmode%
\llap{\spacer}%
${L_{136.48}}=\pdualwitharg{2.3.7}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{r}{2}{840}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{168}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
449904840\\-2655660\\111384000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2655660\\15708\\-657468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
111384000\\-657468\\27575599
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1350\\-2\\-5453
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4991\\-10\\-20160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\0\\-206
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1123\\2\\-4536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2269\\0\\-9165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1733\\-1\\-7000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4382\\-5\\-17700
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-14\\-168\\-105\\-28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\840\\0\\-840\\-840\\-420\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\4\\0\\-30\\-28\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-840\\0\\168\\0\\-84\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-840\\-30\\0\\15\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-420\\-28\\-84\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-420\\-60\\-420\\-180\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[449904840, -2655660, 111384000; -2655660, 15708, -657468; 111384000, -657468, 27575599],[420, 84, 1],[1350, 4991, 51, 1123, 2269, 1733, 4382; -2, -10, 0, 2, 0, -1, -5; -5453, -20160, -206, -4536, -9165, -7000, -17700],135,48]
%
% END OF LATTICE L136.48
%
\hbox{}\par\smallskip%
%
% LATTICE L136.49
%
\leavevmode%
\llap{\spacer}%
${L_{136.49}}=\pdualwitharg{3.5.7}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${140}\above{1pt}{1pt}{r}{2}{42}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{b}{2}{210}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{35}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93660\\-4620\\-2100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4620\\210\\105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2100\\105\\47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
140\\0\\-140\\-840\\-420\\-140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-210\\-168\\-105\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\20\\0\\-60\\-70\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-210\\0\\210\\0\\-105\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-168\\-60\\0\\12\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-105\\-70\\-105\\0\\35\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-30\\-105\\-36\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[93660, -4620, -2100; -4620, 210, 105; -2100, 105, 47],[420, 105, 1],[-3, -2, -1, 0, 1, 1, 0; 0, -3, -2, 1, 4, 4, 1; -140, -84, -40, 0, 36, 35, -3],135,49]
%
% END OF LATTICE L136.49
%
\hbox{}\par\smallskip%
%
% LATTICE L136.50
%
\leavevmode%
\llap{\spacer}%
${L_{136.50}}=\pdualwitharg{2.3.5.7}(\pfillwitharg{2}({L_{136.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{}{2}{42}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{210}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72110791410\\28419879390\\13731070500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28419879390\\11200675080\\5411608440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13731070500\\5411608440\\2614619717
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-11\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-8363\\17094
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-4237\\8675
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69\\-17525\\35910
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-233\\477
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\12123\\-24850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\4115\\-8454
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-35\\-420\\-105\\-140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\42\\0\\-210\\-84\\-210\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\0\\5\\0\\-15\\-70\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-210\\0\\210\\0\\-210\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-84\\-15\\0\\3\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-210\\-70\\-210\\0\\140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-30\\-210\\-36\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[72110791410, 28419879390, 13731070500; 28419879390, 11200675080, 5411608440; 13731070500, 5411608440, 2614619717],[210, 210, 1],[11, 41, 18, 69, 1, -46, -12; -11, -8363, -4237, -17525, -233, 12123, 4115; -35, 17094, 8675, 35910, 477, -24850, -8454],135,50]
%
% END OF LATTICE L136.50
%
\hbox{}\par\smallskip%
%
% LATTICE L136.51
%
\leavevmode%
\llap{\spacer}%
${L_{136.51}}=\pdualwitharg{2.7}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${21}\above{1pt}{1pt}{}{2}{280}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{336}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14280\\5040\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\-1736\\280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\280\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\9\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\24\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\10\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-15\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\-42\\-168\\-105\\-168\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\280\\0\\-280\\-280\\-840\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\0\\12\\0\\-30\\-168\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-280\\0\\56\\0\\-168\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-280\\-30\\0\\5\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-840\\-168\\-168\\0\\336\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-120\\-280\\-120\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14280, 5040, -840; 5040, -1736, 280; -840, 280, -43],[840, 56, 1],[-1, 3, 2, 5, 2, -1, -3; -6, 10, 9, 24, 10, -3, -15; -21, 0, 18, 56, 25, 0, -40],135,51]
%
% END OF LATTICE L136.51
%
\hbox{}\par\smallskip%
%
% LATTICE L136.52
%
\leavevmode%
\llap{\spacer}%
${L_{136.52}}=\pdualwitharg{2.7}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${84}\above{1pt}{1pt}{b}{2}{280}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
333480\\-107520\\-51240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-107520\\34664\\16520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51240\\16520\\7873
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\57\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\70\\-140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-4\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-5\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\35\\-80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
84\\0\\-84\\-336\\-420\\-336\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\280\\0\\-280\\-560\\-840\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\0\\12\\0\\-60\\-168\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-280\\0\\56\\0\\-168\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-560\\-60\\0\\20\\0\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-840\\-168\\-168\\0\\336\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-280\\-120\\-280\\-240\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[333480, -107520, -51240; -107520, 34664, 16520; -51240, 16520, 7873],[840, 56, 1],[-1, 1, 1, 3, 3, 1, -1; 57, 70, 6, -4, -5, 3, 35; -126, -140, -6, 28, 30, 0, -80],135,52]
%
% END OF LATTICE L136.52
%
\hbox{}\par\smallskip%
%
% LATTICE L136.53
%
\leavevmode%
\llap{\spacer}%
${L_{136.53}}=\pdualwitharg{2.5.7}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{2}{280}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{420}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19277160\\179340\\-4639740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
179340\\1540\\-43260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4639740\\-43260\\1116649
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-78\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-617\\1850\\-2492
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1285\\3852\\-5190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2877\\8622\\-11620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231\\692\\-933
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\-627\\840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
103\\-309\\416
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\0\\-210\\-840\\-105\\-420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-280\\-56\\-420\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\60\\0\\-30\\-420\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-280\\0\\280\\0\\-420\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-56\\-30\\0\\1\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-420\\-420\\-420\\0\\420\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-60\\-140\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19277160, 179340, -4639740; 179340, 1540, -43260; -4639740, -43260, 1116649],[420, 140, 1],[26, -617, -1285, -2877, -231, 208, 103; -78, 1850, 3852, 8622, 692, -627, -309; 105, -2492, -5190, -11620, -933, 840, 416],135,53]
%
% END OF LATTICE L136.53
%
\hbox{}\par\smallskip%
%
% LATTICE L136.54
%
\leavevmode%
\llap{\spacer}%
${L_{136.54}}=\pdualwitharg{3.5.7}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${280}\above{1pt}{1pt}{*}{2}{84}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{s}{2}{420}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{b}{2}{70}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
187320\\89040\\-4200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89040\\42315\\-1995
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4200\\-1995\\94
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\0\\-280\\-1680\\-840\\-280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\84\\0\\-420\\-336\\-210\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\0\\40\\0\\-120\\-140\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-420\\0\\420\\0\\-210\\-210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-336\\-120\\0\\24\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-210\\-140\\-210\\0\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-42\\-60\\-210\\-72\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[187320, 89040, -4200; 89040, 42315, -1995; -4200, -1995, 94],[840, 105, 1],[-3, 1, 1, -1, -3, -3, -1; 0, -6, -4, 2, 8, 8, 2; -140, -84, -40, 0, 36, 35, -3],135,54]
%
% END OF LATTICE L136.54
%
\hbox{}\par\smallskip%
%
% LATTICE L136.55
%
\leavevmode%
\llap{\spacer}%
${L_{136.55}}=\pdualwitharg{3.5.7}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${280}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{l}{2}{105}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{70}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2309160\\0\\5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\105\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\0\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\-1400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-462
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-460
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\455
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\0\\-280\\-840\\-840\\-280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\21\\0\\-105\\-168\\-105\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\0\\40\\0\\-120\\-140\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-105\\0\\105\\0\\-105\\-105
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-168\\-120\\0\\24\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-105\\-140\\-105\\0\\70\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\-60\\-105\\-72\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2309160, 0, 5040; 0, 105, 0; 5040, 0, -11],[840, 105, 1],[-3, -1, -1, 0, 1, 1, 0; -8, -1, 0, 1, 0, -1, -1; -1400, -462, -460, 0, 456, 455, -3],135,55]
%
% END OF LATTICE L136.55
%
\hbox{}\par\smallskip%
%
% LATTICE L136.56
%
\leavevmode%
\llap{\spacer}%
${L_{136.56}}=\pdualwitharg{2.3.5}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{140}\above{1pt}{1pt}{l}{2}{120}\above{1pt}{1pt}{}{2}{21}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{*}{2}{336}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-959280\\-24360\\-7560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24360\\2040\\840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7560\\840\\341
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-196\\505
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-587\\1512
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-1169\\3010
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-187\\480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\399\\-1029
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\590\\-1520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\196\\-504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-70\\-120\\-105\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-120\\-168\\-120\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\140\\0\\-210\\-280\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-120\\0\\120\\0\\-120\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-168\\-210\\0\\21\\0\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-120\\-280\\-120\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-840\\-840\\-504\\0\\336
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-959280, -24360, -7560; -24360, 2040, 840; -7560, 840, 341],[840, 120, 1],[1, 3, 6, 1, -2, -3, -1; -196, -587, -1169, -187, 399, 590, 196; 505, 1512, 3010, 480, -1029, -1520, -504],135,56]
%
% END OF LATTICE L136.56
%
\hbox{}\par\smallskip%
%
% LATTICE L136.57
%
\leavevmode%
\llap{\spacer}%
${L_{136.57}}=\pdualwitharg{2.3.5}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{s}{2}{140}\above{1pt}{1pt}{s}{2}{120}\above{1pt}{1pt}{b}{2}{84}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{s}{2}{336}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2709840\\506520\\247800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
506520\\94440\\46200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
247800\\46200\\22601
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-294\\590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-293\\588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-245\\490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\387\\-780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\1064\\-2142
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\338\\-680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-252\\504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-140\\-240\\-420\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-120\\-336\\-120\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\0\\140\\0\\-420\\-280\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-120\\0\\120\\0\\-120\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-336\\-420\\0\\84\\0\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-120\\-280\\-120\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-840\\-840\\-1008\\0\\336
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2709840, 506520, 247800; 506520, 94440, 46200; 247800, 46200, 22601],[840, 120, 1],[1, 1, 1, -1, -3, -1, 1; -294, -293, -245, 387, 1064, 338, -252; 590, 588, 490, -780, -2142, -680, 504],135,57]
%
% END OF LATTICE L136.57
%
\hbox{}\par\smallskip%
%
% LATTICE L136.58
%
\leavevmode%
\llap{\spacer}%
${L_{136.58}}=\pdualwitharg{2.3.7}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{b}{2}{840}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{168}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{s}{2}{240}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\69720\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69720\\-5497296\\66192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\66192\\-797
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\5\\406
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-5\\-420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-82
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\15\\1230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\13\\1064
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\25\\2040
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\-28\\-336\\-420\\-112\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\840\\0\\-840\\-1680\\-840\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\4\\0\\-60\\-56\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-840\\0\\168\\0\\-168\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-1680\\-60\\0\\60\\0\\-720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-840\\-56\\-168\\0\\112\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-120\\-840\\-720\\0\\240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-840, 69720, -840; 69720, -5497296, 66192; -840, 66192, -797],[840, 168, 1],[8, 3, -1, -1, 14, 14, 32; 5, -5, -1, 1, 15, 13, 25; 406, -420, -82, 84, 1230, 1064, 2040],135,58]
%
% END OF LATTICE L136.58
%
\hbox{}\par\smallskip%
%
% LATTICE L136.59
%
\leavevmode%
\llap{\spacer}%
${L_{136.59}}=\pdualwitharg{2.3.7}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{840}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{168}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36515640\\-7162680\\87360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7162680\\-1404816\\17136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
87360\\17136\\-209
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\1589
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-5\\840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-418
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\2925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-1\\5768
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-5\\12960
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-14\\-168\\-105\\-56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\840\\0\\-840\\-840\\-840\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\0\\4\\0\\-30\\-56\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-840\\0\\168\\0\\-168\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-840\\-30\\0\\15\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-840\\-56\\-168\\0\\112\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-840\\-120\\-840\\-360\\0\\240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36515640, -7162680, 87360; -7162680, -1404816, 17136; 87360, 17136, -209],[840, 168, 1],[4, 3, -1, -1, 7, 14, 32; -1, -5, 0, 1, 0, -1, -5; 1589, 840, -418, -336, 2925, 5768, 12960],135,59]
%
% END OF LATTICE L136.59
%
\hbox{}\par\smallskip%
%
% LATTICE L136.60
%
\leavevmode%
\llap{\spacer}%
${L_{136.60}}=\pdualwitharg{2.3.5.7}(\main({L_{136.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{r}{2}{168}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{840}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{140}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7980\\-118860\\-29400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-118860\\1341480\\331800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29400\\331800\\82067
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-736\\2975
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-665\\2688
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\47\\-190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1039\\-4200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\23\\-93
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-381\\1540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-380\\1536
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-70\\-840\\-105\\-140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-840\\-168\\-420\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\20\\0\\-30\\-140\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-840\\0\\840\\0\\-420\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-168\\-30\\0\\3\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-420\\-140\\-420\\0\\140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-84\\-60\\-420\\-36\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7980, -118860, -29400; -118860, 1341480, 331800; -29400, 331800, 82067],[420, 420, 1],[-2, -2, 0, 2, 0, -1, -1; -736, -665, 47, 1039, 23, -381, -380; 2975, 2688, -190, -4200, -93, 1540, 1536],135,60]
%
% END OF LATTICE L136.60
%
\hbox{}\par\smallskip%
%
% LATTICE L136.61
%
\leavevmode%
\llap{\spacer}%
${L_{136.61}}=\pdualwitharg{2.5.7}({L_{136.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${420}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{s}{2}{280}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{1680}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\-88200\\-43680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88200\\3160360\\1565200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43680\\1565200\\775181
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\312\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-513\\1036
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-1233\\2490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-2981\\6020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-514\\1038
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\206\\-416
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
420\\0\\-420\\-1680\\-420\\-1680\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-280\\-112\\-840\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\0\\60\\0\\-60\\-840\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-280\\0\\280\\0\\-840\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-112\\-60\\0\\4\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-840\\-840\\-840\\0\\1680\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-120\\-280\\-48\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1680, -88200, -43680; -88200, 3160360, 1565200; -43680, 1565200, 775181],[840, 280, 1],[-1, 3, 7, 17, 3, 1, -1; 312, -513, -1233, -2981, -514, 0, 206; -630, 1036, 2490, 6020, 1038, 0, -416],135,61]
%
% END OF LATTICE L136.61
%
\hbox{}\par\smallskip%
%
% LATTICE L136.62
%
\leavevmode%
\llap{\spacer}%
${L_{136.62}}=\pdualwitharg{2.5.7}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${105}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{l}{2}{280}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{1680}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1039080\\-115080\\-44520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-115080\\-9520\\-3640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44520\\-3640\\-1391
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-546\\1365
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\269\\-672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\276\\-690
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-785\\1960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-267\\667
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-5379\\13440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-541\\1352
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\0\\-210\\-840\\-105\\-840\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-280\\-56\\-840\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210\\0\\60\\0\\-30\\-840\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-280\\0\\280\\0\\-840\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-56\\-30\\0\\1\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-840\\-840\\-840\\0\\1680\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\-120\\-280\\-24\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1039080, -115080, -44520; -115080, -9520, -3640; -44520, -3640, -1391],[840, 280, 1],[2, -1, -1, 3, 1, 20, 2; -546, 269, 276, -785, -267, -5379, -541; 1365, -672, -690, 1960, 667, 13440, 1352],135,62]
%
% END OF LATTICE L136.62
%
\hbox{}\par\smallskip%
%
% LATTICE L136.63
%
\leavevmode%
\llap{\spacer}%
${L_{136.63}}=\pdualwitharg{2.3.5.7}({L_{136.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${35}\above{1pt}{1pt}{}{2}{168}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{840}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{560}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2306640\\-2307480\\9240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2307480\\-2307480\\9240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9240\\9240\\-37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\245
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-250
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\249
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\2240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\744
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-70\\-840\\-105\\-280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\168\\0\\-840\\-168\\-840\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\0\\20\\0\\-30\\-280\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-840\\0\\840\\0\\-840\\-840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105\\-168\\-30\\0\\3\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-840\\-280\\-840\\0\\560\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-168\\-120\\-840\\-72\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2306640, -2307480, 9240; -2307480, -2307480, 9240; 9240, 9240, -37],[840, 840, 1],[-1, -1, 0, 1, 0, -1, -1; 2, -1, -1, -1, 1, 10, 4; 245, -504, -250, 0, 249, 2240, 744],135,63]
%
% END OF LATTICE L136.63
%
\hbox{}\par\smallskip%
%
% LATTICE L136.64
%
\leavevmode%
\llap{\spacer}%
${L_{136.64}}=\pdualwitharg{2.3.5.7}({L_{136.2}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26481840, 11980920, -35280; 11980920, 5418840, -15960; -35280, -15960, 47],[840, 840, 1],[5, -1, -1, 1, 3, 13, 5; 4, -1, -1, -1, 2, 10, 4; 5110, -1092, -1090, 420, 2934, 13160, 5112],135,64]
%
% END OF LATTICE L136.64
%

\medskip%
%------------------------------WEYL-GROUP-W137--------------------------------
%
\leavevmode\llap{}%
$W_{137}$%
\qquad\llap{64} lattices, $\chi=72$%
\hfill%
$2222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L137.1
%
\leavevmode%
\llap{\spacer}%
${L_{137.1}}$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2\rightarrow N'_{35}}%
\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-37\\-5620\\-24480
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[8617560, -9240, -10920; -9240, 13, 11; -10920, 11, 14],[840, 1, 1],[-1, -5, -19, -4, -37, -83, -179, -221, -23, -23, -29, -2, -3, -1, -1, -3; -150, -756, -2880, -607, -5620, -12614, -27210, -33600, -3498, -3500, -4416, -305, -460, -154, -150, -448; -663, -3311, -12576, -2647, -24480, -54908, -118410, -146188, -15213, -15211, -19176, -1322, -1980, -658, -660, -1988],136,1]
%
% END OF LATTICE L137.1
%
\hbox{}\par\smallskip%
%
% LATTICE L137.2
%
\leavevmode%
\llap{\spacer}%
${L_{137.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[152066040, 533400, -47040; 533400, 1871, -165; -47040, -165, 14],[840, 1, 1],[-2, 5, 37, 19, 101, 122, 271, 683, 74, 79, 107, 17, 19, 4, -1, -11; 570, -1428, -10560, -5422, -28820, -34811, -77325, -194880, -21114, -22540, -30528, -4850, -5420, -1141, 285, 3136; -3, -35, -156, -70, -340, -392, -855, -2128, -225, -231, -300, -44, -40, -7, 0, 0],136,2]
%
% END OF LATTICE L137.2
%
\hbox{}\par\smallskip%
%
% LATTICE L137.3
%
\leavevmode%
\llap{\spacer}%
${L_{137.3}}=\pfillwitharg{2}({L_{137.1}})$%
{${}={}$}Nikulin $35'$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[6090, 630, 630; 630, 65, 63; 630, 63, 38],[210, 1, 1],[4, 8, -13, -12, -87, -243, -569, -743, -172, -204, -155, -30, -53, -51, -61, -41; -42, -84, 138, 127, 920, 2569, 6015, 7854, 1818, 2156, 1638, 317, 560, 539, 645, 434; 3, 7, -12, -11, -80, -224, -525, -686, -159, -189, -144, -28, -50, -49, -60, -42],136,3]
%
% END OF LATTICE L137.3
%
\hbox{}\par\smallskip%
%
% LATTICE L137.4
%
\leavevmode%
\llap{\spacer}%
${L_{137.4}}=\main({L_{137.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[76033020, 533400, -23520; 533400, 3742, -165; -23520, -165, 7],[420, 1, 1],[74, 79, 107, 17, 19, 8, -2, -11, -2, 5, 37, 19, 101, 244, 542, 683; -10557, -11270, -15264, -2425, -2710, -1141, 285, 1568, 285, -714, -5280, -2711, -14410, -34811, -77325, -97440; -225, -231, -300, -44, -40, -14, 0, 0, -3, -35, -156, -70, -340, -784, -1710, -2128],136,4]
%
% END OF LATTICE L137.4
%
\hbox{}\par\smallskip%
%
% LATTICE L137.5
%
\leavevmode%
\llap{\spacer}%
${L_{137.5}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{137.1}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-206430, 14490, -102480; 14490, -548, 7186; -102480, 7186, -50875],[210, 2, 1],[850, -208, -1217, -1377, -3762, -18231, -40493, -25495, -11026, -11664, -3871, -1167, -478, 423, 2333, 1751; -27, 7, 39, 44, 120, 581, 1290, 812, 351, 371, 123, 37, 15, -14, -75, -56; -1716, 420, 2457, 2780, 7595, 36806, 81750, 51471, 22260, 23548, 7815, 2356, 965, -854, -4710, -3535],136,5]
%
% END OF LATTICE L137.5
%
\hbox{}\par\smallskip%
%
% LATTICE L137.6
%
\leavevmode%
\llap{\spacer}%
${L_{137.6}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{137.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-173670, 15540, -57330; 15540, -822, 5121; -57330, 5121, -18925],[210, 3, 1],[190, -138, -543, -922, -5039, -12211, -9041, -34155, -2462, -7814, -1729, -782, -641, 283, 521, 2347; -9, 7, 26, 44, 240, 581, 430, 1624, 117, 371, 82, 37, 30, -14, -25, -112; -578, 420, 1652, 2805, 15330, 37149, 27505, 103908, 7490, 23772, 5260, 2379, 1950, -861, -1585, -7140],136,6]
%
% END OF LATTICE L137.6
%
\hbox{}\par\smallskip%
%
% LATTICE L137.7
%
\leavevmode%
\llap{\spacer}%
${L_{137.7}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{137.1}}))$%
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{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-152250, 17640, 61320; 17640, -1370, -7115; 61320, -7115, -24697],[210, 5, 1],[-716, 170, 2039, 1155, 1263, 15306, 6800, 42817, 9260, 9798, 6505, 981, 161, -354, -392, -2945; -27, 7, 78, 44, 48, 581, 258, 1624, 351, 371, 246, 37, 6, -14, -15, -112; -1770, 420, 5040, 2855, 3122, 37835, 16809, 105840, 22890, 24220, 16080, 2425, 398, -875, -969, -7280],136,7]
%
% END OF LATTICE L137.7
%
\hbox{}\par\smallskip%
%
% LATTICE L137.8
%
\leavevmode%
\llap{\spacer}%
${L_{137.8}}=\pdualwitharg{3}(\main({L_{137.2}}))$%
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{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[166740, 66780, -420; 66780, 26745, -168; -420, -168, 1],[420, 3, 1],[0, 19, 31, 43, 213, 496, 362, 1355, 48, 149, 65, 29, 27, 8, -2, -11; 0, -49, -80, -111, -550, -1281, -935, -3500, -124, -385, -168, -75, -70, -21, 5, 28; -1, -252, -412, -573, -2850, -6657, -4865, -18228, -647, -2016, -884, -399, -390, -147, 5, 84],136,8]
%
% END OF LATTICE L137.8
%
\hbox{}\par\smallskip%
%
% LATTICE L137.9
%
\leavevmode%
\llap{\spacer}%
${L_{137.9}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{137.1}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9492992040, -2559270, 4719221850; -2559270, 690, -1272282; 4719221850, -1272282, 2346052201],[210, 6, 1],[347, 2777, 954, 2601, 6674, 32238, 23917, 45319, 6585, 21485, 2512, 2598, 1991, 4155, 2078, 3205; 20, 182, 50, 119, 265, 1169, 835, 1540, 212, 630, 66, 55, 25, 49, 35, 84; -698, -5586, -1919, -5232, -13425, -64848, -48110, -91161, -13246, -43218, -5053, -5226, -4005, -8358, -4180, -6447],136,9]
%
% END OF LATTICE L137.9
%
\hbox{}\par\smallskip%
%
% LATTICE L137.10
%
\leavevmode%
\llap{\spacer}%
${L_{137.10}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{137.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-147630, 19740, 63630; 19740, -1918, -8519; 63630, -8519, -27425],[210, 7, 1],[-782, 26, 2219, 1258, 6881, 2383, 37057, 6667, 10094, 1526, 7093, 1070, 879, -55, -2137, -459; -27, 1, 78, 44, 240, 83, 1290, 232, 351, 53, 246, 37, 30, -2, -75, -16; -1806, 60, 5124, 2905, 15890, 5503, 85575, 15396, 23310, 3524, 16380, 2471, 2030, -127, -4935, -1060],136,10]
%
% END OF LATTICE L137.10
%
\hbox{}\par\smallskip%
%
% LATTICE L137.11
%
\leavevmode%
\llap{\spacer}%
${L_{137.11}}=\pdualwitharg{2}(\main({L_{137.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2349480, -707700, 586740; -707700, 199612, -176640; 586740, -176640, 146527],[420, 4, 1],[-156266, -154293, -47717, -25041, -6703, 3945, 11633, -494, -5818, -34805, -33325, -56001, -128367, -571239, -1227263, -377702; 4431, 4375, 1353, 710, 190, -112, -330, 14, 165, 987, 945, 1588, 3640, 16198, 34800, 10710; 631080, 623112, 192705, 101128, 27070, -15932, -46980, 1995, 23496, 140560, 134583, 226160, 518410, 2306948, 4956300, 1525349],136,11]
%
% END OF LATTICE L137.11
%
\hbox{}\par\smallskip%
%
% LATTICE L137.12
%
\leavevmode%
\llap{\spacer}%
${L_{137.12}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{137.1}}))$%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[78532440, -300510, 37307550; -300510, 1150, -142760; 37307550, -142760, 17723291],[210, 10, 1],[157, 433, 421, 371, 185, 4394, 1943, 6108, 2639, 2829, 977, 328, 48, 499, 154, 416; 60, 182, 150, 119, 53, 1169, 501, 1540, 636, 630, 198, 55, 5, 49, 21, 84; -330, -910, -885, -780, -389, -9240, -4086, -12845, -5550, -5950, -2055, -690, -101, -1050, -324, -875],136,12]
%
% END OF LATTICE L137.12
%
\hbox{}\par\smallskip%
%
% LATTICE L137.13
%
\leavevmode%
\llap{\spacer}%
${L_{137.13}}=\pdualwitharg{5}(\main({L_{137.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8494500, 215460, -29400; 215460, -5465, 745; -29400, 745, -94],[420, 5, 1],[16, -10, -113, -60, -63, -747, -329, -2059, -220, -228, -295, -42, -5, 33, 23, 155; 639, -399, -4512, -2396, -2516, -29834, -13140, -82236, -8787, -9107, -11784, -1678, -200, 1316, 918, 6188; 60, -35, -420, -225, -238, -2835, -1251, -7840, -840, -875, -1140, -165, -22, 105, 81, 560],136,13]
%
% END OF LATTICE L137.13
%
\hbox{}\par\smallskip%
%
% LATTICE L137.14
%
\leavevmode%
\llap{\spacer}%
${L_{137.14}}=\pdualwitharg{3}({L_{137.1}})$%
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{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-207480, 0, 840; 0, 15, -9; 840, -9, 2],[840, 3, 1],[0, 2, 3, 2, 19, 43, 31, 115, 4, 12, 5, 1, 1, -1, -1, -3; 0, 294, 440, 293, 2780, 6286, 4530, 16800, 584, 1750, 728, 145, 140, -154, -150, -448; -1, 483, 728, 486, 4620, 10458, 7540, 27972, 973, 2919, 1216, 243, 240, -252, -250, -756],136,14]
%
% END OF LATTICE L137.14
%
\hbox{}\par\smallskip%
%
% LATTICE L137.15
%
\leavevmode%
\llap{\spacer}%
${L_{137.15}}=\pdualwitharg{3}({L_{137.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[333480, 68040, -840; 68040, 13881, -171; -840, -171, 2],[840, 3, 1],[0, 19, 31, 43, 213, 248, 181, 1355, 48, 149, 65, 29, 27, 4, -1, -11; 0, -98, -160, -222, -1100, -1281, -935, -7000, -248, -770, -336, -150, -140, -21, 5, 56; -1, -399, -652, -906, -4500, -5250, -3835, -28728, -1019, -3171, -1388, -624, -600, -105, 10, 168],136,15]
%
% END OF LATTICE L137.15
%
\hbox{}\par\smallskip%
%
% LATTICE L137.16
%
\leavevmode%
\llap{\spacer}%
${L_{137.16}}=\pdualwitharg{7}(\main({L_{137.2}}))$%
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{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[47460, 2940, 1260; 2940, 182, 77; 1260, 77, 25],[420, 7, 1],[-8, -4, -89, -34, -143, -43, -631, -109, -76, -10, -79, -8, 3, 1, 1, -3; 138, 69, 1536, 587, 2470, 743, 10905, 1884, 1314, 173, 1368, 139, -50, -17, -15, 52; -21, -11, -252, -98, -420, -128, -1890, -328, -231, -31, -252, -28, 0, 2, 0, -8],136,16]
%
% END OF LATTICE L137.16
%
\hbox{}\par\smallskip%
%
% LATTICE L137.17
%
\leavevmode%
\llap{\spacer}%
${L_{137.17}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{137.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28668360, -214830, 13315050; -214830, 1610, -99778; 13315050, -99778, 6184189],[210, 14, 1],[137, 55, 362, 313, 766, 514, 7919, 709, 2131, 323, 772, 254, 179, 53, 586, 47; 60, 26, 150, 119, 265, 167, 2505, 220, 636, 90, 198, 55, 25, 7, 105, 12; -294, -118, -777, -672, -1645, -1104, -17010, -1523, -4578, -694, -1659, -546, -385, -114, -1260, -101],136,17]
%
% END OF LATTICE L137.17
%
\hbox{}\par\smallskip%
%
% LATTICE L137.18
%
\leavevmode%
\llap{\spacer}%
${L_{137.18}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{137.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[65532810, 336210, 23305170; 336210, 1725, 119565; 23305170, 119565, 8287924],[210, 15, 1],[-41, -337, -221, -294, -295, -3516, -519, -9799, -707, -2281, -527, -267, -79, -411, -42, -673; 20, 182, 100, 119, 106, 1169, 167, 3080, 212, 630, 132, 55, 10, 49, 7, 168; 115, 945, 620, 825, 828, 9870, 1457, 27510, 1985, 6405, 1480, 750, 222, 1155, 118, 1890],136,18]
%
% END OF LATTICE L137.18
%
\hbox{}\par\smallskip%
%
% LATTICE L137.19
%
\leavevmode%
\llap{\spacer}%
${L_{137.19}}=\pdualwitharg{5}({L_{137.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-136840200, -1748040, -189840; -1748040, -22330, -2425; -189840, -2425, -263],[840, 5, 1],[-7, -5, 41, 15, 39, 519, 239, 1543, 175, 201, 295, 27, 17, 69, 13, 25; 561, 399, -3288, -1202, -3124, -41566, -19140, -123564, -14013, -16093, -23616, -2161, -1360, -5516, -1038, -1988; -120, -70, 720, 255, 652, 8610, 3954, 25480, 2880, 3290, 4800, 435, 268, 1050, 186, 280],136,19]
%
% END OF LATTICE L137.19
%
\hbox{}\par\smallskip%
%
% LATTICE L137.20
%
\leavevmode%
\llap{\spacer}%
${L_{137.20}}=\pdualwitharg{5}({L_{137.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-729960, 190680, 7560; 190680, -49805, -1970; 7560, -1970, -73],[840, 5, 1],[-37, 30, 281, 145, 149, 872, 382, 4763, 505, 516, 655, 89, 7, -53, -31, -395; -147, 119, 1116, 576, 592, 3465, 1518, 18928, 2007, 2051, 2604, 354, 28, -210, -123, -1568; 135, -105, -1020, -530, -548, -3220, -1413, -17640, -1875, -1925, -2460, -340, -32, 175, 108, 1400],136,20]
%
% END OF LATTICE L137.20
%
\hbox{}\par\smallskip%
%
% LATTICE L137.21
%
\leavevmode%
\llap{\spacer}%
${L_{137.21}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{137.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[95268390, 479640, 33273030; 479640, 2415, 167517; 33273030, 167517, 11620796],[210, 21, 1],[-71, -85, -377, -492, -2423, -816, -4197, -2257, -1133, -517, -827, -411, -587, -87, -318, -151; 20, 26, 100, 119, 530, 167, 835, 440, 212, 90, 132, 55, 50, 7, 35, 24; 203, 243, 1078, 1407, 6930, 2334, 12005, 6456, 3241, 1479, 2366, 1176, 1680, 249, 910, 432],136,21]
%
% END OF LATTICE L137.21
%
\hbox{}\par\smallskip%
%
% LATTICE L137.22
%
\leavevmode%
\llap{\spacer}%
${L_{137.22}}=\pdualwitharg{2.3}(\main({L_{137.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8457540, -50400, 2100840; -50400, 276, -12516; 2100840, -12516, 521845],[420, 12, 1],[-103, -416, 24, 505, 1983, 11209, 8769, 8594, 1327, 4700, 588, 1331, 1077, 1643, 411, -26; 56, 217, -17, -297, -1135, -6363, -4965, -4858, -748, -2639, -329, -741, -595, -903, -225, 14; 416, 1680, -97, -2040, -8010, -45276, -35420, -34713, -5360, -18984, -2375, -5376, -4350, -6636, -1660, 105],136,22]
%
% END OF LATTICE L137.22
%
\hbox{}\par\smallskip%
%
% LATTICE L137.23
%
\leavevmode%
\llap{\spacer}%
${L_{137.23}}=\pdualwitharg{7}({L_{137.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11031720, -35280, -23520; -35280, 91, 77; -23520, 77, 50],[840, 7, 1],[8, 4, 89, 17, 143, 43, 631, 109, 76, 10, 79, 4, -3, -1, -1, 3; 276, 138, 3072, 587, 4940, 1486, 21810, 3768, 2628, 346, 2736, 139, -100, -34, -30, 104; 3339, 1669, 37128, 7091, 59640, 17932, 263130, 45452, 31689, 4169, 32928, 1666, -1260, -418, -420, 1252],136,23]
%
% END OF LATTICE L137.23
%
\hbox{}\par\smallskip%
%
% LATTICE L137.24
%
\leavevmode%
\llap{\spacer}%
${L_{137.24}}=\pdualwitharg{7}({L_{137.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[316680, -28560, -840; -28560, 2569, 77; -840, 77, 2],[840, 7, 1],[-7, 0, 5, 1, -11, -4, -76, -31, -29, -6, -77, -19, -49, -5, -59, -17; -66, 0, 48, 10, -100, -37, -705, -288, -270, -56, -720, -178, -460, -47, -555, -160; -399, -1, 252, 42, -700, -240, -4515, -1832, -1701, -349, -4452, -1092, -2800, -285, -3360, -968],136,24]
%
% END OF LATTICE L137.24
%
\hbox{}\par\smallskip%
%
% LATTICE L137.25
%
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${L_{137.25}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{137.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-142311750, 1332030, -70760130; 1332030, -8220, 662310; -70760130, 662310, -35183293],[210, 30, 1],[22514, -16080, -32073, -108995, -59590, -1444277, -213881, -2020077, -291250, -924496, -102295, -92557, -7594, 33413, 12329, 138925; -9, 7, 13, 44, 24, 581, 86, 812, 117, 371, 41, 37, 3, -14, -5, -56; -45280, 32340, 64505, 219210, 119847, 2904720, 430156, 4062765, 585760, 1859340, 205735, 186150, 15273, -67200, -24796, -279405],136,25]
%
% END OF LATTICE L137.25
%
\hbox{}\par\smallskip%
%
% LATTICE L137.26
%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8969100, -1283940, -53760; -1283940, -183795, -7695; -53760, -7695, -322],[420, 15, 1],[-109, -365, -173, -90, -25, -81, -3, 23, 5, 1, -35, -66, -79, -1011, -153, -2935; 893, 2989, 1416, 736, 204, 658, 24, -196, -41, -7, 288, 542, 648, 8288, 1254, 24052; -3145, -10500, -4960, -2565, -702, -2205, -73, 840, 145, 0, -1040, -1935, -2298, -29295, -4427, -84840],136,26]
%
% END OF LATTICE L137.26
%
\hbox{}\par\smallskip%
%
% LATTICE L137.27
%
\leavevmode%
\llap{\spacer}%
${L_{137.27}}=\pdualwitharg{2}({L_{137.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[867720, -249480, 52080; -249480, 71728, -14976; 52080, -14976, 3115],[840, 8, 1],[32, 194, 70, 95, 159, 521, 941, 498, 136, 2, -28, -53, -89, -227, -311, -106; 111, 651, 234, 317, 530, 1736, 3135, 1659, 453, 7, -93, -176, -295, -749, -1020, -343; 0, -112, -45, -64, -110, -364, -660, -350, -96, 0, 21, 40, 70, 196, 300, 126],136,27]
%
% END OF LATTICE L137.27
%
\hbox{}\par\smallskip%
%
% LATTICE L137.28
%
\leavevmode%
\llap{\spacer}%
${L_{137.28}}=\pdualwitharg{2}({L_{137.2}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{}{2}{120}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2324280, 6720, 0; 6720, 56, 16; 0, 16, 7],[840, 8, 1],[-11, -13, -5, -2, -2, -5, -8, -2, -1, -1, -1, -1, -3, -16, -37, -12; 3819, 4515, 1737, 695, 695, 1736, 2775, 693, 345, 343, 345, 346, 1040, 5551, 12840, 4165; -8712, -10304, -3966, -1588, -1590, -3976, -6360, -1589, -792, -784, -786, -788, -2370, -12656, -29280, -9499],136,28]
%
% END OF LATTICE L137.28
%
\hbox{}\par\smallskip%
%
% LATTICE L137.29
%
\leavevmode%
\llap{\spacer}%
${L_{137.29}}=\pdualwitharg{5.7}(\pfillwitharg{2}({L_{137.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[867548010, 1868580, -343473480; 1868580, 4025, -739795; -343473480, -739795, 135985594],[210, 35, 1],[956, 382, 5071, 2203, 2167, 3646, 11248, 10079, 15172, 2306, 11057, 1829, 521, 386, 848, 673; 60, 26, 300, 119, 106, 167, 501, 440, 636, 90, 396, 55, 10, 7, 21, 24; 2415, 965, 12810, 5565, 5474, 9210, 28413, 25460, 38325, 5825, 27930, 4620, 1316, 975, 2142, 1700],136,29]
%
% END OF LATTICE L137.29
%
\hbox{}\par\smallskip%
%
% LATTICE L137.30
%
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\llap{\spacer}%
${L_{137.30}}=\pdualwitharg{2.5}(\main({L_{137.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[78540, 477540, 21840; 477540, 3137000, 133960; 21840, 133960, 6079],[420, 20, 1],[-743, -1387, -4454, -10738, -6680, -176992, -81494, -131617, -59821, -69271, -25828, -19544, -3414, -34982, -9352, -9699; -15, -28, -90, -217, -135, -3577, -1647, -2660, -1209, -1400, -522, -395, -69, -707, -189, -196; 3000, 5600, 17985, 43360, 26974, 714700, 329076, 531475, 241560, 279720, 104295, 78920, 13786, 141260, 37764, 39165],136,30]
%
% END OF LATTICE L137.30
%
\hbox{}\par\smallskip%
%
% LATTICE L137.31
%
\leavevmode%
\llap{\spacer}%
${L_{137.31}}=\pdualwitharg{3.7}(\main({L_{137.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1196580, 301980, -8820; 301980, -76209, 2226; -8820, 2226, -65],[420, 21, 1],[-37, -15, -41, -14, -1, 1, 1, -3, -3, -5, -39, -46, -199, -61, -301, -157; -127, -52, -144, -51, -10, 3, 5, -8, -9, -16, -128, -153, -670, -207, -1025, -536; 658, 249, 616, 147, -210, -33, 35, 132, 98, 129, 896, 987, 3990, 1167, 5635, 2892],136,31]
%
% END OF LATTICE L137.31
%
\hbox{}\par\smallskip%
%
% LATTICE L137.32
%
\leavevmode%
\llap{\spacer}%
${L_{137.32}}=\pdualwitharg{2.3.7}(\pfillwitharg{2}({L_{137.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-268973670, 2075430, -133733040; 2075430, -11508, 1031898; -133733040, 1031898, -66491735],[210, 42, 1],[31922, -3192, -45311, -154111, -421456, -291907, -1513093, -408333, -412146, -186920, -144801, -131057, -53824, 6739, 87253, 28109; -9, 1, 13, 44, 120, 83, 430, 116, 117, 53, 41, 37, 15, -2, -25, -8; -64204, 6420, 91133, 309960, 847665, 587106, 3043250, 821271, 828940, 375948, 291235, 263592, 108255, -13554, -175490, -56535],136,32]
%
% END OF LATTICE L137.32
%
\hbox{}\par\smallskip%
%
% LATTICE L137.33
%
\leavevmode%
\llap{\spacer}%
${L_{137.33}}=\pdualwitharg{2.7}(\main({L_{137.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[66360, -44100, 16800; -44100, 26740, -11144; 16800, -11144, 4253],[420, 28, 1],[83, 210, 1628, 2899, 6953, 4537, 68977, 3056, 8989, 1302, 2908, 1637, 607, -1, -937, -32; -3, -7, -54, -96, -230, -150, -2280, -101, -297, -43, -96, -54, -20, 0, 30, 1; -336, -848, -6573, -11704, -28070, -18316, -278460, -12337, -36288, -5256, -11739, -6608, -2450, 4, 3780, 129],136,33]
%
% END OF LATTICE L137.33
%
\hbox{}\par\smallskip%
%
% LATTICE L137.34
%
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\llap{\spacer}%
${L_{137.34}}=\pdualwitharg{3.5}({L_{137.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-22877400, -3270960, -136080; -3270960, -467670, -19455; -136080, -19455, -809],[840, 15, 1],[-259, -869, -413, -108, -61, -207, -9, 23, 11, 1, -83, -78, -187, -2397, -363, -6967; 2093, 7021, 3336, 872, 492, 1666, 72, -196, -89, -7, 672, 631, 1512, 19376, 2934, 56308; -6770, -22680, -10760, -2805, -1572, -5250, -218, 840, 290, 0, -2200, -2055, -4908, -62790, -9502, -182280],136,34]
%
% END OF LATTICE L137.34
%
\hbox{}\par\smallskip%
%
% LATTICE L137.35
%
\leavevmode%
\llap{\spacer}%
${L_{137.35}}=\pdualwitharg{3.5}({L_{137.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26040, 26880, 3360; 26880, -20670, -2295; 3360, -2295, -239],[840, 15, 1],[344, 1139, 533, 271, 71, 101, 2, -143, -16, 9, 123, 221, 257, 1621, 244, 9327; 1355, 4487, 2100, 1068, 280, 399, 8, -560, -63, 35, 484, 870, 1012, 6384, 961, 36736; -8180, -27090, -12680, -6450, -1692, -2415, -49, 3360, 380, -210, -2920, -5250, -6108, -38535, -5801, -221760],136,35]
%
% END OF LATTICE L137.35
%
\hbox{}\par\smallskip%
%
% LATTICE L137.36
%
\leavevmode%
\llap{\spacer}%
${L_{137.36}}=\pdualwitharg{5.7}(\main({L_{137.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40740, -1680, -840; -1680, 70, 35; -840, 35, 17],[420, 35, 1],[-2, 0, 1, 0, -1, -3, -11, -11, -10, -2, -25, -6, -3, -3, -7, -5; -3, 2, 24, 1, -10, -29, -105, -104, -93, -18, -216, -49, -22, -19, -39, -24; -105, -5, 0, 0, -28, -90, -336, -340, -315, -65, -840, -210, -112, -120, -294, -220],136,36]
%
% END OF LATTICE L137.36
%
\hbox{}\par\smallskip%
%
% LATTICE L137.37
%
\leavevmode%
\llap{\spacer}%
${L_{137.37}}=\pdualwitharg{2.5.7}(\pfillwitharg{2}({L_{137.1}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-37666650, 999390, -17972010; 999390, -19180, 476840; -17972010, 476840, -8575043],[210, 70, 1],[43886, -1460, -62272, -70605, -38619, -133743, -415955, -187088, -566510, -85644, -199040, -60051, -4933, 3087, 23987, 12880; -27, 1, 39, 44, 24, 83, 258, 116, 351, 53, 123, 37, 3, -2, -15, -8; -91980, 3060, 130515, 147980, 80941, 280310, 871794, 392115, 1187340, 179500, 417165, 125860, 10339, -6470, -50274, -26995],136,37]
%
% END OF LATTICE L137.37
%
\hbox{}\par\smallskip%
%
% LATTICE L137.38
%
\leavevmode%
\llap{\spacer}%
${L_{137.38}}=\pdualwitharg{3.7}({L_{137.1}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2157960, 110880, -16800; 110880, -5691, 861; -16800, 861, -130],[840, 21, 1],[37, 15, 41, 7, 1, -1, -1, 3, 3, 5, 39, 23, 199, 61, 301, 157; 1226, 496, 1352, 229, 20, -34, -30, 104, 102, 168, 1304, 767, 6620, 2026, 9990, 5208; 3325, 1341, 3640, 609, 0, -96, -70, 300, 287, 465, 3584, 2100, 18060, 5514, 27160, 14148],136,38]
%
% END OF LATTICE L137.38
%
\hbox{}\par\smallskip%
%
% LATTICE L137.39
%
\leavevmode%
\llap{\spacer}%
${L_{137.39}}=\pdualwitharg{3.7}({L_{137.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[76440, -19320, 3360; -19320, -210, 21; 3360, 21, -1],[840, 21, 1],[-3, -2, -9, -7, -19, -2, -8, -7, -1, 0, 1, 1, -1, -1, -7, -9; 667, 445, 2004, 1560, 4240, 447, 1790, 1568, 225, 1, -220, -222, 220, 222, 1555, 2000; 3892, 2598, 11704, 9114, 24780, 2613, 10465, 9168, 1316, 6, -1288, -1302, 1260, 1293, 9065, 11664],136,39]
%
% END OF LATTICE L137.39
%
\hbox{}\par\smallskip%
%
% LATTICE L137.40
%
\leavevmode%
\llap{\spacer}%
${L_{137.40}}=\pdualwitharg{2.3}({L_{137.1}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[151026960, 10804920, -105000; 10804920, 773016, -7512; -105000, -7512, 73],[840, 24, 1],[1, 1, 0, 0, 1, 9, 8, 17, 3, 13, 1, 3, 4, 12, 7, 11; -16, -14, 1, 7, 5, -35, -45, -112, -24, -126, -11, -37, -55, -175, -105, -168; -208, 0, 103, 720, 1950, 9324, 6860, 12894, 1840, 5712, 305, 504, 90, -756, -740, -1470],136,40]
%
% END OF LATTICE L137.40
%
\hbox{}\par\smallskip%
%
% LATTICE L137.41
%
\leavevmode%
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${L_{137.41}}=\pdualwitharg{2.3}({L_{137.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2004240, -4200, -5880; -4200, -120, -48; -5880, -48, -11],[840, 24, 1],[-1, -1, 1, 4, 12, 61, 46, 44, 13, 43, 5, 5, 3, 2, -1, -2; 240, 238, -240, -959, -2875, -14609, -11015, -10535, -3112, -10290, -1196, -1195, -715, -469, 245, 483; -512, -504, 514, 2052, 6150, 31248, 23560, 22533, 6656, 22008, 2558, 2556, 1530, 1008, -520, -1029],136,41]
%
% END OF LATTICE L137.41
%
\hbox{}\par\smallskip%
%
% LATTICE L137.42
%
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\llap{\spacer}%
${L_{137.42}}=\pdualwitharg{3.5.7}(\pfillwitharg{2}({L_{137.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16898070, 843150, 6666450; 843150, -28770, -323085; 6666450, -323085, -2623127],[210, 105, 1],[9918, -990, -28147, -47870, -52367, -90677, -94005, -253689, -128030, -58066, -89965, -40714, -6689, 2093, 5421, 17465; -19845, 1981, 56320, 95784, 104782, 181437, 188096, 507610, 256177, 116185, 180012, 81465, 13384, -4188, -10847, -34946; 27650, -2760, -78470, -133455, -145992, -252795, -262073, -707250, -356930, -161880, -250810, -113505, -18648, 5835, 15113, 48690],136,42]
%
% END OF LATTICE L137.42
%
\hbox{}\par\smallskip%
%
% LATTICE L137.43
%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L137.43
%
\hbox{}\par\smallskip%
%
% LATTICE L137.44
%
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${L_{137.44}}=\pdualwitharg{5.7}({L_{137.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7490280, 3728760, -15960; 3728760, 1856225, -7945; -15960, -7945, 34],[840, 35, 1],[1, -2, -23, -1, 9, 13, 47, 93, 83, 16, 191, 43, 19, 8, 16, 19; -6, 4, 48, 2, -20, -29, -105, -208, -186, -36, -432, -98, -44, -19, -39, -48; -945, -5, 420, 0, -448, -675, -2478, -4960, -4515, -905, -11340, -2730, -1372, -690, -1617, -2320],136,44]
%
% END OF LATTICE L137.44
%
\hbox{}\par\smallskip%
%
% LATTICE L137.45
%
\leavevmode%
\llap{\spacer}%
${L_{137.45}}=\pdualwitharg{5.7}({L_{137.1}})$%
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{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[840, 0, 0; 0, 35, 0; 0, 0, -1],[840, 35, 1],[-2, 0, 1, 0, -1, -3, -11, -11, -10, -2, -25, -3, -3, -3, -7, -5; -15, -1, 0, 1, 0, -4, -18, -20, -21, -5, -72, -10, -12, -14, -36, -28; -105, -5, 0, 0, -28, -90, -336, -340, -315, -65, -840, -105, -112, -120, -294, -220],136,45]
%
% END OF LATTICE L137.45
%
\hbox{}\par\smallskip%
%
% LATTICE L137.46
%
\leavevmode%
\llap{\spacer}%
${L_{137.46}}=\pdualwitharg{2.5}({L_{137.1}})$%
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{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14280, -1680, 0; -1680, 200, 0; 0, 0, -1],[840, 40, 1],[13, 13, 2, 1, 0, -2, -1, -3, -1, 1, 1, 2, 1, 23, 10, 31; 141, 147, 24, 14, 1, -7, -6, -21, -9, 7, 9, 19, 10, 238, 105, 329; -480, -560, -105, -80, -14, -140, -36, -70, 0, 0, -15, -40, -26, -700, -324, -1050],136,46]
%
% END OF LATTICE L137.46
%
\hbox{}\par\smallskip%
%
% LATTICE L137.47
%
\leavevmode%
\llap{\spacer}%
${L_{137.47}}=\pdualwitharg{2.5}({L_{137.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21000, 3360, 0; 3360, -520, -40; 0, -40, 91],[840, 40, 1],[-113, -181, -89, -47, -13, -194, -67, -86, -43, -1, 11, 8, 0, -79, -50, -96; -681, -1099, -543, -288, -80, -1197, -414, -532, -267, -7, 69, 51, 1, -462, -297, -574; -360, -560, -270, -140, -38, -560, -192, -245, -120, 0, 30, 20, -2, -280, -168, -315],136,47]
%
% END OF LATTICE L137.47
%
\hbox{}\par\smallskip%
%
% LATTICE L137.48
%
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${L_{137.48}}=\pdualwitharg{2.3.7}(\main({L_{137.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10659180, 141960, 2641800; 141960, 2940, 35196; 2641800, 35196, 654751],[420, 84, 1],[-30196, -12635, -9021, -13919, -3693, -107, -1179, -616, -1928, -3427, -7041, -34267, -76617, -48079, -239751, -31508; -1429, -598, -427, -659, -175, -5, -55, -29, -91, -162, -333, -1621, -3625, -2275, -11345, -1491; 121912, 51012, 36421, 56196, 14910, 432, 4760, 2487, 7784, 13836, 28427, 138348, 309330, 194112, 967960, 127209],136,48]
%
% END OF LATTICE L137.48
%
\hbox{}\par\smallskip%
%
% LATTICE L137.49
%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[87164700, 64826160, 618660; 64826160, 48212535, 460110; 618660, 460110, 4391],[420, 105, 1],[95, 59, 255, 192, 103, 111, 91, 205, 33, 5, 1, 0, 25, 81, 101, 307; -96, -59, -252, -187, -98, -103, -83, -184, -28, -3, 4, 1, -26, -83, -103, -312; -3325, -2130, -9520, -7455, -4242, -4845, -4123, -9600, -1715, -390, -560, -105, -798, -2715, -3437, -10560],136,49]
%
% END OF LATTICE L137.49
%
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%
% LATTICE L137.50
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[115771937610, 45585307950, 22024537920; 45585307950, 17949257340, 8672182260; 22024537920, 8672182260, 4189964171],[210, 210, 1],[20, 26, 50, 119, 53, 167, 167, 220, 212, 90, 66, 55, 5, 7, 7, 12; 2249, 2485, 6282, 17555, 9210, 32174, 33579, 45599, 46675, 21977, 18180, 19138, 3021, 4519, 3114, 3325; -4760, -5280, -13265, -36960, -19341, -67470, -70378, -95535, -97720, -45960, -37975, -39900, -6279, -9390, -6482, -6945],136,50]
%
% END OF LATTICE L137.50
%
\hbox{}\par\smallskip%
%
% LATTICE L137.51
%
\leavevmode%
\llap{\spacer}%
${L_{137.51}}=\pdualwitharg{2.7}({L_{137.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[840, 0, 0; 0, -1400, -616; 0, -616, -271],[840, 56, 1],[-1, -1, -7, -6, -14, -9, -136, -6, -35, -5, -11, -3, -1, 0, 1, 0; -75, -29, -135, -91, -165, -88, -1215, -50, -249, -25, -27, 10, 30, 7, 0, -4; 168, 64, 294, 196, 350, 184, 2520, 103, 504, 48, 42, -28, -70, -16, 0, 9],136,51]
%
% END OF LATTICE L137.51
%
\hbox{}\par\smallskip%
%
% LATTICE L137.52
%
\leavevmode%
\llap{\spacer}%
${L_{137.52}}=\pdualwitharg{2.7}({L_{137.1}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[35810040, 3604440, -85680; 3604440, 362768, -8624; -85680, -8624, 205],[840, 56, 1],[20, 10, 28, 43, 91, 55, 809, 70, 196, 26, 26, 11, -1, -1, 1, 2; 9, 3, 6, 7, 10, 4, 45, 3, 3, -1, -3, -4, -5, -1, 0, 1; 8736, 4304, 11949, 18256, 38430, 23140, 339780, 29362, 81984, 10816, 10731, 4424, -630, -460, 420, 878],136,52]
%
% END OF LATTICE L137.52
%
\hbox{}\par\smallskip%
%
% LATTICE L137.53
%
\leavevmode%
\llap{\spacer}%
${L_{137.53}}=\pdualwitharg{2.5.7}(\main({L_{137.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20580, 24360, 13020; 24360, 4340, -2800; 13020, -2800, -5303],[420, 140, 1],[2702, 193, 26, -381, -45, 495, 2453, 709, 3118, 777, 2884, 3291, 1015, 2415, 6277, 1231; -8121, -580, -78, 1145, 135, -1489, -7377, -2132, -9375, -2336, -8670, -9893, -3051, -7259, -18867, -3700; 10920, 780, 105, -1540, -182, 2000, 9912, 2865, 12600, 3140, 11655, 13300, 4102, 9760, 25368, 4975],136,53]
%
% END OF LATTICE L137.53
%
\hbox{}\par\smallskip%
%
% LATTICE L137.54
%
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\llap{\spacer}%
${L_{137.54}}=\pdualwitharg{3.5.7}({L_{137.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1168440, 1643880, 45360; 1643880, 2294670, 63315; 45360, 63315, 1747],[840, 105, 1],[-1, 0, 3, 5, 5, 4, 4, 21, 5, 2, 5, 1, -1, -1, -1, -5; 219, 53, -172, -496, -576, -485, -494, -2624, -639, -263, -668, -134, 156, 170, 179, 944; -7910, -1920, 6160, 17850, 20748, 17475, 17801, 94560, 23030, 9480, 24080, 4830, -5628, -6135, -6461, -34080],136,54]
%
% END OF LATTICE L137.54
%
\hbox{}\par\smallskip%
%
% LATTICE L137.55
%
\leavevmode%
\llap{\spacer}%
${L_{137.55}}=\pdualwitharg{3.5.7}({L_{137.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[21560280, 0, -77280; 0, 105, 0; -77280, 0, 277],[840, 105, 1],[95, 59, 255, 96, 103, 111, 91, 205, 33, 5, 1, 0, 25, 81, 101, 307; -7, -5, -24, -10, -12, -14, -12, -28, -5, -1, 0, 1, 0, -4, -6, -20; 26495, 16455, 71120, 26775, 28728, 30960, 25382, 57180, 9205, 1395, 280, 0, 6972, 22590, 28168, 85620],136,55]
%
% END OF LATTICE L137.55
%
\hbox{}\par\smallskip%
%
% LATTICE L137.56
%
\leavevmode%
\llap{\spacer}%
${L_{137.56}}=\pdualwitharg{2.3.5}({L_{137.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1680, -7560, 1680; -7560, 30840, -6840; 1680, -6840, 1517],[840, 120, 1],[21, 69, 4, 8, 1, 5, 0, -3, -1, 1, 1, 7, 4, 100, 15, 143; 130, 392, 20, 29, -1, -91, -15, -140, -18, 0, 8, 55, 29, 679, 99, 924; 560, 1680, 85, 120, -6, -420, -68, -630, -80, 0, 35, 240, 126, 2940, 428, 3990],136,56]
%
% END OF LATTICE L137.56
%
\hbox{}\par\smallskip%
%
% LATTICE L137.57
%
\leavevmode%
\llap{\spacer}%
${L_{137.57}}=\pdualwitharg{2.3.5}({L_{137.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1680, -5880, 1680; -5880, 19560, -5520; 1680, -5520, 1553],[840, 120, 1],[149, 491, 57, 57, 7, 30, -1, -16, -9, -1, 13, 48, 28, 705, 106, 506; 190, 616, 70, 67, 7, 7, -5, -35, -14, 0, 18, 65, 37, 917, 137, 651; 520, 1680, 190, 180, 18, 0, -16, -105, -40, 0, 50, 180, 102, 2520, 376, 1785],136,57]
%
% END OF LATTICE L137.57
%
\hbox{}\par\smallskip%
%
% LATTICE L137.58
%
\leavevmode%
\llap{\spacer}%
${L_{137.58}}=\pdualwitharg{2.3.7}({L_{137.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2136164520, -3010560, 7628880; -3010560, -4200, 10752; 7628880, 10752, -27245],[840, 168, 1],[-203, -63, -25, 19, 67, 16, 1, -9, -57, -67, -105, -214, -392, -211, -976, -121; 421, 119, 33, -74, -180, -41, 0, 23, 143, 163, 249, 497, 885, 464, 2115, 259; -56672, -17592, -6986, 5292, 18690, 4464, 280, -2511, -15904, -18696, -29302, -59724, -109410, -58896, -272440, -33777],136,58]
%
% END OF LATTICE L137.58
%
\hbox{}\par\smallskip%
%
% LATTICE L137.59
%
\leavevmode%
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${L_{137.59}}=\pdualwitharg{2.3.7}({L_{137.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1039080, -1362480, 7560; -1362480, 1786344, -9912; 7560, -9912, 55],[840, 168, 1],[-41, -17, -6, -9, -2, 0, -1, -1, -3, -5, -5, -24, -53, -33, -164, -43; 1, -1, -1, -4, -5, -1, 0, 1, 3, 3, 2, 7, 10, 4, 15, 3; 5768, 2136, 637, 504, -630, -180, 140, 318, 952, 1224, 1043, 4536, 9030, 5220, 25060, 6402],136,59]
%
% END OF LATTICE L137.59
%
\hbox{}\par\smallskip%
%
% LATTICE L137.60
%
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\llap{\spacer}%
${L_{137.60}}=\pdualwitharg{2.3.5.7}(\main({L_{137.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2100, 2092440, 518280; 2092440, -149038260, -36928920; 518280, -36928920, -9150301],[420, 420, 1],[2058, 1615, 2059, 7225, 2283, 5521, 4825, 2858, 2130, 479, 35, -943, -189, 761, 1457, 1330; -74033, -58092, -74059, -259861, -82109, -198559, -173525, -102783, -76599, -17224, -1257, 33913, 6793, -27389, -52423, -47849; 298900, 234540, 299005, 1049160, 331506, 801660, 700588, 414975, 309260, 69540, 5075, -136920, -27426, 110580, 211652, 193185],136,60]
%
% END OF LATTICE L137.60
%
\hbox{}\par\smallskip%
%
% LATTICE L137.61
%
\leavevmode%
\llap{\spacer}%
${L_{137.61}}=\pdualwitharg{2.5.7}({L_{137.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-590520, 1903440, -5040; 1903440, -6133400, 16240; -5040, 16240, -43],[840, 280, 1],[-7, -1, -1, 1, 1, 4, 13, 3, 19, 3, 7, 2, 0, -1, -4, -1; 9, 5, 27, 20, 8, 23, 66, 14, 75, 9, 15, 1, -1, -2, -3, 0; 4200, 2000, 10290, 7420, 2898, 8200, 23352, 4925, 26040, 3040, 4830, 140, -378, -640, -672, 115],136,61]
%
% END OF LATTICE L137.61
%
\hbox{}\par\smallskip%
%
% LATTICE L137.62
%
\leavevmode%
\llap{\spacer}%
${L_{137.62}}=\pdualwitharg{2.5.7}({L_{137.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9939720, -1472520, 15120; -1472520, -217840, 2240; 15120, 2240, -23],[840, 280, 1],[16, 6, 14, 19, 7, 19, 53, 22, 56, 6, 4, -1, -1, -1, 1, 2; -21, -5, -9, -10, -3, -7, -18, -7, -15, -1, 0, 1, 0, -2, -9, -5; 8400, 3440, 8295, 11480, 4298, 11780, 33012, 13750, 35280, 3840, 2625, -560, -658, -860, -252, 810],136,62]
%
% END OF LATTICE L137.62
%
\hbox{}\par\smallskip%
%
% LATTICE L137.63
%
\leavevmode%
\llap{\spacer}%
${L_{137.63}}=\pdualwitharg{2.3.5.7}({L_{137.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[840, 0, 0; 0, -654360, -215880; 0, -215880, -71221],[840, 840, 1],[-5, -1, 0, 1, 0, -2, -3, -5, -7, -5, -3, -10, -3, -7, -6, -7; -1293, -475, -127, 0, 97, 297, 268, 307, 185, -79, -150, -831, -374, -1128, -1099, -1415; 3920, 1440, 385, 0, -294, -900, -812, -930, -560, 240, 455, 2520, 1134, 3420, 3332, 4290],136,63]
%
% END OF LATTICE L137.63
%
\hbox{}\par\smallskip%
%
% LATTICE L137.64
%
\leavevmode%
\llap{\spacer}%
${L_{137.64}}=\pdualwitharg{2.3.5.7}({L_{137.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26871600, 8785560, -43680; 8785560, 2871960, -14280; -43680, -14280, 71],[840, 840, 1],[25, 9, 5, 1, -1, -2, -1, 0, 3, 5, 9, 20, 8, 23, 22, 14; 46, 16, 8, -1, -3, -7, -5, -2, 2, 8, 16, 37, 15, 43, 41, 26; 24640, 8760, 4690, 420, -1218, -2640, -1624, -405, 2240, 4680, 8750, 19740, 7938, 22800, 21784, 13845],136,64]
%
% END OF LATTICE L137.64
%

\medskip%
%------------------------------WEYL-GROUP-W138--------------------------------
%
\leavevmode\llap{}%
$W_{138}$%
\qquad\llap{64} lattices, $\chi=72$%
\hfill%
$2222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L138.1
%
\leavevmode%
\llap{\spacer}%
${L_{138.1}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1611480, -106080, 0; -106080, -6982, -1; 0, -1, 1],[1560, 1, 1],[11, 6, -5, -77, 0, 113, 27, 363, 38, 631, 1083, 19187, 245, 3709, 218, 911; -167, -91, 76, 1170, 0, -1716, -410, -5512, -577, -9581, -16444, -291330, -3720, -56316, -3310, -13832; -160, -78, 80, 1170, -1, -1716, -405, -5408, -564, -9334, -16008, -283530, -3619, -54756, -3215, -13416],137,1]
%
% END OF LATTICE L138.1
%
\hbox{}\par\smallskip%
%
% LATTICE L138.2
%
\leavevmode%
\llap{\spacer}%
${L_{138.2}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[69243720, -118560, 0; -118560, 203, 0; 0, 0, -1],[1560, 1, 1],[1, -1, -1, 1, 1, 31, 11, 63, 6, 92, 155, 1364, 69, 515, 59, 119; 583, -585, -584, 585, 584, 18096, 6420, 36764, 3501, 53677, 90432, 795795, 40256, 300456, 34420, 69420; -15, -13, 0, 0, -2, -156, -70, -468, -49, -819, -1408, -12480, -638, -4836, -570, -1196],137,2]
%
% END OF LATTICE L138.2
%
\hbox{}\par\smallskip%
%
% LATTICE L138.3
%
\leavevmode%
\llap{\spacer}%
${L_{138.3}}=\pfillwitharg{2}({L_{138.1}})$%
{${}={}$}Nikulin $37'$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[76830, -17940, 390; -17940, 4189, -91; 390, -91, 2],[390, 1, 1],[0, 6, -7, -227, -10, -145, -36, -163, -26, -344, -279, -4843, -120, -869, -94, -175; 0, 26, -30, -975, -43, -624, -155, -702, -112, -1482, -1202, -20865, -517, -3744, -405, -754; 1, 13, -4, -195, -10, -156, -40, -182, -29, -377, -304, -5265, -130, -936, -100, -182],137,3]
%
% END OF LATTICE L138.3
%
\hbox{}\par\smallskip%
%
% LATTICE L138.4
%
\leavevmode%
\llap{\spacer}%
${L_{138.4}}=\main({L_{138.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-150540, -74880, 780; -74880, -37246, 389; 780, 389, 97],[780, 1, 1],[-289, -582, -195, -97, 98, 427, -276, -3595, -476, -9363, -16635, -298253, -7748, -60097, -7374, -16295; 581, 1170, 392, 195, -197, -858, 555, 7228, 957, 18824, 33444, 599625, 15577, 120822, 14825, 32760; -7, -13, -4, 0, 2, 0, -10, -104, -13, -247, -436, -7800, -202, -1560, -190, -416],137,4]
%
% END OF LATTICE L138.4
%
\hbox{}\par\smallskip%
%
% LATTICE L138.5
%
\leavevmode%
\llap{\spacer}%
${L_{138.5}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{138.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14754090, -76440, 7377240; -76440, 394, -38220; 7377240, -38220, 3688717],[390, 2, 1],[-567, 207, 97, -194, -193, -3356, -2487, -7352, -2861, -44771, -18951, -668266, -16947, -63490, -14653, -14930; 581, -195, -97, 195, 194, 3393, 2520, 7462, 2907, 45539, 19281, 679965, 17246, 64623, 14920, 15210; 1140, -416, -195, 390, 388, 6747, 5000, 14781, 5752, 90012, 38101, 1343550, 34072, 127647, 29460, 30017],137,5]
%
% END OF LATTICE L138.5
%
\hbox{}\par\smallskip%
%
% LATTICE L138.6
%
\leavevmode%
\llap{\spacer}%
${L_{138.6}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{138.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22194510, -114660, 7398300; -114660, 591, -38220; 7398300, -38220, 2466143],[390, 3, 1],[-372, 142, 129, -43, -128, -1479, -1642, -9699, -1886, -29496, -24967, -146727, -11162, -27875, -9648, -19655; 581, -195, -194, 65, 194, 2262, 2520, 14924, 2907, 45539, 38562, 226655, 17246, 43082, 14920, 30420; 1125, -429, -390, 130, 387, 4472, 4965, 29328, 5703, 89193, 75498, 443690, 33753, 84292, 29175, 59436],137,6]
%
% END OF LATTICE L138.6
%
\hbox{}\par\smallskip%
%
% LATTICE L138.7
%
\leavevmode%
\llap{\spacer}%
${L_{138.7}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{138.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[37044930, -191100, 14818050; -191100, 985, -76440; 14818050, -76440, 5927251],[390, 5, 1],[-435, 181, 155, -31, -153, -5269, -389, -11467, -2227, -34787, -29437, -103787, -13157, -98549, -2273, -23139; 581, -195, -194, 39, 194, 6786, 504, 14924, 2907, 45539, 38562, 135993, 17246, 129246, 2984, 30420; 1095, -455, -390, 78, 385, 13260, 979, 28860, 5605, 87555, 74090, 261222, 33115, 248040, 5721, 58240],137,7]
%
% END OF LATTICE L138.7
%
\hbox{}\par\smallskip%
%
% LATTICE L138.8
%
\leavevmode%
\llap{\spacer}%
${L_{138.8}}=\pdualwitharg{3}(\main({L_{138.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[224491020, -3507660, -87360; -3507660, 54807, 1365; -87360, 1365, 34],[780, 3, 1],[22, 350, 595, 3502, 267, 669, 233, 479, 5, 1, -1, 8, 3, 33, 37, 223; 1421, 22607, 38432, 226200, 17246, 43212, 15050, 30940, 323, 65, -64, 520, 194, 2132, 2390, 14404; -522, -8307, -14124, -83135, -6339, -15886, -5535, -11388, -120, -39, 0, -325, -81, -806, -885, -5304],137,8]
%
% END OF LATTICE L138.8
%
\hbox{}\par\smallskip%
%
% LATTICE L138.9
%
\leavevmode%
\llap{\spacer}%
${L_{138.9}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{138.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5376236970, -24766560, 2679821820; -24766560, 114078, -12345060; 2679821820, -12345060, 1335775381],[390, 6, 1],[-93327, -90085, 486, 31428, -649, -64975, -191843, -666423, -284521, -4821939, -2078650, -24591548, -1888591, -2391037, -1697397, -1789589; -203, -195, 1, 65, -2, -143, -420, -1456, -621, -10517, -4533, -53625, -4118, -5213, -3700, -3900; 187230, 180726, -975, -63050, 1302, 130351, 384870, 1336959, 570798, 9673638, 4170129, 49334870, 3788838, 4796831, 3405270, 3590223],137,9]
%
% END OF LATTICE L138.9
%
\hbox{}\par\smallskip%
%
% LATTICE L138.10
%
\leavevmode%
\llap{\spacer}%
${L_{138.10}}=\pdualwitharg{2}(\main({L_{138.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11625308760, -30113460, -2906251140; -30113460, 78004, 7528168; -2906251140, 7528168, 726543773],[780, 4, 1],[-1266, -1463, -49, 391, 1, -4565, -4667, -8317, -3600, -61795, -26714, -1898131, -48661, -185209, -43993, -23312; -2329, -2717, -97, 390, 0, -8346, -8540, -15236, -6599, -113347, -49007, -3482310, -89280, -339846, -80740, -42796; -5040, -5824, -195, 1560, 4, -18174, -18580, -33111, -14332, -246012, -106351, -7556640, -193724, -737334, -175140, -92807],137,10]
%
% END OF LATTICE L138.10
%
\hbox{}\par\smallskip%
%
% LATTICE L138.11
%
\leavevmode%
\llap{\spacer}%
${L_{138.11}}=\pdualwitharg{5}(\main({L_{138.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[432774420, 10819380, 163800; 10819380, 270485, 4095; 163800, 4095, 62],[780, 5, 1],[32, 506, 859, 3032, 385, 2891, 67, 687, 7, 1, -1, 10, 5, 151, 11, 327; -1257, -19877, -33744, -119106, -15124, -113568, -2632, -26988, -275, -39, 40, -390, -196, -5928, -432, -12844; -1520, -23985, -40700, -143637, -18235, -136890, -3171, -32500, -330, -65, 0, -663, -265, -7410, -529, -15600],137,11]
%
% END OF LATTICE L138.11
%
\hbox{}\par\smallskip%
%
% LATTICE L138.12
%
\leavevmode%
\llap{\spacer}%
${L_{138.12}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{138.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3763437990, -26754000, 1876342650; -26754000, 190130, -13338780; 1876342650, -13338780, 935490833],[390, 10, 1],[-171515, -165213, 875, 33950, -1401, -360014, -70707, -1227042, -523649, -8871919, -3824277, -27145346, -3474419, -13195684, -624457, -3291524; -203, -195, 1, 39, -2, -429, -84, -1456, -621, -10517, -4533, -32175, -4118, -15639, -740, -3900; 344010, 331370, -1755, -68094, 2810, 722085, 141818, 2461095, 1050290, 17794530, 7670405, 54445794, 6968690, 26466765, 1252482, 6601855],137,12]
%
% END OF LATTICE L138.12
%
\hbox{}\par\smallskip%
%
% LATTICE L138.13
%
\leavevmode%
\llap{\spacer}%
${L_{138.13}}=\pdualwitharg{3}({L_{138.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-678991560, -6456840, 174720; -6456840, -61401, 1662; 174720, 1662, -37],[1560, 3, 1],[-310, -5121, -8779, -51823, -1984, -10003, -1761, -7343, -87, -40, 45, 213, -1, -319, -224, -2979; 32543, 537589, 921596, 5440240, 208275, 1050088, 184865, 770848, 9133, 4199, -4724, -22360, 105, 33488, 23515, 312728; -2079, -34359, -58908, -347750, -13314, -67132, -11820, -49296, -585, -273, 300, 1430, -6, -2132, -1500, -19968],137,13]
%
% END OF LATTICE L138.13
%
\hbox{}\par\smallskip%
%
% LATTICE L138.14
%
\leavevmode%
\llap{\spacer}%
${L_{138.14}}=\pdualwitharg{3}({L_{138.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1107926040, -17311320, -262080; -17311320, 270489, 4095; -262080, 4095, 62],[1560, 3, 1],[49, 809, 1387, 4094, 627, 1581, 557, 1163, 14, 10, -1, 1, 3, 57, 73, 475; 3176, 52442, 89912, 265395, 40646, 102492, 36110, 75400, 908, 650, -64, 65, 194, 3692, 4730, 30784; -2649, -44109, -75768, -223795, -34308, -86632, -30600, -64116, -795, -663, 0, -65, -132, -2912, -3840, -25428],137,14]
%
% END OF LATTICE L138.14
%
\hbox{}\par\smallskip%
%
% LATTICE L138.15
%
\leavevmode%
\llap{\spacer}%
${L_{138.15}}=\pdualwitharg{13}(\pfillwitharg{2}({L_{138.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[96384210, -496860, -22242480; -496860, 2561, 114660; -22242480, 114660, 5132873],[390, 13, 1],[228, -10, -91, 7, 88, 227, 1078, 485, 1218, 1456, 15997, 21683, 7142, 4111, 6152, 961; 581, -15, -194, 15, 194, 522, 2520, 1148, 2907, 3503, 38562, 52305, 17246, 9942, 14920, 2340; 975, -43, -390, 30, 377, 972, 4615, 2076, 5213, 6231, 68458, 92790, 30563, 17592, 26325, 4112],137,15]
%
% END OF LATTICE L138.15
%
\hbox{}\par\smallskip%
%
% LATTICE L138.16
%
\leavevmode%
\llap{\spacer}%
${L_{138.16}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{138.1}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3743083110, -32678100, 1243316490; -32678100, 285195, -10854480; 1243316490, -10854480, 412984657],[390, 15, 1],[-139665, -134533, 1425, 9215, -1141, -195441, -57577, -1998369, -426409, -7224429, -6228239, -7368177, -2829229, -7163521, -508497, -5360593; -203, -195, 2, 13, -2, -286, -84, -2912, -621, -10517, -9066, -10725, -4118, -10426, -740, -7800; 420465, 405015, -4290, -27742, 3435, 588380, 173337, 6016140, 1283715, 21749325, 18750270, 22182082, 8517465, 21565960, 1530843, 16138200],137,16]
%
% END OF LATTICE L138.16
%
\hbox{}\par\smallskip%
%
% LATTICE L138.17
%
\leavevmode%
\llap{\spacer}%
${L_{138.17}}=\pdualwitharg{5}({L_{138.1}})$%
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{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5637273720, -54671760, 10840440; -54671760, -530195, 105135; 10840440, 105135, -20846],[1560, 5, 1],[428, 7099, 12181, 43157, 2755, 41699, 490, 10233, 123, 64, -59, -179, 0, 1279, 61, 4093; -10276, -170430, -292432, -1036074, -66139, -1001052, -11763, -245648, -2952, -1534, 1416, 4290, -1, -30732, -1465, -98280; 170745, 2832115, 4859580, 17217408, 1099105, 16635840, 195487, 4082520, 49075, 25545, -23540, -71448, -5, 510120, 24333, 1632800],137,17]
%
% END OF LATTICE L138.17
%
\hbox{}\par\smallskip%
%
% LATTICE L138.18
%
\leavevmode%
\llap{\spacer}%
${L_{138.18}}=\pdualwitharg{5}({L_{138.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[87732840, -2193360, -54600; -2193360, 54835, 1365; -54600, 1365, 34],[1560, 5, 1],[-43, -707, -1211, -2144, -547, -4135, -97, -1011, -12, -8, 1, -1, -3, -155, -13, -419; -1686, -27716, -47472, -84045, -21442, -162084, -3802, -39624, -470, -312, 40, -39, -118, -6084, -510, -16432; -1375, -22815, -39160, -69381, -17720, -134160, -3156, -33020, -405, -325, 0, -39, -80, -4680, -404, -13260],137,18]
%
% END OF LATTICE L138.18
%
\hbox{}\par\smallskip%
%
% LATTICE L138.19
%
\leavevmode%
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${L_{138.19}}=\pdualwitharg{2.3}(\main({L_{138.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-51479220, -1120860, 12930060; -1120860, -14232, 281580; 12930060, 281580, -3247649],[780, 12, 1],[4897, -28303, -22523, -618062, -56892, -89070, -84688, -59320, -9001, -25049, -49, 84542, 15852, 35718, 43648, 32644; -101, 598, 472, 12935, 1189, 1859, 1765, 1235, 187, 520, 1, -1755, -329, -741, -905, -676; 19488, -112632, -89631, -2459600, -226404, -354458, -337020, -236067, -35820, -99684, -195, 336440, 63084, 142142, 173700, 129909],137,19]
%
% END OF LATTICE L138.19
%
\hbox{}\par\smallskip%
%
% LATTICE L138.20
%
\leavevmode%
\llap{\spacer}%
${L_{138.20}}=\pdualwitharg{13}(\main({L_{138.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-187980, -88140, 0; -88140, -41158, -13; 0, -13, 1],[780, 13, 1],[100, 123, 2721, 3697, 1222, 707, 1068, 169, 23, 0, -15, -7, 8, 31, 162, 77; -213, -262, -5796, -7875, -2603, -1506, -2275, -360, -49, 0, 32, 15, -17, -66, -345, -164; -2743, -3379, -74776, -101610, -33592, -19440, -29380, -4652, -637, -1, 416, 210, -208, -840, -4420, -2108],137,20]
%
% END OF LATTICE L138.20
%
\hbox{}\par\smallskip%
%
% LATTICE L138.21
%
\leavevmode%
\llap{\spacer}%
${L_{138.21}}=\pdualwitharg{2.13}(\pfillwitharg{2}({L_{138.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[127558470, -7949760, 63165960; -7949760, 494338, -3936660; 63165960, -3936660, 31279291],[390, 26, 1],[-135007, -9985, 676, 10088, -1249, -21895, -278963, -74421, -412721, -537743, -3013180, -8226008, -2737391, -799697, -2459677, -199443; -203, -15, 1, 15, -2, -33, -420, -112, -621, -809, -4533, -12375, -4118, -1203, -3700, -300; 272610, 20162, -1365, -20370, 2522, 44211, 563290, 150273, 833378, 1085826, 6084299, 16610190, 5527418, 1614771, 4966650, 402721],137,21]
%
% END OF LATTICE L138.21
%
\hbox{}\par\smallskip%
%
% LATTICE L138.22
%
\leavevmode%
\llap{\spacer}%
${L_{138.22}}=\pdualwitharg{3.5}(\main({L_{138.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[38220, -3120, 0; -3120, 255, 0; 0, 0, -1],[780, 15, 1],[2, 1, -1, -1, 0, 7, 2, 67, 7, 116, 199, 235, 90, 227, 16, 167; 24, 13, -12, -13, -1, 78, 23, 780, 82, 1365, 2344, 2769, 1061, 2678, 189, 1976; -15, 0, 0, -13, -15, -130, -27, -780, -75, -1170, -1980, -2327, -885, -2210, -153, -1560],137,22]
%
% END OF LATTICE L138.22
%
\hbox{}\par\smallskip%
%
% LATTICE L138.23
%
\leavevmode%
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${L_{138.23}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{138.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L138.23
%
\hbox{}\par\smallskip%
%
% LATTICE L138.24
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1560, 0, 0; 0, -8, 8; 0, 8, -7],[1560, 8, 1],[-3, -51, -11, -781, -20, -76, -18, -19, -1, -1, 0, 1, 0, -2, -2, -7; -23, -325, -67, -4680, -117, -429, -95, -91, -1, 13, 1, 0, -3, -39, -25, -65; -72, -1144, -243, -17160, -436, -1638, -380, -390, -16, 0, 1, 0, -4, -78, -60, -182],137,24]
%
% END OF LATTICE L138.24
%
\hbox{}\par\smallskip%
%
% LATTICE L138.25
%
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${L_{138.25}}=\pdualwitharg{2}({L_{138.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-606840, 0, -302640; 0, 8, 0; -302640, 0, -150931],[1560, 8, 1],[1229, 20905, 9019, 320189, 8200, 31163, 7382, 3897, 411, 415, 1, -389, 0, 817, 818, 1433; -13, -247, -109, -3900, -101, -390, -95, -52, -7, -13, -1, 0, 1, 0, -5, -13; -2464, -41912, -18082, -641940, -16440, -62478, -14800, -7813, -824, -832, -2, 780, 0, -1638, -1640, -2873],137,25]
%
% END OF LATTICE L138.25
%
\hbox{}\par\smallskip%
%
% LATTICE L138.26
%
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${L_{138.26}}=\pdualwitharg{3.13}(\pfillwitharg{2}({L_{138.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26904930, -4471740, 8738340; -4471740, 741507, -1452360; 8738340, -1452360, 2838089],[390, 39, 1],[-50637, -3745, 507, 1261, -469, -5475, -104633, -55827, -154801, -201693, -2260325, -1028451, -1026721, -199963, -922557, -149611; -203, -15, 2, 5, -2, -22, -420, -224, -621, -809, -9066, -4125, -4118, -802, -3700, -600; 155805, 11523, -1560, -3880, 1443, 16846, 321945, 171774, 476307, 620589, 6954792, 3164440, 3159117, 615266, 2838615, 460338],137,26]
%
% END OF LATTICE L138.26
%
\hbox{}\par\smallskip%
%
% LATTICE L138.27
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-237167580, 3356340, 59578740; 3356340, -40360, -843180; 59578740, -843180, -14966743],[780, 20, 1],[-39059, -428453, -163047, -2207572, -262794, -890912, -33166, -56660, 9423, 43185, 49, -104036, -33566, -264892, -26106, -135974; -777, -8528, -3246, -43953, -5233, -17745, -661, -1131, 187, 858, 1, -2067, -667, -5265, -519, -2704; -155440, -1705080, -648865, -8785296, -1045820, -3545490, -131988, -225485, 37500, 171860, 195, -414024, -133580, -1054170, -103892, -541125],137,27]
%
% END OF LATTICE L138.27
%
\hbox{}\par\smallskip%
%
% LATTICE L138.28
%
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\llap{\spacer}%
${L_{138.28}}=\pdualwitharg{13}({L_{138.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17160, -4680, -1560; -4680, 1274, 403; -1560, 403, -71],[1560, 13, 1],[-6, -5, 1, 71, 30, 79, 137, 97, 101, 100, 2089, 2779, 445, 491, 338, 93; -23, -19, 4, 270, 114, 300, 520, 368, 383, 379, 7916, 10530, 1686, 1860, 1280, 352; 0, 2, 0, -30, -13, -36, -65, -48, -52, -54, -1144, -1530, -247, -276, -195, -56],137,28]
%
% END OF LATTICE L138.28
%
\hbox{}\par\smallskip%
%
% LATTICE L138.29
%
\leavevmode%
\llap{\spacer}%
${L_{138.29}}=\pdualwitharg{13}({L_{138.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-76440, -4680, -3120; -4680, -286, -195; -3120, -195, -97],[1560, 13, 1],[1, -1, -1, 7, 13, 19, 71, 27, 30, 32, 683, 458, 297, 167, 239, 35; -17, 15, 16, -105, -196, -288, -1080, -412, -459, -491, -10488, -7035, -4564, -2568, -3680, -540; 0, 2, 0, -15, -26, -36, -130, -48, -52, -54, -1144, -765, -494, -276, -390, -56],137,29]
%
% END OF LATTICE L138.29
%
\hbox{}\par\smallskip%
%
% LATTICE L138.30
%
\leavevmode%
\llap{\spacer}%
${L_{138.30}}=\pdualwitharg{3.5}({L_{138.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-410280, 1560, 1560; 1560, 15, -15; 1560, -15, -2],[1560, 15, 1],[-7, -116, -199, -235, -45, -227, -8, -167, -2, -1, 1, 1, 0, -7, -1, -67; -564, -9334, -16008, -18902, -3619, -18252, -643, -13416, -160, -78, 80, 78, -1, -572, -81, -5408; -1305, -21645, -37140, -43862, -8400, -42380, -1494, -31200, -375, -195, 180, 182, 0, -1300, -186, -12480],137,30]
%
% END OF LATTICE L138.30
%
\hbox{}\par\smallskip%
%
% LATTICE L138.31
%
\leavevmode%
\llap{\spacer}%
${L_{138.31}}=\pdualwitharg{3.5}({L_{138.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30081480, 751920, -7800; 751920, -18795, 195; -7800, 195, -2],[1560, 15, 1],[-5, -89, -155, -92, -71, -181, -13, -139, -2, -2, 1, 1, 1, -1, -1, -43; -204, -3614, -6288, -3731, -2878, -7332, -526, -5616, -80, -78, 40, 39, 38, -52, -42, -1768; -465, -6435, -10560, -6136, -4590, -11180, -738, -7020, -15, 195, 0, -104, -210, -1300, -222, -5460],137,31]
%
% END OF LATTICE L138.31
%
\hbox{}\par\smallskip%
%
% LATTICE L138.32
%
\leavevmode%
\llap{\spacer}%
${L_{138.32}}=\pdualwitharg{5.13}(\pfillwitharg{2}({L_{138.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4984590, 2484300, 2070510; 2484300, 1235845, 1031940; 2070510, 1031940, 860053],[390, 65, 1],[32420, 2398, -325, -485, 298, 10513, 13396, 35739, 99102, 129124, 1447067, 395051, 657312, 384053, 118126, 95783; -203, -15, 2, 3, -2, -66, -84, -224, -621, -809, -9066, -2475, -4118, -2406, -740, -600; -77805, -5755, 780, 1164, -715, -25230, -32149, -85770, -237835, -309885, -3472820, -948084, -1577485, -921690, -283491, -229870],137,32]
%
% END OF LATTICE L138.32
%
\hbox{}\par\smallskip%
%
% LATTICE L138.33
%
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${L_{138.33}}=\pdualwitharg{3.13}(\main({L_{138.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16491540, -4123080, 31980; -4123080, -1030809, 7995; 31980, 7995, -62],[780, 39, 1],[1, 1, 1, -4, -11, -5, -53, -19, -20, -20, -419, -186, -179, -33, -137, -19; -7, -5, -4, 20, 54, 24, 250, 88, 91, 89, 1852, 820, 786, 144, 590, 80; -390, -129, 0, 515, 1287, 514, 4875, 1536, 1404, 1143, 22308, 9625, 8853, 1514, 5265, 492],137,33]
%
% END OF LATTICE L138.33
%
\hbox{}\par\smallskip%
%
% LATTICE L138.34
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[629778240390, 98378280, 313910443860; 98378280, 15366, 49036260; 313910443860, 49036260, 156467404627],[390, 78, 1],[93309, -4169, -18759, 962, 36235, 15552, 442829, 99564, 499903, 597381, 3281425, 2965098, 2929825, 281054, 2523231, 197042; 581, -15, -97, 5, 194, 87, 2520, 574, 2907, 3503, 19281, 17435, 17246, 1657, 14920, 1170; -187200, 8364, 37635, -1930, -72696, -31201, -888420, -199749, -1002924, -1198488, -6583317, -5948690, -5877924, -563861, -5062200, -395313],137,34]
%
% END OF LATTICE L138.34
%
\hbox{}\par\smallskip%
%
% LATTICE L138.35
%
\leavevmode%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1283880, -40560, 4680; -40560, -1032, 144; 4680, 144, -17],[1560, 24, 1],[27, 419, 177, 2079, 158, 197, 136, 69, 5, -3, -1, 1, 2, 11, 24, 35; 131, 2041, 863, 10140, 771, 962, 665, 338, 25, -13, -5, 0, 9, 52, 115, 169; 8520, 132288, 55890, 656500, 49896, 62218, 42960, 21801, 1584, -936, -318, 260, 624, 3458, 7560, 11037],137,35]
%
% END OF LATTICE L138.35
%
\hbox{}\par\smallskip%
%
% LATTICE L138.36
%
\leavevmode%
\llap{\spacer}%
${L_{138.36}}=\pdualwitharg{2.3}({L_{138.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[140878920, 0, -271440; 0, -24, 0; -271440, 0, 523],[1560, 24, 1],[63, 991, 210, 4939, 376, 470, 326, 333, 13, -3, -1, 1, 4, 24, 54, 161; 49, 819, 176, 4160, 319, 403, 285, 299, 15, 13, 0, 0, 1, 13, 35, 117; 32688, 514176, 108957, 2562560, 195084, 243854, 169140, 172770, 6744, -1560, -519, 520, 2076, 12454, 28020, 83538],137,36]
%
% END OF LATTICE L138.36
%
\hbox{}\par\smallskip%
%
% LATTICE L138.37
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[624098280, -2796300, -156723840; -2796300, 12532, 702208; -156723840, 702208, 39356561],[780, 52, 1],[-6050, -7225, -39694, -215227, -70901, -20413, -61153, -2394, -1176, -1, -49, -1553, -1359, -1265, -11107, -1219; -899, -1069, -5867, -31800, -10470, -3012, -9010, -352, -169, 1, -7, -240, -210, -192, -1670, -182; -24076, -28752, -157963, -856500, -282152, -81234, -243360, -9527, -4680, -4, -195, -6180, -5408, -5034, -44200, -4851],137,37]
%
% END OF LATTICE L138.37
%
\hbox{}\par\smallskip%
%
% LATTICE L138.38
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-99060, 12480, 24960; 12480, 2940, -3120; 24960, -3120, -6289],[780, 60, 1],[407, -588, -49, 601, 799, 3103, 1203, 8182, 3015, 45074, 18870, 88371, 33421, 41383, 5641, 14061; -8, 13, 1, -13, -17, -65, -25, -169, -62, -923, -386, -1807, -683, -845, -115, -286; 1620, -2340, -195, 2392, 3180, 12350, 4788, 32565, 12000, 179400, 75105, 351728, 133020, 164710, 22452, 55965],137,38]
%
% END OF LATTICE L138.38
%
\hbox{}\par\smallskip%
%
% LATTICE L138.39
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13042380, 3260400, -21060; 3260400, -815035, 5265; -21060, 5265, -34],[780, 65, 1],[-10, -8, -155, -40, -61, -31, -7, -3, 3, 1, 1, -2, -9, -11, -7, -11; -39, -35, -708, -186, -292, -156, -40, -24, 7, 3, 4, -6, -28, -36, -24, -40; 130, -495, -14300, -4209, -7735, -5130, -1911, -1900, -780, -155, 0, 309, 1235, 1230, 611, 600],137,39]
%
% END OF LATTICE L138.39
%
\hbox{}\par\smallskip%
%
% LATTICE L138.40
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[116622482730, -54654600, 58148133030; -54654600, 25610, -27250860; 58148133030, -27250860, 28992740471],[390, 130, 1],[-58725, 2493, 11570, -356, -22427, -29011, -55173, -62113, -312163, -373381, -2051454, -1112308, -1832013, -527321, -315715, -123331; 581, -15, -97, 3, 194, 261, 504, 574, 2907, 3503, 19281, 10461, 17246, 4971, 2984, 1170; 117780, -5000, -23205, 714, 44980, 58185, 110656, 124575, 626080, 748860, 4114435, 2230866, 3674320, 1057605, 633204, 247355],137,40]
%
% END OF LATTICE L138.40
%
\hbox{}\par\smallskip%
%
% LATTICE L138.41
%
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${L_{138.41}}=\pdualwitharg{3.13}({L_{138.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11632920, -2015520, -23400; -2015520, 349167, 4056; -23400, 4056, 47],[1560, 39, 1],[-5, -2, 1, 9, 11, 9, 44, 29, 28, 25, 505, 221, 104, 37, 71, 17; -23, -9, 4, 40, 49, 40, 195, 128, 123, 109, 2196, 960, 451, 160, 305, 72; -507, -219, 156, 1030, 1248, 1028, 5070, 3384, 3315, 3027, 61620, 27050, 12792, 4588, 8970, 2232],137,41]
%
% END OF LATTICE L138.41
%
\hbox{}\par\smallskip%
%
% LATTICE L138.42
%
\leavevmode%
\llap{\spacer}%
${L_{138.42}}=\pdualwitharg{3.13}({L_{138.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-31329480, 1957800, -31200; 1957800, -122343, 1950; -31200, 1950, -31],[1560, 39, 1],[1, 1, 1, -2, -11, -5, -53, -19, -20, -20, -419, -93, -179, -33, -137, -19; 13, 15, 16, -30, -166, -76, -810, -292, -309, -311, -6528, -1450, -2794, -516, -2150, -300; -195, -63, 0, 125, 624, 248, 2340, 732, 663, 531, 10296, 2215, 4056, 688, 2340, 204],137,42]
%
% END OF LATTICE L138.42
%
\hbox{}\par\smallskip%
%
% LATTICE L138.43
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8687640, -90480, 7800; -90480, -440, 80; 7800, 80, -7],[1560, 40, 1],[21, 317, 133, 935, 118, 439, 20, 50, 3, -5, -1, 1, 2, 29, 4, 28; 59, 897, 377, 2652, 335, 1248, 57, 143, 9, -13, -3, 0, 5, 78, 11, 78; 24040, 362960, 152290, 1070628, 135120, 502710, 22904, 57265, 3440, -5720, -1150, 1092, 2280, 33150, 4576, 32045],137,43]
%
% END OF LATTICE L138.43
%
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%
% LATTICE L138.44
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16353480, -74880, -26520; -74880, -40, -120; -26520, -120, -43],[1560, 40, 1],[57, 889, 188, 2651, 336, 1258, 58, 295, 11, -5, -1, 1, 4, 68, 10, 147; 167, 2613, 553, 7800, 989, 3705, 171, 871, 33, -13, -3, 0, 11, 195, 29, 429; -35640, -555880, -117555, -1657656, -210100, -786630, -36268, -184470, -6880, 3120, 625, -624, -2500, -42510, -6252, -91910],137,44]
%
% END OF LATTICE L138.44
%
\hbox{}\par\smallskip%
%
% LATTICE L138.45
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L138.45
%
\hbox{}\par\smallskip%
%
% LATTICE L138.46
%
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${L_{138.46}}=\pdualwitharg{5.13}({L_{138.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10419240, 4815720, -34320; 4815720, 2225665, -15860; -34320, -15860, 113],[1560, 65, 1],[7, 2, -1, -5, -10, -23, -7, -21, -18, -13, -241, -61, -45, -43, -4, -1; -23, -7, 4, 18, 36, 84, 26, 80, 71, 55, 1052, 270, 204, 204, 22, 16; -1105, -375, 260, 1008, 2015, 4800, 1521, 4840, 4485, 3755, 74100, 19272, 14885, 15480, 1859, 1920],137,46]
%
% END OF LATTICE L138.46
%
\hbox{}\par\smallskip%
%
% LATTICE L138.47
%
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${L_{138.47}}=\pdualwitharg{5.13}({L_{138.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14272440, -1784640, 15600; -1784640, -223145, 1950; 15600, 1950, -17],[1560, 65, 1],[3, 1, 1, -1, -9, -11, -7, -11, -10, -8, -155, -20, -61, -31, -7, -3; -29, -9, -8, 9, 80, 96, 60, 92, 81, 61, 1152, 147, 440, 216, 44, 12; -585, -115, 0, 114, 910, 900, 442, 420, 65, -405, -11440, -1674, -6110, -4020, -1482, -1460],137,47]
%
% END OF LATTICE L138.47
%
\hbox{}\par\smallskip%
%
% LATTICE L138.48
%
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${L_{138.48}}=\pdualwitharg{2.3.13}(\main({L_{138.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[854830860, -6432660, -214770660; -6432660, 48984, 1616160; -214770660, 1616160, 53959723],[780, 156, 1],[-25481, -389, 49, -6418, -20228, -7294, -203456, -23098, -117095, -142187, -784217, -1419342, -1405532, -135282, -1222304, -48190; -653, -10, 1, -165, -519, -187, -5215, -592, -3001, -3644, -20098, -36375, -36021, -3467, -31325, -1235; -101400, -1548, 195, -25540, -80496, -29026, -809640, -91917, -465972, -565824, -3120741, -5648180, -5593224, -538346, -4864080, -191769],137,48]
%
% END OF LATTICE L138.48
%
\hbox{}\par\smallskip%
%
% LATTICE L138.49
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L138.49
%
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%
% LATTICE L138.50
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[278513916330, 141419240040, 69629180010; 141419240040, 71807544510, 35355164130; 69629180010, 35355164130, 17407469357],[390, 390, 1],[-203, -15, 1, 1, -2, -11, -84, -112, -621, -809, -4533, -825, -4118, -401, -740, -300; 11321347, 837401, -56744, -56454, 104079, 611875, 4678013, 6240197, 34607333, 45091279, 252664484, 45985134, 229539321, 22352465, 41250667, 16724143; -22993230, -1700730, 115245, 114656, -211380, -1242695, -9500868, -12673605, -70286190, -91578690, -513152055, -93394076, -466185720, -45397015, -83778552, -33966105],137,50]
%
% END OF LATTICE L138.50
%
\hbox{}\par\smallskip%
%
% LATTICE L138.51
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30758520, 0, 63960; 0, -104, 0; 63960, 0, 133],[1560, 104, 1],[33, 37, 100, 539, 176, 50, 146, 11, 3, -1, -1, 1, 4, 4, 34, 7; 49, 63, 176, 960, 319, 93, 285, 23, 15, 1, 0, 0, 1, 3, 35, 9; -15912, -17848, -48243, -260040, -84916, -24126, -70460, -5310, -1456, 480, 481, -480, -1924, -1926, -16380, -3374],137,51]
%
% END OF LATTICE L138.51
%
\hbox{}\par\smallskip%
%
% LATTICE L138.52
%
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${L_{138.52}}=\pdualwitharg{2.13}({L_{138.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26520, 26520, -3120; 26520, -26416, 3120; -3120, 3120, -367],[1560, 104, 1],[446, 584, 3276, 8947, 2979, 871, 2683, 109, 150, 12, 2, -7, 1, 23, 297, 40; -13, -19, -109, -300, -101, -30, -95, -4, -7, -1, -1, 0, 1, 0, -5, -1; -3952, -5192, -29146, -79620, -26520, -7758, -23920, -973, -1352, -112, -26, 60, 0, -198, -2600, -353],137,52]
%
% END OF LATTICE L138.52
%
\hbox{}\par\smallskip%
%
% LATTICE L138.53
%
\leavevmode%
\llap{\spacer}%
${L_{138.53}}=\pdualwitharg{2.3.5}({L_{138.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}13\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3539640, 408720, -6240; 408720, 46680, -720; -6240, -720, 11],[1560, 120, 1],[-9, -193, -87, -209, -82, -107, -16, -45, -7, -15, -1, 1, 2, 3, 0, -7; -13, -247, -109, -260, -101, -130, -19, -52, -7, -13, -1, 0, 1, 0, -1, -13; -6000, -126360, -56790, -136292, -53400, -69550, -10368, -29055, -4440, -9360, -630, 572, 1200, 1690, -72, -4875],137,53]
%
% END OF LATTICE L138.53
%
\hbox{}\par\smallskip%
%
% LATTICE L138.54
%
\leavevmode%
\llap{\spacer}%
${L_{138.54}}=\pdualwitharg{2.3.5}({L_{138.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-151320, -23400, 1560; -23400, 240, 0; 1560, 0, -1],[1560, 120, 1],[-4, -54, -11, -51, -19, -23, -3, -14, 0, 2, 0, -1, -1, -3, -1, -12; -393, -5291, -1077, -4992, -1859, -2249, -293, -1365, 1, 195, -1, -104, -101, -299, -99, -1183; -6360, -85800, -17475, -81016, -30180, -36530, -4764, -22230, 0, 3120, -15, -1664, -1620, -4810, -1596, -19110],137,54]
%
% END OF LATTICE L138.54
%
\hbox{}\par\smallskip%
%
% LATTICE L138.55
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[644170020, -4133220, 164962980; -4133220, 27560, -1059240; 164962980, -1059240, 42245309],[780, 260, 1],[-134125, -164303, -908055, -986804, -1630406, -471508, -284818, -56362, -30967, -789, 49, -3748, -20514, -23768, -45366, -26184; 400326, 490399, 2710293, 2945337, 4866311, 1407321, 850103, 168225, 92428, 2355, -146, 11187, 61229, 70941, 135405, 78152; 533780, 653880, 3613805, 3927204, 6488560, 1876470, 1133496, 224305, 123240, 3140, -195, 14916, 81640, 94590, 180544, 104205],137,55]
%
% END OF LATTICE L138.55
%
\hbox{}\par\smallskip%
%
% LATTICE L138.56
%
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${L_{138.56}}=\pdualwitharg{3.5.13}({L_{138.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15183480, -3754920, -1227720; -3754920, 928590, 303615; -1227720, 303615, 99271],[1560, 195, 1],[-2, -1, 1, 1, 6, 5, 5, 17, 17, 16, 329, 29, 69, 25, 10, 13; 247, 45, 4, -28, -167, -104, -69, -88, 69, 271, 6948, 660, 1747, 736, 385, 720; -780, -150, 0, 98, 585, 380, 273, 480, 0, -630, -17160, -1658, -4485, -1940, -1053, -2040],137,56]
%
% END OF LATTICE L138.56
%
\hbox{}\par\smallskip%
%
% LATTICE L138.57
%
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${L_{138.57}}=\pdualwitharg{3.5.13}({L_{138.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L138.57
%
\hbox{}\par\smallskip%
%
% LATTICE L138.58
%
\leavevmode%
\llap{\spacer}%
${L_{138.58}}=\pdualwitharg{2.3.13}({L_{138.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16495440, -563160, -124800; -563160, -18408, -4056; -124800, -4056, -893],[1560, 312, 1],[5, -1, -1, 2, 7, 2, 49, 5, 47, 53, 287, 258, 253, 24, 211, 8; -1252, 258, 254, -515, -1795, -511, -12495, -1273, -11952, -13462, -72876, -65505, -64225, -6091, -53525, -2028; 4992, -1032, -1014, 2060, 7176, 2042, 49920, 5085, 47736, 53760, 291018, 261580, 256464, 24322, 213720, 8097],137,58]
%
% END OF LATTICE L138.58
%
\hbox{}\par\smallskip%
%
% LATTICE L138.59
%
\leavevmode%
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${L_{138.59}}=\pdualwitharg{2.3.13}({L_{138.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3527160, 10893480, 102960; 10893480, -30001296, -283608; 102960, -283608, -2681],[1560, 312, 1],[38, 4, 1, -1, -3, 1, 53, 16, 94, 128, 362, 661, 663, 65, 607, 50; -8451, -887, -219, 225, 664, -224, -11830, -3567, -20945, -28509, -80620, -147205, -147644, -14474, -135150, -11131; 895440, 93984, 23205, -23840, -70356, 23734, 1253460, 377946, 2219256, 3020712, 8542209, 15597320, 15643836, 1533614, 14320020, 1179402],137,59]
%
% END OF LATTICE L138.59
%
\hbox{}\par\smallskip%
%
% LATTICE L138.60
%
\leavevmode%
\llap{\spacer}%
${L_{138.60}}=\pdualwitharg{2.3.5.13}(\main({L_{138.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[391560, 1673100, -420420; 1673100, 7151820, -1797120; -420420, -1797120, 451583],[780, 780, 1],[30, 1, 0, 1, 17, 7, 41, 24, 124, 153, 847, 307, 1523, 147, 267, 53; 973, 45, 49, 46, 584, 222, 1284, 752, 3891, 4819, 26703, 9682, 48056, 4642, 8444, 1680; 3900, 180, 195, 184, 2340, 890, 5148, 3015, 15600, 19320, 107055, 38816, 192660, 18610, 33852, 6735],137,60]
%
% END OF LATTICE L138.60
%
\hbox{}\par\smallskip%
%
% LATTICE L138.61
%
\leavevmode%
\llap{\spacer}%
${L_{138.61}}=\pdualwitharg{2.5.13}({L_{138.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13044720, -1449240, 372840; -1449240, -159640, 41080; 372840, 41080, -10571],[1560, 520, 1],[3, -1, -1, 1, 6, 5, 8, 4, 37, 41, 221, 119, 194, 55, 32, 6; 776, -310, -292, 309, 1819, 1491, 2363, 1172, 10772, 11858, 63806, 34335, 55921, 15831, 9185, 1715; 3120, -1240, -1170, 1236, 7280, 5970, 9464, 4695, 43160, 47520, 255710, 137604, 224120, 63450, 36816, 6875],137,61]
%
% END OF LATTICE L138.61
%
\hbox{}\par\smallskip%
%
% LATTICE L138.62
%
\leavevmode%
\llap{\spacer}%
${L_{138.62}}=\pdualwitharg{2.5.13}({L_{138.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1789320, -1014000, 15600; -1014000, -308360, 4680; 15600, 4680, -71],[1560, 520, 1],[-5, -1, 0, 1, 4, 2, 2, 1, 1, -3, -11, -13, -24, -8, -6, -3; 569, 115, 1, -114, -459, -231, -233, -119, -137, 317, 1187, 1410, 2619, 879, 665, 335; 36400, 7360, 65, -7296, -29380, -14790, -14924, -7630, -8840, 20200, 75725, 89976, 167180, 56130, 42484, 21410],137,62]
%
% END OF LATTICE L138.62
%
\hbox{}\par\smallskip%
%
% LATTICE L138.63
%
\leavevmode%
\llap{\spacer}%
${L_{138.63}}=\pdualwitharg{2.3.5.13}({L_{138.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}13\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2617680, 45045000, -87360; 45045000, -689300040, 1336920; -87360, 1336920, -2593],[1560, 1560, 1],[-7, -3, -1, 1, 14, 3, 12, 5, 39, 35, 177, 31, 146, 13, 20, 3; 762, 304, 96, -101, -1403, -297, -1175, -483, -3710, -3252, -16310, -2847, -13337, -1177, -1773, -254; 393120, 156840, 49530, -52108, -723840, -153230, -606216, -249195, -1914120, -1677840, -8415030, -1468892, -6881160, -607270, -914784, -131055],137,63]
%
% END OF LATTICE L138.63
%
\hbox{}\par\smallskip%
%
% LATTICE L138.64
%
\leavevmode%
\llap{\spacer}%
${L_{138.64}}=\pdualwitharg{2.3.5.13}({L_{138.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1007760, 192558600, -329160; 192558600, -36726788280, 62780640; -329160, 62780640, -107317],[1560, 1560, 1],[255, 49, 0, -16, -191, -31, -89, -39, 1, 207, 704, 272, 1471, 159, 345, 167; 764, 150, 1, -49, -589, -97, -285, -134, -72, 542, 1902, 741, 4049, 443, 977, 480; 446160, 87600, 585, -28616, -343980, -56650, -166452, -78270, -42120, 316440, 1110525, 432656, 2364180, 258670, 570492, 280290],137,64]
%
% END OF LATTICE L138.64
%

\medskip%
%------------------------------WEYL-GROUP-W139--------------------------------
%
\leavevmode\llap{}%
$W_{139}$%
\qquad\llap{64} lattices, $\chi=108$%
\hfill%
$2222222222222222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L139.1
%
\leavevmode%
\llap{\spacer}%
${L_{139.1}}$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[3472440, 1155960, -2280; 1155960, 384814, -759; -2280, -759, 1],[2280, 1, 1],[0, 1423, 237, 9227, 57957, 9935, 83261, 21997, 35714, 25543, 719213, 7095, 133382, 1655, 4963, 13939, 1417, 6609, 707, 234, -1, -253; 0, -4275, -712, -27720, -174116, -29847, -250135, -66084, -107293, -76737, -2160680, -21315, -400710, -4972, -14910, -41876, -4257, -19855, -2124, -703, 3, 760; -1, -285, -44, -1650, -10336, -1770, -14820, -3912, -6346, -4536, -127680, -1259, -23655, -292, -870, -2432, -246, -1140, -120, -38, 0, 0],138,1]
%
% END OF LATTICE L139.1
%
\hbox{}\par\smallskip%
%
% LATTICE L139.2
%
\leavevmode%
\llap{\spacer}%
${L_{139.2}}$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-40166760, -13390440, 34200; -13390440, -4463986, 11401; 34200, 11401, -29],[2280, 1, 1],[-53, -1699, -49, -89, -347, -10, 291, 169, 425, 379, 11773, 263, 5689, 77, 194, 1411, 178, 1039, 167, 107, -1, -1133; 160, 5130, 148, 270, 1064, 33, -855, -504, -1273, -1137, -35340, -790, -17100, -232, -585, -4256, -537, -3135, -504, -323, 3, 3420; 398, 13110, 396, 1155, 8664, 1110, 6460, 1008, 494, -216, -14440, -518, -15390, -412, -1215, -9272, -1206, -7220, -1200, -798, 0, 8360],138,2]
%
% END OF LATTICE L139.2
%
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%
% LATTICE L139.3
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-106590, -53010, 9120; -53010, -26363, 4560; 9120, 4560, 1489],[570, 1, 1],[-66440, -1201594, -4817, -6943, -8785, 142, 4402, -281, -11464, -16282, -299617, -7930, -211436, -5099, -67427, -368023, -119134, -946422, -118711, -365344, -251450, -3468463; 133349, 2411670, 9668, 13935, 17632, -285, -8835, 564, 23009, 32679, 601350, 15916, 424365, 10234, 135330, 738644, 239109, 1899525, 238260, 733267, 504675, 6961410; -1434, -25935, -104, -150, -190, 3, 95, -6, -247, -351, -6460, -171, -4560, -110, -1455, -7942, -2571, -20425, -2562, -7885, -5427, -74860],138,3]
%
% END OF LATTICE L139.3
%
\hbox{}\par\smallskip%
%
% LATTICE L139.4
%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-14322701220, -38194560, -1053360; -38194560, -101854, -2809; -1053360, -2809, -77],[1140, 1, 1],[148, 2972, 29, 119, 385, 44, 222, 25, -7, -26, -1019, -28, -692, -13, 1, 223, 52, 538, 167, 311, 242, 7099; -55535, -1115205, -10882, -44655, -144476, -16512, -83315, -9384, 2622, 9753, 382280, 10505, 259635, 4878, -375, -83676, -19512, -201875, -62664, -116698, -90807, -2663800; 1302, 26220, 260, 1110, 3724, 441, 2375, 324, 95, -123, -6080, -192, -5130, -112, 0, 1900, 447, 4655, 1452, 2717, 2121, 62320],138,4]
%
% END OF LATTICE L139.4
%
\hbox{}\par\smallskip%
%
% LATTICE L139.5
%
\leavevmode%
\llap{\spacer}%
${L_{139.5}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{139.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1248870, 252510, 623580; 252510, 48008, 126066; 623580, 126066, 311363],[570, 2, 1],[1, -187439, -7799, -606083, -1903198, -1304914, -10935368, -722228, -4690162, -3354332, -23611506, -931681, -17514481, -54313, -325597, -457058, -185774, -865912, -23116, -30350, 284, 8946; 0, -1995, -83, -6450, -20254, -13887, -116375, -7686, -49913, -35697, -251275, -9915, -186390, -578, -3465, -4864, -1977, -9215, -246, -323, 3, 95; -2, 376200, 15653, 1216440, 3819817, 2619030, 21947850, 1449549, 9413398, 6732318, 47389515, 1869932, 35152470, 109009, 653490, 917339, 372858, 1737930, 46395, 60914, -570, -17955],138,5]
%
% END OF LATTICE L139.5
%
\hbox{}\par\smallskip%
%
% LATTICE L139.6
%
\leavevmode%
\llap{\spacer}%
${L_{139.6}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{139.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1413030, 324900, 470250; 324900, 72012, 108111; 470250, 108111, 156497],[570, 3, 1],[1, -41548, -10373, -134356, -2531401, -289273, -7272453, -320207, -3119147, -743589, -31405227, -619606, -3882617, -72241, -72179, -607931, -41183, -575877, -10249, -20185, 63, 11907; 0, -665, -166, -2150, -40508, -4629, -116375, -5124, -49913, -11899, -502550, -9915, -62130, -1156, -1155, -9728, -659, -9215, -164, -323, 1, 190; -3, 125305, 31284, 405205, 7634466, 872420, 21933030, 965714, 9407052, 2242594, 94715190, 1868673, 11709605, 217872, 217685, 1833462, 124204, 1736790, 30910, 60876, -190, -35910],138,6]
%
% END OF LATTICE L139.6
%
\hbox{}\par\smallskip%
%
% LATTICE L139.7
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1806330, 469680, -361950; 469680, 120020, -94125; -361950, -94125, 72527],[570, 5, 1],[-1, 15146, 6303, 48986, 1538247, 527345, 883847, 583739, 1895409, 1355569, 3816801, 376516, 1415611, 43899, 26317, 369429, 75079, 69991, 18685, 12267, -115, -1449; 0, -399, -166, -1290, -40508, -13887, -23275, -15372, -49913, -35697, -100510, -9915, -37278, -1156, -693, -9728, -1977, -1843, -492, -323, 3, 38; -5, 75069, 31240, 242793, 7624130, 2613720, 4380678, 2893230, 9394360, 6718710, 18917502, 1866155, 7016301, 217580, 130437, 1831030, 372120, 346902, 92610, 60800, -570, -7182],138,7]
%
% END OF LATTICE L139.7
%
\hbox{}\par\smallskip%
%
% LATTICE L139.8
%
\leavevmode%
\llap{\spacer}%
${L_{139.8}}=\pdualwitharg{3}(\main({L_{139.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6793260, 2704080, 12540; 2704080, -1076367, -4992; 12540, -4992, -23],[1140, 3, 1],[153, 959, 17, -22, -641, -41, -1108, -103, -520, -126, -10709, -213, -1339, -25, 2, 337, 25, 728, 71, 368, 90, 7669; 379, 2375, 42, -55, -1596, -102, -2755, -256, -1292, -313, -26600, -529, -3325, -62, 5, 836, 62, 1805, 176, 912, 223, 19000; 1149, 7315, 150, -65, -3192, -221, -6270, -604, -3135, -772, -66120, -1329, -8455, -174, 5, 2280, 173, 5130, 508, 2679, 664, 57000],138,8]
%
% END OF LATTICE L139.8
%
\hbox{}\par\smallskip%
%
% LATTICE L139.9
%
\leavevmode%
\llap{\spacer}%
${L_{139.9}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{139.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[999844853460, 33493770, 498867737490; 33493770, 1122, 16711554; 498867737490, 16711554, 248907636667],[570, 6, 1],[-1895, -39342, -3082, -47919, -429644, -96323, -2378727, -51486, -985619, -232055, -4868229, -190300, -1177893, -9731, -16146, -57250, -6181, -55743, 234, 11831, 1421, -711; 189, 5225, 472, 9185, 84455, 19124, 476710, 10411, 201172, 47682, 1003865, 39441, 245765, 2170, 4025, 15941, 2012, 25270, 157, -380, -126, 95; 3798, 78850, 6177, 96040, 861099, 193052, 4767480, 103189, 1975392, 465088, 9756975, 381402, 2360750, 19503, 32360, 114741, 12388, 111720, -469, -23712, -2848, 1425],138,9]
%
% END OF LATTICE L139.9
%
\hbox{}\par\smallskip%
%
% LATTICE L139.10
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[288445080, 82366140, 71638740; 82366140, 23576476, 20456452; 71638740, 20456452, 17792327],[1140, 4, 1],[-905673, -19228241, -123721, -1844597, -2153617, -1293179, -9560776, -556462, -3165619, -2056246, -13741947, -1003407, -17044279, -3551, -64483, -264551, -234085, -2530064, -207170, -1670717, -1380098, -10439733; 9692, 205770, 1324, 19740, 23047, 13839, 102315, 5955, 33877, 22005, 147060, 10738, 182400, 38, 690, 2831, 2505, 27075, 2217, 17879, 14769, 111720; 3635444, 77183700, 496626, 7404360, 8644791, 5190924, 38377720, 2233683, 12707048, 8253936, 55161275, 4027756, 68417100, 14254, 258840, 1061929, 939636, 10155880, 831597, 6706392, 5539824, 41905925],138,10]
%
% END OF LATTICE L139.10
%
\hbox{}\par\smallskip%
%
% LATTICE L139.11
%
\leavevmode%
\llap{\spacer}%
${L_{139.11}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{139.1}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[77138791980, 12010470, 38342516610; 12010470, 1870, 5969910; 38342516610, 5969910, 19058485909],[570, 10, 1],[-1019, -14110, -1956, -20257, -306522, -207193, -342723, -111776, -716651, -507917, -711655, -139456, -519695, -7409, -7838, -49348, -17567, -13147, -604, 4911, 2207, -85; 189, 3135, 472, 5511, 84455, 57372, 95342, 31233, 201172, 143046, 200773, 39441, 147459, 2170, 2415, 15941, 6036, 5054, 471, -380, -378, 19; 2050, 28386, 3935, 40752, 616645, 416820, 689472, 224865, 1441720, 1021800, 1431669, 280550, 1045494, 14905, 15768, 99275, 35340, 26448, 1215, -9880, -4440, 171],138,11]
%
% END OF LATTICE L139.11
%
\hbox{}\par\smallskip%
%
% LATTICE L139.12
%
\leavevmode%
\llap{\spacer}%
${L_{139.12}}=\pdualwitharg{5}(\main({L_{139.2}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9607157340, -127809960, 1406760; -127809960, 1700335, -18715; 1406760, -18715, 206],[1140, 5, 1],[5, 235, 89, 658, 20587, 3524, 5899, 7783, 12621, 9019, 50767, 5005, 18803, 579, 344, 4797, 484, 447, 233, 71, -1, 1; 376, 17670, 6692, 49476, 1547968, 264975, 443555, 585216, 948993, 678153, 3817252, 376334, 1413828, 43536, 25866, 360696, 36393, 33611, 17520, 5339, -75, 76; 15, 513, 190, 1431, 44840, 7680, 12863, 16980, 27550, 19695, 110884, 10935, 41097, 1270, 759, 10640, 1080, 1007, 540, 190, 15, 76],138,12]
%
% END OF LATTICE L139.12
%
\hbox{}\par\smallskip%
%
% LATTICE L139.13
%
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\llap{\spacer}%
${L_{139.13}}=\pdualwitharg{3}({L_{139.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13586520, -4126800, 25080; -4126800, -1253481, 7617; 25080, 7617, -46],[2280, 3, 1],[-113, -708, -25, 33, 955, 61, 1647, 153, 772, 187, 15891, 158, 993, 37, -3, -499, -37, -1077, -105, -544, -133, -11331; 379, 2375, 84, -110, -3192, -204, -5510, -512, -2584, -626, -53200, -529, -3325, -124, 10, 1672, 124, 3610, 352, 1824, 446, 38000; 1143, 7220, 276, -230, -7980, -527, -14535, -1372, -7011, -1711, -145920, -1458, -9215, -360, 20, 4788, 359, 10545, 1036, 5415, 1333, 114000],138,13]
%
% END OF LATTICE L139.13
%
\hbox{}\par\smallskip%
%
% LATTICE L139.14
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4475956920, -14667240, 487920; -14667240, -48063, 1599; 487920, 1599, -46],[2280, 3, 1],[281, 1831, 45, 11, -165, -21, -752, -83, -472, -122, -10681, -221, -1451, -37, -1, 469, 37, 1132, 115, 624, 158, 13721; -85698, -558410, -13724, -3355, 50312, 6404, 229330, 25312, 143944, 37206, 3257360, 67398, 442510, 11284, 305, -143032, -11284, -345230, -35072, -190304, -48186, -4184560; 1608, 10450, 252, 50, -1368, -143, -4845, -520, -2907, -745, -64980, -1338, -8740, -216, -5, 2736, 215, 6555, 664, 3591, 907, 78660],138,14]
%
% END OF LATTICE L139.14
%
\hbox{}\par\smallskip%
%
% LATTICE L139.15
%
\leavevmode%
\llap{\spacer}%
${L_{139.15}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{139.1}}))$%
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{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22520911470, 7948080, 7406876970; 7948080, 2805, 2614035; 7406876970, 2614035, 2436039346],[570, 15, 1],[-449, -2075, -1727, -2984, -270949, -30526, -151488, -32939, -316796, -74844, -629205, -61651, -76585, -6553, -1156, -43691, -2594, -5832, -181, 2156, 324, -75; 189, 1045, 944, 1837, 168910, 19124, 95342, 20822, 201172, 47682, 401546, 39441, 49153, 4340, 805, 31882, 2012, 5054, 314, -380, -126, 38; 1365, 6308, 5250, 9071, 823650, 92795, 460503, 100130, 963015, 227515, 1912692, 187410, 232807, 19920, 3514, 132810, 7885, 17727, 550, -6555, -985, 228],138,15]
%
% END OF LATTICE L139.15
%
\hbox{}\par\smallskip%
%
% LATTICE L139.16
%
\leavevmode%
\llap{\spacer}%
${L_{139.16}}=\pdualwitharg{19}(\pfillwitharg{2}({L_{139.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4814790, 1483140, 1773270; 1483140, 456076, 546231; 1773270, 546231, 653089],[570, 19, 1],[7, -7166, -11341, -440858, -145727, -949219, -418667, -1050745, -179569, -2440091, -1808013, -677752, -670579, -79025, -236887, -35005, -135173, -33163, -33647, -1163, 209, 693; 0, -105, -166, -6450, -2132, -13887, -6125, -15372, -2627, -35697, -26450, -9915, -9810, -1156, -3465, -512, -1977, -485, -492, -17, 3, 10; -19, 19545, 30932, 1202415, 397462, 2588940, 1141890, 2865846, 489764, 6655206, 4931250, 1848529, 1828965, 215536, 646095, 95474, 368676, 90450, 91770, 3172, -570, -1890],138,16]
%
% END OF LATTICE L139.16
%
\hbox{}\par\smallskip%
%
% LATTICE L139.17
%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1349266022040, 8823875880, 26411520; 8823875880, 57706030, 172725; 26411520, 172725, 517],[2280, 5, 1],[5650, 21229, 1309, 779, 10877, 1099, 1017, 533, 166, -1, 1, 5, 260, 199, 1483, 46427, 7949, 13309, 17563, 28486, 20359, 114607; -863341, -3243870, -200020, -119034, -1662044, -167931, -155401, -81444, -25365, 153, -152, -764, -39729, -30408, -226608, -7094220, -1214637, -2033665, -2683692, -4352767, -3110931, -17512376; -202165, -759639, -46860, -27906, -389880, -39420, -36518, -19200, -6080, -30, -304, -185, -9291, -7100, -53034, -1660600, -284340, -476102, -628320, -1019160, -728430, -4100656],138,17]
%
% END OF LATTICE L139.17
%
\hbox{}\par\smallskip%
%
% LATTICE L139.18
%
\leavevmode%
\llap{\spacer}%
${L_{139.18}}=\pdualwitharg{5}({L_{139.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[699960, -234840, -2280; -234840, 78790, 765; -2280, 765, 7],[2280, 5, 1],[-679, -2383, -27, 1, 51, 1, -32, -97, -288, -286, -1869, -221, -1037, -93, -91, -4611, -721, -1108, -1343, -1992, -1334, -7251; -2014, -7068, -80, 3, 152, 3, -95, -288, -855, -849, -5548, -656, -3078, -276, -270, -13680, -2139, -3287, -3984, -5909, -3957, -21508; -1190, -4218, -60, -3, 0, 0, -38, -120, -380, -390, -2584, -310, -1482, -140, -147, -7600, -1200, -1862, -2280, -3420, -2310, -12616],138,18]
%
% END OF LATTICE L139.18
%
\hbox{}\par\smallskip%
%
% LATTICE L139.19
%
\leavevmode%
\llap{\spacer}%
${L_{139.19}}=\pdualwitharg{2.3}(\main({L_{139.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[130768260, -1167360, 32420460; -1167360, 10596, -289392; 32420460, -289392, 8037781],[1140, 12, 1],[28175, 201729, 4085, 21693, 77828, 15777, 354422, 6973, 120867, 26484, 534544, 39445, 226531, 423, 847, 7824, 2255, 73838, 2043, 50437, 14088, 321976; 15113, 108205, 2191, 11635, 41743, 8462, 190095, 3740, 64828, 14205, 286710, 21157, 121505, 227, 455, 4199, 1210, 39615, 1096, 27056, 7557, 172710; -113100, -809780, -16398, -87080, -312417, -63332, -1422720, -27991, -485184, -106312, -2145765, -158340, -909340, -1698, -3400, -31407, -9052, -296400, -8201, -202464, -56552, -1292475],138,19]
%
% END OF LATTICE L139.19
%
\hbox{}\par\smallskip%
%
% LATTICE L139.20
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[42770588970, -177000390, 21340033920; -177000390, 720120, -88312890; 21340033920, -88312890, 10647434573],[570, 30, 1],[-943, 4656675, 2906827, 15060999, 709413128, 162134990, 815230260, 89736822, 1748260682, 416776904, 1760243712, 347285563, 435237177, 20245481, 8091309, 170374576, 23083474, 64557444, 2872410, 11314726, -35360, -668304; 0, -133, -83, -430, -20254, -4629, -23275, -2562, -49913, -11899, -50255, -9915, -12426, -578, -231, -4864, -659, -1843, -82, -323, 1, 19; 1890, -9333104, -5825985, -30185888, -1421835645, -324957770, -1633918794, -179854315, -3503937630, -835321810, -3527954523, -696044340, -872320666, -40576845, -16216942, -341471895, -46264870, -129388746, -5757005, -22677450, 70870, 1339443],138,20]
%
% END OF LATTICE L139.20
%
\hbox{}\par\smallskip%
%
% LATTICE L139.21
%
\leavevmode%
\llap{\spacer}%
${L_{139.21}}=\pdualwitharg{3.5}(\main({L_{139.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[607620, 9120, -1140; 9120, 105, -15; -1140, -15, 2],[1140, 15, 1],[-23, -27, -1, 0, 1, 0, -1, -1, -9, -3, -59, -7, -11, -3, -2, -153, -8, -37, -15, -67, -15, -245; -1456, -1710, -64, 0, 76, 1, -57, -60, -551, -185, -3648, -434, -684, -188, -126, -9652, -505, -2337, -948, -4237, -949, -15504; -23895, -28063, -1050, -1, 1140, 10, -969, -1000, -9120, -3055, -60192, -7155, -11267, -3090, -2069, -158460, -8290, -38361, -15560, -69540, -15575, -254448],138,21]
%
% END OF LATTICE L139.21
%
\hbox{}\par\smallskip%
%
% LATTICE L139.22
%
\leavevmode%
\llap{\spacer}%
${L_{139.22}}=\pdualwitharg{2}({L_{139.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-101776920, -36480, 70680; -36480, 8, 24; 70680, 24, -49],[2280, 8, 1],[1, 46, 1, 37, 107, 71, 577, 37, 233, 163, 567, 44, 809, 2, 8, 7, 1, -7, -1, -5, -1, 3; 94, 4275, 92, 3330, 9595, 6357, 51585, 3303, 20767, 14511, 50445, 3911, 71820, 175, 675, 551, 51, -855, -99, -475, -93, 285; 1488, 68400, 1486, 54900, 158726, 105312, 855760, 54870, 345496, 241680, 840655, 65232, 1199280, 2962, 11820, 10298, 1440, -10640, -1494, -7448, -1488, 4465],138,22]
%
% END OF LATTICE L139.22
%
\hbox{}\par\smallskip%
%
% LATTICE L139.23
%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2396280, -802560, -139080; -802560, 268792, 46592; -139080, 46592, 7883],[2280, 8, 1],[-22, -31639, -1299, -100588, -157893, -216497, -1814133, -59903, -777967, -556363, -3916201, -154523, -2904716, -9004, -53957, -37864, -30775, -143437, -1915, -5061, 1, 1061; -65, -93480, -3838, -297195, -466507, -639657, -5359995, -176988, -2298563, -1643817, -11570715, -456550, -8582205, -26603, -159420, -111872, -90927, -423795, -5658, -14953, 3, 3135; -4, -5700, -234, -18120, -28443, -39000, -326800, -10791, -140144, -100224, -705470, -27836, -523260, -1622, -9720, -6821, -5544, -25840, -345, -912, 0, 190],138,23]
%
% END OF LATTICE L139.23
%
\hbox{}\par\smallskip%
%
% LATTICE L139.24
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[170986038420, 34859490, 85319955810; 34859490, 7106, 17394462; 85319955810, 17394462, 42573621371],[570, 38, 1],[-9879, -41002, -22972, -1303801, -209726, -2703965, -1181289, -1469080, -497517, -6716737, -2479933, -1850256, -1819133, -101037, -556334, -38292, -272251, -58801, -19028, 1481, 20251, -247; 189, 825, 472, 27555, 4445, 57372, 25090, 31233, 10588, 143046, 52835, 39441, 38805, 2170, 12075, 839, 6036, 1330, 471, -20, -378, 5; 19798, 82170, 46037, 2612880, 420301, 5418876, 2367360, 2944107, 997048, 13460664, 4969905, 3708002, 3645630, 202483, 1114920, 76739, 545604, 117840, 38133, -2968, -40584, 495],138,24]
%
% END OF LATTICE L139.24
%
\hbox{}\par\smallskip%
%
% LATTICE L139.25
%
\leavevmode%
\llap{\spacer}%
${L_{139.25}}=\pdualwitharg{19}(\main({L_{139.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18938820, -721620, 47880; -721620, 27493, -1824; 47880, -1824, 121],[1140, 19, 1],[-8, -8, -1, 1, 1, 4, 2, 11, 1, 14, 21, 8, 8, 1, -1, -1, -4, -2, -11, -1, -14, -21; -503, -495, -54, 45, 48, 192, 95, 516, 46, 633, 940, 353, 345, 34, -195, -88, -312, -145, -756, -66, -903, -1340; -4427, -4305, -418, 285, 328, 1311, 640, 3420, 297, 3990, 5840, 2147, 2025, 114, -2565, -936, -3135, -1400, -7068, -601, -8094, -11920],138,25]
%
% END OF LATTICE L139.25
%
\hbox{}\par\smallskip%
%
% LATTICE L139.26
%
\leavevmode%
\llap{\spacer}%
${L_{139.26}}=\pdualwitharg{2.5}(\main({L_{139.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[17244780, 4935060, 4309200; 4935060, -1766920, 1224740; 4309200, 1224740, 1076779],[1140, 20, 1],[-85534, -411176, -4402, 742282, 6287135, 4429229, 7604363, 2566921, 17036819, 12368317, 17547684, 6999604, 26684642, 465340, 640532, 2470687, 1102027, 1153459, 198707, 478825, 77009, -32040; -913, -4389, -47, 7923, 67108, 47277, 81168, 27399, 181849, 132018, 187302, 74713, 284829, 4967, 6837, 26372, 11763, 12312, 2121, 5111, 822, -342; 343340, 1650492, 17670, -2979576, -25237035, -17779260, -30524488, -10303815, -68387080, -49647360, -70437731, -28096940, -107114172, -1867910, -2571144, -9917525, -4423620, -4630072, -797625, -1922040, -309120, 128611],138,26]
%
% END OF LATTICE L139.26
%
\hbox{}\par\smallskip%
%
% LATTICE L139.27
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[167318158530, 42233580, 55632937650; 42233580, 10659, 14042577; 55632937650, 14042577, 18497835374],[570, 57, 1],[-7979, -11039, -37109, -351032, -338797, -728010, -954144, -791065, -401852, -1808404, -4006161, -1494481, -489781, -163219, -149788, -61859, -73302, -47496, -10247, 1196, 5452, -399; 189, 275, 944, 9185, 8890, 19124, 25090, 20822, 10588, 47682, 105670, 39441, 12935, 4340, 4025, 1678, 2012, 1330, 314, -20, -126, 10; 23997, 33200, 111606, 1055735, 1018938, 2189503, 2869605, 2379142, 1208577, 5438807, 12048600, 4494678, 1473025, 490884, 450490, 186042, 220457, 142845, 30818, -3597, -16397, 1200],138,27]
%
% END OF LATTICE L139.27
%
\hbox{}\par\smallskip%
%
% LATTICE L139.28
%
\leavevmode%
\llap{\spacer}%
${L_{139.28}}=\pdualwitharg{3.5}({L_{139.2}})$%
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{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4854120, 2280, -2280; 2280, -15, 0; -2280, 0, 1],[2280, 15, 1],[-7, -11, -3, -1, -153, -8, -37, -15, -67, -15, -245, -23, -27, -1, 0, 1, 0, -1, -1, -9, -3, -59; -966, -1520, -416, -139, -21280, -1113, -5149, -2088, -9329, -2089, -34124, -3204, -3762, -140, 0, 152, 1, -133, -136, -1235, -413, -8132; -15780, -24814, -6780, -2264, -346560, -18125, -83847, -34000, -151905, -34015, -555636, -52170, -61256, -2280, -1, 2280, 5, -2223, -2240, -20235, -6755, -132924],138,28]
%
% END OF LATTICE L139.28
%
\hbox{}\par\smallskip%
%
% LATTICE L139.29
%
\leavevmode%
\llap{\spacer}%
${L_{139.29}}=\pdualwitharg{3.5}({L_{139.1}})$%
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{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-118165560, -519840, 102600; -519840, -2265, 450; 102600, 450, -89],[2280, 15, 1],[4, 8, 7, 7, 587, 32, 154, 65, 303, 70, 1163, 56, 68, 9, 1, 21, 0, -2, -1, 1, 2, 53; 395, 798, 708, 714, 59964, 3271, 15751, 6652, 31027, 7171, 119168, 5740, 6973, 928, 104, 2204, 1, -209, -108, 57, 191, 5168; 6600, 13243, 11640, 11672, 979260, 53395, 257013, 108500, 505875, 116885, 1942104, 93525, 113582, 15060, 1678, 35340, 5, -3363, -1700, 1425, 3265, 87096],138,29]
%
% END OF LATTICE L139.29
%
\hbox{}\par\smallskip%
%
% LATTICE L139.30
%
\leavevmode%
\llap{\spacer}%
${L_{139.30}}=\pdualwitharg{19}({L_{139.1}})$%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[38760, -13680, -2280; -13680, 4826, 779; -2280, 779, -161],[2280, 19, 1],[3933, 3838, 791, 1943, 241, 371, 61, -29, -8, 1, 153, 72, 167, 277, 6067, 1895, 6037, 2609, 12845, 1076, 14417, 21207; 11298, 11025, 2272, 5580, 692, 1065, 175, -84, -23, 3, 440, 207, 480, 796, 17430, 5444, 17343, 7495, 36900, 3091, 41415, 60920; -1121, -1095, -228, -570, -72, -114, -20, 0, 2, 0, -40, -19, -45, -76, -1710, -536, -1710, -740, -3648, -306, -4104, -6040],138,30]
%
% END OF LATTICE L139.30
%
\hbox{}\par\smallskip%
%
% LATTICE L139.31
%
\leavevmode%
\llap{\spacer}%
${L_{139.31}}=\pdualwitharg{19}({L_{139.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${76}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{s}{2}{152}\above{1pt}{1pt}{l}{2}{285}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{114}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{456}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{114}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-973560\\25080\\27360
\end{tabular}\endgroup%
\kern3pt%
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25080\\-646\\-703
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\kern3pt%
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27360\\-703\\-731
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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175439\\7135920\\-310080
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1271\\51692\\-2242
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1243\\50550\\-2190
\end{tabular}\endgroup%
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131\\5324\\-228
\end{tabular}\endgroup%
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167\\6780\\-285
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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43\\1744\\-72
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70\\2835\\-114
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13\\525\\-20
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5\\192\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-973560, 25080, 27360; 25080, -646, -703; 27360, -703, -731],[2280, 19, 1],[1271, 1243, 131, 167, 43, 70, 13, 5, -1, -1, 19, 19, 47, 41, 478, 301, 962, 417, 2059, 173, 2323, 3421; 51692, 50550, 5324, 6780, 1744, 2835, 525, 192, -41, -39, 780, 778, 1920, 1672, 19455, 12248, 39141, 16965, 83760, 7037, 94485, 139140; -2242, -2190, -228, -285, -72, -114, -20, 0, 2, 0, -40, -38, -90, -76, -855, -536, -1710, -740, -3648, -306, -4104, -6040],138,31]
%
% END OF LATTICE L139.31
%
\hbox{}\par\smallskip%
%
% LATTICE L139.32
%
\leavevmode%
\llap{\spacer}%
${L_{139.32}}=\pdualwitharg{5.19}(\pfillwitharg{2}({L_{139.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[182746972110, 56981760, -36662551050; 56981760, 17765, -11431635; -36662551050, -11431635, 7355211602],[570, 95, 1],[6499, 5395, 30227, 171562, 275971, 1779028, 155442, 1933117, 327334, 4419182, 652655, 1217351, 239375, 132953, 73208, 50389, 179132, 7738, 25043, -974, -13322, 65; 189, 165, 944, 5511, 8890, 57372, 5018, 62466, 10588, 143046, 21134, 39441, 7761, 4340, 2415, 1678, 6036, 266, 942, -20, -378, 2; 32395, 26892, 150670, 855171, 1375610, 8867775, 774819, 9635850, 1631635, 22027935, 3253236, 6068030, 1193193, 662720, 364914, 251170, 892905, 38571, 124830, -4855, -66405, 324],138,32]
%
% END OF LATTICE L139.32
%
\hbox{}\par\smallskip%
%
% LATTICE L139.33
%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[380760, 20520, 4560; 20520, 1104, 240; 4560, 240, 37],[2280, 24, 1],[-19, -159, -5, -33, -250, -52, -1194, -24, -424, -94, -954, -71, -411, -1, 3, 22, 4, 54, 0, -32, -14, -186; 387, 3230, 101, 665, 5035, 1047, 24035, 483, 8531, 1891, 19190, 1428, 8265, 20, -60, -437, -79, -1045, 1, 665, 287, 3800; -192, -1520, -42, -260, -1938, -400, -9120, -182, -3192, -704, -7125, -528, -3040, -6, 20, 114, 16, 0, -10, -456, -160, -1995],138,33]
%
% END OF LATTICE L139.33
%
\hbox{}\par\smallskip%
%
% LATTICE L139.34
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-576035160, -29259240, 228000; -29259240, -1486128, 11544; 228000, 11544, -71],[2280, 24, 1],[-1691, -11293, -163, -421, -961, -274, -3588, -8, 754, 336, 8694, 431, 3313, 79, 1, -635, -398, -12372, -160, -7138, -1848, -40614; 33799, 225720, 3258, 8415, 19209, 5477, 71725, 160, -15067, -6715, -173755, -8614, -66215, -1579, -20, 12692, 7955, 247285, 3198, 142671, 36937, 811775; 65148, 435100, 6282, 16240, 37107, 10592, 139080, 317, -28728, -12872, -333450, -16548, -127300, -3042, -40, 24453, 15328, 476520, 6163, 274968, 71192, 1564650],138,34]
%
% END OF LATTICE L139.34
%
\hbox{}\par\smallskip%
%
% LATTICE L139.35
%
\leavevmode%
\llap{\spacer}%
${L_{139.35}}=\pdualwitharg{3.19}(\main({L_{139.2}}))$%
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{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[35340, 4560, -1140; 4560, 399, -114; -1140, -114, 31],[1140, 57, 1],[-26, -8, -1, 1, 1, 1, 1, 1, 0, -1, -7, -4, -2, -3, -11, -9, -9, -11, -17, -4, -17, -73; -535, -165, -22, 15, 16, 16, 15, 12, -2, -29, -180, -95, -45, -62, -225, -184, -184, -225, -348, -82, -349, -1500; -2907, -895, -114, 95, 96, 95, 90, 76, -9, -152, -960, -513, -245, -342, -1235, -1008, -1007, -1230, -1900, -447, -1900, -8160],138,35]
%
% END OF LATTICE L139.35
%
\hbox{}\par\smallskip%
%
% LATTICE L139.36
%
\leavevmode%
\llap{\spacer}%
${L_{139.36}}=\pdualwitharg{2.3.19}(\pfillwitharg{2}({L_{139.1}}))$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L139.36
%
\hbox{}\par\smallskip%
%
% LATTICE L139.37
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[365517060, -1277940, 91057500; -1277940, 4440, -318360; 91057500, -318360, 22684217],[1140, 60, 1],[3138, 1968, -710, -282, -71, 563, 5623, 913, 23855, 6731, 30770, 13452, 19046, 1816, 1388, 21085, 3861, 15391, 1299, 18173, 3223, 11258; -281, -323, -3, 1, 0, -3, -38, -7, -209, -64, -304, -139, -209, -25, -29, -532, -109, -494, -49, -855, -188, -760; -12600, -7904, 2850, 1132, 285, -2260, -22572, -3665, -95760, -27020, -123519, -54000, -76456, -7290, -5572, -84645, -15500, -61788, -5215, -72960, -12940, -45201],138,37]
%
% END OF LATTICE L139.37
%
\hbox{}\par\smallskip%
%
% LATTICE L139.38
%
\leavevmode%
\llap{\spacer}%
${L_{139.38}}=\pdualwitharg{2.19}(\main({L_{139.2}}))$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[741174420, -6689520, 184636680; -6689520, 59812, -1666452; 184636680, -1666452, 45995521],[1140, 76, 1],[125911, 108701, -1865, -28117, -359, 6305, 8316, 17566, 8891, 151150, 62167, 106409, 123619, 17353, 260437, 15847, 179551, 69124, 75362, 22085, 267026, 92713; 991, 855, -15, -225, -3, 48, 65, 138, 70, 1191, 490, 839, 975, 137, 2055, 125, 1416, 545, 594, 174, 2103, 730; -505400, -436320, 7486, 112860, 1441, -25308, -33380, -70509, -35688, -606708, -249535, -427120, -496200, -69654, -1045380, -63609, -720708, -277460, -302499, -88648, -1071828, -372145],138,38]
%
% END OF LATTICE L139.38
%
\hbox{}\par\smallskip%
%
% LATTICE L139.39
%
\leavevmode%
\llap{\spacer}%
${L_{139.39}}=\pdualwitharg{2.5}({L_{139.1}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}19\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6808080, -15960, 20520; -15960, 40, 40; 20520, 40, -61],[2280, 40, 1],[-436, -1643, -26, -32, -229, -95, -91, -13, -21, -1, 1, 1, -10, -3, -55, -873, -601, -1011, -335, -2183, -1565, -1103; -16837, -63441, -1003, -1233, -8816, -3654, -3496, -498, -798, -36, 38, 37, -399, -117, -2127, -33744, -23226, -39064, -12942, -84322, -60444, -42598; -158160, -595992, -9430, -11604, -83030, -34440, -32984, -4710, -7600, -360, 361, 360, -3648, -1090, -19956, -316730, -218040, -366776, -121530, -791920, -567720, -400121],138,39]
%
% END OF LATTICE L139.39
%
\hbox{}\par\smallskip%
%
% LATTICE L139.40
%
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${L_{139.40}}=\pdualwitharg{2.5}({L_{139.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[579120, -15960, 0; -15960, 440, 0; 0, 0, -1],[2280, 40, 1],[-105, -394, -6, -7, -24, -19, -17, -1, -1, 1, 1, 0, -5, -1, -14, -109, -149, -249, -41, -531, -379, -533; -3721, -13965, -213, -249, -855, -678, -608, -36, -38, 36, 38, 1, -171, -35, -495, -3857, -5274, -8816, -1452, -18810, -13428, -18886; -2380, -8892, -130, -144, -475, -360, -304, -15, 0, 0, -38, -20, -228, -30, -336, -2565, -3480, -5776, -945, -12160, -8640, -12122],138,40]
%
% END OF LATTICE L139.40
%
\hbox{}\par\smallskip%
%
% LATTICE L139.41
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L139.41
%
\hbox{}\par\smallskip%
%
% LATTICE L139.42
%
\leavevmode%
\llap{\spacer}%
${L_{139.42}}=\pdualwitharg{5.19}(\main({L_{139.2}}))$%
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{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10712580, -3592140, 47880; -3592140, 1204505, -16055; 47880, -16055, 214],[1140, 95, 1],[-326, -64, -35, -19, -13, -23, -1, -7, 0, -1, -1, -4, -2, -9, -47, -75, -241, -21, -521, -44, -593, -175; -452, -90, -56, -36, -28, -57, -3, -36, -1, -3, 0, 2, 0, -4, -54, -92, -303, -27, -684, -59, -807, -240; 38855, 7533, 3610, 1539, 800, 855, -2, -1140, -75, 0, 224, 1045, 447, 1710, 6441, 9840, 31065, 2662, 64980, 5395, 71820, 21056],138,42]
%
% END OF LATTICE L139.42
%
\hbox{}\par\smallskip%
%
% LATTICE L139.43
%
\leavevmode%
\llap{\spacer}%
${L_{139.43}}=\pdualwitharg{3.19}({L_{139.1}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[85844280, 2213880, -72960; 2213880, 57057, -1881; -72960, -1881, 62],[2280, 57, 1],[-2, -1, -3, -11, -9, -9, -11, -17, -4, -17, -73, -13, -4, -1, 1, 1, 1, 1, 1, 0, -1, -7; -95, -45, -124, -450, -368, -368, -450, -696, -164, -698, -3000, -535, -165, -44, 30, 32, 32, 30, 24, -4, -58, -360; -5244, -2545, -7296, -26600, -21756, -21755, -26595, -41116, -9681, -41173, -176880, -31521, -9710, -2508, 2090, 2148, 2147, 2085, 1900, -123, -2945, -19200],138,43]
%
% END OF LATTICE L139.43
%
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%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[223984920, -14012880, -143640; -14012880, 875463, 9177; -143640, 9177, 62],[2280, 57, 1],[-3457, -1481, -1677, -2959, -4833, -4833, -5918, -9173, -2170, -9272, -39949, -14303, -4439, -691, -1, 97, 97, -2, -299, -198, -1384, -7411; -51890, -22230, -25172, -44415, -72544, -72544, -88830, -137688, -32572, -139174, -599640, -214690, -66630, -10372, -15, 1456, 1456, -30, -4488, -2972, -20774, -111240; -328548, -140750, -159372, -281200, -459288, -459287, -562395, -871720, -206217, -881125, -3796380, -1359222, -421840, -65664, -95, 9216, 9215, -195, -28424, -18819, -131537, -704340],138,44]
%
% END OF LATTICE L139.44
%
\hbox{}\par\smallskip%
%
% LATTICE L139.45
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L139.45
%
\hbox{}\par\smallskip%
%
% LATTICE L139.46
%
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${L_{139.46}}=\pdualwitharg{5.19}({L_{139.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[23504520, 7790760, -50160; 7790760, 2582290, -16625; -50160, -16625, 107],[2280, 95, 1],[-689, -134, -133, -61, -35, -47, -1, 23, 2, -1, -7, -16, -7, -55, -221, -341, -1081, -93, -2279, -190, -2537, -745; 2330, 453, 448, 204, 116, 153, 3, -84, -7, 3, 24, 55, 24, 188, 750, 1156, 3663, 315, 7716, 643, 8583, 2520; 38855, 7533, 7220, 3078, 1600, 1710, -4, -2280, -150, 0, 448, 1045, 447, 3420, 12882, 19680, 62130, 5324, 129960, 10790, 143640, 42112],138,46]
%
% END OF LATTICE L139.46
%
\hbox{}\par\smallskip%
%
% LATTICE L139.47
%
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${L_{139.47}}=\pdualwitharg{5.19}({L_{139.2}})$%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2280, 0, -2280; 0, -190, -665; -2280, -665, -47],[2280, 95, 1],[-4871, -929, -363, -55, -39, -1, 2, 7, -18, -524, -207, -529, -151, -357, -485, -1401, -4319, -362, -8647, -702, -9196, -2673; 17020, 3246, 1268, 192, 136, 3, -7, -24, 63, 1833, 724, 1850, 528, 1248, 1695, 4896, 15093, 1265, 30216, 2453, 32133, 9340; -5130, -978, -380, -57, -40, 0, 2, 0, -20, -570, -224, -570, -162, -380, -513, -1480, -4560, -382, -9120, -740, -9690, -2816],138,47]
%
% END OF LATTICE L139.47
%
\hbox{}\par\smallskip%
%
% LATTICE L139.48
%
\leavevmode%
\llap{\spacer}%
${L_{139.48}}=\pdualwitharg{2.3.19}(\main({L_{139.2}}))$%
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0\\0\\0
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653\\-3\\-2622
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1041\\-5\\-4180
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30915660, -1124040, 7700700; -1124040, 40812, -279984; 7700700, -279984, 1918147],[1140, 228, 1],[10563, 3507, 653, 1041, 68, 19, -224, -175, -233, -1154, -1318, -2043, -667, -85, 1799, 500, 2253, 3064, 1311, 1369, 6266, 6998; -11, -5, -3, -5, 0, 2, 5, 3, 4, 21, 25, 41, 15, 5, 15, 2, 6, 5, 1, 0, -3, -5; -42408, -14080, -2622, -4180, -273, -76, 900, 703, 936, 4636, 5295, 8208, 2680, 342, -7220, -2007, -9044, -12300, -5263, -5496, -25156, -28095],138,48]
%
% END OF LATTICE L139.48
%
\hbox{}\par\smallskip%
%
% LATTICE L139.49
%
\leavevmode%
\llap{\spacer}%
${L_{139.49}}=\pdualwitharg{2.3.5}({L_{139.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-78117360, -3586440, 1231200; -3586440, -164280, 56400; 1231200, 56400, -19363],[2280, 120, 1],[173, 208, 6, 1, 3, -1, -5, 0, 31, 15, 83, 22, 39, 7, 12, 244, 105, 499, 26, 957, 219, 905; 122519, 147345, 4267, 717, 2185, -706, -3648, -12, 21242, 10412, 57798, 15361, 27303, 4925, 8475, 172463, 74242, 352944, 18396, 677350, 155044, 640794; 367860, 442396, 12810, 2152, 6555, -2120, -10944, -35, 63840, 31280, 173622, 46140, 82004, 14790, 25448, 517845, 222920, 1059744, 55235, 2033760, 465520, 1923978],138,49]
%
% END OF LATTICE L139.49
%
\hbox{}\par\smallskip%
%
% LATTICE L139.50
%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38760, 2280, -4560; 2280, 1680, 4320; -4560, 4320, 8513],[2280, 120, 1],[4207, 4951, 97, 1, -142, -20, 226, 72, 2832, 974, 2422, 1163, 1851, 261, 357, 13746, 2884, 13378, 1360, 24376, 5470, 11182; 37779, 44460, 871, 9, -1273, -179, 2033, 647, 25441, 8749, 21755, 10446, 16625, 2344, 3206, 123443, 25899, 120137, 12213, 218899, 49121, 100415; -16920, -19912, -390, -4, 570, 80, -912, -290, -11400, -3920, -9747, -4680, -7448, -1050, -1436, -55290, -11600, -53808, -5470, -98040, -22000, -44973],138,50]
%
% END OF LATTICE L139.50
%
\hbox{}\par\smallskip%
%
% LATTICE L139.51
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[678849380022570, 353070296163690, 174724747760580; 353070296163690, 183632242588140, 90874530130530; 174724747760580, 90874530130530, 44971297578533],[570, 570, 1],[189, 55, 472, 1837, 4445, 19124, 5018, 10411, 10588, 47682, 10567, 39441, 2587, 2170, 805, 839, 2012, 266, 157, -20, -126, 1; -1298483, -360070, -3028866, -11475295, -27691944, -119015459, -31198273, -64667850, -65704129, -295689855, -65505383, -244371112, -16017903, -13349107, -4902678, -5064002, -12008325, -1557673, -844042, 192237, 886341, -6509; 2623140, 727388, 6118665, 23181254, 55940475, 240422960, 63023568, 130635355, 132728700, 597322000, 132327171, 493653060, 32357692, 26966415, 9903826, 10229685, 24257680, 3146592, 1704965, -388380, -1790560, 13149],138,51]
%
% END OF LATTICE L139.51
%
\hbox{}\par\smallskip%
%
% LATTICE L139.52
%
\leavevmode%
\llap{\spacer}%
${L_{139.52}}=\pdualwitharg{3.5.19}(\main({L_{139.2}}))$%
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{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1140, 216600, -1140; 216600, -37652205, 198075; -1140, 198075, -1042],[1140, 285, 1],[2407, 153, 179, 18, 19, 0, -1, -1, 9, 87, 103, 263, 25, 177, 160, 693, 712, 179, 1425, 347, 1515, 1321; 2262, 144, 172, 18, 20, 1, -1, -4, 7, 73, 88, 228, 22, 160, 148, 644, 663, 167, 1332, 325, 1421, 1240; 427215, 27197, 32490, 3401, 3780, 190, -189, -760, 1320, 13775, 16608, 43035, 4153, 30210, 27949, 121620, 125210, 31539, 251560, 61380, 268375, 234192],138,52]
%
% END OF LATTICE L139.52
%
\hbox{}\par\smallskip%
%
% LATTICE L139.53
%
\leavevmode%
\llap{\spacer}%
${L_{139.53}}=\pdualwitharg{2.19}({L_{139.1}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-185594280, -17150160, 129960; -17150160, -1584296, 12008; 129960, 12008, -91],[2280, 152, 1],[457, 446, 23, 113, 7, 43, 7, -1, -1, -1, 2, 8, 19, 8, 352, 55, 701, 303, 373, 125, 1675, 308; 2016, 1965, 100, 480, 29, 171, 25, -9, -5, -3, 10, 39, 90, 37, 1575, 245, 3117, 1345, 1653, 553, 7401, 1360; 918232, 895800, 46018, 224580, 13814, 83904, 13280, -2622, -2088, -1824, 4175, 16568, 39000, 16302, 710220, 110826, 1411776, 609920, 750462, 251368, 3367104, 619025],138,53]
%
% END OF LATTICE L139.53
%
\hbox{}\par\smallskip%
%
% LATTICE L139.54
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-432757680, 72255480, 35613600; 72255480, -12064088, -5946392; 35613600, -5946392, -2930407],[2280, 152, 1],[250077, 243844, 12460, 60307, 1840, 22073, 3393, -427, -567, 1, 2551, 4818, 11051, 4533, 194588, 15153, 385759, 166537, 102385, 68539, 917621, 337309; 1250811, 1219635, 62321, 301635, 9203, 110400, 16970, -2136, -2836, 6, 12760, 24099, 55275, 22673, 973275, 75791, 1929456, 832970, 512100, 342812, 4589670, 1687120; 501068, 488580, 24966, 120840, 3687, 44232, 6800, -855, -1136, 0, 5110, 9652, 22140, 9082, 389880, 30361, 772920, 333680, 205143, 137328, 1838592, 675850],138,54]
%
% END OF LATTICE L139.54
%
\hbox{}\par\smallskip%
%
% LATTICE L139.55
%
\leavevmode%
\llap{\spacer}%
${L_{139.55}}=\pdualwitharg{2.5.19}(\main({L_{139.2}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}19\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[350633220, 1211820, -86446200; 1211820, 760, -301340; -86446200, -301340, 21310781],[1140, 380, 1],[-2098890, -413996, -118388, -135590, -24068, -177511, -8095, -16118, 289, 16189, 795, 0, -5782, -21538, -284188, -115858, -1501601, -131831, -823438, -280141, -3794191, -280647; 6325979, 1247769, 356817, 408663, 72540, 535011, 24398, 48579, -871, -48792, -2396, 1, 17427, 64915, 856533, 349192, 4525773, 397334, 2481813, 844335, 11435556, 845860; -8424600, -1661712, -475190, -544236, -96605, -712500, -32492, -64695, 1160, 64980, 3191, 0, -23208, -86450, -1140684, -465035, -6027180, -529148, -3305145, -1124440, -15229260, -1126471],138,55]
%
% END OF LATTICE L139.55
%
\hbox{}\par\smallskip%
%
% LATTICE L139.56
%
\leavevmode%
\llap{\spacer}%
${L_{139.56}}=\pdualwitharg{3.5.19}({L_{139.2}})$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-298618440, -37544760, 394440; -37544760, -4719885, 49590; 394440, 49590, -521],[2280, 285, 1],[2407, 153, 179, 9, 19, 0, -1, -1, 9, 87, 103, 263, 25, 177, 80, 693, 712, 179, 1425, 347, 1515, 1321; 11890, 756, 888, 45, 96, 1, -5, -8, 43, 421, 500, 1280, 122, 868, 394, 3416, 3511, 883, 7032, 1713, 7481, 6524; 2953740, 187774, 220020, 11096, 23520, 95, -1233, -1520, 10905, 105925, 125556, 320910, 30536, 216600, 98059, 849720, 873145, 219543, 1748000, 425715, 1858865, 1620924],138,56]
%
% END OF LATTICE L139.56
%
\hbox{}\par\smallskip%
%
% LATTICE L139.57
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2280, 216600, -2280; 216600, -20152635, 212040; -2280, 212040, -2231],[2280, 285, 1],[2486, 158, 369, 37, 39, 0, -2, -1, 19, 182, 215, 274, 26, 367, 331, 1433, 1472, 370, 2945, 717, 3130, 2729; 1131, 72, 172, 18, 20, 1, -1, -4, 7, 73, 88, 114, 11, 160, 148, 644, 663, 167, 1332, 325, 1421, 1240; 104880, 6677, 15960, 1672, 1860, 95, -93, -380, 645, 6745, 8136, 10545, 1018, 14820, 13718, 59700, 61465, 15483, 123500, 30135, 131765, 114984],138,57]
%
% END OF LATTICE L139.57
%
\hbox{}\par\smallskip%
%
% LATTICE L139.58
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L139.58
%
\hbox{}\par\smallskip%
%
% LATTICE L139.59
%
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${L_{139.59}}=\pdualwitharg{2.3.19}({L_{139.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-825360, 6828600, -45600; 6828600, -47720856, 318744; -45600, 318744, -2129],[2280, 456, 1],[-90, -29, -4, -2, 1, 7, 11, 5, 5, 21, 11, 15, 4, -1, -23, -11, -47, -61, -25, -25, -111, -61; 8975, 2885, 387, 185, -96, -664, -1030, -462, -452, -1854, -955, -1265, -315, 127, 2385, 1124, 4776, 6170, 2518, 2508, 11106, 6095; 1345656, 432560, 58026, 27740, -14394, -99560, -154440, -69274, -67776, -278008, -143205, -189696, -47240, 19038, 357580, 168522, 716072, 925080, 377530, 376032, 1665160, 913845],138,59]
%
% END OF LATTICE L139.59
%
\hbox{}\par\smallskip%
%
% LATTICE L139.60
%
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\llap{\spacer}%
${L_{139.60}}=\pdualwitharg{2.3.5.19}(\main({L_{139.2}}))$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[111720, -20685300, -5152800; -20685300, 3830024940, 954076260; -5152800, 954076260, 237664643],[1140, 1140, 1],[-95861, -6077, -3423, -659, -180, -63, -2, -127, -551, -4504, -1276, -12559, -1151, -3805, -6569, -7048, -28849, -7226, -14329, -13905, -60548, -13180; -188512, -11946, -6692, -1272, -341, -95, -3, -261, -1109, -9017, -2550, -25058, -2292, -7546, -12966, -13897, -56857, -14235, -28215, -27367, -119125, -25926; 754680, 47824, 26790, 5092, 1365, 380, 12, 1045, 4440, 36100, 10209, 100320, 9176, 30210, 51908, 55635, 227620, 56988, 112955, 109560, 476900, 103791],138,60]
%
% END OF LATTICE L139.60
%
\hbox{}\par\smallskip%
%
% LATTICE L139.61
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4560, -2280, 0; -2280, -9183080, 37240; 0, 37240, -151],[2280, 760, 1],[19, 8, 15, 107, 81, 1019, 87, 529, 175, 2323, 85, 626, 121, 28, 22, 5, 13, -1, -13, -3, -1, 1; 37, 15, 27, 177, 132, 1650, 140, 846, 278, 3672, 134, 983, 189, 41, 27, 4, -18, -4, -30, -6, 0, 2; 9120, 3696, 6650, 43548, 32470, 405840, 34432, 208050, 68360, 902880, 32947, 241680, 46464, 10070, 6612, 970, -4560, -992, -7410, -1480, 0, 493],138,61]
%
% END OF LATTICE L139.61
%
\hbox{}\par\smallskip%
%
% LATTICE L139.62
%
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\llap{\spacer}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4560, -823080, -410400; -823080, 141322760, 70465680; -410400, 70465680, 35135261],[2280, 760, 1],[9, 4, 8, 67, 26, 661, 57, 175, 117, 1565, 115, 426, 83, 21, 20, 3, 35, 1, -1, -1, 1, 1; -3221, -1071, -1421, -909, 247, 12510, 1588, 6396, 5346, 81864, 6422, 25961, 5619, 2937, 5457, 1269, 23874, 1444, 2700, 718, 0, -358; 6460, 2148, 2850, 1824, -495, -25080, -3184, -12825, -10720, -164160, -12878, -52060, -11268, -5890, -10944, -2545, -47880, -2896, -5415, -1440, 0, 718],138,62]
%
% END OF LATTICE L139.62
%
\hbox{}\par\smallskip%
%
% LATTICE L139.63
%
\leavevmode%
\llap{\spacer}%
${L_{139.63}}=\pdualwitharg{2.3.5.19}({L_{139.2}})$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-88049040, -14243160, 4799400; -14243160, -2300520, 775200; 4799400, 775200, -261217],[2280, 2280, 1],[1118, 71, 41, 8, 2, -1, -1, 0, 9, 85, 25, 127, 12, 42, 75, 81, 665, 167, 166, 323, 1409, 307; 624999, 39699, 22987, 4509, 1135, -506, -568, -32, 4918, 46852, 13810, 70281, 6653, 23365, 41843, 45217, 371322, 93272, 92736, 180490, 787484, 171598; 1875300, 119116, 68970, 13528, 3405, -1520, -1704, -95, 14760, 140600, 41442, 210900, 19964, 70110, 125552, 135675, 1114160, 279864, 278255, 541560, 2362840, 514878],138,63]
%
% END OF LATTICE L139.63
%
\hbox{}\par\smallskip%
%
% LATTICE L139.64
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2280\\0\\-2280\\-20520\\-52440\\-230280\\-61560\\-129960\\-134520\\-613320\\-68400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-1216\\-3240\\-14440\\-3912\\-8360\\-8760\\-40280\\-4512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2280\\0\\1140\\0\\-1140\\-6840\\-2280\\-5700\\-6840\\-34200\\-3990
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20520\\-1216\\0\\152\\0\\-760\\-456\\-1520\\-2280\\-12920\\-1596
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52440\\-3240\\-1140\\0\\60\\0\\-240\\-1140\\-2160\\-13680\\-1770
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-230280\\-14440\\-6840\\-760\\0\\1520\\0\\-1520\\-4560\\-33440\\-4560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61560\\-3912\\-2280\\-456\\-240\\0\\48\\0\\-480\\-4560\\-672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-129960\\-8360\\-5700\\-1520\\-1140\\-1520\\0\\380\\0\\-3040\\-570
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-134520\\-8760\\-6840\\-2280\\-2160\\-4560\\-480\\0\\240\\0\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-613320\\-40280\\-34200\\-12920\\-13680\\-33440\\-4560\\-3040\\0\\1520\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68400\\-4512\\-3990\\-1596\\-1770\\-4560\\-672\\-570\\-120\\0\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-515280\\-34200\\-31920\\-13680\\-15960\\-43320\\-6840\\-6840\\-2280\\-2280\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34200\\-2288\\-2280\\-1064\\-1320\\-3800\\-648\\-760\\-360\\-760\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31920\\-2280\\-3420\\-2280\\-3420\\-11400\\-2280\\-3420\\-2280\\-6840\\-570
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13680\\-1064\\-2280\\-2432\\-4560\\-17480\\-4104\\-7600\\-6840\\-28120\\-2964
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15960\\-1320\\-3420\\-4560\\-9180\\-36480\\-8880\\-17100\\-16080\\-68400\\-7350
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43320\\-3800\\-11400\\-17480\\-36480\\-147440\\-36480\\-71440\\-68400\\-294880\\-31920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6840\\-648\\-2280\\-4104\\-8880\\-36480\\-9168\\-18240\\-17760\\-77520\\-8448
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6840\\-760\\-3420\\-7600\\-17100\\-71440\\-18240\\-36860\\-36480\\-161120\\-17670
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2280\\-360\\-2280\\-6840\\-16080\\-68400\\-17760\\-36480\\-36720\\-164160\\-18120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2280\\-760\\-6840\\-28120\\-68400\\-294880\\-77520\\-161120\\-164160\\-740240\\-82080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-570\\-2964\\-7350\\-31920\\-8448\\-17670\\-18120\\-82080\\-9123
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-108587280, -80791800, -40253400; -80791800, -60107640, -29947800; -40253400, -29947800, -14921077],[2280, 2280, 1],[1131, 72, 43, 9, 5, 1, -1, -1, 7, 73, 11, 114, 11, 40, 74, 161, 663, 167, 333, 325, 1421, 155; -1065931, -67809, -40101, -8229, -4446, -380, 946, 664, -7184, -72422, -10826, -111479, -10685, -38393, -70265, -152542, -627572, -157934, -314640, -306792, -1340470, -146162; 2136360, 135904, 80370, 16492, 8910, 760, -1896, -1330, 14400, 145160, 21699, 223440, 21416, 76950, 140828, 305730, 1257800, 316536, 630610, 614880, 2686600, 292941],138,64]
%
% END OF LATTICE L139.64
%

\medskip%
%------------------------------WEYL-GROUP-W140--------------------------------
%
\leavevmode\llap{}%
$W_{140}$%
\qquad\llap{4} lattices, $\chi=6$%
\hfill%
$\infty\slashtwo\infty|\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L140.1
%
\leavevmode%
\llap{\spacer}%
${L_{140.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}4\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty a}{4}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{4,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-16\\12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-5\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\4\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-44, -16, 12; -16, -5, 4; 12, 4, -3],[4, 1, 1],[1, -1, 0; 2, 0, -2; 6, -4, -3],139,1]
%
% END OF LATTICE L140.1
%
\hbox{}\par\smallskip%
%
% LATTICE L140.2
%
\leavevmode%
\llap{\spacer}%
${L_{140.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{1,0}{\infty a}{8}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{4,1}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-8\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[8, 0, 0; 0, 0, 1; 0, 1, -6],[8, 1, 1],[-1, 1, 0; 4, 0, -4; 0, 0, -1],139,2]
%
% END OF LATTICE L140.2
%
\hbox{}\par\smallskip%
%
% LATTICE L140.3
%
\leavevmode%
\llap{\spacer}%
${L_{140.3}}=\pdualwitharg{2}({L_{140.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{0}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty z}{1}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{4,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\36\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12, 16, 0; 16, 36, 4; 0, 4, 1],[4, 4, 1],[-1, 0, -2; 1, 0, 1; -2, -1, -4],139,3]
%
% END OF LATTICE L140.3
%
\hbox{}\par\smallskip%
%
% LATTICE L140.4
%
\leavevmode%
\llap{\spacer}%
${L_{140.4}}=\pdualwitharg{2}({L_{140.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{{\rm II}}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty z}{1}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{8,7}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\56\\104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
104\\-8\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-720, 56, 104; 56, 0, -8; 104, -8, -15],[8, 8, 1],[2, -1, -1; -1, 0, 2; 14, -7, -8],139,4]
%
% END OF LATTICE L140.4
%

\medskip%
%------------------------------WEYL-GROUP-W141--------------------------------
%
\leavevmode\llap{}%
$W_{141}$%
\qquad\llap{17} lattices, $\chi=6$%
\hfill%
$\slashtwo2\slashinfty2\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L141.1
%
\leavevmode%
\llap{\spacer}%
${L_{141.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{1,0}{\infty b}{8}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-696\\72\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-7\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-9\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-18\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-16\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\40\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-8\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-696, 72, 80; 72, -7, -9; 80, -9, -8],[8, 1, 1],[1, -3, -3, 7; 5, -18, -16, 40; 4, -10, -12, 24],140,1]
%
% END OF LATTICE L141.1
%
\hbox{}\par\smallskip%
%
% LATTICE L141.2
%
\leavevmode%
\llap{\spacer}%
${L_{141.2}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{3}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{8,3}{\infty}{8}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-848\\160\\32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
160\\-30\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-6\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\2\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-4\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-848, 160, 32; 160, -30, -6; 32, -6, -1],[16, 2, 1],[-1, -1, 0, 1; -4, -6, -1, 6; -8, 2, 4, -4],140,2]
%
% END OF LATTICE L141.2
%
\hbox{}\par\smallskip%
%
% LATTICE L141.3
%
\leavevmode%
\llap{\spacer}%
${L_{141.3}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{7}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{8,7}{\infty}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1296\\208\\96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\-30\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\-16\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\2\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-4\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1296, 208, 96; 208, -30, -16; 96, -16, -7],[16, 2, 1],[-3, 1, 2, -3; -4, 2, 3, -6; -32, 9, 20, -28],140,3]
%
% END OF LATTICE L141.3
%
\hbox{}\par\smallskip%
%
% LATTICE L141.4
%
\leavevmode%
\llap{\spacer}%
${L_{141.4}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{5}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{8,3}{\infty z}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-304\\32\\32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-2\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-4\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\8\\-4
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-4\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-304, 32, 32; 32, -2, -4; 32, -4, -3],[16, 2, 1],[1, -1, -1, 1; 0, -4, -2, 3; 8, -6, -8, 6],140,4]
%
% END OF LATTICE L141.4
%
\hbox{}\par\smallskip%
%
% LATTICE L141.5
%
\leavevmode%
\llap{\spacer}%
${L_{141.5}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{1}{1}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${64}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{32}\above{1pt}{1pt}{16,7}{\infty z}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
shares genus with {$ {L_{141.6}}$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17344\\512\\512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
512\\-8\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
512\\-16\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\7\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\0\\-64\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\0\\32\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-16\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-17344, 512, 512; 512, -8, -16; 512, -16, -15],[64, 8, 1],[1, -1, -1, 2; 0, -4, -2, 7; 32, -30, -32, 60],140,5]
%
% END OF LATTICE L141.5
%
\hbox{}\par\smallskip%
%
% LATTICE L141.6
%
\leavevmode%
\llap{\spacer}%
${L_{141.6}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${64}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{32}\above{1pt}{1pt}{16,15}{\infty z}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with {$ {L_{141.5}}$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-266176\\2048\\4096
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2048\\-8\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4096\\-32\\-63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\15\\252
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\0\\-64\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\0\\32\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-16\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-266176, 2048, 4096; 2048, -8, -32; 4096, -32, -63],[64, 8, 1],[1, -1, -1, 4; 0, -4, -2, 15; 64, -63, -64, 252],140,6]
%
% END OF LATTICE L141.6
%
\hbox{}\par\smallskip%
%
% LATTICE L141.7
%
\leavevmode%
\llap{\spacer}%
${L_{141.7}}=\pfillwitharg{2}({L_{141.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{4,1}{\infty}{2}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-2\\0
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\kern3pt%
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0\\4\\0\\-4
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-2\\0\\2\\-2
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0\\-4\\-2\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[4, 0, 0; 0, -2, -2; 0, -2, -1],[4, 2, 1],[0, -1, 0, 1; -1, 0, 1, -1; 1, 0, -2, 0],140,7]
%
% END OF LATTICE L141.7
%
\hbox{}\par\smallskip%
%
% LATTICE L141.8
%
\leavevmode%
\llap{\spacer}%
${L_{141.8}}=\main({L_{141.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{7}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-424\\32\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-2\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-6\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1\\1\\-6
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\HardButStrongLineBreak\kern3pt%
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-1\\-4\\-4
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\HardButStrongLineBreak\kern3pt%
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1\\-2\\6
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-4\\0
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0\\8\\0\\-4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-424, 32, 80; 32, -2, -6; 80, -6, -15],[8, 2, 1],[-1, -1, 1, 0; 1, -4, -2, 2; -6, -4, 6, -1],140,8]
%
% END OF LATTICE L141.8
%
\hbox{}\par\smallskip%
%
% LATTICE L141.9
%
\leavevmode%
\llap{\spacer}%
${L_{141.9}}=\main({L_{141.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-6\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\0\\0
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-1\\1\\-4
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\HardButStrongLineBreak\kern3pt%
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-1\\0\\-2
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\3
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-8\\0
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0\\2\\0\\-2
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-8\\0\\4\\-2
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0\\-2\\-2\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[8, 8, 0; 8, -6, -4; 0, -4, -1],[8, 2, 1],[1, -1, -1, 1; 0, 1, 0, -1; 0, -4, -2, 3],140,9]
%
% END OF LATTICE L141.9
%
\hbox{}\par\smallskip%
%
% LATTICE L141.10
%
\leavevmode%
\llap{\spacer}%
${L_{141.10}}=\pfillwitharg{2}({L_{141.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{8,7}{\infty}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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0\\-1\\1
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\HardButStrongLineBreak\kern3pt%
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1\\-1\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-4
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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16\\0\\-16\\0
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0\\1\\0\\-4
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0\\-4\\-8\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[16, 0, 0; 0, -8, -8; 0, -8, -7],[16, 8, 1],[-1, 0, 1, 0; 0, -1, -1, 3; 0, 1, 0, -4],140,10]
%
% END OF LATTICE L141.10
%
\hbox{}\par\smallskip%
%
% LATTICE L141.11
%
\leavevmode%
\llap{\spacer}%
${L_{141.11}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{141.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{8,7}{\infty z}{2}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\7\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\0
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0\\1\\0\\-2
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-16\\0\\8\\-4
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0\\-2\\-4\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16, 0, 16; 0, -2, -8; 16, -8, -15],[16, 2, 1],[1, -1, -1, 2; 0, -4, -2, 7; 0, 1, 0, -2],140,11]
%
% END OF LATTICE L141.11
%
\hbox{}\par\smallskip%
%
% LATTICE L141.12
%
\leavevmode%
\llap{\spacer}%
${L_{141.12}}=\pdualwitharg{2}(\main({L_{141.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{2,1}{\infty b}{8}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-4
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\0\\-8\\0
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0\\4\\0\\-8
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-8\\0\\8\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-8\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8, 0, 0; 0, -12, -4; 0, -4, -1],[8, 4, 1],[0, -1, -1, 1; 1, 0, -2, 0; -4, -2, 4, 0],140,12]
%
% END OF LATTICE L141.12
%
\hbox{}\par\smallskip%
%
% LATTICE L141.13
%
\leavevmode%
\llap{\spacer}%
${L_{141.13}}=\pdualwitharg{2}(\main({L_{141.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{4,3}{\infty}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\16\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-12\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-2\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-8\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8, 16, 8; 16, -12, -12; 8, -12, -7],[8, 4, 1],[-2, 1, 5, -3; 1, 0, -2, 0; -4, 1, 8, -4],140,13]
%
% END OF LATTICE L141.13
%
\hbox{}\par\smallskip%
%
% LATTICE L141.14
%
\leavevmode%
\llap{\spacer}%
${L_{141.14}}=\pdualwitharg{2}({L_{141.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{0}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{2,1}{\infty z}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-152\\-256\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-256\\-424\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\80\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-2\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-152, -256, 48; -256, -424, 80; 48, 80, -15],[8, 8, 1],[1, -2, -1, 1; -2, -1, 1, 0; -8, -12, 2, 3],140,14]
%
% END OF LATTICE L141.14
%
\hbox{}\par\smallskip%
%
% LATTICE L141.15
%
\leavevmode%
\llap{\spacer}%
${L_{141.15}}=\pdualwitharg{2}({L_{141.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{8,1}{\infty}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-32\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-16\\-8\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16, -32, 0; -32, 72, 0; 0, 0, -1],[16, 8, 1],[-2, 1, 2, -3; -1, 0, 1, -1; -2, 0, 0, -4],140,15]
%
% END OF LATTICE L141.15
%
\hbox{}\par\smallskip%
%
% LATTICE L141.16
%
\leavevmode%
\llap{\spacer}%
${L_{141.16}}=\pdualwitharg{2}({L_{141.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{8,5}{\infty}{8}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-816\\-96\\64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\8\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\8\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-16\\-8\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-816, -96, 64; -96, 8, 8; 64, 8, -5],[16, 8, 1],[1, -3, -2, 3; -1, 0, 1, -1; 10, -40, -24, 36],140,16]
%
% END OF LATTICE L141.16
%
\hbox{}\par\smallskip%
%
% LATTICE L141.17
%
\leavevmode%
\llap{\spacer}%
${L_{141.17}}=\pdualwitharg{2}({L_{141.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{8,3}{\infty}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3888\\224\\272
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
224\\-8\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
272\\-16\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\3\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-8\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3888, 224, 272; 224, -8, -16; 272, -16, -19],[16, 8, 1],[-3, 1, 3, -2; -4, 0, 3, -1; -40, 14, 40, -28],140,17]
%
% END OF LATTICE L141.17
%

\medskip%
%------------------------------WEYL-GROUP-W142--------------------------------
%
\leavevmode\llap{}%
$W_{142}$%
\qquad\llap{22} lattices, $\chi=9$%
\hfill%
$4\infty22$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L142.1
%
\leavevmode%
\llap{\spacer}%
${L_{142.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{3,2}{\infty b}{4}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1116\\288\\108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\-56\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108\\-25\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\6\\-37
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-8\\52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\-4\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-4\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1116, 288, 108; 288, -56, -25; 108, -25, -10],[36, 1, 1],[-2, -1, 3, 1; 6, 2, -8, 0; -37, -16, 52, 9],141,1]
%
% END OF LATTICE L142.1
%
\hbox{}\par\smallskip%
%
% LATTICE L142.2
%
\leavevmode%
\llap{\spacer}%
${L_{142.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{4}{2}\above{1pt}{1pt}{12,11}{\infty a}{8}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47592\\-16128\\1728
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16128\\-5465\\585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1728\\585\\-62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-42\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\32\\23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\68\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-54\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\-4\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-4\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-47592, -16128, 1728; -16128, -5465, 585; 1728, 585, -62],[72, 1, 1],[13, -10, -21, 17; -42, 32, 68, -54; -34, 23, 56, -36],141,2]
%
% END OF LATTICE L142.2
%
\hbox{}\par\smallskip%
%
% LATTICE L142.3
%
\leavevmode%
\llap{\spacer}%
${L_{142.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{12,11}{\infty b}{8}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38088\\576\\648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
576\\-7\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
648\\-10\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\53
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\9\\108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-4\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-4\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-38088, 576, 648; 576, -7, -10; 648, -10, -11],[72, 1, 1],[1, 0, -1, 2; 6, 1, -8, 9; 53, -1, -52, 108],141,3]
%
% END OF LATTICE L142.3
%
\hbox{}\par\smallskip%
%
% LATTICE L142.4
%
\leavevmode%
\llap{\spacer}%
${L_{142.4}}=\pfillwitharg{2}({L_{142.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{6,5}{\infty}{1}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1764\\171\\234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
171\\-14\\-23
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
234\\-23\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-3\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\4\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\2\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-9\\-54
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-1\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1764, 171, 234; 171, -14, -23; 234, -23, -31],[9, 1, 1],[-4, 4, 3, -8; -3, 4, 2, -9; -28, 27, 21, -54],141,4]
%
% END OF LATTICE L142.4
%
\hbox{}\par\smallskip%
%
% LATTICE L142.5
%
\leavevmode%
\llap{\spacer}%
${L_{142.5}}=\pfillwitharg{2}({L_{142.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{6,5}{\infty b}{2}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-22\\11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\11\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-2\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-2\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-234, 72, -36; 72, -22, 11; -36, 11, -5],[18, 1, 1],[1, 0, -1, 1; 3, 1, -4, 0; -1, 2, -2, -9],141,5]
%
% END OF LATTICE L142.5
%
\hbox{}\par\smallskip%
%
% LATTICE L142.6
%
\leavevmode%
\llap{\spacer}%
${L_{142.6}}=\main({L_{142.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{12,11}{\infty}{4}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65700\\1008\\1440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\-14\\-23
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1440\\-23\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-39\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\40\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\58\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-81\\-126
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-2\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-2\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-65700, 1008, 1440; 1008, -14, -23; 1440, -23, -31],[36, 1, 1],[-2, 2, 3, -4; -39, 40, 58, -81; -64, 63, 96, -126],141,6]
%
% END OF LATTICE L142.6
%
\hbox{}\par\smallskip%
%
% LATTICE L142.7
%
\leavevmode%
\llap{\spacer}%
${L_{142.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{142.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{4}\above{1pt}{1pt}{12,5}{\infty z}{1}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\54\\342
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\14\\28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
342\\28\\163
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-5\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-3\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-27\\108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\-2\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-2\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[720, 54, 342; 54, 14, 28; 342, 28, 163],[18, 2, 1],[-10, -1, -5, -49; -5, 0, -3, -27; 22, 2, 11, 108],141,7]
%
% END OF LATTICE L142.7
%
\hbox{}\par\smallskip%
%
% LATTICE L142.8
%
\leavevmode%
\llap{\spacer}%
${L_{142.8}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{142.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{}{4}{9}\above{1pt}{1pt}{6,1}{\infty}{9}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\558\\81
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
558\\4545\\657
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\657\\95
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\7\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-9\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\9\\-9\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-9\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[72, 558, 81; 558, 4545, 657; 81, 657, 95],[9, 9, 1],[2, 7, 2, 0; 1, 7, 5, 1; -9, -54, -36, -7],141,8]
%
% END OF LATTICE L142.8
%
\hbox{}\par\smallskip%
%
% LATTICE L142.9
%
\leavevmode%
\llap{\spacer}%
${L_{142.9}}=\pdualwitharg{2}(\main({L_{142.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{4}{4}\above{1pt}{1pt}{12,5}{\infty}{4}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
662328\\-52092\\144540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52092\\4100\\-11368
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144540\\-11368\\31543
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\2\\-68
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\7\\-332
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\2\\-178
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\0\\-252
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-4\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\-4\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-4\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[662328, -52092, 144540; -52092, 4100, -11368; 144540, -11368, 31543],[36, 4, 1],[15, 73, 39, 55; 2, 7, 2, 0; -68, -332, -178, -252],141,9]
%
% END OF LATTICE L142.9
%
\hbox{}\par\smallskip%
%
% LATTICE L142.10
%
\leavevmode%
\llap{\spacer}%
${L_{142.10}}=\pdualwitharg{2}({L_{142.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{6,5}{\infty z}{1}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
110448\\2484\\23652
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2484\\56\\532
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23652\\532\\5065
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\13\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\63\\-72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-4\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\-2\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-2\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[110448, 2484, 23652; 2484, 56, 532; 23652, 532, 5065],[36, 4, 1],[4, 3, 1, 14; 13, 0, 3, 63; -20, -14, -5, -72],141,10]
%
% END OF LATTICE L142.10
%
\hbox{}\par\smallskip%
%
% LATTICE L142.11
%
\leavevmode%
\llap{\spacer}%
${L_{142.11}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{142.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{4}{18}\above{1pt}{1pt}{6,1}{\infty b}{18}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
738\\18\\306
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\18\\9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
306\\9\\127
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\2\\-27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\2\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\1\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-9\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\18\\-18\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-18\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[738, 18, 306; 18, 18, 9; 306, 9, 127],[18, 9, 1],[11, 0, 15, 7; 2, 1, 2, 1; -27, 0, -36, -17],141,11]
%
% END OF LATTICE L142.11
%
\hbox{}\par\smallskip%
%
% LATTICE L142.12
%
\leavevmode%
\llap{\spacer}%
${L_{142.12}}=\pdualwitharg{3}(\main({L_{142.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{}{4}{9}\above{1pt}{1pt}{12,7}{\infty}{36}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-126\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-63\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-9\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\9\\-18\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-18\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-36, -72, -36; -72, -126, -63; -36, -63, -31],[36, 9, 1],[-2, 0, 3, 0; 1, 4, -2, -1; 0, -9, 0, 2],141,12]
%
% END OF LATTICE L142.12
%
\hbox{}\par\smallskip%
%
% LATTICE L142.13
%
\leavevmode%
\llap{\spacer}%
${L_{142.13}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{142.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{}{4}{36}\above{1pt}{1pt}{12,1}{\infty z}{9}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-2070\\-972
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2070\\-43164\\-20268
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-972\\-20268\\-9517
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\93\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-17\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-38\\81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\16\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-18\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\36\\-18\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-18\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-90, -2070, -972; -2070, -43164, -20268; -972, -20268, -9517],[18, 18, 1],[-1, 2, -1, -1; 93, -17, -38, 16; -198, 36, 81, -34],141,13]
%
% END OF LATTICE L142.13
%
\hbox{}\par\smallskip%
%
% LATTICE L142.14
%
\leavevmode%
\llap{\spacer}%
${L_{142.14}}=\pdualwitharg{3}({L_{142.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{*}{4}{36}\above{1pt}{1pt}{3,1}{\infty a}{36}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12924\\-1152\\720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1152\\-90\\63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\63\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\11\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-12\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-18\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\36\\-36\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-36\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12924, -1152, 720; -1152, -90, 63; 720, 63, -40],[36, 9, 1],[6, -1, -5, 1; 11, 0, -12, 1; 126, -18, -108, 20],141,14]
%
% END OF LATTICE L142.14
%
\hbox{}\par\smallskip%
%
% LATTICE L142.15
%
\leavevmode%
\llap{\spacer}%
${L_{142.15}}=\pdualwitharg{2}({L_{142.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{4}{16}\above{1pt}{1pt}{24,5}{\infty z}{4}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1368\\0\\648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
648\\0\\307
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\3\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-8\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\16\\-8\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-8\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1368, 0, 648; 0, -8, 0; 648, 0, 307],[72, 8, 1],[-2, 15, 1, -17; 1, 3, 0, 0; 4, -32, -2, 36],141,15]
%
% END OF LATTICE L142.15
%
\hbox{}\par\smallskip%
%
% LATTICE L142.16
%
\leavevmode%
\llap{\spacer}%
${L_{142.16}}=\pdualwitharg{2}({L_{142.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{4}{16}\above{1pt}{1pt}{24,17}{\infty z}{4}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\7\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-8\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\16\\-8\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-8\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[72, 0, 0; 0, 8, 8; 0, 8, 7],[72, 8, 1],[0, -1, 0, 1; 1, 7, 1, 0; 0, -8, -2, 0],141,16]
%
% END OF LATTICE L142.16
%
\hbox{}\par\smallskip%
%
% LATTICE L142.17
%
\leavevmode%
\llap{\spacer}%
${L_{142.17}}=\pdualwitharg{3}({L_{142.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{*}{4}{18}\above{1pt}{1pt}{12,7}{\infty b}{72}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75240\\3456\\2160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3456\\-153\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2160\\-99\\-62
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\-153
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-4\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\-18\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\18\\-36\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-36\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-75240, 3456, 2160; 3456, -153, -99; 2160, -99, -62],[72, 9, 1],[-3, -4, 5, 1; 2, 8, -4, -2; -108, -153, 180, 38],141,17]
%
% END OF LATTICE L142.17
%
\hbox{}\par\smallskip%
%
% LATTICE L142.18
%
\leavevmode%
\llap{\spacer}%
${L_{142.18}}=\pdualwitharg{3}({L_{142.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{4}{18}\above{1pt}{1pt}{12,7}{\infty a}{72}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11592\\1152\\-4032
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1152\\-99\\414
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4032\\414\\-1391
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\77\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-9\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-60\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\20\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-9\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\18\\-36\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-36\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11592, 1152, -4032; 1152, -99, 414; -4032, 414, -1391],[72, 9, 1],[39, -4, -31, 10; 77, -9, -60, 20; -90, 9, 72, -23],141,18]
%
% END OF LATTICE L142.18
%
\hbox{}\par\smallskip%
%
% LATTICE L142.19
%
\leavevmode%
\llap{\spacer}%
${L_{142.19}}=\pdualwitharg{2.3}(\main({L_{142.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{}{4}{36}\above{1pt}{1pt}{12,1}{\infty}{36}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\180\\36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\1656\\396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\396\\95
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-9\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-35\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-13\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-36\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\36\\-36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-36\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[180, 180, 36; 180, 1656, 396; 36, 396, 95],[36, 36, 1],[2, 7, 2, 0; -9, -35, -13, -1; 36, 144, 54, 4],141,19]
%
% END OF LATTICE L142.19
%
\hbox{}\par\smallskip%
%
% LATTICE L142.20
%
\leavevmode%
\llap{\spacer}%
${L_{142.20}}=\pdualwitharg{2.3}({L_{142.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{*}{4}{36}\above{1pt}{1pt}{6,1}{\infty z}{9}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1440\\-12348\\-2664
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12348\\107640\\23220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2664\\23220\\5009
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-85\\396
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-29\\135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-55\\256
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-36\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\36\\-18\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-18\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1440, -12348, -2664; -12348, 107640, 23220; -2664, 23220, 5009],[36, 36, 1],[4, 1, 1, 2; -85, 4, -29, -55; 396, -18, 135, 256],141,20]
%
% END OF LATTICE L142.20
%
\hbox{}\par\smallskip%
%
% LATTICE L142.21
%
\leavevmode%
\llap{\spacer}%
${L_{142.21}}=\pdualwitharg{2.3}({L_{142.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{*}{4}{144}\above{1pt}{1pt}{24,13}{\infty z}{36}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1872\\7560\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7560\\30168\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-576\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\2\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-72\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\144\\-72\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-72\\36\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\0\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1872, 7560, -144; 7560, 30168, -576; -144, -576, 11],[72, 72, 1],[1, 3, 0, 0; -1, 2, -1, -1; -36, 144, -54, -52],141,21]
%
% END OF LATTICE L142.21
%
\hbox{}\par\smallskip%
%
% LATTICE L142.22
%
\leavevmode%
\llap{\spacer}%
${L_{142.22}}=\pdualwitharg{2.3}({L_{142.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{}{4}{144}\above{1pt}{1pt}{24,1}{\infty z}{36}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1872\\2376\\792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2376\\2952\\1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
792\\1008\\335
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\2\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-1\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-1\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-72\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\144\\-72\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-72\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1872, 2376, 792; 2376, 2952, 1008; 792, 1008, 335],[72, 72, 1],[1, -33, 9, 8; -1, 2, -1, -1; 0, 72, -18, -16],141,22]
%
% END OF LATTICE L142.22
%

\medskip%
%------------------------------WEYL-GROUP-W143--------------------------------
%
\leavevmode\llap{}%
$W_{143}$%
\qquad\llap{22} lattices, $\chi=18$%
\hfill%
$2\infty\slashtwo\infty2|\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L143.1
%
\leavevmode%
\llap{\spacer}%
${L_{143.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{1,0}{\infty}{36}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{3,1}{\infty a}{4}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2556\\900\\144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
900\\-316\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-51\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\5\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\36\\108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-18\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\-36\\-36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-36\\36\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-36\\0\\4\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-36\\-4\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2556, 900, 144; 900, -316, -51; 144, -51, -8],[36, 1, 1],[3, 19, -1, -3, 1; 5, 36, 0, -6, 0; 21, 108, -18, -16, 16],142,1]
%
% END OF LATTICE L143.1
%
\hbox{}\par\smallskip%
%
% LATTICE L143.2
%
\leavevmode%
\llap{\spacer}%
${L_{143.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{4,1}{\infty z}{18}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{12,7}{\infty b}{8}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52200\\-25488\\576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25488\\-12445\\281
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
576\\281\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\252\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\108\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-52\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-18\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\72\\-36\\-36\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-36\\18\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-36\\0\\2\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-36\\-4\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-52200, -25488, 576; -25488, -12445, 281; 576, 281, -6],[72, 1, 1],[1, -121, -52, -1, 25; -2, 252, 108, 2, -52; 2, 180, 63, -3, -36],142,2]
%
% END OF LATTICE L143.2
%
\hbox{}\par\smallskip%
%
% LATTICE L143.3
%
\leavevmode%
\llap{\spacer}%
${L_{143.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{4,3}{\infty z}{18}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{12,7}{\infty a}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-334728\\3456\\1656
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3456\\-35\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1656\\-18\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-71\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\360\\252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\783\\585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\213\\161
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-72\\-52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-9\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\72\\-36\\-36\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-36\\18\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-36\\0\\2\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-36\\-4\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-334728, 3456, 1656; 3456, -35, -18; 1656, -18, -7],[72, 1, 1],[-1, 5, 11, 3, -1; -71, 360, 783, 213, -72; -54, 252, 585, 161, -52],142,3]
%
% END OF LATTICE L143.3
%
\hbox{}\par\smallskip%
%
% LATTICE L143.4
%
\leavevmode%
\llap{\spacer}%
${L_{143.4}}=\pfillwitharg{2}({L_{143.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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9\\0\\0
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\kern3pt%
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0\\-1\\-3
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\kern3pt%
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0\\-3\\-8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\5\\-3
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0\\-3\\1
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1\\18\\-9
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[9, 0, 0; 0, -1, -3; 0, -3, -8],[9, 1, 1],[1, 1, 0, -1, 1; 5, 0, -3, 0, 18; -3, -1, 1, 0, -9],142,4]
%
% END OF LATTICE L143.4
%
\hbox{}\par\smallskip%
%
% LATTICE L143.5
%
\leavevmode%
\llap{\spacer}%
${L_{143.5}}=\pfillwitharg{2}({L_{143.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{6,5}{\infty}{2}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{2,1}{\infty b}{18}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-90\\36\\-36
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\kern3pt%
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36\\-14\\15
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\kern3pt%
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-36\\15\\-13
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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3\\5\\-3
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3\\6\\-2
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-2\\-3\\2
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13\\18\\-18
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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1\\0\\-3\\-9\\0
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0\\2\\-2\\-18\\-18
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0\\-18\\-18\\-18\\18
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-90, 36, -36; 36, -14, 15; -36, 15, -13],[18, 1, 1],[3, 3, -2, -4, 13; 5, 6, -3, -9, 18; -3, -2, 2, 0, -18],142,5]
%
% END OF LATTICE L143.5
%
\hbox{}\par\smallskip%
%
% LATTICE L143.6
%
\leavevmode%
\llap{\spacer}%
${L_{143.6}}=\main({L_{143.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{2,1}{\infty z}{9}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{12,7}{\infty}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-9252\\360\\180
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\kern3pt%
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360\\-14\\-7
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180\\-7\\-3
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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0\\-1\\2
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5\\126\\0
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2\\54\\-9
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0\\1\\-3
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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2\\0\\-9\\-3\\0
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0\\36\\-18\\-18\\-36
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-9252, 360, 180; 360, -14, -7; 180, -7, -3],[36, 1, 1],[0, 5, 2, 0, -1; -1, 126, 54, 1, -26; 2, 0, -9, -3, 0],142,6]
%
% END OF LATTICE L143.6
%
\hbox{}\par\smallskip%
%
% LATTICE L143.7
%
\leavevmode%
\llap{\spacer}%
${L_{143.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{143.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{6,5}{\infty}{4}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{4,1}{\infty z}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3168\\198\\1530
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\kern3pt%
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198\\14\\96
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\kern3pt%
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1530\\96\\739
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-14\\-7\\30
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-8\\-3\\17
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-1\\0\\2
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-46\\-27\\99
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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2\\0\\-6\\-18\\0
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0\\1\\-2\\-18\\-9
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-6\\-2\\4\\0\\-18
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0\\-9\\-18\\-18\\9
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3168, 198, 1530; 198, 14, 96; 1530, 96, 739],[18, 2, 1],[-14, -8, -1, -25, -46; -7, -3, 0, -18, -27; 30, 17, 2, 54, 99],142,7]
%
% END OF LATTICE L143.7
%
\hbox{}\par\smallskip%
%
% LATTICE L143.8
%
\leavevmode%
\llap{\spacer}%
${L_{143.8}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{143.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{3,1}{\infty}{9}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{2,1}{\infty}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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72\\-27\\0
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\kern3pt%
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-27\\9\\0
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\kern3pt%
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0\\0\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\9\\9
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3\\8\\9
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0\\1\\0
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\HardButStrongLineBreak\kern3pt%
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0\\0\\-1
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1\\2\\1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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18\\0\\-27\\-9\\0
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0\\9\\-9\\-9\\-9
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-27\\-9\\9\\0\\-9
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0\\-9\\-9\\-1\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[72, -27, 0; -27, 9, 0; 0, 0, 1],[9, 9, 1],[4, 3, 0, 0, 1; 9, 8, 1, 0, 2; 9, 9, 0, -1, 1],142,8]
%
% END OF LATTICE L143.8
%
\hbox{}\par\smallskip%
%
% LATTICE L143.9
%
\leavevmode%
\llap{\spacer}%
${L_{143.9}}=\pdualwitharg{2}({L_{143.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{1,0}{\infty}{36}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{6,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2736\\-180\\504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\32\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\-24\\97
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\49\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\54\\-117
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\9\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\6\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\21\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-36\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\-18\\-18\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-18\\36\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-18\\0\\4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-18\\-2\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2736, -180, 504; -180, 32, -24; 504, -24, 97],[36, 4, 1],[23, 25, 4, 3, 10; 49, 54, 9, 6, 21; -108, -117, -18, -14, -47],142,9]
%
% END OF LATTICE L143.9
%
\hbox{}\par\smallskip%
%
% LATTICE L143.10
%
\leavevmode%
\llap{\spacer}%
${L_{143.10}}=\pdualwitharg{2}(\main({L_{143.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{1,0}{\infty b}{36}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{12,1}{\infty}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1656\\756\\396
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
756\\516\\188
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
396\\188\\95
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91\\18\\-414
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143\\27\\-648
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\7\\-172
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\2\\-50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-36\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\-36\\-36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-36\\36\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-36\\0\\4\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-36\\-4\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1656, 756, 396; 756, 516, 188; 396, 188, 95],[36, 4, 1],[-1, 91, 143, 38, 11; 0, 18, 27, 7, 2; 4, -414, -648, -172, -50],142,10]
%
% END OF LATTICE L143.10
%
\hbox{}\par\smallskip%
%
% LATTICE L143.11
%
\leavevmode%
\llap{\spacer}%
${L_{143.11}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{143.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{6,1}{\infty}{18}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{2,1}{\infty b}{2}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4626\\270\\1998
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
270\\18\\117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1998\\117\\863
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\6\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\8\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\1\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\4\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-27\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\-18\\-18\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-18\\18\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-18\\0\\2\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-18\\-2\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4626, 270, 1998; 270, 18, 117; 1998, 117, 863],[18, 9, 1],[19, 15, 0, 6, 17; 6, 8, 1, 1, 4; -45, -36, 0, -14, -40],142,11]
%
% END OF LATTICE L143.11
%
\hbox{}\par\smallskip%
%
% LATTICE L143.12
%
\leavevmode%
\llap{\spacer}%
${L_{143.12}}=\pdualwitharg{3}(\main({L_{143.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{2,1}{\infty z}{1}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{12,11}{\infty}{36}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6228\\-612\\468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-612\\-54\\45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
468\\45\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\12\\81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\6\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-9\\-27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-2\\-18\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-2\\1\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-18\\0\\9\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-18\\-18\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6228, -612, 468; -612, -54, 45; 468, 45, -35],[36, 9, 1],[-2, -1, 1, 5, 5; -7, -2, 3, 12, 6; -36, -16, 17, 81, 72],142,12]
%
% END OF LATTICE L143.12
%
\hbox{}\par\smallskip%
%
% LATTICE L143.13
%
\leavevmode%
\llap{\spacer}%
${L_{143.13}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{143.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{6,1}{\infty}{36}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-234\\-3294\\-1548
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3294\\-35244\\-16560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1548\\-16560\\-7781
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\347\\-738
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\165\\-351
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-237\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-47\\100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\87\\-185
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-54\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\-18\\-18\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-18\\36\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-18\\0\\4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-18\\-2\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-234, -3294, -1548; -3294, -35244, -16560; -1548, -16560, -7781],[18, 18, 1],[-3, -1, 2, 0, -1; 347, 165, -237, -47, 87; -738, -351, 504, 100, -185],142,13]
%
% END OF LATTICE L143.13
%
\hbox{}\par\smallskip%
%
% LATTICE L143.14
%
\leavevmode%
\llap{\spacer}%
${L_{143.14}}=\pdualwitharg{3}({L_{143.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{1,0}{\infty}{4}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{3,2}{\infty b}{36}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15012\\1476\\1188
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1476\\144\\117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1188\\117\\94
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\20\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\32\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-18\\-54\\0
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0\\4\\-4\\-36\\-36
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-18\\-4\\4\\0\\-36
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-54\\-36\\0\\36\\-36
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0\\-36\\-36\\-36\\36
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15012, 1476, 1188; 1476, 144, 117; 1188, 117, 94],[36, 9, 1],[-7, -3, 1, 3, -9; 20, 8, -2, -2, 32; 63, 28, -10, -36, 72],142,14]
%
% END OF LATTICE L143.14
%
\hbox{}\par\smallskip%
%
% LATTICE L143.15
%
\leavevmode%
\llap{\spacer}%
${L_{143.15}}=\pdualwitharg{2}({L_{143.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{2,1}{\infty a}{144}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{24,1}{\infty z}{4}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-113904\\-4968\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4968\\-216\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\16\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\21\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\81\\-162
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\18\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-20\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-72\\-24\\0
\end{tabular}\endgroup%
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0\\36\\-72\\-72\\-36
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-72\\-72\\144\\0\\-72
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-24\\-72\\0\\16\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-72\\-8\\4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-113904, -4968, 360; -4968, -216, 16; 360, 16, -1],[72, 8, 1],[-1, -4, -1, 1, 0; 21, 81, 18, -20, 1; -32, -162, -72, 40, 10],142,15]
%
% END OF LATTICE L143.15
%
\hbox{}\par\smallskip%
%
% LATTICE L143.16
%
\leavevmode%
\llap{\spacer}%
${L_{143.16}}=\pdualwitharg{2}({L_{143.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{2,1}{\infty b}{144}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{24,13}{\infty z}{4}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140904\\-1008\\2664
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1008\\24\\32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2664\\32\\-45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-227\\548
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-531\\1278
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\207\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\133\\-320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-113\\274
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-72\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\-72\\-72\\-36
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-72\\144\\0\\-72
\end{tabular}\endgroup%
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-24\\-72\\0\\16\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-72\\-8\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-140904, -1008, 2664; -1008, 24, 32; 2664, 32, -45],[72, 8, 1],[12, 28, -11, -7, 6; -227, -531, 207, 133, -113; 548, 1278, -504, -320, 274],142,16]
%
% END OF LATTICE L143.16
%
\hbox{}\par\smallskip%
%
% LATTICE L143.17
%
\leavevmode%
\llap{\spacer}%
${L_{143.17}}=\pdualwitharg{3}({L_{143.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{12,1}{\infty z}{18}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{4,3}{\infty a}{8}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12456\\-6840\\936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6840\\-3753\\513
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
936\\513\\-70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-14\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\12\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\24\\81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\6\\17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-54\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\72\\-36\\-36\\-72
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-36\\18\\0\\-36
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-36\\0\\2\\-4
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-36\\-4\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12456, -6840, 936; -6840, -3753, 513; 936, 513, -70],[72, 9, 1],[5, -1, -7, -2, 1; -14, 12, 24, 6, -4; -36, 72, 81, 17, -16],142,17]
%
% END OF LATTICE L143.17
%
\hbox{}\par\smallskip%
%
% LATTICE L143.18
%
\leavevmode%
\llap{\spacer}%
${L_{143.18}}=\pdualwitharg{3}({L_{143.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{12,7}{\infty z}{18}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{4,3}{\infty b}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78984\\1584\\792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1584\\-27\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
792\\-18\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-23\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\120\\252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\117\\261
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\23\\53
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\-52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-27\\-9\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\72\\-36\\-36\\-72
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-36\\18\\0\\-36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-36\\0\\2\\-4
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-36\\-4\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-78984, 1584, 792; 1584, -27, -18; 792, -18, -7],[72, 9, 1],[-1, 5, 5, 1, -1; -23, 120, 117, 23, -24; -54, 252, 261, 53, -52],142,18]
%
% END OF LATTICE L143.18
%
\hbox{}\par\smallskip%
%
% LATTICE L143.19
%
\leavevmode%
\llap{\spacer}%
${L_{143.19}}=\pdualwitharg{2.3}({L_{143.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{1,0}{\infty}{4}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{6,5}{\infty z}{9}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\-3924\\-900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3924\\32256\\7416
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\7416\\1705
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\214\\-936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-27\\118
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\103\\-450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\175\\-765
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\0\\-36\\-108\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2\\-18\\-9
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-2\\4\\0\\-18
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-18\\0\\36\\-18
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-18\\-18\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[216, -3924, -900; -3924, 32256, 7416; -900, 7416, 1705],[36, 36, 1],[-9, 0, 1, -4, -7; 214, -3, -27, 103, 175; -936, 13, 118, -450, -765],142,19]
%
% END OF LATTICE L143.19
%
\hbox{}\par\smallskip%
%
% LATTICE L143.20
%
\leavevmode%
\llap{\spacer}%
${L_{143.20}}=\pdualwitharg{2.3}(\main({L_{143.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{1,0}{\infty b}{4}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{12,5}{\infty}{36}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1044\\-4716\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4716\\22248\\4752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1008\\4752\\1015
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\15\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\16\\-76
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\83\\-396
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\79\\-378
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\0\\-36\\-108\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-4\\-36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-4\\4\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-36\\0\\36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-36\\-36\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1044, -4716, -1008; -4716, 22248, 4752; -1008, 4752, 1015],[36, 36, 1],[-2, 0, -1, -7, -8; 15, 3, 16, 83, 79; -72, -14, -76, -396, -378],142,20]
%
% END OF LATTICE L143.20
%
\hbox{}\par\smallskip%
%
% LATTICE L143.21
%
\leavevmode%
\llap{\spacer}%
${L_{143.21}}=\pdualwitharg{2.3}({L_{143.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{2,1}{\infty a}{16}\above{1pt}{1pt}{s}{2}{144}\above{1pt}{1pt}{24,17}{\infty z}{36}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-432\\-9576\\-4608
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9576\\-201240\\-96840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4608\\-96840\\-46601
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-277\\576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-51\\106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\50\\-104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\104\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-199\\414
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\0\\-72\\-216\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-8\\-72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-8\\16\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-72\\0\\144\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-72\\-72\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-432, -9576, -4608; -9576, -201240, -96840; -4608, -96840, -46601],[72, 72, 1],[-3, 0, 1, -1, -4; -277, -51, 50, 104, -199; 576, 106, -104, -216, 414],142,21]
%
% END OF LATTICE L143.21
%
\hbox{}\par\smallskip%
%
% LATTICE L143.22
%
\leavevmode%
\llap{\spacer}%
${L_{143.22}}=\pdualwitharg{2.3}({L_{143.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{2,1}{\infty b}{16}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{24,5}{\infty z}{36}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21096\\-24192\\-10872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24192\\-27720\\-12456
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10872\\-12456\\-5597
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-93\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-17\\34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\17\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\35\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-67\\126
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\0\\-72\\-216\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-8\\-72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-8\\16\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-72\\0\\144\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-72\\-72\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21096, -24192, -10872; -24192, -27720, -12456; -10872, -12456, -5597],[72, 72, 1],[14, 2, -3, -3, 12; -93, -17, 17, 35, -67; 180, 34, -32, -72, 126],142,22]
%
% END OF LATTICE L143.22
%

\medskip%
%------------------------------WEYL-GROUP-W144--------------------------------
%
\leavevmode\llap{}%
$W_{144}$%
\qquad\llap{16} lattices, $\chi=12$%
\hfill%
$2|2\infty|\infty\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L144.1
%
\leavevmode%
\llap{\spacer}%
${L_{144.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{12,7}{\infty}{12}\above{1pt}{1pt}{3,1}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-612\\-180\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-33\\-177
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-792\\-177\\-868
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\100\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-27\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-85\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-16\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-18\\-6\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-612, -180, -792; -180, -33, -177; -792, -177, -868],[36, 3, 1],[17, -4, -14, -3; 100, -27, -85, -16; -36, 9, 30, 6],143,1]
%
% END OF LATTICE L144.1
%
\hbox{}\par\smallskip%
%
% LATTICE L144.2
%
\leavevmode%
\llap{\spacer}%
${L_{144.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{3}%
\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{12,7}{\infty b}{24}\above{1pt}{1pt}{3,2}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19368\\-3672\\-1944
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3672\\-696\\-369
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1944\\-369\\-194
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-56\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-6\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\32\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\20\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\6\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-36\\-12\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-19368, -3672, -1944; -3672, -696, -369; -1944, -369, -194],[72, 3, 1],[13, 2, -7, -5; -56, -6, 32, 20; -24, -9, 9, 12],143,2]
%
% END OF LATTICE L144.2
%
\hbox{}\par\smallskip%
%
% LATTICE L144.3
%
\leavevmode%
\llap{\spacer}%
${L_{144.3}}=\pfillwitharg{3}({L_{144.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{2,1}{\infty}{12}\above{1pt}{1pt}{1,0}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-12\\-204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204\\-5\\-97
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\13\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\24\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-30\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-6\\-12\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-420, -12, -204; -12, 0, -5; -204, -5, -97],[12, 1, 1],[1, 2, 5, -5; 12, 13, 24, -30; -3, -5, -12, 12],143,3]
%
% END OF LATTICE L144.3
%
\hbox{}\par\smallskip%
%
% LATTICE L144.4
%
\leavevmode%
\llap{\spacer}%
${L_{144.4}}=\pfillwitharg{3}({L_{144.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{4,3}{\infty a}{24}\above{1pt}{1pt}{1,0}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21288\\-2232\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2232\\-234\\5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\5\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-48\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-19\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-9\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\48\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\6\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\-12\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-21288, -2232, 48; -2232, -234, 5; 48, 5, 0],[24, 1, 1],[5, 2, 1, -5; -48, -19, -9, 48; -24, -2, 12, 12],143,4]
%
% END OF LATTICE L144.4
%
\hbox{}\par\smallskip%
%
% LATTICE L144.5
%
\leavevmode%
\llap{\spacer}%
${L_{144.5}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{144.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{4,3}{\infty}{4}\above{1pt}{1pt}{1,0}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-96\\36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-81\\30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\30\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-4\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\1\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-6\\-2\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-108, -96, 36; -96, -81, 30; 36, 30, -11],[12, 3, 1],[1, 1, 0, -1; -4, -1, 2, 2; -8, 0, 5, 2],143,5]
%
% END OF LATTICE L144.5
%
\hbox{}\par\smallskip%
%
% LATTICE L144.6
%
\leavevmode%
\llap{\spacer}%
${L_{144.6}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{144.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{2,1}{\infty}{12}\above{1pt}{1pt}{1,0}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
372\\-108\\168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\48\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-56\\79
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\27\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\12\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\6\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\-6\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[372, -108, 168; -108, 48, -56; 168, -56, 79],[12, 4, 1],[-19, -9, -5, -2; 27, 12, 6, 3; 60, 28, 15, 6],143,6]
%
% END OF LATTICE L144.6
%
\hbox{}\par\smallskip%
%
% LATTICE L144.7
%
\leavevmode%
\llap{\spacer}%
${L_{144.7}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{144.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{1,0}{\infty}{8}\above{1pt}{1pt}{1,0}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1656\\24\\192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
192\\-3\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-8\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-12\\-8\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1656, 24, 192; 24, 0, -3; 192, -3, -22],[24, 3, 1],[0, -1, -1, 1; 4, -2, -8, 4; -1, -9, -8, 8],143,7]
%
% END OF LATTICE L144.7
%
\hbox{}\par\smallskip%
%
% LATTICE L144.8
%
\leavevmode%
\llap{\spacer}%
${L_{144.8}}=\pdualwitharg{3}({L_{144.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{12,11}{\infty}{12}\above{1pt}{1pt}{3,2}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1332\\144\\-324
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-15\\36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-324\\36\\-77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\20\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-14\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-6\\-6\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1332, 144, -324; 144, -15, 36; -324, 36, -77],[36, 3, 1],[5, 1, -1, -3; 20, 5, -2, -14; -12, -2, 3, 6],143,8]
%
% END OF LATTICE L144.8
%
\hbox{}\par\smallskip%
%
% LATTICE L144.9
%
\leavevmode%
\llap{\spacer}%
${L_{144.9}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{144.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{4,1}{\infty}{4}\above{1pt}{1pt}{1,0}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108\\312\\96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
312\\900\\276
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\276\\85
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-2\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-1\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\4\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-12\\-4\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[108, 312, 96; 312, 900, 276; 96, 276, 85],[12, 12, 1],[-1, -5, -4, -1; 0, -2, -1, 1; 1, 12, 8, -2],143,9]
%
% END OF LATTICE L144.9
%
\hbox{}\par\smallskip%
%
% LATTICE L144.10
%
\leavevmode%
\llap{\spacer}%
${L_{144.10}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{144.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{8,5}{\infty z}{12}\above{1pt}{1pt}{1,0}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1392\\-24\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\32\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\16\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-15\\33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-8\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\54\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-24\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\48\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-24\\-24\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1392, -24, 72; -24, 32, 16; 72, 16, 3],[24, 8, 1],[2, 1, -7, -1; -15, -8, 54, 9; 33, 16, -120, -18],143,10]
%
% END OF LATTICE L144.10
%
\hbox{}\par\smallskip%
%
% LATTICE L144.11
%
\leavevmode%
\llap{\spacer}%
${L_{144.11}}=\pdualwitharg{3}({L_{144.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{12,11}{\infty a}{24}\above{1pt}{1pt}{3,1}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-576\\288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-576\\-66\\45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\45\\-28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-56\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\17\\38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\71\\156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\6\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\-12\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-936, -576, 288; -576, -66, 45; 288, 45, -28],[72, 3, 1],[-3, 1, 4, -1; -56, 17, 71, -16; -120, 38, 156, -36],143,11]
%
% END OF LATTICE L144.11
%
\hbox{}\par\smallskip%
%
% LATTICE L144.12
%
\leavevmode%
\llap{\spacer}%
${L_{144.12}}=\pdualwitharg{2}({L_{144.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{12,1}{\infty}{12}\above{1pt}{1pt}{3,2}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252\\-2412\\-1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2412\\32016\\23664
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\23664\\17495
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-80\\111
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
175\\-519\\720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
117\\-346\\480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\13\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-36\\-12\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[252, -2412, -1800; -2412, 32016, 23664; -1800, 23664, 17495],[36, 12, 1],[27, 175, 117, -4; -80, -519, -346, 13; 111, 720, 480, -18],143,12]
%
% END OF LATTICE L144.12
%
\hbox{}\par\smallskip%
%
% LATTICE L144.13
%
\leavevmode%
\llap{\spacer}%
${L_{144.13}}=\pdualwitharg{2.3}({L_{144.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{12,5}{\infty}{12}\above{1pt}{1pt}{3,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2448\\-1188\\-1440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1188\\804\\864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1440\\864\\967
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-37\\51
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-55\\76
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-113\\156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-22\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-12\\-12\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2448, -1188, -1440; -1188, 804, 864; -1440, 864, 967],[36, 12, 1],[12, 18, 37, 7; -37, -55, -113, -22; 51, 76, 156, 30],143,13]
%
% END OF LATTICE L144.13
%
\hbox{}\par\smallskip%
%
% LATTICE L144.14
%
\leavevmode%
\llap{\spacer}%
${L_{144.14}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{144.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{1,0}{\infty}{4}\above{1pt}{1pt}{1,0}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6816\\-7848\\-3528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7848\\-9024\\-4056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3528\\-4056\\-1823
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\17\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\35\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-17\\34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\1\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-24\\-2\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6816, -7848, -3528; -7848, -9024, -4056; -3528, -4056, -1823],[24, 24, 1],[-3, -3, 1, 2; 17, 35, -4, -17; -32, -72, 7, 34],143,14]
%
% END OF LATTICE L144.14
%
\hbox{}\par\smallskip%
%
% LATTICE L144.15
%
\leavevmode%
\llap{\spacer}%
${L_{144.15}}=\pdualwitharg{2}({L_{144.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{3,1}{\infty}{12}\above{1pt}{1pt}{3,1}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80640\\-25848\\1728
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25848\\-8160\\552
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1728\\552\\-37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-7\\-432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-9\\-648
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\2\\123
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\5\\354
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\3\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-72\\-6\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-80640, -25848, 1728; -25848, -8160, 552; 1728, 552, -37],[72, 24, 1],[-7, -11, 2, 6; -7, -9, 2, 5; -432, -648, 123, 354],143,15]
%
% END OF LATTICE L144.15
%
\hbox{}\par\smallskip%
%
% LATTICE L144.16
%
\leavevmode%
\llap{\spacer}%
${L_{144.16}}=\pdualwitharg{2.3}({L_{144.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{3,2}{\infty}{12}\above{1pt}{1pt}{3,2}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12672\\-2376\\-936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2376\\-384\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\-144\\-53
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\73\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-21\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-23\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\22\\-42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\3\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\-6\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12672, -2376, -936; -2376, -384, -144; -936, -144, -53],[72, 24, 1],[-3, 1, 1, -1; 73, -21, -23, 22; -144, 40, 45, -42],143,16]
%
% END OF LATTICE L144.16
%

\medskip%
%------------------------------WEYL-GROUP-W145--------------------------------
%
\leavevmode\llap{}%
$W_{145}$%
\qquad\llap{2} lattices, $\chi=6$%
\hfill%
$\slashinfty4|4\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L145.1
%
\leavevmode%
\llap{\spacer}%
${L_{145.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}16\above{1pt}{1pt}{1}{7}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{8,7}{\infty}{4}\above{1pt}{1pt}{*}{4}{2}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-656\\-48\\64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-3\\5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\5\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-656, -48, 64; -48, -3, 5; 64, 5, -6],[16, 1, 1],[0, -1, 1; -1, 6, -4; -1, -6, 7],144,1]
%
% END OF LATTICE L145.1
%
\hbox{}\par\smallskip%
%
% LATTICE L145.2
%
\leavevmode%
\llap{\spacer}%
${L_{145.2}}=\pdualwitharg{2}({L_{145.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{2}{2}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{4,1}{\infty b}{16}\above{1pt}{1pt}{}{4}{32}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-16\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\16\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-16\\32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[32, 16, 0; 16, 16, 0; 0, 0, -1],[16, 16, 1],[-1, 1, -1; -1, -1, 2; -8, 0, 0],144,2]
%
% END OF LATTICE L145.2
%

\medskip%
%------------------------------WEYL-GROUP-W146--------------------------------
%
\leavevmode\llap{}%
$W_{146}$%
\qquad\llap{9} lattices, $\chi=12$%
\hfill%
$\slashtwo\slashinfty\slashtwo\slashinfty\rtimes D_{4}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L146.1
%
\leavevmode%
\llap{\spacer}%
${L_{146.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{-}{5}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{8,5}{\infty}{4}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{4,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3632\\64\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\-1\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-6\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-2\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-8\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3632, 64, 336; 64, -1, -6; 336, -6, -31],[16, 1, 1],[-3, 0, 1, -1; -8, 4, 4, -16; -32, -1, 10, -8],145,1]
%
% END OF LATTICE L146.1
%
\hbox{}\par\smallskip%
%
% LATTICE L146.2
%
\leavevmode%
\llap{\spacer}%
${L_{146.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{8,1}{\infty}{4}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{1,0}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1392\\112\\208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\-8\\-17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\-17\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\16\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\4\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-2\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-8\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1392, 112, 208; 112, -8, -17; 208, -17, -31],[16, 1, 1],[11, 2, -3, -3; 16, 4, -4, -8; 64, 11, -18, -16],145,2]
%
% END OF LATTICE L146.2
%
\hbox{}\par\smallskip%
%
% LATTICE L146.3
%
\leavevmode%
\llap{\spacer}%
${L_{146.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}16\above{1pt}{1pt}{-}{3}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{8,3}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\48\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-15\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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15\\-16\\112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\3\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-80, 48, 16; 48, -15, -8; 16, -8, -3],[16, 1, 1],[-1, -1, 3, 2; 0, 1, -2, -2; -6, -8, 20, 15],145,3]
%
% END OF LATTICE L146.3
%
\hbox{}\par\smallskip%
%
% LATTICE L146.4
%
\leavevmode%
\llap{\spacer}%
${L_{146.4}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{4,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2672\\112\\288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\-4\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\-12\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\-32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-8\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-2\\-16
\end{tabular}\endgroup%
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-8\\-2\\1\\0
\end{tabular}\endgroup%
\kern3pt%
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-16\\-16\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2672, 112, 288; 112, -4, -12; 288, -12, -31],[16, 4, 1],[-1, 1, 0, -3; -4, -2, 2, 8; -8, 10, -1, -32],145,4]
%
% END OF LATTICE L146.4
%
\hbox{}\par\smallskip%
%
% LATTICE L146.5
%
\leavevmode%
\llap{\spacer}%
${L_{146.5}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{8,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\48\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-12\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-8\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-4\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-2\\-8
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-2\\1\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-8\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-80, 48, 16; 48, -12, -8; 16, -8, -3],[16, 4, 1],[-1, -1, 1, 2; 1, 0, -1, -1; -8, -6, 7, 12],145,5]
%
% END OF LATTICE L146.5
%
\hbox{}\par\smallskip%
%
% LATTICE L146.6
%
\leavevmode%
\llap{\spacer}%
${L_{146.6}}=\pdualwitharg{2}({L_{146.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{2,1}{\infty a}{16}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{8,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-16\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\-16\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-16\\16\\0
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\kern3pt%
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-4\\-16\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16, 0, 0; 0, -12, -4; 0, -4, -1],[16, 4, 1],[1, 1, -1, 0; -1, -4, 0, 1; 0, 8, 0, -4],145,6]
%
% END OF LATTICE L146.6
%
\hbox{}\par\smallskip%
%
% LATTICE L146.7
%
\leavevmode%
\llap{\spacer}%
${L_{146.7}}=\pdualwitharg{2}({L_{146.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{2}{0}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{2,1}{\infty z}{1}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{8,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1616\\-2080\\224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2080\\-2672\\288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
224\\288\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-8\\-16
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-2\\-16
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-2\\1\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-16\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1616, -2080, 224; -2080, -2672, 288; 224, 288, -31],[16, 16, 1],[-2, -1, 1, 4; -1, 1, 0, -3; -24, 2, 7, 0],145,7]
%
% END OF LATTICE L146.7
%
\hbox{}\par\smallskip%
%
% LATTICE L146.8
%
\leavevmode%
\llap{\spacer}%
${L_{146.8}}=\pdualwitharg{2}({L_{146.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{-2}{4}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{2,1}{\infty a}{4}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{4,3}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-80\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-64\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\16\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-4\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-4\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-16\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-80, -80, 16; -80, -64, 16; 16, 16, -3],[16, 16, 1],[-1, -1, 1, 3; 1, 0, -1, -1; 0, -6, -2, 8],145,8]
%
% END OF LATTICE L146.8
%
\hbox{}\par\smallskip%
%
% LATTICE L146.9
%
\leavevmode%
\llap{\spacer}%
${L_{146.9}}=\pdualwitharg{2}({L_{146.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{-2}{6}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{8,5}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-192\\32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\-752\\128
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\128\\-21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\-14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\2\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-1\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-2\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-32\\-16
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\-16\\-32
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
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16\\0
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0\\16
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-48, -192, 32; -192, -752, 128; 32, 128, -21],[16, 16, 1],[10, 9, -5, -4; -1, -2, 0, 1; 8, 0, -8, 0],145,9]
%
% END OF LATTICE L146.9
%

\medskip%
%------------------------------WEYL-GROUP-W147--------------------------------
%
\leavevmode\llap{}%
$W_{147}$%
\qquad\llap{4} lattices, $\chi=12$%
\hfill%
$2|2\infty|\infty\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L147.1
%
\leavevmode%
\llap{\spacer}%
${L_{147.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}16\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{2,1}{\infty z}{16}\above{1pt}{1pt}{1,0}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\11
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-16\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\24\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-48\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\16\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-8\\-16\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[16, 0, 0; 0, -4, 11; 0, 11, -30],[16, 1, 1],[1, 0, -3, -1; 0, -8, -16, 24; 0, -3, -8, 8],146,1]
%
% END OF LATTICE L147.1
%
\hbox{}\par\smallskip%
%
% LATTICE L147.2
%
\leavevmode%
\llap{\spacer}%
${L_{147.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-}{3}64\above{1pt}{1pt}{-}{3}$}\EasyButWeakLineBreak%
{${32}\above{1pt}{1pt}{*}{2}{64}\above{1pt}{1pt}{s}{2}{32}\above{1pt}{1pt}{16,1}{\infty z}{8}\above{1pt}{1pt}{8,3}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3392\\384\\64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
384\\-40\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\-8\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-18\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\0\\-96\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\64\\0\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\0\\32\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-32\\-16\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3392, 384, 64; 384, -40, -8; 64, -8, -1],[64, 8, 1],[-3, -1, 1, 0; -18, -4, 6, -1; -64, -32, 16, 4],146,2]
%
% END OF LATTICE L147.2
%
\hbox{}\par\smallskip%
%
% LATTICE L147.3
%
\leavevmode%
\llap{\spacer}%
${L_{147.3}}=\pdualwitharg{2}({L_{147.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{2}{{\rm II}}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{32}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{1,0}{\infty b}{4}\above{1pt}{1pt}{1,0}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-320\\-16\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\32\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\32\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-16\\-4\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-320, -16, 48; -16, 32, 0; 48, 0, -7],[16, 16, 1],[2, 0, -2, 1; 1, -1, -1, 1; 13, 0, -14, 6],146,3]
%
% END OF LATTICE L147.3
%
\hbox{}\par\smallskip%
%
% LATTICE L147.4
%
\leavevmode%
\llap{\spacer}%
${L_{147.4}}=\pdualwitharg{2}({L_{147.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-}{3}64\above{1pt}{1pt}{1}{7}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{8,7}{\infty b}{32}\above{1pt}{1pt}{16,5}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2624\\1152\\-192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1152\\-488\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\80\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\13\\52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-24\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-16\\-16\\32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2624, 1152, -192; 1152, -488, 80; -192, 80, -13],[64, 8, 1],[-1, 0, 2, 1; -7, -1, 13, 10; -28, -6, 52, 48],146,4]
%
% END OF LATTICE L147.4
%

\medskip%
%------------------------------WEYL-GROUP-W148--------------------------------
%
\leavevmode\llap{}%
$W_{148}$%
\qquad\llap{16} lattices, $\chi=12$%
\hfill%
$2\infty2\infty\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L148.1
%
\leavevmode%
\llap{\spacer}%
${L_{148.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{6,1}{\infty z}{2}\above{1pt}{1pt}{b}{2}{72}\above{1pt}{1pt}{2,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
936\\288\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\82\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-81\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-36\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\39\\17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-77\\252\\108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-9\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\-4\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-4\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-72\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[936, 288, 0; 288, 82, 15; 0, 15, -34],[72, 1, 1],[25, 11, -12, -77; -81, -36, 39, 252; -36, -16, 17, 108],147,1]
%
% END OF LATTICE L148.1
%
\hbox{}\par\smallskip%
%
% LATTICE L148.2
%
\leavevmode%
\llap{\spacer}%
${L_{148.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18648\\432\\216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\-10\\-5
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\-5\\-2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-18648, 432, 216; 432, -10, -5; 216, -5, -2],[72, 1, 1],[0, 1, 0, -1; 1, 44, -1, -44; -3, -4, 2, 0],147,2]
%
% END OF LATTICE L148.2
%
\hbox{}\par\smallskip%
%
% LATTICE L148.3
%
\leavevmode%
\llap{\spacer}%
${L_{148.3}}=\pfillwitharg{2}({L_{148.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{1,0}{\infty}{18}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{3,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
450\\-158\\-51
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-51\\-16
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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2\\3\\8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\36\\54
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-9
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-8
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
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2\\0\\-9\\-2
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0\\18\\-18\\-18
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0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1278, 450, 144; 450, -158, -51; 144, -51, -16],[18, 1, 1],[2, 19, -1, -3; 3, 36, 0, -6; 8, 54, -9, -8],147,3]
%
% END OF LATTICE L148.3
%
\hbox{}\par\smallskip%
%
% LATTICE L148.4
%
\leavevmode%
\llap{\spacer}%
${L_{148.4}}=\pfillwitharg{2}({L_{148.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{3,2}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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72\\-9\\-4
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-990, 126, 72; 126, -16, -9; 72, -9, -4],[18, 1, 1],[2, 3, -1, -1; 17, 24, -9, -8; -3, -2, 2, 0],147,4]
%
% END OF LATTICE L148.4
%
\hbox{}\par\smallskip%
%
% LATTICE L148.5
%
\leavevmode%
\llap{\spacer}%
${L_{148.5}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{148.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\kern3pt%
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252\\-24\\97
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8\\9\\-18
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3\\3\\-7
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{$\left[\!\llap{\phantom{%
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-18\\-18\\36\\0
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-2\\-9\\0\\1
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[684, -90, 252; -90, 32, -24; 252, -24, 97],[18, 2, 1],[17, 50, 8, 3; 18, 54, 9, 3; -40, -117, -18, -7],147,5]
%
% END OF LATTICE L148.5
%
\hbox{}\par\smallskip%
%
% LATTICE L148.6
%
\leavevmode%
\llap{\spacer}%
${L_{148.6}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{148.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{3,2}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1584\\450\\756
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
450\\136\\216
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
756\\216\\361
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-2\\-3
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1584, 450, 756; 450, 136, 216; 756, 216, 361],[18, 2, 1],[21, 16, -1, 4; 7, 6, 0, 1; -48, -37, 2, -9],147,6]
%
% END OF LATTICE L148.6
%
\hbox{}\par\smallskip%
%
% LATTICE L148.7
%
\leavevmode%
\llap{\spacer}%
${L_{148.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{148.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{1,0}{\infty}{2}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{3,2}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7218\\-1404\\3096
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1404\\288\\-603
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3096\\-603\\1328
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\7\\108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\4\\74
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\2\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-9\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-2\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-2\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-18\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7218, -1404, 3096; -1404, 288, -603; 3096, -603, 1328],[18, 9, 1],[-45, -31, -3, -15; 7, 4, 0, 2; 108, 74, 7, 36],147,7]
%
% END OF LATTICE L148.7
%
\hbox{}\par\smallskip%
%
% LATTICE L148.8
%
\leavevmode%
\llap{\spacer}%
${L_{148.8}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{148.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{3,1}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-27\\-18
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\kern3pt%
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0\\18\\-18\\-54
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
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18\\0
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0\\18
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-27\\-18
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-18\\-54
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[18, 0, 0; 0, 18, -9; 0, -9, 4],[18, 9, 1],[-1, -3, 0, 1; 3, 8, 1, 0; 9, 18, 0, 0],147,8]
%
% END OF LATTICE L148.8
%
\hbox{}\par\smallskip%
%
% LATTICE L148.9
%
\leavevmode%
\llap{\spacer}%
${L_{148.9}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{148.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{1,0}{\infty}{4}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{3,2}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\10422\\4896
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10422\\-363132\\-170586
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4896\\-170586\\-80135
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\237\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\118\\-251
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-165\\351
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-18\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-2\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-9\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-288, 10422, 4896; 10422, -363132, -170586; 4896, -170586, -80135],[18, 18, 1],[8, 3, -1, -4; 237, 118, -16, -165; -504, -251, 34, 351],147,9]
%
% END OF LATTICE L148.9
%
\hbox{}\par\smallskip%
%
% LATTICE L148.10
%
\leavevmode%
\llap{\spacer}%
${L_{148.10}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{148.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{3,1}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\162\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
162\\-288\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-126\\-55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-32\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\87\\-198
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\40\\-91
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\12\\-27
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-54\\-18
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\-18\\-27
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
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36\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9
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-54\\-18
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-27
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{$\left.\llap{\phantom{%
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0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-72, 162, 72; 162, -288, -126; 72, -126, -55],[18, 18, 1],[-3, -1, 2, 0; -1, 12, 9, -4; 0, -27, -18, 9],147,10]
%
% END OF LATTICE L148.10
%
\hbox{}\par\smallskip%
%
% LATTICE L148.11
%
\leavevmode%
\llap{\spacer}%
${L_{148.11}}=\pdualwitharg{2}({L_{148.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${144}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{3,1}{\infty a}{16}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{1,0}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\216\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\-48\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-40\\13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\81\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\15\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-18\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-18\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\0\\-72\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-8\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-8\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-36\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-720, 216, 72; 216, -48, -40; 72, -40, 13],[72, 8, 1],[32, 6, -7, -7; 81, 15, -18, -18; 72, 14, -16, -18],147,11]
%
% END OF LATTICE L148.11
%
\hbox{}\par\smallskip%
%
% LATTICE L148.12
%
\leavevmode%
\llap{\spacer}%
${L_{148.12}}=\pdualwitharg{2}({L_{148.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{3,2}{\infty b}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\216\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\-304\\104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\104\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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24\\-13\\-96
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\right.$}%
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-1\\2\\8
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
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16\\0
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0\\4
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-24\\-8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-144, 216, -72; 216, -304, 104; -72, 104, -35],[72, 8, 1],[-1, -1, 3, 3; 2, 0, -3, -1; 8, 2, -16, -10],147,12]
%
% END OF LATTICE L148.12
%
\hbox{}\par\smallskip%
%
% LATTICE L148.13
%
\leavevmode%
\llap{\spacer}%
${L_{148.13}}=\pdualwitharg{3}({L_{148.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{2,1}{\infty z}{2}\above{1pt}{1pt}{b}{2}{72}\above{1pt}{1pt}{6,5}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
936\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-54\\45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\45\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-27\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\13\\17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\84\\108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-9\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\-4\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-4\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-72\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[936, -72, 0; -72, -54, 45; 0, 45, -34],[72, 9, 1],[-2, -1, 1, 7; -27, -12, 13, 84; -36, -16, 17, 108],147,13]
%
% END OF LATTICE L148.13
%
\hbox{}\par\smallskip%
%
% LATTICE L148.14
%
\leavevmode%
\llap{\spacer}%
${L_{148.14}}=\pdualwitharg{3}({L_{148.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{b}{2}{72}\above{1pt}{1pt}{6,1}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11160\\-504\\-2592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-18\\-117
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2592\\-117\\-602
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\13\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\2\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\-108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-27\\-36
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0\\72\\-36\\-216
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
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18\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\72
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-216
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11160, -504, -2592; -504, -18, -117; -2592, -117, -602],[72, 9, 1],[2, 25, -2, -25; 2, 0, -3, -4; -9, -108, 9, 108],147,14]
%
% END OF LATTICE L148.14
%
\hbox{}\par\smallskip%
%
% LATTICE L148.15
%
\leavevmode%
\llap{\spacer}%
${L_{148.15}}=\pdualwitharg{2.3}({L_{148.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{3,2}{\infty b}{144}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{1,0}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\360\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-432\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-72\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\17\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-72\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\-72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-72\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-36\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-144, 360, 72; 360, -432, -72; 72, -72, -11],[72, 72, 1],[-1, -1, 5, 1; -2, -4, 17, 5; 8, 18, -72, -22],147,15]
%
% END OF LATTICE L148.15
%
\hbox{}\par\smallskip%
%
% LATTICE L148.16
%
\leavevmode%
\llap{\spacer}%
${L_{148.16}}=\pdualwitharg{2.3}({L_{148.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${144}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{3,1}{\infty a}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2736\\-7272\\-1512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7272\\-12816\\-2592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1512\\-2592\\-523
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-229\\1704\\-7776
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-551\\4117\\-18792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114\\852\\-3889
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\221\\-1008
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\209\\-954
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\0\\-216\\-72
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\-72\\-108
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-108
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2736, -7272, -1512; -7272, -12816, -2592; -1512, -2592, -523],[72, 72, 1],[-30, -28, 11, 9; 221, 209, -79, -67; -1008, -954, 360, 306],147,16]
%
% END OF LATTICE L148.16
%

\medskip%
%------------------------------WEYL-GROUP-W149--------------------------------
%
\leavevmode\llap{}%
$W_{149}$%
\qquad\llap{27} lattices, $\chi=18$%
\hfill%
$\slashtwo2\infty|\infty2\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L149.1
%
\leavevmode%
\llap{\spacer}%
${L_{149.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{5}%
\EasyButWeakLineBreak%
{${100}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{10,9}{\infty}{20}\above{1pt}{1pt}{5,4}{\infty z}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11100\\-800\\-5200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800\\-55\\-375
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5200\\-375\\-2436
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\0\\100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-4\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\4\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\3\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\0\\-100\\-100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\0\\20\\-20\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-10\\-20\\20\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-40\\-10\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-11100, -800, -5200; -800, -55, -375; -5200, -375, -2436],[100, 5, 1],[-47, 1, 19, -5, -19; 0, -1, -4, 4, 3; 100, -2, -40, 10, 40],148,1]
%
% END OF LATTICE L149.1
%
\hbox{}\par\smallskip%
%
% LATTICE L149.2
%
\leavevmode%
\llap{\spacer}%
${L_{149.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${100}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{5,4}{\infty b}{20}\above{1pt}{1pt}{5,3}{\infty}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1900\\200\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\-20\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\28\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\102\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-48\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\0\\-100\\-100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\0\\20\\-20\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-20\\-20\\20\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-80\\-20\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1900, 200, 0; 200, -20, -5; 0, -5, 24],[100, 5, 1],[-1, 3, 11, 1, -5; -10, 28, 102, 8, -48; 0, 6, 20, 0, -10],148,2]
%
% END OF LATTICE L149.2
%
\hbox{}\par\smallskip%
%
% LATTICE L149.3
%
\leavevmode%
\llap{\spacer}%
${L_{149.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${50}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{20,19}{\infty b}{40}\above{1pt}{1pt}{20,13}{\infty z}{10}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143800\\-3600\\-1000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3600\\90\\25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1000\\25\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-25\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-23\\-60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\0\\-50\\-100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-20\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\0\\10\\-20\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-20\\-20\\40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-40\\-20\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[143800, -3600, -1000; -3600, 90, 25; -1000, 25, 7],[200, 5, 1],[-1, 0, 0, -1, -1; -25, -1, -3, -24, -23; -50, 4, 10, -60, -60],148,3]
%
% END OF LATTICE L149.3
%
\hbox{}\par\smallskip%
%
% LATTICE L149.4
%
\leavevmode%
\llap{\spacer}%
${L_{149.4}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${50}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{20,19}{\infty a}{40}\above{1pt}{1pt}{20,3}{\infty z}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
442200\\-6400\\-2400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6400\\90\\35
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2400\\35\\13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-25\\-300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\17\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-13\\-150
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\0\\-50\\-100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-20\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\0\\10\\-20\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-20\\-20\\40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-40\\-20\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[442200, -6400, -2400; -6400, 90, 35; -2400, 35, 13],[200, 5, 1],[-2, 0, 1, 1, -1; -25, 1, 17, 16, -13; -300, -2, 140, 140, -150],148,4]
%
% END OF LATTICE L149.4
%
\hbox{}\par\smallskip%
%
% LATTICE L149.5
%
\leavevmode%
\llap{\spacer}%
${L_{149.5}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{5}%
\EasyButWeakLineBreak%
{${200}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{5,4}{\infty}{40}\above{1pt}{1pt}{5,2}{\infty z}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2200\\600\\-2600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
600\\-110\\425
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2600\\425\\-1568
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\1060\\200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-21\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-424\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\104\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\423\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\0\\-200\\-200\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-20\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\0\\40\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-20\\-40\\40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-80\\-20\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2200, 600, -2600; 600, -110, 425; -2600, 425, -1568],[200, 5, 1],[53, -1, -21, 5, 21; 1060, -21, -424, 104, 423; 200, -4, -80, 20, 80],148,5]
%
% END OF LATTICE L149.5
%
\hbox{}\par\smallskip%
%
% LATTICE L149.6
%
\leavevmode%
\llap{\spacer}%
${L_{149.6}}=\pfillwitharg{5}({L_{149.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{2,1}{\infty}{20}\above{1pt}{1pt}{1,0}{\infty z}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4780\\1060\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1060\\-235\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\-9\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-40\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\40\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\35\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-4780, 1060, 40; 1060, -235, -9; 40, -9, 0],[20, 1, 1],[1, -2, -9, 9, 8; 4, -9, -40, 40, 35; 8, -2, -20, 10, 20],148,6]
%
% END OF LATTICE L149.6
%
\hbox{}\par\smallskip%
%
% LATTICE L149.7
%
\leavevmode%
\llap{\spacer}%
${L_{149.7}}=\pfillwitharg{5}({L_{149.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{1,0}{\infty}{40}\above{1pt}{1pt}{1,0}{\infty z}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6360\\-1400\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1400\\-308\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-9\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\12
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-9\\-5
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\HardButStrongLineBreak\kern3pt%
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9\\-40\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-9\\40\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-8\\35\\35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8\\0\\-40\\-40\\0
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0\\2\\0\\-20\\-10
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-40\\0\\40\\-40\\-80
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0\\-10\\-80\\-20\\10
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-6360, -1400, -40; -1400, -308, -9; -40, -9, 0],[40, 1, 1],[-1, 2, 9, -9, -8; 4, -9, -40, 40, 35; 12, -5, -40, 20, 35],148,7]
%
% END OF LATTICE L149.7
%
\hbox{}\par\smallskip%
%
% LATTICE L149.8
%
\leavevmode%
\llap{\spacer}%
${L_{149.8}}=\pfillwitharg{2}({L_{149.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${25}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{10,9}{\infty}{5}\above{1pt}{1pt}{5,3}{\infty}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
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\HardButStrongLineBreak\kern3pt%
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0\\-1\\3
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\HardButStrongLineBreak\kern3pt%
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1\\-4\\10
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\HardButStrongLineBreak\kern3pt%
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1\\-1\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\-25\\-25\\0
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0\\1\\0\\-5\\-5
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-25\\-5\\-5\\5\\-5
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-20\\-5\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[25, 0, 0; 0, -20, -5; 0, -5, -1],[25, 5, 1],[-1, 0, 1, 1, 0; 0, -1, -4, -1, 1; 0, 3, 10, 0, -5],148,8]
%
% END OF LATTICE L149.8
%
\hbox{}\par\smallskip%
%
% LATTICE L149.9
%
\leavevmode%
\llap{\spacer}%
${L_{149.9}}=\pfillwitharg{2}({L_{149.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{50}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{10,1}{\infty b}{10}\above{1pt}{1pt}{10,7}{\infty}{10}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-950\\200\\100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\-40\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\-25\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\11\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-19\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\41\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-10\\-10\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\50\\0\\-50\\-50
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-10\\0\\10\\-10\\-40
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-10\\-50\\-10\\10\\-10
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-50\\-40\\-10\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-950, 200, 100; 200, -40, -25; 100, -25, -3],[50, 5, 1],[3, -1, -5, 1, 11; 11, -5, -19, 4, 41; 6, 0, -10, 0, 20],148,9]
%
% END OF LATTICE L149.9
%
\hbox{}\par\smallskip%
%
% LATTICE L149.10
%
\leavevmode%
\llap{\spacer}%
${L_{149.10}}=\main({L_{149.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${25}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{20,19}{\infty}{20}\above{1pt}{1pt}{10,3}{\infty z}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17900\\-900\\-500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\45\\25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-500\\25\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-4\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\-25\\-50\\0
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0\\1\\0\\-10\\-5
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-25\\0\\5\\-10\\-20
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-50\\-10\\-10\\20\\-10
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-20\\-10\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[17900, -900, -500; -900, 45, 25; -500, 25, 14],[100, 5, 1],[-1, 0, 0, -1, -1; -5, -1, -3, -4, -3; -25, 2, 5, -30, -30],148,10]
%
% END OF LATTICE L149.10
%
\hbox{}\par\smallskip%
%
% LATTICE L149.11
%
\leavevmode%
\llap{\spacer}%
${L_{149.11}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{149.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{2,1}{\infty}{20}\above{1pt}{1pt}{1,0}{\infty a}{20}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2380\\820\\660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
820\\320\\236
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
660\\236\\185
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\12\\-51
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\13\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\10\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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22\\25\\-110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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63\\75\\-320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\right.$}%
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1\\0\\-5\\-10\\0
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0\\4\\0\\-20\\-20
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-5\\0\\5\\-10\\-40
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-20\\-10\\20\\-20
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-40\\-20\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2380, 820, 660; 820, 320, 236; 660, 236, 185],[20, 4, 1],[10, 11, 9, 22, 63; 12, 13, 10, 25, 75; -51, -56, -45, -110, -320],148,11]
%
% END OF LATTICE L149.11
%
\hbox{}\par\smallskip%
%
% LATTICE L149.12
%
\leavevmode%
\llap{\spacer}%
${L_{149.12}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{149.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{4,1}{\infty}{4}\above{1pt}{1pt}{1,0}{\infty a}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\-20\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-5\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\8\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-5\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\1\\-2\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-20\\-2\\4\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-8\\-4\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-300, -20, -60; -20, 0, -5; -60, -5, -11],[20, 5, 1],[1, -1, -1, -1, 1; -2, 8, 4, 2, -4; -5, 0, 3, 4, -4],148,12]
%
% END OF LATTICE L149.12
%
\hbox{}\par\smallskip%
%
% LATTICE L149.13
%
\leavevmode%
\llap{\spacer}%
${L_{149.13}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{149.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{4,1}{\infty b}{8}\above{1pt}{1pt}{1,0}{\infty a}{8}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14200\\120\\560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
560\\-5\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-10\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\40\\0\\-40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\2\\-4\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-40\\-4\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-16\\-8\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14200, 120, 560; 120, 0, -5; 560, -5, -22],[40, 5, 1],[-1, 1, 1, 1, -1; -4, 16, 8, 4, -8; -25, 20, 23, 24, -24],148,13]
%
% END OF LATTICE L149.13
%
\hbox{}\par\smallskip%
%
% LATTICE L149.14
%
\leavevmode%
\llap{\spacer}%
${L_{149.14}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{149.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{1,0}{\infty}{20}\above{1pt}{1pt}{1,0}{\infty a}{80}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35680\\760\\1880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
760\\-16\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1880\\-40\\-99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\3\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\15\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\25\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-10\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\5\\-10\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-40\\-10\\20\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-80\\-40\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35680, 760, 1880; 760, -16, -40; 1880, -40, -99],[40, 8, 1],[0, -2, -2, -1, 2; -5, 3, 15, 25, -5; 2, -40, -45, -30, 40],148,14]
%
% END OF LATTICE L149.14
%
\hbox{}\par\smallskip%
%
% LATTICE L149.15
%
\leavevmode%
\llap{\spacer}%
${L_{149.15}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{149.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{4,3}{\infty}{4}\above{1pt}{1pt}{1,0}{\infty z}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
300\\-620\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-620\\1600\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\360\\81
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-13\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-12\\55
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-15\\68
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\4\\-4\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-10\\-4\\4\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-8\\-2\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[300, -620, -140; -620, 1600, 360; -140, 360, 81],[20, 20, 1],[1, 1, 1, 0, 0; -13, -12, -15, -5, -2; 60, 55, 68, 22, 9],148,15]
%
% END OF LATTICE L149.15
%
\hbox{}\par\smallskip%
%
% LATTICE L149.16
%
\leavevmode%
\llap{\spacer}%
${L_{149.16}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{149.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{100}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{20,1}{\infty z}{5}\above{1pt}{1pt}{10,7}{\infty}{20}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\50\\50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\1430\\100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\100\\29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\2\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\2\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\4\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\12\\-220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\100\\0\\-50\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\-10\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-50\\-10\\5\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-100\\-80\\-10\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[100, 50, 50; 50, 1430, 100; 50, 100, 29],[50, 10, 1],[17, -1, 19, 36, 105; 2, 0, 2, 4, 12; -36, 0, -40, -75, -220],148,16]
%
% END OF LATTICE L149.16
%
\hbox{}\par\smallskip%
%
% LATTICE L149.17
%
\leavevmode%
\llap{\spacer}%
${L_{149.17}}=\pdualwitharg{5}({L_{149.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${25}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{20,9}{\infty}{20}\above{1pt}{1pt}{5,1}{\infty b}{20}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111100\\-2500\\2000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2500\\-55\\45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2000\\45\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\-25\\-50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\0\\5\\-10\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-20\\-10\\20\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-40\\-20\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-111100, -2500, 2000; -2500, -55, 45; 2000, 45, -36],[100, 5, 1],[1, 1, 1, -1, -1; -5, 0, 3, 4, -4; 50, 56, 60, -50, -60],148,17]
%
% END OF LATTICE L149.17
%
\hbox{}\par\smallskip%
%
% LATTICE L149.18
%
\leavevmode%
\llap{\spacer}%
${L_{149.18}}=\pdualwitharg{5}({L_{149.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${50}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{20,9}{\infty b}{40}\above{1pt}{1pt}{5,2}{\infty a}{40}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66200\\-2800\\-60000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2800\\-110\\-2455
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60000\\-2455\\-53568
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-985\\100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-1100\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59\\-1177\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\984\\-100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\1176\\-120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\0\\-50\\-100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\0\\10\\-20\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-40\\-20\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-80\\-40\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-66200, -2800, -60000; -2800, -110, -2455; -60000, -2455, -53568],[200, 5, 1],[-49, -55, -59, 49, 59; -985, -1100, -1177, 984, 1176; 100, 112, 120, -100, -120],148,18]
%
% END OF LATTICE L149.18
%
\hbox{}\par\smallskip%
%
% LATTICE L149.19
%
\leavevmode%
\llap{\spacer}%
${L_{149.19}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{149.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{1,0}{\infty}{4}\above{1pt}{1pt}{1,0}{\infty a}{16}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\160\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\40\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\10\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-10\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\1\\-2\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-40\\-2\\4\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-16\\-8\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-80, 40, 0; 40, 160, 40; 0, 40, 9],[40, 40, 1],[-1, 3, 1, 1, -1; -2, 10, 3, 3, -2; 10, -40, -13, -14, 8],148,19]
%
% END OF LATTICE L149.19
%
\hbox{}\par\smallskip%
%
% LATTICE L149.20
%
\leavevmode%
\llap{\spacer}%
${L_{149.20}}=\pdualwitharg{2}({L_{149.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${100}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{10,9}{\infty z}{5}\above{1pt}{1pt}{5,3}{\infty}{20}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10400\\9700\\2700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9700\\9120\\2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2700\\2520\\701
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\1\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75\\6\\-310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\2\\-115
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\1\\-50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\0\\-100\\-50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-10\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\0\\20\\-10\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-10\\-10\\5\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-80\\-10\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10400, 9700, 2700; 9700, 9120, 2520; 2700, 2520, 701],[100, 20, 1],[-13, 10, 75, 28, 12; 0, 1, 6, 2, 1; 50, -42, -310, -115, -50],148,20]
%
% END OF LATTICE L149.20
%
\hbox{}\par\smallskip%
%
% LATTICE L149.21
%
\leavevmode%
\llap{\spacer}%
${L_{149.21}}=\pdualwitharg{2.5}({L_{149.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${100}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{20,19}{\infty}{20}\above{1pt}{1pt}{5,3}{\infty z}{5}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
601100\\-500\\143800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-500\\80\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143800\\-120\\34401
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-311\\5\\1300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\1\\209
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-311\\7\\1300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-122\\3\\510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\1\\255
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\0\\-100\\-100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-10\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\0\\20\\-20\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-10\\-20\\20\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-40\\-10\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[601100, -500, 143800; -500, 80, -120; 143800, -120, 34401],[100, 20, 1],[-311, -50, -311, -122, -61; 5, 1, 7, 3, 1; 1300, 209, 1300, 510, 255],148,21]
%
% END OF LATTICE L149.21
%
\hbox{}\par\smallskip%
%
% LATTICE L149.22
%
\leavevmode%
\llap{\spacer}%
${L_{149.22}}=\pdualwitharg{2}({L_{149.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${25}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{10,9}{\infty}{20}\above{1pt}{1pt}{5,3}{\infty b}{20}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1239100\\-7300\\305500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7300\\80\\-1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
305500\\-1800\\75321
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-302\\5\\1225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71\\1\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\1\\215
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-106\\3\\430
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-355\\7\\1440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\-25\\-50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\0\\5\\-10\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-20\\-10\\20\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-40\\-20\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1239100, -7300, 305500; -7300, 80, -1800; 305500, -1800, 75321],[100, 20, 1],[-302, -71, -53, -106, -355; 5, 1, 1, 3, 7; 1225, 288, 215, 430, 1440],148,22]
%
% END OF LATTICE L149.22
%
\hbox{}\par\smallskip%
%
% LATTICE L149.23
%
\leavevmode%
\llap{\spacer}%
${L_{149.23}}=\pdualwitharg{2}(\main({L_{149.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${100}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{20,9}{\infty}{20}\above{1pt}{1pt}{5,3}{\infty a}{20}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22100\\2100\\5300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2100\\-120\\500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5300\\500\\1271
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
191\\10\\-800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\1\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\-2\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74\\-3\\310
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\3\\-180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\0\\-100\\-100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\0\\20\\-20\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-20\\-20\\20\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-80\\-20\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22100, 2100, 5300; 2100, -120, 500; 5300, 500, 1271],[100, 20, 1],[191, 21, -43, -74, 43; 10, 1, -2, -3, 3; -800, -88, 180, 310, -180],148,23]
%
% END OF LATTICE L149.23
%
\hbox{}\par\smallskip%
%
% LATTICE L149.24
%
\leavevmode%
\llap{\spacer}%
${L_{149.24}}=\pdualwitharg{2}({L_{149.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${400}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{40,9}{\infty z}{20}\above{1pt}{1pt}{10,3}{\infty a}{80}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34600\\-13600\\4600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13600\\-5320\\1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4600\\1800\\-609
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-65\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-19\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-19\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\17\\50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\11\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\0\\-400\\-200\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\0\\80\\-40\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-40\\-40\\20\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-320\\-40\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-34600, -13600, 4600; -13600, -5320, 1800; 4600, 1800, -609],[200, 40, 1],[-1, -1, -3, 0, 1; -65, -19, -19, 17, 11; -200, -64, -80, 50, 40],148,24]
%
% END OF LATTICE L149.24
%
\hbox{}\par\smallskip%
%
% LATTICE L149.25
%
\leavevmode%
\llap{\spacer}%
${L_{149.25}}=\pdualwitharg{2}({L_{149.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${400}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{40,29}{\infty z}{20}\above{1pt}{1pt}{10,3}{\infty b}{80}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1028600\\-275200\\10200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-275200\\-73080\\2720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10200\\2720\\-101
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-55\\-3400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\21\\1272
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\127\\7760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\29\\1790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-33\\-2000
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\0\\-400\\-200\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\0\\80\\-40\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-40\\-40\\20\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-320\\-40\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1028600, -275200, 10200; -275200, -73080, 2720; 10200, 2720, -101],[200, 40, 1],[-19, 7, 43, 10, -11; -55, 21, 127, 29, -33; -3400, 1272, 7760, 1790, -2000],148,25]
%
% END OF LATTICE L149.25
%
\hbox{}\par\smallskip%
%
% LATTICE L149.26
%
\leavevmode%
\llap{\spacer}%
${L_{149.26}}=\pdualwitharg{2}({L_{149.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${100}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{5,4}{\infty}{20}\above{1pt}{1pt}{5,4}{\infty a}{80}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64400\\-5800\\26200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5800\\-480\\2360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26200\\2360\\-10659
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\0\\-150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-2\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\-2\\115
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\-1\\130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\2\\-120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\0\\-50\\-100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\0\\5\\-10\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-40\\-10\\20\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-80\\-40\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-64400, -5800, 26200; -5800, -480, 2360; 26200, 2360, -10659],[200, 40, 1],[-61, 23, 47, 53, -49; 0, -2, -2, -1, 2; -150, 56, 115, 130, -120],148,26]
%
% END OF LATTICE L149.26
%
\hbox{}\par\smallskip%
%
% LATTICE L149.27
%
\leavevmode%
\llap{\spacer}%
${L_{149.27}}=\pdualwitharg{2.5}({L_{149.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{400}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{5,1}{\infty}{20}\above{1pt}{1pt}{5,1}{\infty b}{80}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\3800\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3800\\-35680\\1880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\1880\\-99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-10\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-1\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-10\\-20\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\400\\0\\-200\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\5\\-10\\-80
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-200\\-10\\20\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-400\\-80\\-40\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-400, 3800, -200; 3800, -35680, 1880; -200, 1880, -99],[200, 40, 1],[-1, 3, 3, 5, -1; 0, -10, -2, -1, 2; 2, -200, -45, -30, 40],148,27]
%
% END OF LATTICE L149.27
%

\medskip%
%------------------------------WEYL-GROUP-W150--------------------------------
%
\leavevmode\llap{}%
$W_{150}$%
\qquad\llap{60} lattices, $\chi=12$%
\hfill%
$22|22\slashinfty\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L150.1
%
\leavevmode%
\llap{\spacer}%
${L_{150.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3}%
\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{3,1}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28584\\144\\1080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\15\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1080\\-9\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\88\\384
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\75\\333
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\28\\128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-42\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-40\\-180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-72\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-36\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-36\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-28584, 144, 1080; 144, 15, -9; 1080, -9, -40],[72, 3, 1],[15, 13, 5, -7, -7; 88, 75, 28, -42, -40; 384, 333, 128, -180, -180],149,1]
%
% END OF LATTICE L150.1
%
\hbox{}\par\smallskip%
%
% LATTICE L150.2
%
\leavevmode%
\llap{\spacer}%
${L_{150.2}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3m,3,2}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{24,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75024\\4608\\288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4608\\-282\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\-18\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\14\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-12\\-72\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-72\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-36\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-75024, 4608, 288; 4608, -282, -18; 288, -18, -1],[144, 6, 1],[0, -1, -1, -1, 1; -1, -15, -14, -12, 14; 12, -27, -40, -72, 36],149,2]
%
% END OF LATTICE L150.2
%
\hbox{}\par\smallskip%
%
% LATTICE L150.3
%
\leavevmode%
\llap{\spacer}%
${L_{150.3}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{32,3,m}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{144}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{24,13}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-118224\\5904\\2304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5904\\-282\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2304\\-120\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\59\\156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\228\\594
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\50\\128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-108\\-288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-70\\-180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-12\\-72\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-144\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-144\\0\\144\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-72\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-118224, 5904, 2304; 5904, -282, -120; 2304, -120, -43],[144, 6, 1],[6, 23, 5, -11, -7; 59, 228, 50, -108, -70; 156, 594, 128, -288, -180],149,3]
%
% END OF LATTICE L150.3
%
\hbox{}\par\smallskip%
%
% LATTICE L150.4
%
\leavevmode%
\llap{\spacer}%
${L_{150.4}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3m,3,m}%
\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{144}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{24,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10224\\576\\1296
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
576\\-30\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1296\\-84\\-115
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-106\\-24
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-84\\-18
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\35\\8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\648\\144
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\83\\18
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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24\\0\\-12\\-144\\-12
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-144\\-36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\2\\0\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-144\\0\\144\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-36\\-6\\0\\6
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-10224, 576, 1296; 576, -30, -84; 1296, -84, -115],[144, 6, 1],[-9, -7, 3, 55, 7; -106, -84, 35, 648, 83; -24, -18, 8, 144, 18],149,4]
%
% END OF LATTICE L150.4
%
\hbox{}\par\smallskip%
%
% LATTICE L150.5
%
\leavevmode%
\llap{\spacer}%
${L_{150.5}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3}%
\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{24,13}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60336\\1440\\-13392
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1440\\-30\\336
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13392\\336\\-2911
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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15\\182\\-48
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14\\168\\-45
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-61\\16
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-157\\-1896\\504
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-337\\90
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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24\\0\\-12\\-144\\-12
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-72\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-72\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-18\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-60336, 1440, -13392; 1440, -30, 336; -13392, 336, -2911],[144, 6, 1],[15, 14, -5, -157, -28; 182, 168, -61, -1896, -337; -48, -45, 16, 504, 90],149,5]
%
% END OF LATTICE L150.5
%
\hbox{}\par\smallskip%
%
% LATTICE L150.6
%
\leavevmode%
\llap{\spacer}%
${L_{150.6}}=\pfillwitharg{3.2}({L_{150.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{4,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\6\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-9\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-6\\-6\\-6
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-4\\-12
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-6\\0\\2\\0\\-6
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-6\\-4\\0\\1\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-12\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[60, -24, 0; -24, 6, -2; 0, -2, -1],[12, 2, 1],[1, 1, -1, -2, -4; 3, 2, -3, -5, -9; -6, -4, 4, 7, 12],149,6]
%
% END OF LATTICE L150.6
%
\hbox{}\par\smallskip%
%
% LATTICE L150.7
%
\leavevmode%
\llap{\spacer}%
${L_{150.7}}=\pfillwitharg{3}({L_{150.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{2,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31080\\1176\\1248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1176\\-43\\-49
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1248\\-49\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-96\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-98\\-82
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\68\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\125\\106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\600\\504
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-12\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-4\\-24
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-24\\0\\8\\0\\-24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-4\\0\\1\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-31080, 1176, 1248; 1176, -43, -49; 1248, -49, -48],[24, 1, 1],[-7, -7, 5, 9, 43; -96, -98, 68, 125, 600; -84, -82, 60, 106, 504],149,7]
%
% END OF LATTICE L150.7
%
\hbox{}\par\smallskip%
%
% LATTICE L150.8
%
\leavevmode%
\llap{\spacer}%
${L_{150.8}}=\main(\pfillwitharg{3}({L_{150.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{4,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\24\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-2\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-6\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-8\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-6
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-8\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-12\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-168, 24, 48; 24, -2, -6; 48, -6, -13],[24, 2, 1],[1, -1, -1, 0, 1; -6, 0, 4, 3, 0; 6, -4, -6, -2, 3],149,8]
%
% END OF LATTICE L150.8
%
\hbox{}\par\smallskip%
%
% LATTICE L150.9
%
\leavevmode%
\llap{\spacer}%
${L_{150.9}}=\main(\pfillwitharg{3}({L_{150.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{4,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-96\\96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-186\\172
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\172\\-157
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-15\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\60\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\42\\39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-24\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-8\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\1\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-8\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-6\\-3\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-24, -96, 96; -96, -186, 172; 96, 172, -157],[24, 2, 1],[1, 4, 0, -15, -11; -6, -15, 1, 60, 42; -6, -14, 1, 56, 39],149,9]
%
% END OF LATTICE L150.9
%
\hbox{}\par\smallskip%
%
% LATTICE L150.10
%
\leavevmode%
\llap{\spacer}%
${L_{150.10}}=\pfillwitharg{3}({L_{150.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{8,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2064\\-240\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-26\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-32\\-39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-21\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-12\\-24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-8\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-8\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-2064, -240, -288; -240, -26, -32; -288, -32, -39],[48, 2, 1],[-1, 0, 1, 1, -1; -21, -4, 10, 20, -6; 24, 3, -16, -24, 12],149,10]
%
% END OF LATTICE L150.10
%
\hbox{}\par\smallskip%
%
% LATTICE L150.11
%
\leavevmode%
\llap{\spacer}%
${L_{150.11}}=\pfillwitharg{3}({L_{150.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{8,5}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10704\\528\\720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\-26\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-36\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-21\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\52\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\90\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-126\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-12\\-24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-16\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-16\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-10704, 528, 720; 528, -26, -36; 720, -36, -43],[48, 2, 1],[-1, 3, 5, 1, -7; -21, 52, 90, 20, -126; 0, 6, 8, 0, -12],149,11]
%
% END OF LATTICE L150.11
%
\hbox{}\par\smallskip%
%
% LATTICE L150.12
%
\leavevmode%
\llap{\spacer}%
${L_{150.12}}=\pfillwitharg{3}({L_{150.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{8,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23664\\-7584\\912
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7584\\-2430\\292
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
912\\292\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-12\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\48\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\33\\66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-12\\-48\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-16\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-16\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-23664, -7584, 912; -7584, -2430, 292; 912, 292, -35],[48, 2, 1],[1, 3, 0, -11, -8; -6, -12, 1, 48, 33; -24, -22, 8, 112, 66],149,12]
%
% END OF LATTICE L150.12
%
\hbox{}\par\smallskip%
%
% LATTICE L150.13
%
\leavevmode%
\llap{\spacer}%
${L_{150.13}}=\pfillwitharg{3}({L_{150.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{8,5}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\24\\-143
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-12\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\96\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\69\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-12\\-48\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-8\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-8\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-6\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-48, 0, 144; 0, 2, 24; 144, 24, -143],[48, 2, 1],[1, 3, 0, -23, -17; -6, -12, 1, 96, 69; 0, 1, 0, -8, -6],149,13]
%
% END OF LATTICE L150.13
%
\hbox{}\par\smallskip%
%
% LATTICE L150.14
%
\leavevmode%
\llap{\spacer}%
${L_{150.14}}=\pfillwitharg{2}({L_{150.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{12,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-6\\-36\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-36\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-36\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-18\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[36, 0, 0; 0, -30, -6; 0, -6, -1],[36, 6, 1],[1, 1, 0, -1, 0; -1, -3, -1, 0, 1; 0, 9, 4, 0, -6],149,14]
%
% END OF LATTICE L150.14
%
\hbox{}\par\smallskip%
%
% LATTICE L150.15
%
\leavevmode%
\llap{\spacer}%
${L_{150.15}}=\main({L_{150.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{12,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37512\\5472\\2592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5472\\-798\\-378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2592\\-378\\-179
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\3\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-36\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\72\\0\\-72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-72\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-36\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-37512, 5472, 2592; 5472, -798, -378; 2592, -378, -179],[72, 6, 1],[1, -1, -1, -2, 0; 4, -24, -6, 3, 7; 6, 36, -2, -36, -15],149,15]
%
% END OF LATTICE L150.15
%
\hbox{}\par\smallskip%
%
% LATTICE L150.16
%
\leavevmode%
\llap{\spacer}%
${L_{150.16}}=\main({L_{150.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{12,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5112\\-2808\\-648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2808\\-1482\\-330
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-648\\-330\\-71
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\28\\-66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\39\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-8\\19
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\-216\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-35\\81
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-72\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-72\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\1\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-72\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-18\\-3\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-5112, -2808, -648; -2808, -1482, -330; -648, -330, -71],[72, 6, 1],[-7, -10, 2, 55, 9; 28, 39, -8, -216, -35; -66, -90, 19, 504, 81],149,16]
%
% END OF LATTICE L150.16
%
\hbox{}\par\smallskip%
%
% LATTICE L150.17
%
\leavevmode%
\llap{\spacer}%
${L_{150.17}}=\pdualwitharg{3}(\pfillwitharg{3.2}({L_{150.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{4,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-174\\66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\66\\-25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-6\\-12\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-6\\-6\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12, 0, 0; 0, -174, 66; 0, 66, -25],[12, 6, 1],[1, 1, 0, -1, 0; 1, 6, 7, 0, -3; 2, 15, 18, 0, -8],149,17]
%
% END OF LATTICE L150.17
%
\hbox{}\par\smallskip%
%
% LATTICE L150.18
%
\leavevmode%
\llap{\spacer}%
${L_{150.18}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{150.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{1,0}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2232\\240\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-24\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-27\\-25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\24\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\22\\33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\28\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-12\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-24\\-24\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-12\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2232, 240, 240; 240, -24, -27; 240, -27, -25],[24, 3, 1],[7, 6, 7, -3, -3; 24, 22, 28, -10, -12; 40, 33, 36, -18, -16],149,18]
%
% END OF LATTICE L150.18
%
\hbox{}\par\smallskip%
%
% LATTICE L150.19
%
\leavevmode%
\llap{\spacer}%
${L_{150.19}}=\pdualwitharg{2}(\main(\pfillwitharg{3}({L_{150.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{4,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-140\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-8\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24, 0, 0; 0, -140, -12; 0, -12, -1],[24, 4, 1],[-1, -1, -1, 0, 1; -6, -2, 0, 1, 0; 60, 18, -4, -12, 0],149,19]
%
% END OF LATTICE L150.19
%
\hbox{}\par\smallskip%
%
% LATTICE L150.20
%
\leavevmode%
\llap{\spacer}%
${L_{150.20}}=\pdualwitharg{2}(\main(\pfillwitharg{3}({L_{150.4}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{2,1}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\144\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-140\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-84\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-6\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-1\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\1\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-4\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-120, 144, 72; 144, -140, -84; 72, -84, -43],[24, 4, 1],[37, 8, 5, -6, -7; -6, -1, 0, 1, 0; 72, 15, 8, -12, -12],149,20]
%
% END OF LATTICE L150.20
%
\hbox{}\par\smallskip%
%
% LATTICE L150.21
%
\leavevmode%
\llap{\spacer}%
${L_{150.21}}=\pdualwitharg{3}(\main(\pfillwitharg{3}({L_{150.2}})))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{4,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8280\\240\\624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-6\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\-18\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\4\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\0\\-6\\-12\\-2
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0\\6\\0\\-24\\-12
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-6\\0\\12\\0\\-12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8280, 240, 624; 240, -6, -18; 624, -18, -47],[24, 6, 1],[0, -2, -3, -1, 1; 2, 7, 4, -4, -2; -1, -30, -42, -12, 14],149,21]
%
% END OF LATTICE L150.21
%
\hbox{}\par\smallskip%
%
% LATTICE L150.22
%
\leavevmode%
\llap{\spacer}%
${L_{150.22}}=\pdualwitharg{3}(\main(\pfillwitharg{3}({L_{150.4}})))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{4,3}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-456\\120\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-18\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-12\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-4\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-6\\-2
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0\\24\\0\\-24\\-24
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-3\\0\\3\\0\\-6
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-6\\-24\\0\\6\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-24\\-6\\0\\4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-456, 120, 48; 120, -18, -12; 48, -12, -5],[24, 6, 1],[1, 7, 1, -1, -1; -1, -4, 0, 1, 0; 11, 72, 9, -12, -10],149,22]
%
% END OF LATTICE L150.22
%
\hbox{}\par\smallskip%
%
% LATTICE L150.23
%
\leavevmode%
\llap{\spacer}%
${L_{150.23}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{150.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{2,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-144\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-120\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\40\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-24\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-16\\-12
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-6
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-16\\0\\8\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-12\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-168, -144, 48; -144, -120, 40; 48, 40, -13],[24, 8, 1],[1, -1, -1, 0, 1; -3, 0, 2, 3, 0; -6, -4, 2, 8, 3],149,23]
%
% END OF LATTICE L150.23
%
\hbox{}\par\smallskip%
%
% LATTICE L150.24
%
\leavevmode%
\llap{\spacer}%
${L_{150.24}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{150.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{12,5}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1980\\288\\-864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\-30\\126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-864\\126\\-377
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-1\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-1\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-3\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\1\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-18\\-12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4\\-6
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\18\\0\\-18
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-4\\0\\4\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-6\\-18\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1980, 288, -864; 288, -30, 126; -864, 126, -377],[36, 6, 1],[-21, -11, -16, 7, 8; -1, -1, -3, 0, 1; 48, 25, 36, -16, -18],149,24]
%
% END OF LATTICE L150.24
%
\hbox{}\par\smallskip%
%
% LATTICE L150.25
%
\leavevmode%
\llap{\spacer}%
${L_{150.25}}=\pdualwitharg{3}({L_{150.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{3,2}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136872\\3312\\3312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3312\\-75\\-81
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3312\\-81\\-80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\112\\672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\23\\142
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\36\\252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-18\\-106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-16\\-108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-72\\-24\\-24
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4\\-12
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\72\\0\\-72
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-4\\0\\4\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\-72\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-136872, 3312, 3312; 3312, -75, -81; 3312, -81, -80],[72, 3, 1],[19, 4, 7, -3, -3; 112, 23, 36, -18, -16; 672, 142, 252, -106, -108],149,25]
%
% END OF LATTICE L150.25
%
\hbox{}\par\smallskip%
%
% LATTICE L150.26
%
\leavevmode%
\llap{\spacer}%
${L_{150.26}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{150.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{8,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16560\\-4224\\1488
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4224\\-1074\\378
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1488\\378\\-133
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-18\\-40
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\50\\108
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-1\\22\\51
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\right.$}%
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8\\0\\-24\\-12\\-4
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-24\\-24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-12
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\0\\3\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-24\\-12\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16560, -4224, 1488; -4224, -1074, 378; 1488, 378, -133],[48, 6, 1],[1, -1, -3, -1, 0; -18, 4, 50, 22, 5; -40, 0, 108, 51, 14],149,26]
%
% END OF LATTICE L150.26
%
\hbox{}\par\smallskip%
%
% LATTICE L150.27
%
\leavevmode%
\llap{\spacer}%
${L_{150.27}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{150.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{8,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4464\\-1344\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1344\\-402\\102
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\102\\-25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\6\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\4\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-14\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-16\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-24\\-24\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-48\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-24\\-12\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4464, -1344, 336; -1344, -402, 102; 336, 102, -25],[48, 6, 1],[-3, -3, 7, 9, 2; 6, 4, -14, -16, -3; -16, -24, 36, 54, 14],149,27]
%
% END OF LATTICE L150.27
%
\hbox{}\par\smallskip%
%
% LATTICE L150.28
%
\leavevmode%
\llap{\spacer}%
${L_{150.28}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{150.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{8,3}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3408\\-960\\384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-960\\-270\\108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
384\\108\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-24\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-6\\-12\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-48\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-48\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-48\\-12\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3408, -960, 384; -960, -270, 108; 384, 108, -43],[48, 6, 1],[1, 7, 1, -1, -1; -5, -24, -1, 6, 2; -4, 0, 6, 6, -4],149,28]
%
% END OF LATTICE L150.28
%
\hbox{}\par\smallskip%
%
% LATTICE L150.29
%
\leavevmode%
\llap{\spacer}%
${L_{150.29}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{150.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{8,7}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-912\\-768\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-768\\-558\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\240\\-103
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-25\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-128\\-264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-9\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\13\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-6\\-6\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-24\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-24\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-48\\-12\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-912, -768, 336; -768, -558, 240; 336, 240, -103],[48, 6, 1],[2, 11, 1, -1, -1; -25, -128, -9, 13, 10; -52, -264, -18, 27, 20],149,29]
%
% END OF LATTICE L150.29
%
\hbox{}\par\smallskip%
%
% LATTICE L150.30
%
\leavevmode%
\llap{\spacer}%
${L_{150.30}}=\pdualwitharg{2.3}(\main(\pfillwitharg{3}({L_{150.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{2,1}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\24\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\36\\12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\12\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\1\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-2\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-2\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-24\\-24\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-24\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-120, 24, -48; 24, 36, 12; -48, 12, -19],[24, 12, 1],[3, 5, -5, -13, -9; 0, 1, 0, -2, -2; -8, -12, 12, 30, 20],149,30]
%
% END OF LATTICE L150.30
%
\hbox{}\par\smallskip%
%
% LATTICE L150.31
%
\leavevmode%
\llap{\spacer}%
${L_{150.31}}=\pdualwitharg{2.3}(\main(\pfillwitharg{3}({L_{150.4}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{4,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7464\\624\\432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\-36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\-36\\-25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-1\\87
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-2\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-24\\-12\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-24\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7464, 624, 432; 624, -36, -36; 432, -36, -25],[24, 12, 1],[-3, -2, 7, 5, 5; 0, 1, 0, -1, -2; -52, -36, 120, 87, 88],149,31]
%
% END OF LATTICE L150.31
%
\hbox{}\par\smallskip%
%
% LATTICE L150.32
%
\leavevmode%
\llap{\spacer}%
${L_{150.32}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{150.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{8,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17040\\6576\\-1344
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6576\\-2536\\520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1344\\520\\-105
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-63\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-31\\38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-13\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\22\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\21\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-48\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-16\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-16\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17040, 6576, -1344; 6576, -2536, 520; -1344, 520, -105],[48, 8, 1],[-31, -15, -6, 11, 10; -63, -31, -13, 22, 21; 84, 38, 12, -32, -24],149,32]
%
% END OF LATTICE L150.32
%
\hbox{}\par\smallskip%
%
% LATTICE L150.33
%
\leavevmode%
\llap{\spacer}%
${L_{150.33}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{150.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{8,5}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62544\\26496\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26496\\-11224\\712
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\712\\-45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-45\\108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-23\\46
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-11\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\14\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\15\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-48\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-16\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-16\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-62544, 26496, -1680; 26496, -11224, 712; -1680, 712, -45],[48, 8, 1],[-22, -11, -5, 7, 7; -45, -23, -11, 14, 15; 108, 46, 12, -40, -24],149,33]
%
% END OF LATTICE L150.33
%
\hbox{}\par\smallskip%
%
% LATTICE L150.34
%
\leavevmode%
\llap{\spacer}%
${L_{150.34}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{150.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{8,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53904\\17088\\-2928
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17088\\-5416\\928
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2928\\928\\-159
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\27\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-12\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-8\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-8\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-53904, 17088, -2928; 17088, -5416, 928; -2928, 928, -159],[48, 8, 1],[1, -1, -1, 0, 2; -3, -10, 1, 5, 27; -36, -40, 24, 29, 120],149,34]
%
% END OF LATTICE L150.34
%
\hbox{}\par\smallskip%
%
% LATTICE L150.35
%
\leavevmode%
\llap{\spacer}%
${L_{150.35}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{150.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{8,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\432\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\-232\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\40\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\3\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\4\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-4\\50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-9\\96
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-16\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-16\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-624, 432, -48; 432, -232, 40; -48, 40, -3],[48, 8, 1],[5, 5, -3, -7, -14; 3, 4, -1, -4, -9; -36, -32, 24, 50, 96],149,35]
%
% END OF LATTICE L150.35
%
\hbox{}\par\smallskip%
%
% LATTICE L150.36
%
\leavevmode%
\llap{\spacer}%
${L_{150.36}}=\pdualwitharg{3}(\main({L_{150.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{12,5}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\72\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-42\\30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\30\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-24\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\10\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-18\\-12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-8\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\36\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-8\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-12\\-36\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-72, 72, -72; 72, -42, 30; -72, 30, -13],[72, 6, 1],[1, -1, -7, -1, 3; 4, -3, -24, -4, 10; 3, -2, -18, -4, 6],149,36]
%
% END OF LATTICE L150.36
%
\hbox{}\par\smallskip%
%
% LATTICE L150.37
%
\leavevmode%
\llap{\spacer}%
${L_{150.37}}=\pdualwitharg{3}(\main({L_{150.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{12,11}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1368\\-1368\\504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1368\\-1122\\420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\420\\-157
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\38\\111
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\60\\176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-13\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-9\\-6\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-8\\-24
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\9\\0\\-18
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-6\\-8\\0\\2\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-24\\-18\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1368, -1368, 504; -1368, -1122, 420; 504, 420, -157],[72, 6, 1],[3, 5, 1, -1, -1; 38, 60, 9, -13, -10; 111, 176, 27, -38, -30],149,37]
%
% END OF LATTICE L150.37
%
\hbox{}\par\smallskip%
%
% LATTICE L150.38
%
\leavevmode%
\llap{\spacer}%
${L_{150.38}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{150.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{2,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5928\\-7008\\528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7008\\-8280\\624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\624\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-4\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-12\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-48\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\12\\0\\-12
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-48\\0\\24\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-24\\-12\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5928, -7008, 528; -7008, -8280, 624; 528, 624, -47],[24, 24, 1],[1, 7, 2, -2, -1; 0, -4, -3, -1, 1; 11, 24, -18, -36, 2],149,38]
%
% END OF LATTICE L150.38
%
\hbox{}\par\smallskip%
%
% LATTICE L150.39
%
\leavevmode%
\llap{\spacer}%
${L_{150.39}}=\pdualwitharg{3}({L_{150.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{24,17}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7056\\288\\720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\6\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-36\\-71
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\6\\17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\54\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-22\\-60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-36\\-24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-8\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\72\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-8\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\-72\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7056, 288, 720; 288, 6, -36; 720, -36, -71],[144, 6, 1],[0, 2, 17, 1, -7; -1, 6, 54, 4, -22; 0, 17, 144, 8, -60],149,39]
%
% END OF LATTICE L150.39
%
\hbox{}\par\smallskip%
%
% LATTICE L150.40
%
\leavevmode%
\llap{\spacer}%
${L_{150.40}}=\pdualwitharg{3}({L_{150.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{24,5}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60624\\-8208\\3456
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8208\\-1110\\468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3456\\468\\-197
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-6\\108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-36\\-24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-16\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\72\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-16\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\-72\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-60624, -8208, 3456; -8208, -1110, 468; 3456, 468, -197],[144, 6, 1],[-1, 1, 7, 1, -3; 5, 2, -6, -4, 2; -6, 22, 108, 8, -48],149,40]
%
% END OF LATTICE L150.40
%
\hbox{}\par\smallskip%
%
% LATTICE L150.41
%
\leavevmode%
\llap{\spacer}%
${L_{150.41}}=\pdualwitharg{3}({L_{150.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{24,17}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-236016\\2880\\2880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2880\\-30\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2880\\-36\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\56\\352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\35\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-36\\-48\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-16\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\18\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-16\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\-18\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-236016, 2880, 2880; 2880, -30, -36; 2880, -36, -35],[144, 6, 1],[-1, -1, 1, 5, 3; -10, -12, 9, 56, 35; -72, -70, 72, 352, 210],149,41]
%
% END OF LATTICE L150.41
%
\hbox{}\par\smallskip%
%
% LATTICE L150.42
%
\leavevmode%
\llap{\spacer}%
${L_{150.42}}=\pdualwitharg{3}({L_{150.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{24,5}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2736\\288\\720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\-30\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-72\\-143
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\104\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\71\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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24\\0\\-36\\-48\\-12
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2736, 288, 720; 288, -30, -72; 720, -72, -143],[144, 6, 1],[-1, -1, 1, 9, 6; -10, -12, 9, 104, 71; 0, 1, 0, -8, -6],149,42]
%
% END OF LATTICE L150.42
%
\hbox{}\par\smallskip%
%
% LATTICE L150.43
%
\leavevmode%
\llap{\spacer}%
${L_{150.43}}=\pdualwitharg{2}(\main({L_{150.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-12
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\HardButStrongLineBreak\kern3pt%
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-1\\0\\-18
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-4
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-72\\-24
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0\\36\\0\\-72\\-72
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-24\\0\\8\\0\\-24
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-24\\-72\\-24\\0\\24
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[360, 144, 0; 144, 60, 0; 0, 0, -1],[72, 12, 1],[1, -1, -1, -1, 1; -4, 0, 2, 3, -2; -12, -18, -4, 0, 0],149,43]
%
% END OF LATTICE L150.43
%
\hbox{}\par\smallskip%
%
% LATTICE L150.44
%
\leavevmode%
\llap{\spacer}%
${L_{150.44}}=\pdualwitharg{2}(\main({L_{150.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{6,1}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26856\\-12528\\1080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12528\\-5844\\504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1080\\504\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-32\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-27\\9
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-10\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\15\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\14\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-72\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-36\\-36
\end{tabular}\endgroup%
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-24\\0\\8\\0\\-24
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-72\\-36\\0\\36\\0
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-24\\-36\\-24\\0\\24
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26856, -12528, 1080; -12528, -5844, 504; 1080, 504, -43],[72, 12, 1],[15, 13, 5, -7, -7; -32, -27, -10, 15, 14; 0, 9, 8, 0, -12],149,44]
%
% END OF LATTICE L150.44
%
\hbox{}\par\smallskip%
%
% LATTICE L150.45
%
\leavevmode%
\llap{\spacer}%
${L_{150.45}}=\pdualwitharg{2.3}(\main({L_{150.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{12,5}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-577944\\225792\\-11304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
225792\\-88212\\4416
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11304\\4416\\-221
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\92\\252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\38\\94
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\30\\36
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-15\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-14\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-72\\-24\\-24
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-24
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-72\\0\\72\\0\\-72
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-24\\-8\\0\\4\\0
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-24\\-24\\-72\\0\\24
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-577944, 225792, -11304; 225792, -88212, 4416; -11304, 4416, -221],[72, 12, 1],[31, 13, 11, -5, -5; 92, 38, 30, -15, -14; 252, 94, 36, -44, -24],149,45]
%
% END OF LATTICE L150.45
%
\hbox{}\par\smallskip%
%
% LATTICE L150.46
%
\leavevmode%
\llap{\spacer}%
${L_{150.46}}=\pdualwitharg{2.3}(\main({L_{150.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{6,5}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22392\\-10800\\3168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10800\\-5052\\1476
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3168\\1476\\-431
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-238\\-720
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-54\\-163
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-120\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\37\\112
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\52\\156
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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24\\0\\-72\\-24\\-24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4\\-12
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-72\\0\\72\\0\\-72
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-24\\-4\\0\\4\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\-72\\0\\24
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-22392, -10800, 3168; -10800, -5052, 1476; 3168, 1476, -431],[72, 12, 1],[13, 3, 7, -2, -3; -238, -54, -120, 37, 52; -720, -163, -360, 112, 156],149,46]
%
% END OF LATTICE L150.46
%
\hbox{}\par\smallskip%
%
% LATTICE L150.47
%
\leavevmode%
\llap{\spacer}%
${L_{150.47}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{150.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{8,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-48\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-4
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\3\\6
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\5\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-3\\-8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-24\\-48\\-8
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-48\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-48\\0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-24\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[48, -48, 0; -48, -72, 48; 0, 48, -19],[48, 24, 1],[0, 4, 5, -1, -3; -1, 3, 5, 0, -3; -4, 6, 12, 0, -8],149,47]
%
% END OF LATTICE L150.47
%
\hbox{}\par\smallskip%
%
% LATTICE L150.48
%
\leavevmode%
\llap{\spacer}%
${L_{150.48}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{150.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{8,5}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2448\\672\\288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
672\\264\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\72\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-39\\116
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-79\\234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-53\\156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\40\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\27\\-80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8\\0\\-24\\-48\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-48\\-24
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-24\\0\\24\\0\\-24
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-48\\-48\\0\\48\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-24\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2448, 672, 288; 672, 264, 72; 288, 72, 17],[48, 24, 1],[3, 6, 4, -3, -2; -39, -79, -53, 40, 27; 116, 234, 156, -120, -80],149,48]
%
% END OF LATTICE L150.48
%
\hbox{}\par\smallskip%
%
% LATTICE L150.49
%
\leavevmode%
\llap{\spacer}%
${L_{150.49}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{150.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{8,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\672\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
672\\-3048\\432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\432\\-61
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\2\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\3\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-24\\-12\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-24\\-48
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\0\\3\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-48\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-144, 672, -96; 672, -3048, 432; -96, 432, -61],[48, 24, 1],[-2, -3, 4, 3, 3; -1, -4, 1, 2, 3; -4, -24, 0, 9, 16],149,49]
%
% END OF LATTICE L150.49
%
\hbox{}\par\smallskip%
%
% LATTICE L150.50
%
\leavevmode%
\llap{\spacer}%
${L_{150.50}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{150.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{8,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29808\\44736\\-864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44736\\-67128\\1296
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-864\\1296\\-25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-3\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\7\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\8\\102
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\3\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-24\\-24\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-48\\-48
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-48\\-24\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-29808, 44736, -864; 44736, -67128, 1296; -864, 1296, -25],[48, 24, 1],[-3, -3, 7, 9, 4; -3, -2, 7, 8, 3; -52, 0, 120, 102, 16],149,50]
%
% END OF LATTICE L150.50
%
\hbox{}\par\smallskip%
%
% LATTICE L150.51
%
\leavevmode%
\llap{\spacer}%
${L_{150.51}}=\pdualwitharg{2}({L_{150.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{72}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{6,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-576\\1512\\216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1512\\-3912\\-552
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\-552\\-77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-2\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\9\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\3\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-2\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-72\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\72\\0\\-144\\-36
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-6
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-144\\0\\72\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-36\\-6\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-576, 1512, 216; 1512, -3912, -552; 216, -552, -77],[72, 24, 1],[-3, 10, 4, 7, -2; -2, 9, 3, 3, -2; 6, -36, -10, 0, 9],149,51]
%
% END OF LATTICE L150.51
%
\hbox{}\par\smallskip%
%
% LATTICE L150.52
%
\leavevmode%
\llap{\spacer}%
${L_{150.52}}=\pdualwitharg{2.3}({L_{150.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{6,5}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14328\\-1584\\432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1584\\-168\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\48\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-18\\-12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-16\\-24
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\36\\0\\-36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-16\\0\\8\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-24\\-36\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14328, -1584, 432; -1584, -168, 48; 432, 48, -13],[72, 24, 1],[0, 1, 2, 0, -1; 1, 0, -3, -1, 1; 3, 32, 54, -4, -30],149,52]
%
% END OF LATTICE L150.52
%
\hbox{}\par\smallskip%
%
% LATTICE L150.53
%
\leavevmode%
\llap{\spacer}%
${L_{150.53}}=\pdualwitharg{2}({L_{150.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{2}{144}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{12,7}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-288\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\600\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-3\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-72\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-144\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-144\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-144\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[144, -288, 0; -288, 600, 0; 0, 0, -1],[144, 24, 1],[2, 1, -2, -7, -3; 1, 0, -1, -3, -1; 0, 0, -4, -18, -12],149,53]
%
% END OF LATTICE L150.53
%
\hbox{}\par\smallskip%
%
% LATTICE L150.54
%
\leavevmode%
\llap{\spacer}%
${L_{150.54}}=\pdualwitharg{2}({L_{150.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{12,7}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-229680\\-9504\\1728
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9504\\24\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1728\\72\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\1\\-792
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\-936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\524
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-3\\2106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-1\\924
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-72\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-144\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-144\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-144\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-229680, -9504, 1728; -9504, 24, 72; 1728, 72, -13],[144, 24, 1],[-6, -7, 4, 16, 7; 1, 0, -1, -3, -1; -792, -936, 524, 2106, 924],149,54]
%
% END OF LATTICE L150.54
%
\hbox{}\par\smallskip%
%
% LATTICE L150.55
%
\leavevmode%
\llap{\spacer}%
${L_{150.55}}=\pdualwitharg{2}({L_{150.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{b}{2}{144}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{24,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2469168\\19872\\17712
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19872\\-120\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17712\\-144\\-127
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-29\\-804
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-36\\-936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\19\\536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\39\\1071
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\35\\936
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-36\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-72\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-72\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-144\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2469168, 19872, 17712; 19872, -120, -144; 17712, -144, -127],[144, 24, 1],[-6, -7, 4, 8, 7; -29, -36, 19, 39, 35; -804, -936, 536, 1071, 936],149,55]
%
% END OF LATTICE L150.55
%
\hbox{}\par\smallskip%
%
% LATTICE L150.56
%
\leavevmode%
\llap{\spacer}%
${L_{150.56}}=\pdualwitharg{2}({L_{150.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{l}{2}{144}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{24,19}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-120\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-12\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-72\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-144\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-144\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-144\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[144, 0, 0; 0, -120, -72; 0, -72, -43],[144, 24, 1],[0, -1, 0, 1, 1; 7, 0, -5, -12, -1; -12, 0, 8, 18, 0],149,56]
%
% END OF LATTICE L150.56
%
\hbox{}\par\smallskip%
%
% LATTICE L150.57
%
\leavevmode%
\llap{\spacer}%
${L_{150.57}}=\pdualwitharg{2.3}({L_{150.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{12,11}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7920\\-1872\\-1728
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\-408\\-408
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1728\\-408\\-377
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\1\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-3\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-1\\38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-1\\84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-72\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-16\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\72\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-16\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\-72\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7920, -1872, -1728; -1872, -408, -408; -1728, -408, -377],[144, 24, 1],[5, 7, -7, -8, -18; 1, 0, -3, -1, -1; -24, -32, 36, 38, 84],149,57]
%
% END OF LATTICE L150.57
%
\hbox{}\par\smallskip%
%
% LATTICE L150.58
%
\leavevmode%
\llap{\spacer}%
${L_{150.58}}=\pdualwitharg{2.3}({L_{150.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{b}{2}{72}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{12,11}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1208880\\-102240\\9648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102240\\-8616\\816
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9648\\816\\-77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-376
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\468
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-1\\1116
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-72\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-16\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\72\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-16\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\-72\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1208880, -102240, 9648; -102240, -8616, 816; 9648, 816, -77],[144, 24, 1],[-2, -3, 4, 4, 9; 1, 0, -3, -1, -1; -240, -376, 468, 490, 1116],149,58]
%
% END OF LATTICE L150.58
%
\hbox{}\par\smallskip%
%
% LATTICE L150.59
%
\leavevmode%
\llap{\spacer}%
${L_{150.59}}=\pdualwitharg{2.3}({L_{150.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{24,23}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3518352\\21456\\27504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21456\\-120\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27504\\-168\\-215
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-376
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\9\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\6\\251
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\29\\1128
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-72\\-12\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-8\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\72\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-8\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\-72\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3518352, 21456, 27504; 21456, -120, -168; 27504, -168, -215],[144, 24, 1],[-2, -3, 4, 2, 9; -5, -10, 9, 6, 29; -252, -376, 504, 251, 1128],149,59]
%
% END OF LATTICE L150.59
%
\hbox{}\par\smallskip%
%
% LATTICE L150.60
%
\leavevmode%
\llap{\spacer}%
${L_{150.60}}=\pdualwitharg{2.3}({L_{150.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{24,11}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7920\\1152\\-1152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1152\\-120\\192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1152\\192\\-155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\19\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\16\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-39\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-26\\50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-49\\96
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-72\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-16\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\72\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-16\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\-72\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7920, 1152, -1152; 1152, -120, 192; -1152, 192, -155],[144, 24, 1],[8, 7, -16, -11, -21; 19, 16, -39, -26, -49; -36, -32, 72, 50, 96],149,60]
%
% END OF LATTICE L150.60
%

\medskip%
%------------------------------WEYL-GROUP-W151--------------------------------
%
\leavevmode\llap{}%
$W_{151}$%
\qquad\llap{30} lattices, $\chi=6$%
\hfill%
$\slashtwo22|22\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L151.1
%
\leavevmode%
\llap{\spacer}%
${L_{151.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-456\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-1\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-8\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-12\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-456, 48, 0; 48, -1, -2; 0, -2, 1],[24, 1, 1],[0, -1, -1, 1, 1; 0, -10, -12, 8, 9; -1, -20, -18, 20, 18],150,1]
%
% END OF LATTICE L151.1
%
\hbox{}\par\smallskip%
%
% LATTICE L151.2
%
\leavevmode%
\llap{\spacer}%
${L_{151.2}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3984\\144\\144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-2\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-6\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-48\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-4\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-4\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-24\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3984, 144, 144; 144, -2, -6; 144, -6, -5],[48, 2, 1],[-1, -1, 1, 1, 1; -4, -6, 0, 5, 6; -24, -22, 24, 22, 21],150,2]
%
% END OF LATTICE L151.2
%
\hbox{}\par\smallskip%
%
% LATTICE L151.3
%
\leavevmode%
\llap{\spacer}%
${L_{151.3}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6000\\144\\288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-2\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\-8\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\11\\15
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\60\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
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16\\0\\-48\\-16\\0
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0\\1\\0\\-4\\-3
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-6000, 144, 288; 144, -2, -8; 288, -8, -13],[48, 2, 1],[1, 1, 5, -1, -1; 8, 11, 60, -10, -12; 16, 15, 72, -16, -15],150,3]
%
% END OF LATTICE L151.3
%
\hbox{}\par\smallskip%
%
% LATTICE L151.4
%
\leavevmode%
\llap{\spacer}%
${L_{151.4}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-12\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\0
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\HardButStrongLineBreak\kern3pt%
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-1\\-6\\1
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\HardButStrongLineBreak\kern3pt%
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1\\0\\0
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\HardButStrongLineBreak\kern3pt%
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2\\11\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\30\\-6
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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16\\0\\-48\\-8\\0
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0\\1\\0\\-2\\-6
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-48\\0\\48\\0\\-48
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-8\\-2\\0\\2\\0
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0\\-6\\-48\\0\\12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[48, 0, 48; 0, -2, -12; 48, -12, -23],[48, 2, 1],[-1, -1, 1, 2, 5; -4, -6, 0, 11, 30; 0, 1, 0, -2, -6],150,4]
%
% END OF LATTICE L151.4
%
\hbox{}\par\smallskip%
%
% LATTICE L151.5
%
\leavevmode%
\llap{\spacer}%
${L_{151.5}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-816\\48\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-2\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-4\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\8\\8
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\6
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\HardButStrongLineBreak\kern3pt%
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1\\12\\0
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\HardButStrongLineBreak\kern3pt%
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-1\\-10\\-8
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-48\\-16\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-12
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-48\\0\\48\\0\\-48
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-8\\0\\8\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-48\\0\\12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-816, 48, 48; 48, -2, -4; 48, -4, -1],[48, 2, 1],[1, 1, 1, -1, -1; 8, 10, 12, -10, -12; 8, 6, 0, -8, -6],150,5]
%
% END OF LATTICE L151.5
%
\hbox{}\par\smallskip%
%
% LATTICE L151.6
%
\leavevmode%
\llap{\spacer}%
${L_{151.6}}=\pfillwitharg{2}({L_{151.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\-1\\-1
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-4
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\HardButStrongLineBreak\kern3pt%
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-1\\0\\-3
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0\\1\\0
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1\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-2\\0
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0\\4\\0\\-4\\-12
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-3\\0\\3\\0\\-12
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-2\\-4\\0\\2\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-12\\0\\12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[12, 0, 0; 0, 2, 0; 0, 0, -1],[12, 2, 1],[0, -1, -1, 0, 1; -1, -2, 0, 1, 0; -1, -4, -3, 0, 0],150,6]
%
% END OF LATTICE L151.6
%
\hbox{}\par\smallskip%
%
% LATTICE L151.7
%
\leavevmode%
\llap{\spacer}%
${L_{151.7}}=\main({L_{151.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\24\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-10\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\1\\-6
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\HardButStrongLineBreak\kern3pt%
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-1\\0\\-4
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2\\-3\\12
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\HardButStrongLineBreak\kern3pt%
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1\\-1\\5
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1\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\right.$}%
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2\\0\\-6\\-2\\0
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0\\8\\0\\-4\\-24
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-6\\0\\6\\0\\-24
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-2\\-4\\0\\1\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[24, 24, 0; 24, -10, -6; 0, -6, -1],[24, 2, 1],[-1, -1, 2, 1, 1; 1, 0, -3, -1, 0; -6, -4, 12, 5, 0],150,7]
%
% END OF LATTICE L151.7
%
\hbox{}\par\smallskip%
%
% LATTICE L151.8
%
\leavevmode%
\llap{\spacer}%
${L_{151.8}}=\main({L_{151.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-408\\48\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-2\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-4\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\8
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\10
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\HardButStrongLineBreak\kern3pt%
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1\\0\\12
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\HardButStrongLineBreak\kern3pt%
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-1\\-3\\-12
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\right.$}%
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8\\0\\-24\\-8\\0
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0\\2\\0\\-4\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\24\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-4\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-24\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-408, 48, 24; 48, -2, -4; 24, -4, -1],[24, 2, 1],[1, 1, 1, -1, -1; 4, 3, 0, -4, -3; 8, 10, 12, -10, -12],150,8]
%
% END OF LATTICE L151.8
%
\hbox{}\par\smallskip%
%
% LATTICE L151.9
%
\leavevmode%
\llap{\spacer}%
${L_{151.9}}=\pdualwitharg{3}({L_{151.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\48\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-3\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-3\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-6\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-4\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24, 48, -24; 48, -3, -3; -24, -3, 4],[24, 3, 1],[1, -1, -1, 1, 1; 5, -6, -6, 4, 5; 9, -12, -10, 12, 10],150,9]
%
% END OF LATTICE L151.9
%
\hbox{}\par\smallskip%
%
% LATTICE L151.10
%
\leavevmode%
\llap{\spacer}%
${L_{151.10}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{151.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-4\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12, 0, 0; 0, -6, 6; 0, 6, -5],[12, 6, 1],[1, 0, -1, -1, 0; 0, -2, -2, 1, 1; 0, -3, -4, 0, 1],150,10]
%
% END OF LATTICE L151.10
%
\hbox{}\par\smallskip%
%
% LATTICE L151.11
%
\leavevmode%
\llap{\spacer}%
${L_{151.11}}=\pdualwitharg{2}(\main({L_{151.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
552\\-264\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-264\\124\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-12\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-4\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-24\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[552, -264, 24; -264, 124, -12; 24, -12, 1],[24, 4, 1],[1, 0, -2, -1, 1; 1, 0, -3, -2, 0; -12, -1, 12, 4, -18],150,11]
%
% END OF LATTICE L151.11
%
\hbox{}\par\smallskip%
%
% LATTICE L151.12
%
\leavevmode%
\llap{\spacer}%
${L_{151.12}}=\pdualwitharg{2}(\main({L_{151.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-8\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-12\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[120, 48, 0; 48, 20, 0; 0, 0, -1],[24, 4, 1],[0, -1, -1, 1, 1; -1, 2, 3, -2, -3; -4, -2, 0, 0, -3],150,12]
%
% END OF LATTICE L151.12
%
\hbox{}\par\smallskip%
%
% LATTICE L151.13
%
\leavevmode%
\llap{\spacer}%
${L_{151.13}}=\pdualwitharg{3}(\main({L_{151.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1032\\336\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\102\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-24\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-6\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-6\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-8\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1032, 336, -72; 336, 102, -24; -72, -24, 5],[24, 6, 1],[-1, -2, -1, 1, 1; 0, 1, 0, -1, -1; -12, -24, -16, 9, 10],150,13]
%
% END OF LATTICE L151.13
%
\hbox{}\par\smallskip%
%
% LATTICE L151.14
%
\leavevmode%
\llap{\spacer}%
${L_{151.14}}=\pdualwitharg{3}(\main({L_{151.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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24\\0\\0
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0\\-30\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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\HardButStrongLineBreak\kern3pt%
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0\\-1\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-12\\-6
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-24\\0\\8\\0\\-8
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-24\\-12\\0\\12\\0
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0\\-6\\-8\\0\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24, 0, 0; 0, -30, 6; 0, 6, -1],[24, 6, 1],[1, 0, -1, -1, 0; 0, -1, 0, 2, 1; 0, -6, -4, 6, 4],150,14]
%
% END OF LATTICE L151.14
%
\hbox{}\par\smallskip%
%
% LATTICE L151.15
%
\leavevmode%
\llap{\spacer}%
${L_{151.15}}=\pdualwitharg{2}({L_{151.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-24\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-8\\-24
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-24\\0\\24\\0\\-48
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-8\\-8\\0\\4\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\-48\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24, 0, 0; 0, 8, 0; 0, 0, -1],[24, 8, 1],[-1, 0, 1, 0, -2; 0, 1, 0, -1, -3; -4, 0, 0, -2, -12],150,15]
%
% END OF LATTICE L151.15
%
\hbox{}\par\smallskip%
%
% LATTICE L151.16
%
\leavevmode%
\llap{\spacer}%
${L_{151.16}}=\pdualwitharg{3}({L_{151.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-96\\96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-42\\30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\30\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\20\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\8\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-13\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-7\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12\\-6
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\16\\0\\-8
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\0\\6\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-8\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-48, -96, 96; -96, -42, 30; 96, 30, -17],[48, 6, 1],[-1, -3, -1, 2, 1; 4, 20, 8, -13, -7; 0, 18, 8, -12, -7],150,16]
%
% END OF LATTICE L151.16
%
\hbox{}\par\smallskip%
%
% LATTICE L151.17
%
\leavevmode%
\llap{\spacer}%
${L_{151.17}}=\pdualwitharg{2.3}(\main({L_{151.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-984\\144\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-12\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-8\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-984, 144, 72; 144, -12, -12; 72, -12, -5],[24, 12, 1],[-1, 1, 2, 1, -1; -1, 2, 3, 0, -2; -12, 9, 20, 12, -10],150,17]
%
% END OF LATTICE L151.17
%
\hbox{}\par\smallskip%
%
% LATTICE L151.18
%
\leavevmode%
\llap{\spacer}%
${L_{151.18}}=\pdualwitharg{3}({L_{151.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1392\\-192\\96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\6\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\6\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-6\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\16\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-16\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1392, -192, 96; -192, 6, 6; 96, 6, -5],[48, 6, 1],[-1, -1, -1, 1, 1; -4, -6, -8, 5, 6; -24, -27, -32, 24, 26],150,18]
%
% END OF LATTICE L151.18
%
\hbox{}\par\smallskip%
%
% LATTICE L151.19
%
\leavevmode%
\llap{\spacer}%
${L_{151.19}}=\pdualwitharg{3}({L_{151.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112848\\1152\\2304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1152\\-6\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2304\\-24\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\11\\141
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\28\\328
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-12\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\16\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-12\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-8\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-112848, 1152, 2304; 1152, -6, -24; 2304, -24, -47],[48, 6, 1],[1, 3, 7, -1, -1; 0, 11, 28, -2, -4; 48, 141, 328, -48, -47],150,19]
%
% END OF LATTICE L151.19
%
\hbox{}\par\smallskip%
%
% LATTICE L151.20
%
\leavevmode%
\llap{\spacer}%
${L_{151.20}}=\pdualwitharg{2.3}(\main({L_{151.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-72\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-516\\156
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\156\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-11\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-4\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24, -72, 24; -72, -516, 156; 24, 156, -47],[24, 12, 1],[2, -1, -1, 1, 1; -11, 2, 5, 0, -4; -36, 6, 16, 0, -13],150,20]
%
% END OF LATTICE L151.20
%
\hbox{}\par\smallskip%
%
% LATTICE L151.21
%
\leavevmode%
\llap{\spacer}%
${L_{151.21}}=\pdualwitharg{3}({L_{151.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2928\\384\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
384\\42\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-18\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\16\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-16\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2928, 384, -144; 384, 42, -18; -144, -18, 7],[48, 6, 1],[1, -1, -1, 1, 1; 0, -4, -4, 2, 2; 24, -30, -32, 24, 26],150,21]
%
% END OF LATTICE L151.21
%
\hbox{}\par\smallskip%
%
% LATTICE L151.22
%
\leavevmode%
\llap{\spacer}%
${L_{151.22}}=\pdualwitharg{2}({L_{151.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2064\\48\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-48\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-4\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-48\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2064, 48, -48; 48, -8, 0; -48, 0, 1],[48, 8, 1],[-1, 0, 1, 0, -1; -4, 0, 6, 1, -3; -48, -1, 48, 4, -42],150,22]
%
% END OF LATTICE L151.22
%
\hbox{}\par\smallskip%
%
% LATTICE L151.23
%
\leavevmode%
\llap{\spacer}%
${L_{151.23}}=\pdualwitharg{2}({L_{151.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5232\\1056\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1056\\200\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-48\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-16\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\3\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-16\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-24\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5232, 1056, -240; 1056, 200, -48; -240, -48, 11],[48, 8, 1],[-1, -1, 1, 2, 1; -1, -2, 0, 1, 0; -26, -32, 21, 48, 24],150,23]
%
% END OF LATTICE L151.23
%
\hbox{}\par\smallskip%
%
% LATTICE L151.24
%
\leavevmode%
\llap{\spacer}%
${L_{151.24}}=\pdualwitharg{2}({L_{151.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-16\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-16\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-48\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[48, 0, 0; 0, 8, 0; 0, 0, -1],[48, 8, 1],[0, -1, -1, 0, 1; -1, -2, 0, 1, 0; -2, -8, -6, 0, 0],150,24]
%
% END OF LATTICE L151.24
%
\hbox{}\par\smallskip%
%
% LATTICE L151.25
%
\leavevmode%
\llap{\spacer}%
${L_{151.25}}=\pdualwitharg{2}({L_{151.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-672\\144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672\\-200\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\48\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\24\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-9\\-66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-48\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-8\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-48\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1680, -672, 144; -672, -200, 48; 144, 48, -11],[48, 8, 1],[-1, 1, 5, 0, -2; -6, 4, 24, 1, -9; -40, 30, 168, 4, -66],150,25]
%
% END OF LATTICE L151.25
%
\hbox{}\par\smallskip%
%
% LATTICE L151.26
%
\leavevmode%
\llap{\spacer}%
${L_{151.26}}=\pdualwitharg{2.3}({L_{151.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\456\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-24\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\8\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\-16\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24, 0, 0; 0, 456, 48; 0, 48, 5],[24, 24, 1],[1, 0, -1, -1, 0; 0, 1, 2, 2, 1; 0, -12, -20, -18, -8],150,26]
%
% END OF LATTICE L151.26
%
\hbox{}\par\smallskip%
%
% LATTICE L151.27
%
\leavevmode%
\llap{\spacer}%
${L_{151.27}}=\pdualwitharg{2.3}({L_{151.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1776\\-1440\\96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1440\\-984\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\72\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\4\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\16\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-12\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-16\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1776, -1440, 96; -1440, -984, 72; 96, 72, -5],[48, 24, 1],[-1, 1, 3, 0, -1; -2, 1, 4, 1, -1; -48, 33, 112, 12, -34],150,27]
%
% END OF LATTICE L151.27
%
\hbox{}\par\smallskip%
%
% LATTICE L151.28
%
\leavevmode%
\llap{\spacer}%
${L_{151.28}}=\pdualwitharg{2.3}({L_{151.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1968\\96\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-48\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\1\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-8\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1968, 96, -48; 96, -24, 0; -48, 0, 1],[48, 24, 1],[0, 1, 0, -1, -1; 1, 4, 0, -3, -2; 6, 48, -1, -48, -40],150,28]
%
% END OF LATTICE L151.28
%
\hbox{}\par\smallskip%
%
% LATTICE L151.29
%
\leavevmode%
\llap{\spacer}%
${L_{151.29}}=\pdualwitharg{2.3}({L_{151.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2928\\2496\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2496\\2040\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-120\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\38
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\16\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-24\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-16\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2928, 2496, -144; 2496, 2040, -120; -144, -120, 7],[48, 24, 1],[1, -1, -1, 1, 1; 0, -2, -2, 1, 1; 24, -54, -56, 36, 38],150,29]
%
% END OF LATTICE L151.29
%
\hbox{}\par\smallskip%
%
% LATTICE L151.30
%
\leavevmode%
\llap{\spacer}%
${L_{151.30}}=\pdualwitharg{2.3}({L_{151.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-528\\96\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\168\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\48\\13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-8\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-8\\26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-7\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\-32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-48\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-16\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-528, 96, 48; 96, 168, 48; 48, 48, 13],[48, 24, 1],[-1, 1, 1, 1, -1; 9, -8, -8, -7, 10; -30, 24, 26, 24, -32],150,30]
%
% END OF LATTICE L151.30
%

\medskip%
%------------------------------WEYL-GROUP-W152--------------------------------
%
\leavevmode\llap{}%
$W_{152}$%
\qquad\llap{22} lattices, $\chi=90$%
\hfill%
$4\infty\infty2\infty24\infty\infty2\infty2\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L152.1
%
\leavevmode%
\llap{\spacer}%
${L_{152.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{5,4}{\infty b}{4}\above{1pt}{1pt}{5,3}{\infty}{4}\above{1pt}{1pt}{*}{2}{100}\above{1pt}{1pt}{1,0}{\infty b}{100}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47100\\-22000\\-700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22000\\-10276\\-327
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-700\\-327\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3001\\6000\\16500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1402\\2803\\7711
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\72\\197
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\32\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-32\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-78\\-46
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
263\\-550\\-500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
387\\-800\\-1100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-8\\-62\\-650\\-1400\\-198\\-362\\-48\\-22\\-50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\-4\\-96\\-1100\\-2500\\-362\\-680\\-100\\-60\\-200\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-4\\4\\-4\\-100\\-300\\-48\\-100\\-20\\-20\\-100\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62\\-96\\-4\\4\\0\\-100\\-22\\-60\\-20\\-40\\-300\\-500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-650\\-1100\\-100\\0\\100\\-100\\-50\\-200\\-100\\-300\\-2600\\-4900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1400\\-2500\\-300\\-100\\-100\\100\\0\\-100\\-100\\-500\\-4900\\-10100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-8\\-62\\-650\\-1400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\-4\\-96\\-1100\\-2500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-4\\4\\-4\\-100\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62\\-96\\-4\\4\\0\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-650\\-1100\\-100\\0\\100\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1400\\-2500\\-300\\-100\\-100\\100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198\\-362\\-48\\-22\\-50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-362\\-680\\-100\\-60\\-200\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-100\\-20\\-20\\-100\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-60\\-20\\-40\\-300\\-500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-200\\-100\\-300\\-2600\\-4900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-100\\-100\\-500\\-4900\\-10100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-47100, -22000, -700; -22000, -10276, -327; -700, -327, -10],[100, 1, 1],[-15, -1, 15, 37, 263, 387, 43, 53, -7, -25, -163, -187; 32, 2, -32, -78, -550, -800, -88, -106, 16, 54, 350, 400; 3, 4, -4, -46, -500, -1100, -157, -290, -40, -20, -50, 0],151,1]
%
% END OF LATTICE L152.1
%
\hbox{}\par\smallskip%
%
% LATTICE L152.2
%
\leavevmode%
\llap{\spacer}%
${L_{152.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{20,19}{\infty b}{8}\above{1pt}{1pt}{20,3}{\infty z}{2}\above{1pt}{1pt}{s}{2}{50}\above{1pt}{1pt}{4,3}{\infty b}{200}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-448200\\2000\\2600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2000\\-7\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2600\\-12\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11699\\335400\\1747200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-947\\-4928
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-2064\\-10753
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\31\\148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-32\\-148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\111\\599
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\1525\\8075
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
267\\7600\\39900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-8\\-31\\-325\\-1400\\-99\\-181\\-48\\-11\\-25\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-4\\-48\\-550\\-2500\\-181\\-340\\-100\\-30\\-100\\-100
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-4\\8\\-4\\-100\\-600\\-48\\-100\\-40\\-20\\-100\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-48\\-4\\2\\0\\-100\\-11\\-30\\-20\\-20\\-150\\-500
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-325\\-550\\-100\\0\\50\\-100\\-25\\-100\\-100\\-150\\-1300\\-4900
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1400\\-2500\\-600\\-100\\-100\\200\\0\\-100\\-200\\-500\\-4900\\-20200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-8\\-31\\-325\\-1400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-4\\-48\\-550\\-2500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-4\\8\\-4\\-100\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-48\\-4\\2\\0\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-325\\-550\\-100\\0\\50\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1400\\-2500\\-600\\-100\\-100\\200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\-181\\-48\\-11\\-25\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-181\\-340\\-100\\-30\\-100\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-100\\-40\\-20\\-100\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-30\\-20\\-20\\-150\\-500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-100\\-100\\-150\\-1300\\-4900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-100\\-200\\-500\\-4900\\-20200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-448200, 2000, 2600; 2000, -7, -12; 2600, -12, -15],[200, 1, 1],[1, 0, -1, 4, 54, 267, 20, 39, 13, 5, 21, 33; 31, 1, -32, 111, 1525, 7600, 571, 1117, 376, 147, 625, 1000; 148, -1, -148, 599, 8075, 39900, 2988, 5825, 1940, 745, 3125, 4900],151,2]
%
% END OF LATTICE L152.2
%
\hbox{}\par\smallskip%
%
% LATTICE L152.3
%
\leavevmode%
\llap{\spacer}%
${L_{152.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{4}{2}\above{1pt}{1pt}{20,19}{\infty a}{8}\above{1pt}{1pt}{20,13}{\infty z}{2}\above{1pt}{1pt}{b}{2}{50}\above{1pt}{1pt}{4,3}{\infty a}{200}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-6269800, 2600, 60000; 2600, -1, -25; 60000, -25, -574],[200, 1, 1],[-1, 0, 3, 13, 138, 599, 85, 78, 21, 5, 12, 1; -150, 18, 484, 2028, 21450, 92900, 13170, 12072, 3236, 762, 1800, 100; -98, -1, 292, 1269, 13475, 58500, 8302, 7619, 2052, 489, 1175, 100],151,3]
%
% END OF LATTICE L152.3
%
\hbox{}\par\smallskip%
%
% LATTICE L152.4
%
\leavevmode%
\llap{\spacer}%
${L_{152.4}}=\pfillwitharg{2}({L_{152.1}})$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[350, 500, -275; 500, 715, -393; -275, -393, 216],[25, 1, 1],[1, -1, -3, -24, -253, -547, -155, -142, -19, -9, -22, -3; -4, -1, 1, -1, -25, -75, -24, -25, -5, -5, -25, -25; -6, -3, -2, -33, -375, -850, -246, -231, -34, -21, -75, -50],151,4]
%
% END OF LATTICE L152.4
%
\hbox{}\par\smallskip%
%
% LATTICE L152.5
%
\leavevmode%
\llap{\spacer}%
${L_{152.5}}=\pfillwitharg{2}({L_{152.2}})$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-23550, 1550, -350; 1550, -102, 23; -350, 23, -5],[50, 1, 1],[1, 0, -1, -2, -12, -13, -1, 0, 1, 2, 12, 13; 16, 1, -16, -39, -275, -400, -44, -53, 8, 27, 175, 200; 3, 4, -4, -46, -500, -1100, -157, -290, -40, -20, -50, 0],151,5]
%
% END OF LATTICE L152.5
%
\hbox{}\par\smallskip%
%
% LATTICE L152.6
%
\leavevmode%
\llap{\spacer}%
${L_{152.6}}=\main({L_{152.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-4900, 100, 1300; 100, -2, -25; 1300, -25, -287],[100, 1, 1],[85, 78, 21, 5, 12, 1, -1, 0, 3, 13, 138, 599; 6585, 6036, 1618, 381, 900, 50, -75, 9, 242, 1014, 10725, 46450; -198, -181, -48, -11, -25, 0, 2, -1, -8, -31, -325, -1400],151,6]
%
% END OF LATTICE L152.6
%
\hbox{}\par\smallskip%
%
% LATTICE L152.7
%
\leavevmode%
\llap{\spacer}%
${L_{152.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{152.2}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6900, -250, 3300; -250, 190, -108; 3300, -108, 1579],[50, 2, 1],[-99, -75, -37, -1353, -15423, -17501, -5067, -9519, -701, -861, -3027, -949; -13, -10, -5, -180, -2050, -2325, -673, -1264, -93, -114, -400, -125; 206, 156, 77, 2816, 32100, 36425, 10546, 19812, 1459, 1792, 6300, 1975],151,7]
%
% END OF LATTICE L152.7
%
\hbox{}\par\smallskip%
%
% LATTICE L152.8
%
\leavevmode%
\llap{\spacer}%
${L_{152.8}}=\pdualwitharg{2}({L_{152.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1837600, -37300, 458500; -37300, 760, -9308; 458500, -9308, 114401],[100, 4, 1],[-102, -39, -19, -693, -7899, -8963, -5190, -4875, -359, -441, -1551, -487; 187, 70, 35, 1285, 14650, 16625, 9627, 9043, 666, 818, 2875, 900; 424, 162, 79, 2882, 32850, 37275, 21584, 20274, 1493, 1834, 6450, 2025],151,8]
%
% END OF LATTICE L152.8
%
\hbox{}\par\smallskip%
%
% LATTICE L152.9
%
\leavevmode%
\llap{\spacer}%
${L_{152.9}}=\pdualwitharg{2}(\main({L_{152.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[45400, 22700, 11200; 22700, 11348, 5600; 11200, 5600, 2763],[100, 4, 1],[995, 889, 107, 36, 37, -37, -1, 34, 35, 177, 1738, 3587; 280, 253, 32, 13, 25, 0, 0, 7, 8, 47, 475, 1000; -4596, -4112, -498, -172, -200, 150, 4, -152, -158, -812, -8000, -16550],151,9]
%
% END OF LATTICE L152.9
%
\hbox{}\par\smallskip%
%
% LATTICE L152.10
%
\leavevmode%
\llap{\spacer}%
${L_{152.10}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{152.1}}))$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25, -50, -75; -50, -250, -125; -75, -125, 9],[25, 25, 1],[-2, -36, 1, 254, 120, 277, 2018, 1908, 287, 178, 24, 11; 1, 20, 0, -135, -64, -148, -1079, -1021, -154, -96, -13, -6; 0, -25, 0, 175, 83, 192, 1400, 1325, 200, 125, 17, 8],151,10]
%
% END OF LATTICE L152.10
%
\hbox{}\par\smallskip%
%
% LATTICE L152.11
%
\leavevmode%
\llap{\spacer}%
${L_{152.11}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{152.2}}))$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3250, -400, 1450; -400, 200, -175; 1450, -175, 647],[50, 25, 1],[56, 45, 45, 785, 357, 809, 2926, 5493, 807, 493, 69, 43; 3, 3, 2, 33, 15, 34, 123, 231, 34, 21, 3, 2; -125, -100, -100, -1750, -796, -1804, -6525, -12250, -1800, -1100, -154, -96],151,11]
%
% END OF LATTICE L152.11
%
\hbox{}\par\smallskip%
%
% LATTICE L152.12
%
\leavevmode%
\llap{\spacer}%
${L_{152.12}}=\pdualwitharg{2}({L_{152.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[256600, -14800, 308800; -14800, 856, -17840; 308800, -17840, 371979],[200, 8, 1],[57, 59, 5, 277, 3263, 3781, 1105, 2099, 161, 221, 887, 369; -1432, -1481, -122, -6881, -81125, -94050, -27492, -52235, -4010, -5515, -22175, -9250; -116, -120, -10, -560, -6600, -7650, -2236, -4248, -326, -448, -1800, -750],151,12]
%
% END OF LATTICE L152.12
%
\hbox{}\par\smallskip%
%
% LATTICE L152.13
%
\leavevmode%
\llap{\spacer}%
${L_{152.13}}=\pdualwitharg{2}({L_{152.2}})$%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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-88\\-240\\-40\\-160\\-1200\\-1000
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0\\-200\\-100\\-1000\\-9800\\-10100
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1531600, -7800, -11000; -7800, 40, 56; -11000, 56, 79],[200, 8, 1],[-1, -1, 0, -3, -37, -44, -13, -25, -2, -3, -13, -6; 7, 6, -3, -44, -450, -475, -133, -240, -15, -10, 0, 25; -144, -144, 2, -384, -4800, -5750, -1704, -3288, -266, -408, -1800, -850],151,13]
%
% END OF LATTICE L152.13
%
\hbox{}\par\smallskip%
%
% LATTICE L152.14
%
\leavevmode%
\llap{\spacer}%
${L_{152.14}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{152.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${50}\above{1pt}{1pt}{}{4}{100}\above{1pt}{1pt}{20,1}{\infty z}{25}\above{1pt}{1pt}{10,7}{\infty}{100}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-250\\-21350\\-10450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21350\\-1765100\\-863950
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10450\\-863950\\-422871
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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197\\-1764\\3600
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19822\\-176597\\360400
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9702\\-86436\\176399
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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3\\416\\-850
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4\\-49\\100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2\\-208\\425
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\HardButStrongLineBreak\kern3pt%
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-46\\-489\\1000
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\HardButStrongLineBreak\kern3pt%
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-20\\-37\\76
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\HardButStrongLineBreak\kern3pt%
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-22\\80\\-163
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-1550\\-2400\\-50\\100\\0\\-50
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-550\\-1500\\-250\\-1000\\-300\\-250
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-50\\-200\\-50\\-300\\-104\\-98
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-50\\-25\\-250\\-98\\-101
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-250, -21350, -10450; -21350, -1765100, -863950; -10450, -863950, -422871],[50, 50, 1],[3, 4, -2, -46, -20, -22, -157, -290, -20, -20, -2, 0; 416, -49, -208, -489, -37, 80, 956, 2597, 404, 1077, 233, 116; -850, 100, 425, 1000, 76, -163, -1950, -5300, -825, -2200, -476, -237],151,14]
%
% END OF LATTICE L152.14
%
\hbox{}\par\smallskip%
%
% LATTICE L152.15
%
\leavevmode%
\llap{\spacer}%
${L_{152.15}}=\pdualwitharg{5}({L_{152.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${50}\above{1pt}{1pt}{*}{4}{100}\above{1pt}{1pt}{5,1}{\infty a}{100}\above{1pt}{1pt}{5,2}{\infty}{100}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{1,0}{\infty b}{4}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-395100\\-10100\\8800
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8800\\225\\-196
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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1703\\9372\\85200
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36\\197\\1800
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-38\\-209\\-1901
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\50
\end{tabular}\endgroup%
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9\\-4\\400
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11\\-46\\450
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1\\-20\\26
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\HardButStrongLineBreak\kern3pt%
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-3\\-44\\-176
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-4950\\-9050\\-1200\\-550\\-50\\0
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-1200\\-2500\\-500\\-500\\-100\\-100
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-550\\-1500\\-500\\-1000\\-300\\-500
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-50\\-200\\-100\\-300\\-104\\-196
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-100\\-100\\-500\\-196\\-404
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-395100, -10100, 8800; -10100, -250, 225; 8800, 225, -196],[100, 25, 1],[-9, 1, 9, 11, 1, -3, -19, -53, -17, -23, -5, -5; 3, 4, -4, -46, -20, -44, -157, -290, -40, -20, -2, 0; -400, 50, 400, 450, 26, -176, -1000, -2650, -800, -1050, -226, -224],151,15]
%
% END OF LATTICE L152.15
%
\hbox{}\par\smallskip%
%
% LATTICE L152.16
%
\leavevmode%
\llap{\spacer}%
${L_{152.16}}=\pdualwitharg{5}(\main({L_{152.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${50}\above{1pt}{1pt}{}{4}{25}\above{1pt}{1pt}{20,11}{\infty}{100}\above{1pt}{1pt}{10,7}{\infty z}{25}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{4,3}{\infty}{4}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204500\\-5700\\4500
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5700\\-150\\125
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-204500, -5700, 4500; -5700, -150, 125; 4500, 125, -99],[100, 25, 1],[408, 367, 91, 17, 1, -1, -2, 10, 25, 70, 28, 117; 973, 879, 222, 44, 3, -2, -7, 18, 54, 163, 66, 278; 19700, 17725, 4400, 825, 49, -48, -100, 475, 1200, 3375, 1351, 5648],151,16]
%
% END OF LATTICE L152.16
%
\hbox{}\par\smallskip%
%
% LATTICE L152.17
%
\leavevmode%
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${L_{152.17}}=\pdualwitharg{5}({L_{152.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-89800, 6400, -36800; 6400, -375, 2700; -36800, 2700, -15007],[200, 25, 1],[203, -12, -191, 968, 510, 2493, 4648, 9021, 2963, 1111, 183, 279; 404, -25, -380, 1935, 1019, 4980, 9284, 18017, 5916, 2217, 365, 556; -425, 25, 400, -2025, -1067, -5216, -9725, -18875, -6200, -2325, -383, -584],151,17]
%
% END OF LATTICE L152.17
%
\hbox{}\par\smallskip%
%
% LATTICE L152.18
%
\leavevmode%
\llap{\spacer}%
${L_{152.18}}=\pdualwitharg{5}({L_{152.3}})$%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-409000, -210200, 9000; -210200, -108025, 4625; 9000, 4625, -198],[200, 25, 1],[5, -8, -29, -93, -38, -161, -565, -512, -131, -27, -2, 1; -14, 36, 108, 326, 132, 556, 1946, 1758, 444, 88, 6, -4; -100, 475, 1200, 3375, 1351, 5648, 19700, 17725, 4400, 825, 49, -48],151,18]
%
% END OF LATTICE L152.18
%
\hbox{}\par\smallskip%
%
% LATTICE L152.19
%
\leavevmode%
\llap{\spacer}%
${L_{152.19}}=\pdualwitharg{2.5}({L_{152.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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400\\15900\\3800
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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1\\-245\\1025
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\-50\\-25\\-250\\-98\\-101
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[400, 15900, 3800; 15900, 2681000, 640800; 3800, 640800, 153161],[100, 100, 1],[6, 3, 1, 33, 15, 17, 246, 231, 17, 21, 3, 1; -1219, -490, -245, -8545, -3886, -4403, -63699, -59791, -4392, -5366, -751, -234; 5100, 2050, 1025, 35750, 16258, 18421, 266500, 250150, 18375, 22450, 3142, 979],151,19]
%
% END OF LATTICE L152.19
%
\hbox{}\par\smallskip%
%
% LATTICE L152.20
%
\leavevmode%
\llap{\spacer}%
${L_{152.20}}=\pdualwitharg{2.5}(\main({L_{152.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${200}\above{1pt}{1pt}{}{4}{100}\above{1pt}{1pt}{20,1}{\infty}{100}\above{1pt}{1pt}{5,2}{\infty b}{100}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{4,1}{\infty}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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4500\\-80300\\-19200
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-80300\\1527400\\365200
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19200\\365200\\87319
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4500, -80300, -19200; -80300, 1527400, 365200; -19200, 365200, 87319],[100, 100, 1],[-362, -325, -40, -15, -1, 0, -2, -13, -12, -63, -25, -52; 72379, 65232, 8163, 3227, 240, 11, 239, 2199, 2211, 12334, 4947, 10363; -302800, -272900, -34150, -13500, -1004, -46, -1000, -9200, -9250, -51600, -20696, -43354],151,20]
%
% END OF LATTICE L152.20
%
\hbox{}\par\smallskip%
%
% LATTICE L152.21
%
\leavevmode%
\llap{\spacer}%
${L_{152.21}}=\pdualwitharg{2.5}({L_{152.3}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${200}\above{1pt}{1pt}{*}{4}{400}\above{1pt}{1pt}{40,21}{\infty z}{100}\above{1pt}{1pt}{10,7}{\infty a}{400}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{8,5}{\infty z}{4}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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3\\0\\424
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1800400, 1237800, -12800; 1237800, 851000, -8800; -12800, -8800, 91],[200, 200, 1],[123, 231, 17, 21, 3, 1, 3, 3, 1, 33, 15, 17; -133, -240, -15, -10, 0, 1, 7, 6, -3, -44, -18, -19; 4500, 9400, 950, 2000, 424, 238, 1100, 1000, -150, 400, 376, 562],151,21]
%
% END OF LATTICE L152.21
%
\hbox{}\par\smallskip%
%
% LATTICE L152.22
%
\leavevmode%
\llap{\spacer}%
${L_{152.22}}=\pdualwitharg{2.5}({L_{152.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1000\\-42200\\-20800
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-42200\\1800400\\887400
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\kern3pt%
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-20800\\887400\\437391
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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107\\7692\\-15600
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-5094\\-362807\\735800
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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0\\0\\0
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-10\\-1479\\3000
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0\\-209\\424
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1\\-68\\138
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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-2400\\-5000\\-500\\-1000\\-200\\-100
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0\\-200\\-100\\-1000\\-392\\-404
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1000, -42200, -20800; -42200, 1800400, 887400; -20800, 887400, 437391],[200, 200, 1],[-133, -240, -15, -10, 0, 1, 7, 6, -3, -44, -18, -19; -8777, -16469, -1208, -1479, -209, -68, -197, -197, -74, -2367, -1073, -1214; 17800, 33400, 2450, 3000, 424, 138, 400, 400, 150, 4800, 2176, 2462],151,22]
%
% END OF LATTICE L152.22
%

\medskip%
%------------------------------WEYL-GROUP-W153--------------------------------
%
\leavevmode\llap{}%
$W_{153}$%
\qquad\llap{22} lattices, $\chi=60$%
\hfill%
$\infty222\infty\infty222\infty\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L153.1
%
\leavevmode%
\llap{\spacer}%
${L_{153.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{5,2}{\infty b}{4}\above{1pt}{1pt}{r}{2}{50}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{5,4}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
255300\\86500\\-16400
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
86500\\29308\\-5557
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16400\\-5557\\1054
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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179199\\-627200\\-518000
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60608\\-212129\\-175195
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-11392\\39872\\32929
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-9311\\32600\\26975
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\HardButStrongLineBreak\kern3pt%
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-478\\1674\\1387
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-411\\1440\\1196
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\0\\50\\0\\-100
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-6\\-4\\0\\2\\0
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\kern3pt%
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-12\\-28\\-100\\0\\4
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\kern3pt%
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-8\\-52\\-250\\-10\\-4
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-52\\-788\\-4200\\-220\\-196
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-250\\-4200\\-22550\\-1200\\-1100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-220\\-1200\\-66\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-196\\-1100\\-64\\-68
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[255300, 86500, -16400; 86500, 29308, -5557; -16400, -5557, 1054],[100, 1, 1],[-131, -1763, -9311, -478, -411, 3, -29, -289, -34, -101; 458, 6172, 32600, 1674, 1440, -10, 100, 1000, 118, 352; 376, 5104, 26975, 1387, 1196, -6, 76, 775, 93, 284],152,1]
%
% END OF LATTICE L153.1
%
\hbox{}\par\smallskip%
%
% LATTICE L153.2
%
\leavevmode%
\llap{\spacer}%
${L_{153.2}}$%
\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-322600, -97800, 8000; -97800, -29649, 2425; 8000, 2425, -198],[200, 1, 1],[-20, 17, 43, -13, -149, -156, -4945, -26443, -1395, -1259; 70, -60, -150, 46, 524, 548, 17364, 92850, 4898, 4420; 49, -48, -100, 38, 396, 407, 12816, 68500, 3610, 3252],152,2]
%
% END OF LATTICE L153.2
%
\hbox{}\par\smallskip%
%
% LATTICE L153.3
%
\leavevmode%
\llap{\spacer}%
${L_{153.3}}$%
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{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{1}{}$}\spacer%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1566600, -311000, 6200; -311000, -61739, 1230; 6200, 1230, -23],[200, 1, 1],[-54, 55, 54, -22, -447, -457, -14363, -38379, -2022, -3641; 275, -280, -275, 112, 2276, 2327, 73136, 195425, 10296, 18540; 149, -148, -150, 59, 1216, 1247, 39236, 104850, 5525, 9952],152,3]
%
% END OF LATTICE L153.3
%
\hbox{}\par\smallskip%
%
% LATTICE L153.4
%
\leavevmode%
\llap{\spacer}%
${L_{153.4}}=\pfillwitharg{2}({L_{153.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[50, -25, 0; -25, 12, -5; 0, -5, -49],[25, 1, 1],[3, -5, -1, 9, 29, 52, 775, 8251, 431, 191; 5, -10, 0, 18, 56, 99, 1466, 15600, 814, 360; -1, 1, 0, -2, -7, -13, -197, -2100, -110, -49],152,4]
%
% END OF LATTICE L153.4
%
\hbox{}\par\smallskip%
%
% LATTICE L153.5
%
\leavevmode%
\llap{\spacer}%
${L_{153.5}}=\pfillwitharg{2}({L_{153.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{10,7}{\infty a}{2}\above{1pt}{1pt}{}{2}{25}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{10,9}{\infty}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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127650\\-49350\\-8200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49350\\19079\\3170
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8200\\3170\\527
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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-134401\\-313600\\-204400
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52032\\121407\\79132
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8544\\19936\\12993
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-2\\-5\\-1
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21\\50\\26
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211\\500\\275
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25\\59\\34
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75\\176\\108
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-2\\2\\0\\-2\\-14
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-3\\-2\\0\\1\\0
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-6\\-14\\-50\\0\\2
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-4\\-26\\-125\\-5\\-2
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-26\\-394\\-2100\\-110\\-98
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-125\\-2100\\-11275\\-600\\-550
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-5\\-110\\-600\\-33\\-32
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-2\\-98\\-550\\-32\\-34
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[127650, -49350, -8200; -49350, 19079, 3170; -8200, 3170, 527],[50, 1, 1],[-2, 21, 211, 25, 75, 98, 1323, 6989, 359, 309; -5, 50, 500, 59, 176, 229, 3086, 16300, 837, 720; -1, 26, 275, 34, 108, 147, 2018, 10675, 550, 476],152,5]
%
% END OF LATTICE L153.5
%
\hbox{}\par\smallskip%
%
% LATTICE L153.6
%
\leavevmode%
\llap{\spacer}%
${L_{153.6}}=\main({L_{153.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{20,7}{\infty}{4}\above{1pt}{1pt}{b}{2}{50}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{10,9}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-161300\\63500\\4000
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63500\\-24998\\-1575
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\kern3pt%
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4000\\-1575\\-99
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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984199\\2286600\\3374400
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-388094\\-901663\\-1330608
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24073\\-55929\\-82537
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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118\\274\\407
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3737\\8682\\12816
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19982\\46425\\68500
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1054\\2449\\3610
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951\\2210\\3252
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-161300, 63500, 4000; 63500, -24998, -1575; 4000, -1575, -99],[100, 1, 1],[118, 3737, 19982, 1054, 951, 15, -13, -32, 10, 113; 274, 8682, 46425, 2449, 2210, 35, -30, -75, 23, 262; 407, 12816, 68500, 3610, 3252, 49, -48, -100, 38, 396],152,6]
%
% END OF LATTICE L153.6
%
\hbox{}\par\smallskip%
%
% LATTICE L153.7
%
\leavevmode%
\llap{\spacer}%
${L_{153.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{153.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{20,17}{\infty z}{1}\above{1pt}{1pt}{}{2}{50}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{10,9}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-39550\\-7400\\-22150
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-7400\\-1262\\-4090
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-39550, -7400, -22150; -7400, -1262, -4090; -22150, -4090, -12381],[50, 2, 1],[145, 11, -1717, -332, -530, -1311, -8173, -85733, -4332, -1802; 135, 10, -1600, -309, -493, -1219, -7598, -79700, -4027, -1675; -304, -23, 3600, 696, 1111, 2748, 17131, 179700, 9080, 3777],152,7]
%
% END OF LATTICE L153.7
%
\hbox{}\par\smallskip%
%
% LATTICE L153.8
%
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${L_{153.8}}=\pdualwitharg{2}(\main({L_{153.3}}))$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[44200, 6500, 13400; 6500, 1252, 2100; 13400, 2100, 4119],[100, 4, 1],[1379, 22319, 239059, 12653, 5745, 216, 11, 191, 107, 635; 2523, 40832, 437350, 23148, 10510, 395, 20, 350, 196, 1162; -5772, -93418, -1000600, -52960, -24046, -904, -46, -800, -448, -2658],152,8]
%
% END OF LATTICE L153.8
%
\hbox{}\par\smallskip%
%
% LATTICE L153.9
%
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${L_{153.9}}=\pdualwitharg{2}({L_{153.1}})$%
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% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12073200, 246900, -2968400; 246900, -5048, 60704; -2968400, 60704, -729831],[100, 4, 1],[-555, -3463, -72658, -3672, -764, 61, 5, -1442, -280, -224; 1066, 6637, 139225, 7033, 1462, -119, -8, 2825, 543, 432; 2346, 14637, 307100, 15520, 3229, -258, -21, 6100, 1184, 947],152,9]
%
% END OF LATTICE L153.9
%
\hbox{}\par\smallskip%
%
% LATTICE L153.10
%
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${L_{153.10}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{153.1}}))$%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2450, -250, 1225; -250, 25, -125; 1225, -125, 613],[25, 25, 1],[-10, 0, 1, 1, -18, -47, -798, -343, -457, -210; 24, 1, 0, 8, 63, 142, 2323, 996, 1320, 601; 25, 0, -2, 0, 50, 125, 2100, 902, 1200, 550],152,10]
%
% END OF LATTICE L153.10
%
\hbox{}\par\smallskip%
%
% LATTICE L153.11
%
\leavevmode%
\llap{\spacer}%
${L_{153.11}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{153.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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-5884350\\188000\\-2812300
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\kern3pt%
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-50\\-2450\\-550\\-800\\-850
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5884350, 188000, -2812300; 188000, -5925, 89850; -2812300, 89850, -1344079],[50, 25, 1],[-1445, -215, 685, 3308, 10557, 13053, 162727, 34139, 43124, 35875; 23, 2, -11, -52, -164, -201, -2494, -523, -660, -548; 3025, 450, -1434, -6925, -22100, -27325, -340650, -71466, -90275, -75100],152,11]
%
% END OF LATTICE L153.11
%
\hbox{}\par\smallskip%
%
% LATTICE L153.12
%
\leavevmode%
\llap{\spacer}%
${L_{153.12}}=\pdualwitharg{2}({L_{153.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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9600\\21928\\-5024
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4200, 9600, -2200; 9600, 21928, -5024; -2200, -5024, 1151],[200, 8, 1],[-69, -520, -5541, -290, -129, -5, 2, -9, -6, -19; 221, 1657, 17650, 923, 410, 15, -5, 50, 21, 62; 832, 6234, 66400, 3472, 1542, 56, -18, 200, 80, 234],152,12]
%
% END OF LATTICE L153.12
%
\hbox{}\par\smallskip%
%
% LATTICE L153.13
%
\leavevmode%
\llap{\spacer}%
${L_{153.13}}=\pdualwitharg{2}({L_{153.2}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[23454200, 9316000, -231400; 9316000, 3700312, -91912; -231400, -91912, 2283],[200, 8, 1],[-95, -717, -7641, -400, -178, -7, 3, -9, -8, -26; 299, 2248, 23950, 1253, 557, 21, -8, 50, 27, 83; 2408, 17826, 189700, 9900, 4382, 136, -18, 1100, 276, 706],152,13]
%
% END OF LATTICE L153.13
%
\hbox{}\par\smallskip%
%
% LATTICE L153.14
%
\leavevmode%
\llap{\spacer}%
${L_{153.14}}=\pdualwitharg{5}(\main({L_{153.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${25}\above{1pt}{1pt}{20,3}{\infty}{100}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{50}\above{1pt}{1pt}{b}{2}{100}\above{1pt}{1pt}{10,1}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-125\\-5500\\-1200\\-1650\\-1600
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-50\\-4900\\-1100\\-1600\\-1700
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4900, 1300, 100; 1300, -275, -25; 100, -25, -2],[100, 25, 1],[0, -5, -1, -1, 1, 2, 61, 13, 17, 15; -3, -4, 0, 2, 0, -5, -220, -48, -66, -64; 25, -250, -57, -75, 50, 150, 5150, 1107, 1475, 1350],152,14]
%
% END OF LATTICE L153.14
%
\hbox{}\par\smallskip%
%
% LATTICE L153.15
%
\leavevmode%
\llap{\spacer}%
${L_{153.15}}=\pdualwitharg{5}({L_{153.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${100}\above{1pt}{1pt}{5,3}{\infty a}{100}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{50}\above{1pt}{1pt}{l}{2}{100}\above{1pt}{1pt}{5,1}{\infty}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2304700, 523500, 24100; 523500, -118850, -5475; 24100, -5475, -252],[100, 25, 1],[-53, -17, 4, 23, 3, -93, -2027, -442, -607, -587; -56, -24, 3, 23, 4, -94, -2076, -453, -623, -604; -3850, -1100, 318, 1700, 200, -6850, -148700, -32418, -44500, -43000],152,15]
%
% END OF LATTICE L153.15
%
\hbox{}\par\smallskip%
%
% LATTICE L153.16
%
\leavevmode%
\llap{\spacer}%
${L_{153.16}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{153.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26350, -6946500, -3402200; -6946500, 1832333950, 897425550; -3402200, 897425550, 439533753],[50, 50, 1],[-1, 13, 11, 34, 54, 147, 1009, 427, 550, 238; -2155, 11375, 9148, 27109, 40688, 106379, 718193, 303524, 389787, 167760; 4400, -23225, -18678, -55350, -83075, -217200, -1466375, -619722, -795850, -342525],152,16]
%
% END OF LATTICE L153.16
%
\hbox{}\par\smallskip%
%
% LATTICE L153.17
%
\leavevmode%
\llap{\spacer}%
${L_{153.17}}=\pdualwitharg{5}({L_{153.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-39400, 4800, 200; 4800, -550, -25; 200, -25, -1],[200, 25, 1],[0, -5, -1, -1, 1, 2, 61, 13, 17, 15; -3, -24, -4, -2, 4, 3, 24, 4, 2, -4; 50, -500, -114, -150, 100, 300, 10300, 2214, 2950, 2700],152,17]
%
% END OF LATTICE L153.17
%
\hbox{}\par\smallskip%
%
% LATTICE L153.18
%
\leavevmode%
\llap{\spacer}%
${L_{153.18}}=\pdualwitharg{5}({L_{153.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-52710600, -182800, -26549800; -182800, -575, -91900; -26549800, -91900, -13372351],[200, 25, 1],[-7389, -16727, -675, 1838, 1875, -9190, -447485, -49062, -68154, -134507; -44337, -100376, -4051, 11028, 11252, -55141, -2685008, -294383, -408940, -807076; 14975, 33900, 1368, -3725, -3800, 18625, 906900, 99432, 138125, 272600],152,18]
%
% END OF LATTICE L153.18
%
\hbox{}\par\smallskip%
%
% LATTICE L153.19
%
\leavevmode%
\llap{\spacer}%
${L_{153.19}}=\pdualwitharg{2.5}(\main({L_{153.3}}))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[200, 7100, 1700; 7100, 303700, 72700; 1700, 72700, 17403],[100, 100, 1],[8, 17, 5, 1, 1, 9, 221, 97, 135, 67; 311, 634, 178, 0, -12, 287, 7738, 3410, 4784, 2404; -1300, -2650, -744, 0, 50, -1200, -32350, -14256, -20000, -10050],152,19]
%
% END OF LATTICE L153.19
%
\hbox{}\par\smallskip%
%
% LATTICE L153.20
%
\leavevmode%
\llap{\spacer}%
${L_{153.20}}=\pdualwitharg{2.5}({L_{153.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[25600, -1780500, -426300; -1780500, 124041800, 29699000; -426300, 29699000, 7110753],[100, 100, 1],[-223, -246, -144, -22, 1, -197, -2694, -2376, -3338, -841; -5522, -6079, -3559, -551, 6, -4947, -67204, -59255, -83201, -20944; 23050, 25375, 14856, 2300, -25, 20650, 280525, 247344, 347300, 87425],152,20]
%
% END OF LATTICE L153.20
%
\hbox{}\par\smallskip%
%
% LATTICE L153.21
%
\leavevmode%
\llap{\spacer}%
${L_{153.21}}=\pdualwitharg{2.5}({L_{153.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-686200, -57400, -1200; -57400, -4400, 0; -1200, 0, 23],[200, 200, 1],[50, 5, -5, -23, -1, 92, 989, 431, 591, 285; -653, -66, 65, 300, 13, -1201, -12912, -5627, -7716, -3721; 2600, 250, -264, -1200, -50, 4800, 51550, 22464, 30800, 14850],152,21]
%
% END OF LATTICE L153.21
%
\hbox{}\par\smallskip%
%
% LATTICE L153.22
%
\leavevmode%
\llap{\spacer}%
${L_{153.22}}=\pdualwitharg{2.5}({L_{153.2}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\-200\\-200\\-600\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\100\\0\\-200\\-700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\0\\8\\0\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-600\\-200\\0\\200\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-600\\-700\\-200\\0\\100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800\\-2600\\-1000\\-1000\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2600\\-19700\\-8400\\-11000\\-4900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1000\\-8400\\-3608\\-4800\\-2200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1000\\-11000\\-4800\\-6600\\-3200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-4900\\-2200\\-3200\\-1700
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-454623600, 16186200, 8768800; 16186200, -576200, -312200; 8768800, -312200, -169133],[200, 200, 1],[-519, 32, 86, 275, -1, -1073, -11176, -4862, -6643, -3183; 50, 5, -5, -23, -1, 92, 989, 431, 591, 285; -27000, 1650, 4468, 14300, -50, -55800, -581250, -252868, -345500, -165550],152,22]
%
% END OF LATTICE L153.22
%

\medskip%
%------------------------------WEYL-GROUP-W154--------------------------------
%
\leavevmode\llap{}%
$W_{154}$%
\qquad\llap{3} lattices, $\chi=2$%
\hfill%
$\slashthree2\slashtwo2\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L154.1
%
\leavevmode%
\llap{\spacer}%
${L_{154.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{+}{3}{6}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
252\\72\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\6\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-27\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-7\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\17\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-42\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\-6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\6\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[252, 72, -36; 72, 6, -27; -36, -27, -14],[36, 3, 1],[3, -7, -1, 17; -7, 17, 2, -42; 6, -15, -2, 36],153,1]
%
% END OF LATTICE L154.1
%
\hbox{}\par\smallskip%
%
% LATTICE L154.2
%
\leavevmode%
\llap{\spacer}%
${L_{154.2}}=\pfillwitharg{2}({L_{154.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{+}{3}{6}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\3\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\-9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\6\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[9, 0, 9; 0, -3, 3; 9, 3, 7],[9, 3, 1],[1, -3, -1, 1; -1, 2, 0, -1; 0, 3, 0, -1],153,2]
%
% END OF LATTICE L154.2
%
\hbox{}\par\smallskip%
%
% LATTICE L154.3
%
\leavevmode%
\llap{\spacer}%
${L_{154.3}}=\pdualwitharg{2}({L_{154.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{+}{3}{24}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45288\\8532\\-5544
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8532\\1608\\-1044
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5544\\-1044\\679
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-11\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-8\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-15\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-12\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\24\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[45288, 8532, -5544; 8532, 1608, -1044; -5544, -1044, 679],[36, 12, 1],[5, 3, 1, 5; -11, -8, -4, -15; 24, 12, 2, 18],153,3]
%
% END OF LATTICE L154.3
%

\medskip%
%------------------------------WEYL-GROUP-W155--------------------------------
%
\leavevmode\llap{}%
$W_{155}$%
\qquad\llap{3} lattices, $\chi=2$%
\hfill%
$6\slashtwo6|\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L155.1
%
\leavevmode%
\llap{\spacer}%
${L_{155.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{6}{18}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\36\\36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\-30\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\-27\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-9\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-36, 36, 36; 36, -30, -27; 36, -27, -22],[36, 3, 1],[2, -2, -1; 5, -3, -3; -3, 0, 2],154,1]
%
% END OF LATTICE L155.1
%
\hbox{}\par\smallskip%
%
% LATTICE L155.2
%
\leavevmode%
\llap{\spacer}%
${L_{155.2}}=\pfillwitharg{2}({L_{155.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{6}{18}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\18\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-30\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-27\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\5\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-9\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-9, 18, 18; 18, -30, -27; 18, -27, -22],[9, 3, 1],[4, -4, -2; 5, -3, -3; -3, 0, 2],154,2]
%
% END OF LATTICE L155.2
%
\hbox{}\par\smallskip%
%
% LATTICE L155.3
%
\leavevmode%
\llap{\spacer}%
${L_{155.3}}=\pdualwitharg{2}({L_{155.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{6}{72}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1800\\-540\\-828
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\168\\252
\end{tabular}\endgroup%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1800, -540, -828; -540, 168, 252; -828, 252, 383],[36, 12, 1],[3, 1, 1; -8, 3, -3; 12, 0, 4],154,3]
%
% END OF LATTICE L155.3
%

\medskip%
%------------------------------WEYL-GROUP-W156--------------------------------
%
\leavevmode\llap{}%
$W_{156}$%
\qquad\llap{6} lattices, $\chi=18$%
\hfill%
$242242\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L156.1
%
\leavevmode%
\llap{\spacer}%
${L_{156.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}27\above{1pt}{1pt}{-}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-1352484\\8316\\11556
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8316\\-46\\-73
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11556\\-73\\-98
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38879\\1356480\\3572640
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-261\\-9107\\-23983
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1352484, 8316, 11556; 8316, -46, -73; 11556, -73, -98],[108, 1, 1],[-10, -5, 5, 253, 59, 4; -351, -174, 175, 8829, 2058, 139; -918, -460, 459, 23247, 5422, 368],155,1]
%
% END OF LATTICE L156.1
%
\hbox{}\par\smallskip%
%
% LATTICE L156.2
%
\leavevmode%
\llap{\spacer}%
${L_{156.2}}=\pfillwitharg{2}({L_{156.1}})$%
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{$\left[\!\llap{\phantom{%
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-22545\\6723\\-13203
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\kern3pt%
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87749\\640224\\176202
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-25375\\-185137\\-50953
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-22545, 6723, -13203; 6723, -1999, 3916; -13203, 3916, -7655],[27, 1, 1],[-3482, -407, -57, 107, 32, -68; -25407, -2969, -415, 783, 233, -497; -6993, -817, -114, 216, 64, -137],155,2]
%
% END OF LATTICE L156.2
%
\hbox{}\par\smallskip%
%
% LATTICE L156.3
%
\leavevmode%
\llap{\spacer}%
${L_{156.3}}=\pdualwitharg{2}({L_{156.1}})$%
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{$\left[\!\llap{\phantom{%
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171471384\\-2517156\\-44527752
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-2517156\\36952\\653656
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-23150989\\619353\\-89186832
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[171471384, -2517156, -44527752; -2517156, 36952, 653656; -44527752, 653656, 11562983],[108, 4, 1],[-617, -41, -733, -28651, -3211, -351; 0, 2, 23, 783, 85, 6; -2376, -158, -2824, -110376, -12370, -1352],155,3]
%
% END OF LATTICE L156.3
%
\hbox{}\par\smallskip%
%
% LATTICE L156.4
%
\leavevmode%
\llap{\spacer}%
${L_{156.4}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{156.1}}))$%
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-27891\\-66690\\3105
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-66690\\-159462\\7425
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3105\\7425\\-337
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-136219\\56794\\-3942
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-325617\\135760\\-9423
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-27891, -66690, 3105; -66690, -159462, 7425; 3105, 7425, -337],[27, 27, 1],[-993, -2854, -98, 60, 55, -835; 414, 1190, 41, -25, -23, 348; -29, -81, 0, 2, 0, -27],155,4]
%
% END OF LATTICE L156.4
%
\hbox{}\par\smallskip%
%
% LATTICE L156.5
%
\leavevmode%
\llap{\spacer}%
${L_{156.5}}=\pdualwitharg{3}({L_{156.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-57132, 8424, 4212; 8424, -1242, -621; 4212, -621, -310],[108, 27, 1],[4, 29, 4, 0, -5, 0; 41, 274, 27, -1, -34, 13; -29, -162, 0, 2, 0, -27],155,5]
%
% END OF LATTICE L156.5
%
\hbox{}\par\smallskip%
%
% LATTICE L156.6
%
\leavevmode%
\llap{\spacer}%
${L_{156.6}}=\pdualwitharg{2.3}({L_{156.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30024, 19764, -4968; 19764, 15768, -3996; -4968, -3996, 1013],[108, 108, 1],[29, 85, 6, 0, 2, 23; -995, -2911, -199, 1, -71, -795; -3784, -11070, -756, 4, -270, -3024],155,6]
%
% END OF LATTICE L156.6
%

\medskip%
%------------------------------WEYL-GROUP-W157--------------------------------
%
\leavevmode\llap{}%
$W_{157}$%
\qquad\llap{8} lattices, $\chi=48$%
\hfill%
$\infty2|2\infty|\infty2|2\infty|\rtimes D_{4}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L157.1
%
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${L_{157.1}}$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-868, -56, -728; -56, 0, -25; -728, -25, -477],[28, 1, 1],[-25, -47, -13, -151, -101, -16, 4, 25; -350, -644, -177, -2044, -1358, -210, 55, 336; 56, 105, 29, 336, 224, 35, -9, -56],156,1]
%
% END OF LATTICE L157.1
%
\hbox{}\par\smallskip%
%
% LATTICE L157.2
%
\leavevmode%
\llap{\spacer}%
${L_{157.2}}$%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1736\\-672\\-56
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672\\-254\\-25
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-56\\-25\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1736, -672, -56; -672, -254, -25; -56, -25, 0],[56, 1, 1],[-101, -151, -13, -47, -25, 25, 4, -16; 224, 336, 29, 105, 56, -56, -9, 35; 420, 616, 52, 182, 84, -112, -16, 70],156,2]
%
% END OF LATTICE L157.2
%
\hbox{}\par\smallskip%
%
% LATTICE L157.3
%
\leavevmode%
\llap{\spacer}%
${L_{157.3}}=\pdualwitharg{2}({L_{157.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{1,0}{\infty a}{28}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{2,1}{\infty}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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170352\\53116\\-44772
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53116\\16564\\-13960
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-44772\\-13960\\11767
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{$\left.\llap{\phantom{%
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16855\\784\\65072
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351\\7\\1344
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80\\3\\308
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[170352, 53116, -44772; 53116, 16564, -13960; -44772, -13960, 11767],[28, 4, 1],[195, 351, 80, 255, 407, 251, 6, 46; 0, 7, 3, 14, 28, 21, 1, 0; 742, 1344, 308, 987, 1582, 980, 24, 175],156,3]
%
% END OF LATTICE L157.3
%
\hbox{}\par\smallskip%
%
% LATTICE L157.4
%
\leavevmode%
\llap{\spacer}%
${L_{157.4}}=\pdualwitharg{7}({L_{157.1}})$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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196\\56\\0
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0\\14\\-3
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[196, 56, 0; 56, -49, 14; 0, 14, -3],[28, 7, 1],[-5, -7, -4, -2, -1, 1, 1, -1; 18, 24, 13, 6, 2, -4, -3, 4; 78, 104, 56, 25, 6, -20, -14, 17],156,4]
%
% END OF LATTICE L157.4
%
\hbox{}\par\smallskip%
%
% LATTICE L157.5
%
\leavevmode%
\llap{\spacer}%
${L_{157.5}}=\pdualwitharg{7}({L_{157.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18424\\336\\336
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-8\\-28\\-52
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-32\\0\\2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-104\\-28\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-120\\-56\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-56\\-42\\-28
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-24\\-28\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18424, 336, 336; 336, 0, -7; 336, -7, -6],[56, 7, 1],[5, 7, 4, 2, 1, -1, -1, 1; 36, 48, 26, 12, 4, -8, -6, 8; 232, 328, 189, 95, 48, -48, -49, 45],156,5]
%
% END OF LATTICE L157.5
%
\hbox{}\par\smallskip%
%
% LATTICE L157.6
%
\leavevmode%
\llap{\spacer}%
${L_{157.6}}=\pdualwitharg{2}({L_{157.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{1,0}{\infty z}{7}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{8,1}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23408\\-5544\\-1792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5544\\-1312\\-424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1792\\-424\\-137
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2156\\6776
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-505\\1584
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-161\\505
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-63\\238
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-7\\49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\12\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\14\\-56
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-14\\-56\\-728\\-420\\-182\\-56\\-56
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{$\left.\llap{\phantom{%
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\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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28\\-14\\-56\\-728
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\7\\0\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-728\\-224\\0\\112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-182\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-182\\-105\\-56\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-56\\-48\\-224
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-168\\-224\\-1680
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-23408, -5544, -1792; -5544, -1312, -424; -1792, -424, -137],[56, 8, 1],[-3, -2, -1, 1, 3, 2, 1, -1; -63, -7, 12, 14, -35, -42, -40, -210; 238, 49, -24, -56, 70, 105, 112, 672],156,6]
%
% END OF LATTICE L157.6
%
\hbox{}\par\smallskip%
%
% LATTICE L157.7
%
\leavevmode%
\llap{\spacer}%
${L_{157.7}}=\pdualwitharg{2.7}({L_{157.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty z}{1}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{4,1}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
588\\-3416\\952
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3416\\19796\\-5516
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
952\\-5516\\1537
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\-30\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\161\\756
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75\\-45\\-211
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\13\\50
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\4\\17
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-7\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-28\\-52\\-60\\-26\\-28\\-4
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-16\\-26\\-15\\-28\\-12
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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4\\-2\\-28\\-52
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-16
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-28\\0\\28\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52\\-16\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-26\\-28\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-15\\-28\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-28\\-84\\-56
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-12\\-56\\-60
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[588, -3416, 952; -3416, 19796, -5516; 952, -5516, 1537],[28, 28, 1],[-5, -4, -11, -8, -5, -1, 1, -2; 13, 4, -2, -7, -7, -1, 8, 13; 50, 17, 0, -20, -22, -3, 28, 48],156,7]
%
% END OF LATTICE L157.7
%
\hbox{}\par\smallskip%
%
% LATTICE L157.8
%
\leavevmode%
\llap{\spacer}%
${L_{157.8}}=\pdualwitharg{2.7}({L_{157.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty z}{1}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{8,1}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-1848\\-784
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1848\\-9632\\-4088
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-784\\-4088\\-1735
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\-580\\1400
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276\\-2669\\6440
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-117\\-1131\\2729
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\-11\\26
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-1\\-18\\43
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-140\\336
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-126\\304
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-56\\-104\\-60\\-26\\-56\\-8
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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4\\-2\\-56\\-104
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-2\\1\\0\\-32
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-56\\0\\112\\0
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-104\\-32\\0\\16
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-26\\-56\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-15\\-56\\-24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-56\\-336\\-224
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-24\\-224\\-240
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-336, -1848, -784; -1848, -9632, -4088; -784, -4088, -1735],[56, 56, 1],[0, -1, -11, -13, -6, -2, -1, 1; -11, -18, -140, -126, -47, -11, 24, 10; 26, 43, 336, 304, 114, 27, -56, -24],156,8]
%
% END OF LATTICE L157.8
%

\medskip%
%------------------------------WEYL-GROUP-W158--------------------------------
%
\leavevmode\llap{}%
$W_{158}$%
\qquad\llap{2} lattices, $\chi=16$%
\hfill%
$3\infty3\infty\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L158.1
%
\leavevmode%
\llap{\spacer}%
${L_{158.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}32\above{1pt}{1pt}{-}{5}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{+}{3}{2}\above{1pt}{1pt}{8,5}{\infty a}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[160, 32, 128; 32, 6, 25; 128, 25, 102],[32, 1, 1],[1, -1, 2, 2; -2, -3, -1, 2; -1, 2, -2, -3],157,1]
%
% END OF LATTICE L158.1
%
\hbox{}\par\smallskip%
%
% LATTICE L158.2
%
\leavevmode%
\llap{\spacer}%
${L_{158.2}}=\pdualwitharg{2}({L_{158.1}})$%
\nopagebreak\par%
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}}\right.$}%
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\kern3pt%
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32\\128\\5
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[192, 800, 32; 800, 3264, 128; 32, 128, 5],[32, 32, 1],[-2, -3, -1, 2; -1, 2, -2, -3; 32, -32, 64, 64],157,2]
%
% END OF LATTICE L158.2
%

\medskip%
%------------------------------WEYL-GROUP-W159--------------------------------
%
\leavevmode\llap{}%
$W_{159}$%
\qquad\llap{2} lattices, $\chi=12$%
\hfill%
$\slashtwo4\slashinfty4\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L159.1
%
\leavevmode%
\llap{\spacer}%
${L_{159.1}}$%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-47392, 1216, 608; 1216, -31, -16; 608, -16, -7],[32, 1, 1],[-1, -1, 3, 2; -31, -32, 93, 63; -16, -14, 47, 29],158,1]
%
% END OF LATTICE L159.1
%
\hbox{}\par\smallskip%
%
% LATTICE L159.2
%
\leavevmode%
\llap{\spacer}%
${L_{159.2}}=\pdualwitharg{2}({L_{159.1}})$%
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{$1\above{1pt}{1pt}{1}{7}32\above{1pt}{1pt}{2}{2}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\-1
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[64, -96, 0; -96, 160, 0; 0, 0, -1],[32, 32, 1],[-3, -2, 3, 1; -3, -1, 2, -2; -16, 0, 0, -32],158,2]
%
% END OF LATTICE L159.2
%

\medskip%
%------------------------------WEYL-GROUP-W160--------------------------------
%
\leavevmode\llap{}%
$W_{160}$%
\qquad\llap{10} lattices, $\chi=24$%
\hfill%
$2\slashinfty2\infty|\infty\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L160.1
%
\leavevmode%
\llap{\spacer}%
${L_{160.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}32\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26848\\992\\928
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\kern3pt%
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992\\-30\\-35
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928\\-35\\-32
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{$\left.\llap{\phantom{%
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0\\0\\0
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-5\\-16\\-128
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67\\192\\1728
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29\\80\\752
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{$\left.\llap{\phantom{%
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0\\0\\0
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32\\0\\-32\\-224\\-32
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0\\2\\-2\\-32\\-16
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-32\\-2\\2\\0\\-16
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-32\\-16\\-16\\-32\\32
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-26848, 992, 928; 992, -30, -35; 928, -35, -32],[32, 1, 1],[-5, -4, 5, 67, 29; -16, -11, 15, 192, 80; -128, -104, 128, 1728, 752],159,1]
%
% END OF LATTICE L160.1
%
\hbox{}\par\smallskip%
%
% LATTICE L160.2
%
\leavevmode%
\llap{\spacer}%
${L_{160.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{1}64\above{1pt}{1pt}{-}{5}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14528\\-1344\\-1472
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\kern3pt%
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-1344\\-124\\-136
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\kern3pt%
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-1472\\-136\\-149
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{$\left.\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-14528, -1344, -1472; -1344, -124, -136; -1472, -136, -149],[64, 4, 1],[-1, 1, 7, 5, 0; 2, 24, -8, -30, -9; 8, -32, -64, -24, 8],159,2]
%
% END OF LATTICE L160.2
%
\hbox{}\par\smallskip%
%
% LATTICE L160.3
%
\leavevmode%
\llap{\spacer}%
${L_{160.3}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-21824, 576, 576; 576, -12, -16; 576, -16, -15],[64, 4, 1],[-1, -1, 1, 11, 2; -6, -8, 6, 74, 15; -32, -30, 31, 336, 60],159,3]
%
% END OF LATTICE L160.3
%
\hbox{}\par\smallskip%
%
% LATTICE L160.4
%
\leavevmode%
\llap{\spacer}%
${L_{160.4}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-13376, 960, 448; 960, -60, -32; 448, -32, -15],[64, 4, 1],[-1, -1, 2, 9, 5; 1, 0, -2, -7, -2; -32, -30, 63, 280, 152],159,4]
%
% END OF LATTICE L160.4
%
\hbox{}\par\smallskip%
%
% LATTICE L160.5
%
\leavevmode%
\llap{\spacer}%
${L_{160.5}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[64, 0, 0; 0, -1020, -32; 0, -32, -1],[64, 4, 1],[-1, -1, 1, 0, -1; -7, -16, 0, 1, -2; 208, 480, 0, -32, 56],159,5]
%
% END OF LATTICE L160.5
%
\hbox{}\par\smallskip%
%
% LATTICE L160.6
%
\leavevmode%
\llap{\spacer}%
${L_{160.6}}=\pdualwitharg{2}({L_{160.5}})$%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13376, 960, 448; 960, -48, -32; 448, -32, -15],[64, 16, 1],[-1, -1, 1, 11, 4; 1, 0, -1, -7, -1; -32, -30, 31, 336, 120],159,6]
%
% END OF LATTICE L160.6
%
\hbox{}\par\smallskip%
%
% LATTICE L160.7
%
\leavevmode%
\llap{\spacer}%
${L_{160.7}}=\pdualwitharg{2}({L_{160.1}})$%
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-14\\-64\\0\\4\\-4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-64, 480, -32; 480, -3328, 224; -32, 224, -15],[32, 32, 1],[3, 13, -1, -1, 1; -1, -2, 2, 0, -1; -23, -64, 32, 2, -18],159,7]
%
% END OF LATTICE L160.7
%
\hbox{}\par\smallskip%
%
% LATTICE L160.8
%
\leavevmode%
\llap{\spacer}%
${L_{160.8}}=\pdualwitharg{2}({L_{160.4}})$%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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64\\-128\\0
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\kern3pt%
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-128\\272\\0
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0\\0\\-1
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0\\0\\0
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[64, -128, 0; -128, 272, 0; 0, 0, -1],[64, 16, 1],[-16, -17, 1, 2, -3; -7, -8, 0, 1, -1; -32, -32, 0, 0, -8],159,8]
%
% END OF LATTICE L160.8
%
\hbox{}\par\smallskip%
%
% LATTICE L160.9
%
\leavevmode%
\llap{\spacer}%
${L_{160.9}}=\pdualwitharg{2}({L_{160.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{-}{5}64\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-225728\\3840\\3712
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\kern3pt%
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21\\56\\1216
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16\\0\\-128\\-112\\-16
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0\\64\\-64\\-128\\-64
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0\\0\\0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-225728, 3840, 3712; 3840, -48, -64; 3712, -64, -61],[64, 16, 1],[-4, -5, 21, 22, 5; -11, -16, 56, 61, 15; -232, -288, 1216, 1272, 288],159,9]
%
% END OF LATTICE L160.9
%
\hbox{}\par\smallskip%
%
% LATTICE L160.10
%
\leavevmode%
\llap{\spacer}%
${L_{160.10}}=\pdualwitharg{2}({L_{160.2}})$%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-64\\576\\-64
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576\\-4720\\528
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0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-64, 576, -64; 576, -4720, 528; -64, 528, -59],[64, 16, 1],[-1, -1, 2, 11, 4; -1, 1, 1, -1, -3; -8, 10, 6, -24, -32],159,10]
%
% END OF LATTICE L160.10
%

\medskip%
%------------------------------WEYL-GROUP-W161--------------------------------
%
\leavevmode\llap{}%
$W_{161}$%
\qquad\llap{12} lattices, $\chi=24$%
\hfill%
$\slashinfty2|2\slashinfty2|2\rtimes D_{4}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L161.1
%
\leavevmode%
\llap{\spacer}%
${L_{161.1}}$%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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3968\\-16\\-247
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-63712, 256, 3968; 256, -1, -16; 3968, -16, -247],[32, 1, 1],[-3, 3, 1, 7, 1, -1; -16, 224, 44, 240, 0, -32; -48, 32, 13, 96, 16, -14],160,1]
%
% END OF LATTICE L161.1
%
\hbox{}\par\smallskip%
%
% LATTICE L161.2
%
\leavevmode%
\llap{\spacer}%
${L_{161.2}}$%
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160\\-6\\-8
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3488, 160, 160; 160, -6, -8; 160, -8, -7],[32, 2, 1],[5, 1, -1, -1, 1, 5; 36, 8, -6, -8, 6, 34; 70, 13, -16, -14, 15, 72],160,2]
%
% END OF LATTICE L161.2
%
\hbox{}\par\smallskip%
%
% LATTICE L161.3
%
\leavevmode%
\llap{\spacer}%
${L_{161.3}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-32320, 704, 704; 704, -14, -16; 704, -16, -15],[64, 2, 1],[1, 7, 2, 2, -1, -1; 13, 102, 30, 31, -14, -16; 32, 216, 61, 60, -32, -30],160,3]
%
% END OF LATTICE L161.3
%
\hbox{}\par\smallskip%
%
% LATTICE L161.4
%
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\llap{\spacer}%
${L_{161.4}}$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-4000, 992, 64; 992, -244, -16; 64, -16, -1],[32, 4, 1],[-3, -1, 1, 3, 1, -1; -12, -12, -2, 4, 4, -2; -16, 96, 80, 112, 0, -32],160,4]
%
% END OF LATTICE L161.4
%
\hbox{}\par\smallskip%
%
% LATTICE L161.5
%
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${L_{161.5}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[32, 0, 0; 0, -252, -16; 0, -16, -1],[32, 4, 1],[-3, -5, -1, -1, 1, 0; 0, -8, -3, -8, 0, 1; -16, 96, 40, 112, 0, -16],160,5]
%
% END OF LATTICE L161.5
%
\hbox{}\par\smallskip%
%
% LATTICE L161.6
%
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\llap{\spacer}%
${L_{161.6}}=\main({L_{161.3}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1312, 224, 96; 224, -30, -16; 96, -16, -7],[32, 2, 1],[7, 1, -1, -1, 2, 4; -4, 0, 1, 0, -2, -3; 102, 13, -16, -14, 31, 60],160,6]
%
% END OF LATTICE L161.6
%
\hbox{}\par\smallskip%
%
% LATTICE L161.7
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1312, 224, 96; 224, -24, -16; 96, -16, -7],[32, 8, 1],[1, 5, 5, 1, -1, -1; 0, -2, -3, -1, 0, 1; 13, 70, 72, 15, -14, -16],160,7]
%
% END OF LATTICE L161.7
%
\hbox{}\par\smallskip%
%
% LATTICE L161.8
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-32, 160, -32; 160, -712, 144; -32, 144, -29],[32, 8, 1],[2, 5, 5, 2, -1, -1; -3, -3, -1, 1, 1, -1; -18, -22, -12, 2, 6, -4],160,8]
%
% END OF LATTICE L161.8
%
\hbox{}\par\smallskip%
%
% LATTICE L161.9
%
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${L_{161.9}}=\pdualwitharg{2}(\main({L_{161.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[32, 0, 0; 0, -240, -16; 0, -16, -1],[32, 16, 1],[-3, -5, -2, -1, 1, 0; 0, -4, -3, -4, 0, 1; -16, 32, 32, 48, 0, -16],160,9]
%
% END OF LATTICE L161.9
%
\hbox{}\par\smallskip%
%
% LATTICE L161.10
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15328, -10624, 1088; -10624, -7344, 752; 1088, 752, -77],[32, 16, 1],[-1, 1, 0, -3, -5, -2; 36, -8, -9, 16, 60, 35; 336, -64, -88, 112, 512, 312],160,10]
%
% END OF LATTICE L161.10
%
\hbox{}\par\smallskip%
%
% LATTICE L161.11
%
\leavevmode%
\llap{\spacer}%
${L_{161.11}}=\pdualwitharg{2}({L_{161.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}32\above{1pt}{1pt}{2}{0}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{2,1}{\infty b}{4}\above{1pt}{1pt}{l}{2}{32}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{2,1}{\infty}{4}\above{1pt}{1pt}{b}{2}{32}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14624\\-16576\\960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16576\\-18784\\1088
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
960\\1088\\-63
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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2\\-3\\-22
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5\\-4\\6
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10\\-5\\64
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1\\0\\15
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-48
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\-4\\-32\\-10\\-12\\0
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-4\\4\\0\\-6\\-20\\-32
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-32\\0\\32\\0\\-32\\-96
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-10\\-6\\0\\1\\-2\\-16
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-12\\-20\\-32\\-2\\4\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-32\\-96\\-16\\0\\32
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14624, -16576, 960; -16576, -18784, 1088; 960, 1088, -63],[32, 32, 1],[2, 5, 10, 1, -1, -2; -3, -4, -5, 0, 1, -1; -22, 6, 64, 15, 2, -48],160,11]
%
% END OF LATTICE L161.11
%
\hbox{}\par\smallskip%
%
% LATTICE L161.12
%
\leavevmode%
\llap{\spacer}%
${L_{161.12}}=\pdualwitharg{2}({L_{161.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[32\above{1pt}{1pt}{1}{}64\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}$}\EasyButWeakLineBreak%
{${32}\above{1pt}{1pt}{16,9}{\infty}{32}\above{1pt}{1pt}{l}{2}{64}\above{1pt}{1pt}{}{2}{32}\above{1pt}{1pt}{16,1}{\infty}{32}\above{1pt}{1pt}{b}{2}{64}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\-64\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\96\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-3\\-16
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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32\\-32\\-128\\-160\\-96\\0
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\kern3pt%
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-32\\32\\0\\-96\\-160\\-128
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128\\0\\64\\0\\-128\\-192
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-160\\-96\\0\\32\\-32\\-128
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-96\\-160\\-128\\-32\\32\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-128\\-192\\-128\\0\\64
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[64, -64, 0; -64, 96, 0; 0, 0, -1],[64, 32, 1],[-3, -3, -1, 1, 1, -1; -5, -3, 0, 1, -1, -4; -32, -16, 0, 0, -16, -32],160,12]
%
% END OF LATTICE L161.12
%

\medskip%
%------------------------------WEYL-GROUP-W162--------------------------------
%
\leavevmode\llap{}%
$W_{162}$%
\qquad\llap{16} lattices, $\chi=36$%
\hfill%
$\slashtwo\infty|\infty\slashtwo\infty|\infty\rtimes D_{4}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L162.1
%
\leavevmode%
\llap{\spacer}%
${L_{162.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{6,1}{\infty}{4}\above{1pt}{1pt}{3,1}{\infty z}{1}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{2,1}{\infty}{36}\above{1pt}{1pt}{1,0}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-298476\\-100440\\-33696
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100440\\-33799\\-11339
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33696\\-11339\\-3804
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\44\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-56\\34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\-191\\82
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
367\\-1260\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
119\\-396\\126
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-18\\-27\\-144\\-18
\end{tabular}\endgroup%
\kern3pt%
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0\\4\\-4\\-16\\-108\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-4\\4\\-2\\-36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-16\\-2\\1\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-108\\-36\\0\\36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-36\\-36\\-18\\-36\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-298476, -100440, -33696; -100440, -33799, -11339; -33696, -11339, -3804],[36, 1, 1],[-1, -13, 15, 55, 367, 119; 9, 44, -56, -191, -1260, -396; -18, -16, 34, 82, 504, 126],161,1]
%
% END OF LATTICE L162.1
%
\hbox{}\par\smallskip%
%
% LATTICE L162.2
%
\leavevmode%
\llap{\spacer}%
${L_{162.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{12,5}{\infty}{4}\above{1pt}{1pt}{3,2}{\infty a}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-348012\\-147888\\-3492
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-147888\\-62845\\-1484
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3492\\-1484\\-35
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61559\\-139320\\-233280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26182\\-59255\\-99216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
608\\-1376\\-2305
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-159\\360\\596
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-84\\190\\323
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\106\\194
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\right.$}%
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4\\0\\-12\\-36\\-16\\-4
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0\\1\\-2\\-16\\-9\\-6
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-12\\-2\\4\\-4\\-6\\-12
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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4\\0\\-12
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0\\1\\-2
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-12\\-2\\4
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-36\\-16\\-4
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-9\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-6\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-348012, -147888, -3492; -147888, -62845, -1484; -3492, -1484, -35],[36, 1, 1],[-159, -84, -47, 7, 8, -29; 360, 190, 106, -16, -18, 66; 596, 323, 194, -20, -35, 94],161,2]
%
% END OF LATTICE L162.2
%
\hbox{}\par\smallskip%
%
% LATTICE L162.3
%
\leavevmode%
\llap{\spacer}%
${L_{162.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{4,1}{\infty a}{72}\above{1pt}{1pt}{1,0}{\infty a}{72}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{12,1}{\infty a}{8}\above{1pt}{1pt}{3,2}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-741240\\333072\\-99000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
333072\\-149664\\44485
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99000\\44485\\-13222
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\-96\\-16
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\-288\\-63
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
233\\540\\72
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\HardButStrongLineBreak\kern3pt%
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1531\\3600\\648
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
275\\648\\121
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
157\\372\\76
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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0\\18\\-36\\-288\\-54\\-36
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-32\\-54\\-36\\0\\2\\-4
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-741240, 333072, -99000; 333072, -149664, 44485; -99000, 44485, -13222],[72, 1, 1],[-41, -121, 233, 1531, 275, 157; -96, -288, 540, 3600, 648, 372; -16, -63, 72, 648, 121, 76],161,3]
%
% END OF LATTICE L162.3
%
\hbox{}\par\smallskip%
%
% LATTICE L162.4
%
\leavevmode%
\llap{\spacer}%
${L_{162.4}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{3,2}{\infty}{8}\above{1pt}{1pt}{3,1}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1289592\\-282096\\-9504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-282096\\-61708\\-2079
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9504\\-2079\\-70
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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83447\\-377712\\-109800
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\kern3pt%
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18259\\-82647\\-24025
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608\\-2752\\-801
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{$\left.\llap{\phantom{%
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0\\0\\0
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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8\\-36\\-17
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7\\-32\\0
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1289592, -282096, -9504; -282096, -61708, -2079; -9504, -2079, -70],[72, 1, 1],[8, 7, -47, -84, -159, -29; -36, -32, 212, 380, 720, 132; -17, 0, 84, 117, 200, 16],161,4]
%
% END OF LATTICE L162.4
%
\hbox{}\par\smallskip%
%
% LATTICE L162.5
%
\leavevmode%
\llap{\spacer}%
${L_{162.5}}=\pdualwitharg{2}({L_{162.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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-13860\\1968\\-3656
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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-349\\351\\1512
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-3\\-13
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308\\309\\1334
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1605\\1611\\6952
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2591\\2601\\11223
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1658\\1665\\7182
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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0\\1\\-2\\-16\\-27\\-18
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-36\\-2\\4\\-4\\-18\\-36
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-108\\-16\\-4\\4\\0\\-36
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-144\\-27\\-18\\0\\9\\-18
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-36\\-18\\-36\\-36\\-18\\36
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0\\0\\0\\0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[109116, -13860, 28404; -13860, 1968, -3656; 28404, -3656, 7405],[36, 4, 1],[-349, 3, -308, -1605, -2591, -1658; 351, -3, 309, 1611, 2601, 1665; 1512, -13, 1334, 6952, 11223, 7182],161,5]
%
% END OF LATTICE L162.5
%
\hbox{}\par\smallskip%
%
% LATTICE L162.6
%
\leavevmode%
\llap{\spacer}%
${L_{162.6}}=\pdualwitharg{2}({L_{162.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{12,11}{\infty}{4}\above{1pt}{1pt}{3,1}{\infty z}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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1599444\\-49104\\377208
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\kern3pt%
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-49104\\1516\\-11584
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\kern3pt%
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377208\\-11584\\88961
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{$\left.\llap{\phantom{%
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32336\\-59255\\-144824
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1599444, -49104, 377208; -49104, 1516, -11584; 377208, -11584, 88961],[36, 4, 1],[-754, -1479, -347, 2, -25, -405; 1382, 2710, 635, -4, 46, 743; 3377, 6624, 1554, -9, 112, 1814],161,6]
%
% END OF LATTICE L162.6
%
\hbox{}\par\smallskip%
%
% LATTICE L162.7
%
\leavevmode%
\llap{\spacer}%
${L_{162.7}}=\pdualwitharg{3}({L_{162.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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-3060\\-27\\-1359
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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7\\12\\-16
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-1\\-18
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-112\\126
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-108\\-144\\-36\\36\\0\\-36
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-16\\-27\\-18\\0\\1\\-2
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-4\\-18\\-36\\-36\\-2\\4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-345708, -3060, -153540; -3060, -27, -1359; -153540, -1359, -68192],[36, 9, 1],[7, 8, -55, -221, -36, -15; 12, -1, -112, -332, -49, -12; -16, -18, 126, 504, 82, 34],161,7]
%
% END OF LATTICE L162.7
%
\hbox{}\par\smallskip%
%
% LATTICE L162.8
%
\leavevmode%
\llap{\spacer}%
${L_{162.8}}=\pdualwitharg{3}({L_{162.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{12,1}{\infty}{36}\above{1pt}{1pt}{3,1}{\infty b}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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36\\0\\0
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0\\-315\\297
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[36, 0, 0; 0, -315, 297; 0, 297, -280],[36, 9, 1],[1, 0, -3, -9, -4, -1; 0, -17, 16, 200, 117, 84; 0, -18, 18, 216, 126, 90],161,8]
%
% END OF LATTICE L162.8
%
\hbox{}\par\smallskip%
%
% LATTICE L162.9
%
\leavevmode%
\llap{\spacer}%
${L_{162.9}}=\pdualwitharg{2}({L_{162.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-204624\\2808\\-66456
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\kern3pt%
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2808\\-32\\912
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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39\\2\\-120
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\0\\-117
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-193\\-9\\594
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1801\\-54\\5544
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-153\\-4\\471
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{$\left.\llap{\phantom{%
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0\\0\\0
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0\\0\\0\\0\\0\\0
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16\\0\\-72\\-432\\-32\\-8
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0\\9\\-18\\-288\\-27\\-18
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-72\\-18\\36\\-72\\-18\\-36
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-432\\-288\\-72\\144\\0\\-72
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-32\\-27\\-18\\0\\1\\-2
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-204624, 2808, -66456; 2808, -32, 912; -66456, 912, -21583],[72, 8, 1],[39, 38, -193, -1801, -153, -76; 2, 0, -9, -54, -4, -1; -120, -117, 594, 5544, 471, 234],161,9]
%
% END OF LATTICE L162.9
%
\hbox{}\par\smallskip%
%
% LATTICE L162.10
%
\leavevmode%
\llap{\spacer}%
${L_{162.10}}=\pdualwitharg{2}({L_{162.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{3,2}{\infty}{4}\above{1pt}{1pt}{3,2}{\infty a}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-720720\\5112\\10152
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\kern3pt%
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5112\\-32\\-72
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10152\\-72\\-143
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-720720, 5112, 10152; 5112, -32, -72; 10152, -72, -143],[72, 8, 1],[-1, -1, 3, 37, 10, 6; 2, 0, -3, -18, -4, -1; -72, -71, 214, 2632, 711, 426],161,10]
%
% END OF LATTICE L162.10
%
\hbox{}\par\smallskip%
%
% LATTICE L162.11
%
\leavevmode%
\llap{\spacer}%
${L_{162.11}}=\pdualwitharg{3}({L_{162.3}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\kern3pt%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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413\\-1528\\936
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88\\-332\\199
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71\\-276\\160
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72\\0\\-72\\-216\\-288\\-72
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0\\2\\-4\\-32\\-54\\-36
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-72\\-4\\8\\-8\\-36\\-72
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-216\\-32\\-8\\8\\0\\-72
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-288\\-54\\-36\\0\\18\\-36
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-72\\-36\\-72\\-72\\-36\\72
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1222776, -33840, 484200; -33840, -936, 13401; 484200, 13401, -191734],[72, 9, 1],[413, 88, 71, -7, -68, -17; -1528, -332, -276, 24, 268, 100; 936, 199, 160, -16, -153, -36],161,11]
%
% END OF LATTICE L162.11
%
\hbox{}\par\smallskip%
%
% LATTICE L162.12
%
\leavevmode%
\llap{\spacer}%
${L_{162.12}}=\pdualwitharg{3}({L_{162.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{12,1}{\infty a}{72}\above{1pt}{1pt}{3,2}{\infty a}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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72\\0\\0
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[72, 0, 0; 0, -630, 297; 0, 297, -140],[72, 9, 1],[1, 0, -3, -9, -4, -1; 0, -17, 16, 200, 117, 84; 0, -36, 36, 432, 252, 180],161,12]
%
% END OF LATTICE L162.12
%
\hbox{}\par\smallskip%
%
% LATTICE L162.13
%
\leavevmode%
\llap{\spacer}%
${L_{162.13}}=\pdualwitharg{2.3}({L_{162.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{12,11}{\infty}{36}\above{1pt}{1pt}{3,1}{\infty z}{9}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{4,3}{\infty}{4}\above{1pt}{1pt}{1,0}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11664\\41868\\10044
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41868\\224316\\53820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10044\\53820\\12913
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-743\\3096
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-3538\\14742
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-5536\\23067
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-3431\\14296
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-660\\2750
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\0\\-18\\-27\\-16\\-2
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0\\36\\-36\\-144\\-108\\-36
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-36\\36\\-18\\-36\\-36
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-27\\-144\\-18\\9\\0\\-18
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-16\\-108\\-36\\0\\4\\-4
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-36\\-36\\-18\\-4\\4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[11664, 41868, 10044; 41868, 224316, 53820; 10044, 53820, 12913],[36, 36, 1],[0, 1, 5, 8, 5, 1; 6, -743, -3538, -5536, -3431, -660; -25, 3096, 14742, 23067, 14296, 2750],161,13]
%
% END OF LATTICE L162.13
%
\hbox{}\par\smallskip%
%
% LATTICE L162.14
%
\leavevmode%
\llap{\spacer}%
${L_{162.14}}=\pdualwitharg{2.3}({L_{162.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{12,7}{\infty}{36}\above{1pt}{1pt}{3,2}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10080\\-10692\\-2376
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10692\\11484\\2556
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2376\\2556\\569
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{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-801\\-7150\\28800
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
992\\8865\\-35712
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
224\\2002\\-8065
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\-2\\9
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\-36
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\156\\-630
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
\par\leavevmode\begin{center}%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\36\\-36
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-54\\-36\\36
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-81\\-144\\-18
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-144\\-324\\-108
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-18\\-108\\-108
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10080, -10692, -2376; -10692, 11484, 2556; -2376, 2556, 569],[36, 36, 1],[0, 1, 17, 32, 63, 15; -2, 9, 156, 288, 563, 130; 9, -36, -630, -1161, -2268, -522],161,14]
%
% END OF LATTICE L162.14
%
\hbox{}\par\smallskip%
%
% LATTICE L162.15
%
\leavevmode%
\llap{\spacer}%
${L_{162.15}}=\pdualwitharg{2.3}({L_{162.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{8,7}{\infty z}{4}\above{1pt}{1pt}{1,0}{\infty z}{1}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{24,23}{\infty z}{36}\above{1pt}{1pt}{3,2}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-204624\\1648152\\366840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1648152\\-13274496\\-2954592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
366840\\-2954592\\-657623
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\78\\-351
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\94\\-424
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-163\\734
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-342\\1541
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\-4074\\18360
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-459\\2070
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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9\\0\\-18\\-27\\-288\\-18
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0\\16\\-8\\-32\\-432\\-72
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-8\\4\\-2\\-72\\-36
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-27\\-32\\-2\\1\\0\\-18
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-288\\-432\\-72\\0\\144\\-72
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-72\\-36\\-18\\-72\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-204624, 1648152, 366840; 1648152, -13274496, -2954592; 366840, -2954592, -657623],[72, 72, 1],[-1, -3, 3, 8, 101, 14; 78, 94, -163, -342, -4074, -459; -351, -424, 734, 1541, 18360, 2070],161,15]
%
% END OF LATTICE L162.15
%
\hbox{}\par\smallskip%
%
% LATTICE L162.16
%
\leavevmode%
\llap{\spacer}%
${L_{162.16}}=\pdualwitharg{2.3}({L_{162.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${144}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{3,1}{\infty}{36}\above{1pt}{1pt}{3,1}{\infty b}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-25272\\-5688
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25272\\-1956816\\-440424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5688\\-440424\\-99127
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\4991\\-22176
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2648\\472005\\-2097216
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
596\\106237\\-472033
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-2933\\13032
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-4\\-792\\3519
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-474\\2106
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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-72\\-54\\-108
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-288, -25272, -5688; -25272, -1956816, -440424; -5688, -440424, -99127],[72, 72, 1],[-18, -4, -1, 2, 0, -3; -2933, -792, -474, 81, 79, -239; 13032, 3519, 2106, -360, -351, 1062],161,16]
%
% END OF LATTICE L162.16
%

\medskip%
%------------------------------WEYL-GROUP-W163--------------------------------
%
\leavevmode\llap{}%
$W_{163}$%
\qquad\llap{49} lattices, $\chi=36$%
\hfill%
$\slashtwo2\slashinfty2\slashtwo2\slashinfty2\rtimes D_{4}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L163.1
%
\leavevmode%
\llap{\spacer}%
${L_{163.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{5}%
\EasyButWeakLineBreak%
{${25}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{5,4}{\infty b}{40}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{100}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{5,1}{\infty a}{40}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-733400\\1000\\8600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1000\\35\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8600\\-25\\-96
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\1105\\3025
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\746\\2044
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
125\\3728\\10220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91\\2712\\7440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\119\\327
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-330\\-900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-328\\-900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\688\\1880
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\-100\\-200\\-25\\-200\\-100\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-40\\-8\\-100\\-80\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\0\\40\\-40\\-20\\-400\\-440\\-360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-40\\-40\\40\\0\\-200\\-360\\-440
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-8\\-20\\0\\1\\0\\-20\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-100\\-400\\-200\\0\\100\\0\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-80\\-440\\-360\\-20\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-360\\-440\\-40\\-200\\-40\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-733400, 1000, 8600; 1000, 35, -25; 8600, -25, -96],[200, 5, 1],[37, 25, 125, 91, 4, -11, -11, 23; 1105, 746, 3728, 2712, 119, -330, -328, 688; 3025, 2044, 10220, 7440, 327, -900, -900, 1880],162,1]
%
% END OF LATTICE L163.1
%
\hbox{}\par\smallskip%
%
% LATTICE L163.2
%
\leavevmode%
\llap{\spacer}%
${L_{163.2}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{5m,5,2}%
\EasyButWeakLineBreak%
{${400}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{40,19}{\infty}{40}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{100}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{40,11}{\infty}{40}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-587600\\21600\\800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21600\\-790\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
800\\-30\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\-146
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-89\\-560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-202\\-1220
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-68\\-392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-70\\-350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\22\\140
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\0\\-200\\-800\\-400\\-800\\-200\\0
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\kern3pt%
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0\\4\\0\\-40\\-32\\-100\\-40\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\0\\10\\-20\\-40\\-200\\-110\\-180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800\\-40\\-20\\40\\0\\-200\\-180\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-32\\-40\\0\\16\\0\\-40\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800\\-100\\-200\\-200\\0\\100\\0\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-40\\-110\\-180\\-40\\0\\10\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-180\\-440\\-160\\-200\\-20\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-587600, 21600, 800; 21600, -790, -30; 800, -30, -1],[400, 10, 1],[-1, -1, -4, -9, -3, -3, 0, 1; -20, -22, -89, -202, -68, -70, -1, 22; -200, -146, -560, -1220, -392, -350, 20, 140],162,2]
%
% END OF LATTICE L163.2
%
\hbox{}\par\smallskip%
%
% LATTICE L163.3
%
\leavevmode%
\llap{\spacer}%
${L_{163.3}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{52,5,m}%
\EasyButWeakLineBreak%
{${400}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{40,39}{\infty}{40}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{40,31}{\infty}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-931600\\27600\\10400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27600\\-790\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10400\\-320\\-111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-340\\-800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\72\\167
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\791\\1840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
119\\2138\\4980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\844\\1968
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\610\\1425
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\179\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-198\\-460
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\0\\-200\\-800\\-400\\-400\\-200\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-20\\-16\\-25\\-20\\-20
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-200\\0\\10\\-20\\-40\\-100\\-110\\-180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800\\-20\\-20\\40\\0\\-100\\-180\\-440
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-16\\-40\\0\\16\\0\\-40\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-25\\-100\\-100\\0\\25\\0\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-20\\-110\\-180\\-40\\0\\10\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-180\\-440\\-160\\-100\\-20\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-931600, 27600, 10400; 27600, -790, -320; 10400, -320, -111],[400, 10, 1],[-19, 4, 44, 119, 47, 34, 10, -11; -340, 72, 791, 2138, 844, 610, 179, -198; -800, 167, 1840, 4980, 1968, 1425, 420, -460],162,3]
%
% END OF LATTICE L163.3
%
\hbox{}\par\smallskip%
%
% LATTICE L163.4
%
\leavevmode%
\llap{\spacer}%
${L_{163.4}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{5m,5,m}%
\EasyButWeakLineBreak%
{${400}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{40,19}{\infty z}{10}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{s}{2}{100}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{40,11}{\infty z}{10}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-676400\\8800\\201600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8800\\-90\\-2820
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
201600\\-2820\\-58499
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1853\\37040\\4600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
315\\6296\\782
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1579\\31558\\3920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
576\\11511\\1430
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
203\\4056\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-141\\-2820\\-350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145\\-2898\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
141\\2819\\350
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\0\\-400\\-400\\-400\\-800\\-400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20\\-32\\-100\\-80\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\0\\40\\-20\\-80\\-400\\-440\\-180
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-20\\-20\\10\\0\\-100\\-180\\-110
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-32\\-80\\0\\16\\0\\-80\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800\\-100\\-400\\-100\\0\\100\\0\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-80\\-440\\-180\\-80\\0\\40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-180\\-110\\-80\\-100\\-20\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-676400, 8800, 201600; 8800, -90, -2820; 201600, -2820, -58499],[400, 10, 1],[1853, 315, 1579, 576, 203, -141, -145, 141; 37040, 6296, 31558, 11511, 4056, -2820, -2898, 2819; 4600, 782, 3920, 1430, 504, -350, -360, 350],162,4]
%
% END OF LATTICE L163.4
%
\hbox{}\par\smallskip%
%
% LATTICE L163.5
%
\leavevmode%
\llap{\spacer}%
${L_{163.5}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-}{3}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{25,5,2}%
\EasyButWeakLineBreak%
{${1600}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{160}\above{1pt}{1pt}{80,79}{\infty z}{40}\above{1pt}{1pt}{b}{2}{64}\above{1pt}{1pt}{s}{2}{100}\above{1pt}{1pt}{*}{2}{160}\above{1pt}{1pt}{80,71}{\infty z}{40}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 2-dual${}\iso{}$5-dual; isometric to own %
2.5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-958400\\19200\\233600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19200\\-360\\-4880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
233600\\-4880\\-55311
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1301\\26000\\3200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\999\\123
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
943\\18838\\2320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
317\\6331\\780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91\\1816\\224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\-1220\\-150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-65\\-1298\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
122\\2439\\300
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1600\\0\\-1600\\-1600\\-1600\\-1600\\-1600\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-20\\-32\\-50\\-80\\-20
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1600\\0\\160\\-80\\-320\\-800\\-1760\\-720
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1600\\-20\\-80\\40\\0\\-200\\-720\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1600\\-32\\-320\\0\\64\\0\\-320\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1600\\-50\\-800\\-200\\0\\100\\0\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1600\\-80\\-1760\\-720\\-320\\0\\160\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-720\\-440\\-320\\-200\\-80\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-958400, 19200, 233600; 19200, -360, -4880; 233600, -4880, -55311],[1600, 40, 1],[1301, 50, 943, 317, 91, -61, -65, 122; 26000, 999, 18838, 6331, 1816, -1220, -1298, 2439; 3200, 123, 2320, 780, 224, -150, -160, 300],162,5]
%
% END OF LATTICE L163.5
%
\hbox{}\par\smallskip%
%
% LATTICE L163.6
%
\leavevmode%
\llap{\spacer}%
${L_{163.6}}=\pfillwitharg{5.2}({L_{163.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{4,3}{\infty}{10}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
260\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\10\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\180\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\17\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\36\\-57
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-40\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-13\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-25\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-40\\-20\\-8\\-20\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-10\\-10\\10\\0\\-10\\-90\\-110
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-10
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\10\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-10\\-10\\10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-8\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-5\\-20\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-20\\-110\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-90\\-110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[260, -60, 0; -60, 10, -2; 0, -2, -1],[20, 2, 1],[-9, -3, -6, 1, 1, -1, -14, -21; -40, -13, -25, 5, 4, -5, -65, -95; 64, 21, 40, -10, -8, 7, 100, 150],162,6]
%
% END OF LATTICE L163.6
%
\hbox{}\par\smallskip%
%
% LATTICE L163.7
%
\leavevmode%
\llap{\spacer}%
${L_{163.7}}=\pfillwitharg{5}({L_{163.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{2,1}{\infty}{40}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-206360\\4840\\5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4840\\-111\\-121
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\-121\\-120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22909\\502440\\455040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-493\\-10813\\-9792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-609\\-13356\\-12097
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69\\1514\\1370
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\549\\496
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
111\\2440\\2200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-240\\-220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-80\\-20\\-40\\40\\0\\-20\\-360\\-440
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-40\\-80
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-20
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-20\\-40\\40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-8\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-5\\-40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\-440\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-360\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-206360, 4840, 5040; 4840, -111, -121; 5040, -121, -120],[40, 1, 1],[69, 25, 111, -11, -11, 4, 179, 301; 1514, 549, 2440, -240, -242, 87, 3920, 6600; 1370, 496, 2200, -220, -218, 80, 3560, 5980],162,7]
%
% END OF LATTICE L163.7
%
\hbox{}\par\smallskip%
%
% LATTICE L163.8
%
\leavevmode%
\llap{\spacer}%
${L_{163.8}}=\main(\pfillwitharg{5}({L_{163.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{4,1}{\infty z}{5}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33480\\280\\1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\-2\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\-10\\-43
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\-78\\25
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2\\30\\-65
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-10\\-16\\-20\\0
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-16\\-40\\-80\\-20
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-33480, 280, 1200; 280, -2, -10; 1200, -10, -43],[40, 2, 1],[1, 1, -1, -2, -1, -1, 1, 2; 5, 24, 70, 30, 7, 0, -10, 0; 26, 20, -50, -65, -30, -28, 30, 55],162,8]
%
% END OF LATTICE L163.8
%
\hbox{}\par\smallskip%
%
% LATTICE L163.9
%
\leavevmode%
\llap{\spacer}%
${L_{163.9}}=\main(\pfillwitharg{5}({L_{163.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{4,3}{\infty z}{5}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-240\\280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-286\\358
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
280\\358\\-443
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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259\\1040\\1000
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\kern3pt%
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598\\2391\\2300
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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32\\129\\124
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133\\530\\510
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\140\\135
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-40\\-16\\-40\\0
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-16\\-10\\-40\\-10
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-120, -240, 280; -240, -286, 358; 280, 358, -443],[40, 2, 1],[49, 32, 133, 36, 3, -6, -3, 29; 200, 129, 530, 140, 8, -25, -10, 120; 192, 124, 510, 135, 8, -24, -10, 115],162,9]
%
% END OF LATTICE L163.9
%
\hbox{}\par\smallskip%
%
% LATTICE L163.10
%
\leavevmode%
\llap{\spacer}%
${L_{163.10}}=\pfillwitharg{5}({L_{163.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{8,1}{\infty z}{10}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90960\\-2800\\-3120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2800\\-86\\-96
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-96\\-107
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-20\\-32\\-40\\0
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-32\\-80\\-160\\-40
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-40\\-160\\-440\\-180
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-90960, -2800, -3120; -2800, -86, -96; -3120, -96, -107],[80, 2, 1],[1, 1, -1, -2, -1, -1, 1, 2; -4, 12, 10, -25, -24, -68, -150, -45; -26, -40, 20, 80, 50, 88, 100, -20],162,10]
%
% END OF LATTICE L163.10
%
\hbox{}\par\smallskip%
%
% LATTICE L163.11
%
\leavevmode%
\llap{\spacer}%
${L_{163.11}}=\pfillwitharg{5}({L_{163.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{8,5}{\infty z}{10}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12560\\1040\\1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1040\\-86\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\-100\\-111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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959\\9280\\1920
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-755\\-156
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\40\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-110\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-25\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-20\\-20\\-5\\-16\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-40\\-16\\-80\\-160\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\40\\-20\\-20\\-160\\-440\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-40\\-20\\10\\0\\-40\\-180\\-110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-20\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-40\\-20\\10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-16\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-80\\-160\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-160\\-440\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-180\\-110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-12560, 1040, 1200; 1040, -86, -100; 1200, -100, -111],[80, 2, 1],[4, 1, -11, -2, 2, 23, 71, 32; 40, 12, -110, -25, 18, 220, 690, 315; 7, 0, -20, 0, 5, 48, 140, 60],162,11]
%
% END OF LATTICE L163.11
%
\hbox{}\par\smallskip%
%
% LATTICE L163.12
%
\leavevmode%
\llap{\spacer}%
${L_{163.12}}=\pfillwitharg{5}({L_{163.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{8,5}{\infty}{40}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-174640\\35760\\4160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35760\\-7322\\-852
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4160\\-852\\-99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8579\\32760\\78000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1749\\-6679\\-15900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-209\\-798\\-1901
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\104\\238
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\160\\344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\95\\190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-80\\-20\\-32\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-32\\-80\\-80\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\10\\-20\\-20\\-80\\-110\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-80\\-20\\40\\0\\-80\\-180\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\10\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-80\\-20\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-32\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-80\\-80\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-80\\-110\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-180\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-174640, 35760, 4160; 35760, -7322, -852; 4160, -852, -99],[80, 2, 1],[27, 41, 24, -3, -5, 3, 31, 113; 104, 160, 95, -10, -20, 8, 115, 430; 238, 344, 190, -40, -38, 56, 310, 1040],162,12]
%
% END OF LATTICE L163.12
%
\hbox{}\par\smallskip%
%
% LATTICE L163.13
%
\leavevmode%
\llap{\spacer}%
${L_{163.13}}=\pfillwitharg{2.5}({L_{163.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{8,7}{\infty}{40}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-160\\320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\40\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
320\\-24\\-111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
639\\3240\\1120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-163\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-272\\-1377\\-477
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\370\\128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\61\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\235\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-55\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-80\\-160\\-80\\-16\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-20\\-16\\-5\\-40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\40\\-40\\-80\\-40\\-440\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-20\\-40\\40\\0\\-20\\-360\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-80\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-20\\-40\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-16\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-5\\-40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-40\\-440\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-360\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[240, -160, 320; -160, 40, -24; 320, -24, -111],[80, 8, 1],[73, 12, 46, -11, -9, 4, 114, 171; 370, 61, 235, -55, -46, 20, 575, 865; 128, 21, 80, -20, -16, 7, 200, 300],162,13]
%
% END OF LATTICE L163.13
%
\hbox{}\par\smallskip%
%
% LATTICE L163.14
%
\leavevmode%
\llap{\spacer}%
${L_{163.14}}=\pfillwitharg{2}({L_{163.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${25}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{20,9}{\infty}{10}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{100}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{20,1}{\infty}{10}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-11\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-9\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\-50\\-100\\-25\\-200\\-50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20\\-8\\-100\\-40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\0\\10\\-10\\-10\\-200\\-110\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-20\\-10\\10\\0\\-100\\-90\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-8\\-10\\0\\1\\0\\-10\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-100\\-200\\-100\\0\\100\\0\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-40\\-110\\-90\\-10\\0\\10\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-90\\-110\\-20\\-100\\-10\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[100, 0, 0; 0, -90, -10; 0, -10, -1],[100, 10, 1],[2, 1, 2, 1, 0, -1, 0, 1; -5, -4, -11, -9, -1, 0, 1, -1; 25, 24, 70, 60, 7, 0, -10, 0],162,14]
%
% END OF LATTICE L163.14
%
\hbox{}\par\smallskip%
%
% LATTICE L163.15
%
\leavevmode%
\llap{\spacer}%
${L_{163.15}}=\main({L_{163.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{50}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{20,11}{\infty}{20}\above{1pt}{1pt}{s}{2}{200}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{20,19}{\infty}{20}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-293800\\47200\\11600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47200\\-7570\\-1860
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11600\\-1860\\-457
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3\\-128\\444
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-3\\-105\\350
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0\\11\\-45
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1\\48\\-170
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-1\\-80\\300
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-1\\-51\\182
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-20\\-100\\-55\\-90\\-100\\0\\5\\-10
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-100\\-90\\-220\\-400\\-20\\-10\\20
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-293800, 47200, 11600; 47200, -7570, -1860; 11600, -1860, -457],[200, 10, 1],[-3, -3, 0, 1, -1, -1, -4, -9; -128, -105, 11, 48, -80, -51, -191, -408; 444, 350, -45, -170, 300, 182, 675, 1430],162,15]
%
% END OF LATTICE L163.15
%
\hbox{}\par\smallskip%
%
% LATTICE L163.16
%
\leavevmode%
\llap{\spacer}%
${L_{163.16}}=\main({L_{163.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{2}{200}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{20,1}{\infty}{20}\above{1pt}{1pt}{l}{2}{50}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{20,9}{\infty}{20}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110200\\-46000\\-6200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46000\\-18270\\-2360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6200\\-2360\\-293
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88\\-469\\1916
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
533\\-2840\\11600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\-229\\935
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-39\\208\\-850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-46\\245\\-1000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\-272\\1112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
154\\-821\\3355
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
433\\-2308\\9430
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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0\\200\\0\\-200\\-400\\-200\\-200\\-200
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-50\\-400\\-50\\0\\50\\0\\-50\\-200
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\kern3pt%
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-16\\-200\\-40\\-40\\0\\8\\0\\-40
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-200\\-55\\-90\\-50\\0\\5\\-10
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-200\\-90\\-220\\-200\\-40\\-10\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-110200, -46000, -6200; -46000, -18270, -2360; -6200, -2360, -293],[200, 10, 1],[88, 533, 43, -39, -46, 51, 154, 433; -469, -2840, -229, 208, 245, -272, -821, -2308; 1916, 11600, 935, -850, -1000, 1112, 3355, 9430],162,16]
%
% END OF LATTICE L163.16
%
\hbox{}\par\smallskip%
%
% LATTICE L163.17
%
\leavevmode%
\llap{\spacer}%
${L_{163.17}}=\pfillwitharg{5}({L_{163.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-}{3}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${64}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{160}\above{1pt}{1pt}{16,15}{\infty z}{40}\above{1pt}{1pt}{b}{2}{64}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{160}\above{1pt}{1pt}{16,7}{\infty z}{40}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10471360\\2107200\\64640
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2107200\\-424040\\-13008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64640\\-13008\\-399
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\320\\2688
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\51\\443
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
213\\830\\7440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\215\\2060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\8\\224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-20\\-158
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-3\\-10\\-160
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\195\\1580
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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64\\0\\-320\\-320\\-320\\-64\\-320\\0
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-64\\-10\\-160\\-40\\0\\4\\0\\-40
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\kern3pt%
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-320\\-80\\-1760\\-720\\-320\\0\\160\\-80
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\kern3pt%
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0\\-20\\-720\\-440\\-320\\-40\\-80\\40
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-10471360, 2107200, 64640; 2107200, -424040, -13008; 64640, -13008, -399],[320, 8, 1],[81, 13, 213, 56, 3, -5, -3, 49; 320, 51, 830, 215, 8, -20, -10, 195; 2688, 443, 7440, 2060, 224, -158, -160, 1580],162,17]
%
% END OF LATTICE L163.17
%
\hbox{}\par\smallskip%
%
% LATTICE L163.18
%
\leavevmode%
\llap{\spacer}%
${L_{163.18}}=\pfillwitharg{2}({L_{163.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${400}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{40,39}{\infty}{40}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{25}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{40,31}{\infty}{40}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-200\\-111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\0\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-69\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-81\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-28\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-15\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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400\\0\\-400\\-800\\-400\\-400\\-400\\0
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0\\1\\0\\-20\\-16\\-25\\-40\\-20
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-40\\-440\\-360\\-80\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-360\\-440\\-160\\-100\\-40\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[400, 0, 0; 0, -360, -200; 0, -200, -111],[400, 40, 1],[-1, 0, 1, 2, 1, 1, 1, 0; 0, -4, -69, -81, -28, -15, -1, 11; 0, 7, 120, 140, 48, 25, 0, -20],162,18]
%
% END OF LATTICE L163.18
%
\hbox{}\par\smallskip%
%
% LATTICE L163.19
%
\leavevmode%
\llap{\spacer}%
${L_{163.19}}=\pdualwitharg{2}(\pfillwitharg{2.5}({L_{163.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{8,1}{\infty}{40}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\80\\400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-26\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\-120\\-399
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
419\\1640\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-147\\-575\\28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-819\\-3198\\155
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\102\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\320\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\195\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-10\\-40\\-5\\-16\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-16\\-80\\-80\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\-20\\-10\\-80\\-110\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-80\\-20\\40\\0\\-80\\-180\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-10\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\10\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-80\\-20\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-16\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-80\\-80\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-80\\-110\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-180\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-240, 80, 400; 80, -26, -120; 400, -120, -399],[80, 2, 1],[26, 81, 49, -3, -5, 3, 56, 213; 102, 320, 195, -10, -20, 8, 215, 830; -5, -16, -10, 0, 1, 0, -10, -40],162,19]
%
% END OF LATTICE L163.19
%
\hbox{}\par\smallskip%
%
% LATTICE L163.20
%
\leavevmode%
\llap{\spacer}%
${L_{163.20}}=\pdualwitharg{2}(\main(\pfillwitharg{5}({L_{163.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{4,1}{\infty}{40}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-396\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-20\\320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\75\\-1216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\4\\-65
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-40\\-80\\-20\\-16\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-40\\-16\\-20\\-80\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\-40\\-40\\-80\\-440\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-40\\-40\\40\\0\\-40\\-360\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-40\\-40\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-16\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-20\\-80\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-80\\-440\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-360\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40, 0, 0; 0, -396, -20; 0, -20, -1],[40, 4, 1],[-1, 0, 1, -1, -1, -2, -11, -9; 0, 1, 0, -10, -4, -5, -20, -10; -6, -20, 0, 180, 70, 84, 320, 140],162,20]
%
% END OF LATTICE L163.20
%
\hbox{}\par\smallskip%
%
% LATTICE L163.21
%
\leavevmode%
\llap{\spacer}%
${L_{163.21}}=\pdualwitharg{2}(\main(\pfillwitharg{5}({L_{163.4}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{2,1}{\infty a}{40}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\400\\200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\-396\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\-220\\-111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
809\\-60\\1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1026\\75\\-1976
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-459\\34\\-885
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\1\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\-10\\200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-20\\-40\\-5\\-8\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-40\\-8\\-20\\-80\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\40\\-40\\-20\\-80\\-440\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\-40\\40\\0\\-40\\-360\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-20\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\-40\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-8\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-20\\-80\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-80\\-440\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-360\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-360, 400, 200; 400, -396, -220; 200, -220, -111],[40, 4, 1],[4, -10, -11, 101, 23, 64, 281, 169; 0, 1, 0, -10, -2, -5, -20, -10; 7, -20, -20, 200, 45, 124, 540, 320],162,21]
%
% END OF LATTICE L163.21
%
\hbox{}\par\smallskip%
%
% LATTICE L163.22
%
\leavevmode%
\llap{\spacer}%
${L_{163.22}}=\pdualwitharg{5}(\pfillwitharg{5.2}({L_{163.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{4,1}{\infty}{2}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-930\\200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\200\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-44\\-200
\end{tabular}\endgroup%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[20, 0, 0; 0, -930, 200; 0, 200, -43],[20, 10, 1],[2, 5, 2, 1, 0, -1, 0, 1; 10, 44, 25, 21, 12, 0, -3, 1; 45, 200, 114, 96, 55, 0, -14, 4],162,22]
%
% END OF LATTICE L163.22
%
\hbox{}\par\smallskip%
%
% LATTICE L163.23
%
\leavevmode%
\llap{\spacer}%
${L_{163.23}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{163.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48120\\1480\\1440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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2821\\10624\\83000
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
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37\\140\\1088
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27\\104\\792
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-48120, 1480, 1440; 1480, -40, -45; 1440, -45, -43],[40, 5, 1],[11, 37, 37, 27, 6, -3, -3, 7; 40, 138, 140, 104, 24, -10, -12, 24; 325, 1090, 1088, 792, 175, -90, -88, 208],162,23]
%
% END OF LATTICE L163.23
%
\hbox{}\par\smallskip%
%
% LATTICE L163.24
%
\leavevmode%
\llap{\spacer}%
${L_{163.24}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{163.1}}))$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33480\\-4240\\1200
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-33480, -4240, 1200; -4240, -536, 152; 1200, 152, -43],[40, 8, 1],[2, 1, -1, -2, -2, -1, 1, 2; 5, 12, 35, 15, 7, 0, -5, 0; 72, 68, 90, -5, -32, -28, 10, 55],162,24]
%
% END OF LATTICE L163.24
%
\hbox{}\par\smallskip%
%
% LATTICE L163.25
%
\leavevmode%
\llap{\spacer}%
${L_{163.25}}=\pdualwitharg{5}(\main(\pfillwitharg{5}({L_{163.2}})))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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40\\0\\-20\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\1\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-20\\-2\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-80\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-50\\-20\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-20\\-11\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-18\\-44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35720, 640, 1680; 640, -10, -30; 1680, -30, -79],[40, 10, 1],[-1, -5, -4, -9, -15, -3, 0, 1; -4, 7, 8, 22, 44, 13, 2, -2; -20, -110, -89, -202, -340, -70, -1, 22],162,25]
%
% END OF LATTICE L163.25
%
\hbox{}\par\smallskip%
%
% LATTICE L163.26
%
\leavevmode%
\llap{\spacer}%
${L_{163.26}}=\pdualwitharg{5}(\main(\pfillwitharg{5}({L_{163.4}})))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{4,1}{\infty}{4}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2840\\360\\160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-30\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
160\\-20\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
259\\-104\\4680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\11\\-540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\6\\-271
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\71
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-4\\198
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-10\\-40\\-50\\-80\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\40\\0\\-40\\-80\\-200\\-40\\-40
\end{tabular}\endgroup%
\kern3pt%
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-10\\0\\1\\-2\\-10\\-40\\-11\\-18
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-10\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\40\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\1\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-40\\-2\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-80\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-200\\-40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-40\\-11\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-18\\-44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2840, 360, 160; 360, -30, -20; 160, -20, -9],[40, 10, 1],[-1, 7, 4, 11, 11, 13, 1, -1; 1, 0, -1, -4, -5, -8, -1, 0; -20, 120, 71, 198, 200, 240, 19, -18],162,26]
%
% END OF LATTICE L163.26
%
\hbox{}\par\smallskip%
%
% LATTICE L163.27
%
\leavevmode%
\llap{\spacer}%
${L_{163.27}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{163.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{8,1}{\infty}{40}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-149360\\35840\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35840\\-8600\\96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\96\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5179\\21460\\-5920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1246\\-5163\\1424
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\58\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-33\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-120\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76\\-315\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59\\-245\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-40\\-80\\-20\\-32\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-32\\-80\\-160\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\-40\\-40\\-160\\-440\\-360
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-80\\-40\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-32\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-80\\-160\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-160\\-440\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-360\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-149360, 35840, -400; 35840, -8600, 96; -400, 96, -1],[80, 8, 1],[-8, -29, -76, -59, -6, 1, 6, -11; -33, -120, -315, -245, -25, 4, 25, -45; 22, 48, 80, 20, -6, -16, 0, 60],162,27]
%
% END OF LATTICE L163.27
%
\hbox{}\par\smallskip%
%
% LATTICE L163.28
%
\leavevmode%
\llap{\spacer}%
${L_{163.28}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{163.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{8,5}{\infty}{40}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2342960\\528560\\-16560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528560\\-119240\\3736
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16560\\3736\\-117
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51479\\208780\\-617760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11628\\-47159\\139536
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\1460\\-4321
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-28\\-113\\354
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-83\\-336\\1016
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-191\\-775\\2280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-124\\-505\\1420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-40\\-80\\-20\\-32\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-32\\-80\\-160\\-80
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\kern3pt%
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-80\\-80\\-40\\40\\0\\-80\\-360\\-440
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-40\\-80
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-80\\-40\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-32\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-80\\-160\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-160\\-440\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-360\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2342960, 528560, -16560; 528560, -119240, 3736; -16560, 3736, -117],[80, 8, 1],[-28, -83, -191, -124, -8, 11, 11, -56; -113, -336, -775, -505, -33, 44, 45, -225; 354, 1016, 2280, 1420, 78, -152, -120, 740],162,28]
%
% END OF LATTICE L163.28
%
\hbox{}\par\smallskip%
%
% LATTICE L163.29
%
\leavevmode%
\llap{\spacer}%
${L_{163.29}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{163.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{8,3}{\infty}{40}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-38960, -24080, 4080; -24080, -14872, 2520; 4080, 2520, -427],[80, 8, 1],[1, 1, 4, 1, -1, -1, -4, -1; 84, 32, 75, -5, -16, 2, 95, 175; 504, 198, 480, -20, -104, 2, 520, 1020],162,29]
%
% END OF LATTICE L163.29
%
\hbox{}\par\smallskip%
%
% LATTICE L163.30
%
\leavevmode%
\llap{\spacer}%
${L_{163.30}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{163.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-71440\\-19360\\4000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19360\\-5230\\1080
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-71440, -19360, 4000; -19360, -5230, 1080; 4000, 1080, -223],[80, 10, 1],[-15, -3, 0, 1, -1, -5, -4, -9; 428, 96, 5, -26, 12, 124, 105, 246; 1800, 410, 24, -108, 40, 510, 436, 1028],162,30]
%
% END OF LATTICE L163.30
%
\hbox{}\par\smallskip%
%
% LATTICE L163.31
%
\leavevmode%
\llap{\spacer}%
${L_{163.31}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{163.3}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{8,7}{\infty}{8}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35920\\-7520\\1280
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7520\\-1550\\260
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-400\\-80\\-40\\0
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-80\\-25\\-20\\-20
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-40\\-20\\-22\\-36
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-35920, -7520, 1280; -7520, -1550, 260; 1280, 260, -43],[80, 10, 1],[59, 8, 2, -3, -3, 6, 12, 31; -580, -78, -19, 30, 28, -60, -119, -306; -1760, -235, -56, 92, 80, -185, -364, -932],162,31]
%
% END OF LATTICE L163.31
%
\hbox{}\par\smallskip%
%
% LATTICE L163.32
%
\leavevmode%
\llap{\spacer}%
${L_{163.32}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{163.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12400\\-4320\\960
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4320\\-1490\\330
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
960\\330\\-73
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{$\left.\llap{\phantom{%
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12400, -4320, 960; -4320, -1490, 330; 960, 330, -73],[80, 10, 1],[7, -1, -1, 1, 13, 11, 11, 4; -64, 14, 10, -13, -152, -122, -118, -41; -200, 50, 32, -46, -520, -410, -392, -134],162,32]
%
% END OF LATTICE L163.32
%
\hbox{}\par\smallskip%
%
% LATTICE L163.33
%
\leavevmode%
\llap{\spacer}%
${L_{163.33}}=\pdualwitharg{2.5}(\pfillwitharg{2.5}({L_{163.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{8,7}{\infty z}{2}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5680\\-4480\\1040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4480\\-3250\\750
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1040\\750\\-173
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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363\\-6968\\-28080
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\1340\\5399
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\18\\72
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-41\\-166
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-400\\-80\\-80\\0
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-80\\-25\\-40\\-10
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5680, -4480, 1040; -4480, -3250, 750; 1040, 750, -173],[80, 10, 1],[7, -1, -1, 2, 21, 8, 15, 5; -120, 21, 18, -41, -416, -155, -286, -93; -480, 85, 72, -166, -1680, -625, -1152, -374],162,33]
%
% END OF LATTICE L163.33
%
\hbox{}\par\smallskip%
%
% LATTICE L163.34
%
\leavevmode%
\llap{\spacer}%
${L_{163.34}}=\pdualwitharg{2.5}(\main(\pfillwitharg{5}({L_{163.2}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{2,1}{\infty b}{8}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\760\\200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
760\\-500\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\-140\\-37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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29\\-5\\180
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17\\-4\\110
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5\\-2\\36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-16
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-80\\-40\\-8\\8
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-100\\-80\\-40\\0
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-80\\-100\\-80\\-40
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1080, 760, 200; 760, -500, -140; 200, -140, -37],[40, 20, 1],[29, 17, 5, -3, -3, 9, 21, 29; -5, -4, -2, 0, 1, 0, -2, -4; 180, 110, 36, -16, -20, 50, 124, 176],162,34]
%
% END OF LATTICE L163.34
%
\hbox{}\par\smallskip%
%
% LATTICE L163.35
%
\leavevmode%
\llap{\spacer}%
${L_{163.35}}=\pdualwitharg{2.5}(\main(\pfillwitharg{5}({L_{163.4}})))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{4,3}{\infty}{8}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
560\\-60\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\-60\\-43
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3720, 560, 400; 560, -60, -60; 400, -60, -43],[40, 20, 1],[32, 9, 5, -3, -2, 6, 25, 33; -5, -2, -2, 0, 1, 0, -2, -4; 300, 85, 48, -28, -20, 55, 232, 308],162,35]
%
% END OF LATTICE L163.35
%
\hbox{}\par\smallskip%
%
% LATTICE L163.36
%
\leavevmode%
\llap{\spacer}%
${L_{163.36}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{163.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{2,1}{\infty}{4}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-29240\\-32320\\1520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32320\\-35720\\1680
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1520\\1680\\-79
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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859\\-688\\1720
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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4\\-4\\-9
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11\\-9\\18
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-80\\-40\\-2\\4
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-200\\-160\\-20\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-200\\-40\\-40
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-29240, -32320, 1520; -32320, -35720, 1680; 1520, 1680, -79],[40, 40, 1],[-2, 7, 4, 11, 22, 13, 1, -1; -1, -10, -4, -9, -15, -6, 0, 1; -60, -80, -9, 18, 100, 120, 19, 2],162,36]
%
% END OF LATTICE L163.36
%
\hbox{}\par\smallskip%
%
% LATTICE L163.37
%
\leavevmode%
\llap{\spacer}%
${L_{163.37}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{163.3}}))$%
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{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\-80\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-1000\\-200
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-200\\-37
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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23\\28\\-160
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-156\\-183\\1040
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{$\left[\!\llap{\phantom{%
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2\\1\\-4
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3\\3\\-16
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-160\\-40\\-8\\8
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-400\\-160\\-80\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-100\\-80\\-40
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-80\\-80\\-88\\-72
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[80, -80, 0; -80, -1000, -200; 0, -200, -37],[80, 40, 1],[-23, -3, 2, 3, -1, -9, -14, -15; -28, -5, 1, 3, 0, -9, -15, -17; 160, 30, -4, -16, 0, 50, 84, 96],162,37]
%
% END OF LATTICE L163.37
%
\hbox{}\par\smallskip%
%
% LATTICE L163.38
%
\leavevmode%
\llap{\spacer}%
${L_{163.38}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{163.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{4,3}{\infty b}{8}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4080\\2720\\-480
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2720\\200\\-80
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-480\\-80\\23
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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467\\6188\\31200
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4080, 2720, -480; 2720, 200, -80; -480, -80, 23],[80, 40, 1],[39, 11, 3, -2, -3, 7, 15, 20; 516, 145, 39, -27, -40, 93, 199, 265; 2600, 730, 196, -136, -200, 470, 1004, 1336],162,38]
%
% END OF LATTICE L163.38
%
\hbox{}\par\smallskip%
%
% LATTICE L163.39
%
\leavevmode%
\llap{\spacer}%
${L_{163.39}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{163.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\4960\\-160
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4960\\-99400\\3200
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\3200\\-103
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-109\\-60\\-1680
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\24\\680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\8\\230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\3\\88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-160\\-400\\-160\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-40\\-160\\-100\\-80\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\8\\-8\\-80\\-80\\-88\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-40\\-8\\8\\0\\-40\\-72\\-88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-40\\-8\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-160\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-100\\-80\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-80\\-88\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-72\\-88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-240, 4960, -160; 4960, -99400, 3200; -160, 3200, -103],[80, 40, 1],[39, 11, 3, -2, -3, 7, 15, 20; 24, 8, 3, -1, -4, 2, 7, 11; 680, 230, 88, -28, -120, 50, 192, 308],162,39]
%
% END OF LATTICE L163.39
%
\hbox{}\par\smallskip%
%
% LATTICE L163.40
%
\leavevmode%
\llap{\spacer}%
${L_{163.40}}=\pdualwitharg{5}(\pfillwitharg{2.5}({L_{163.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{8,7}{\infty}{8}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14800\\22240\\-800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22240\\-33400\\1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800\\1200\\-43
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1345\\-1248\\-9600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1988\\1845\\14200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-65\\-501
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\54\\400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\7\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\3\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-3\\-28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-160\\-400\\-80\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-20\\-80\\-25\\-40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\8\\-8\\-80\\-40\\-88\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-20\\-8\\8\\0\\-20\\-72\\-88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\8\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-20\\-8\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-80\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-25\\-40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-40\\-88\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-72\\-88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14800, 22240, -800; 22240, -33400, 1200; -800, 1200, -43],[80, 40, 1],[59, 8, 4, -3, -3, 6, 24, 31; 54, 7, 3, -3, -2, 6, 23, 29; 400, 45, 8, -28, 0, 55, 192, 228],162,40]
%
% END OF LATTICE L163.40
%
\hbox{}\par\smallskip%
%
% LATTICE L163.41
%
\leavevmode%
\llap{\spacer}%
${L_{163.41}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{163.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${400}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{40,39}{\infty z}{10}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{25}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{40,31}{\infty z}{10}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-239600\\4800\\-26800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4800\\-90\\560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26800\\560\\-2911
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-299\\-6000\\1600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-462\\123
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-217\\-4362\\1160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73\\-1469\\390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-424\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\280\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\302\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-561\\150
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\0\\-400\\-400\\-400\\-400\\-400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-10\\-16\\-25\\-40\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\0\\40\\-20\\-80\\-200\\-440\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-10\\-20\\10\\0\\-50\\-180\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-16\\-80\\0\\16\\0\\-80\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-25\\-200\\-50\\0\\25\\0\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-40\\-440\\-180\\-80\\0\\40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-180\\-110\\-80\\-50\\-20\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-239600, 4800, -26800; 4800, -90, 560; -26800, 560, -2911],[400, 10, 1],[-299, -23, -217, -73, -21, 14, 15, -28; -6000, -462, -4362, -1469, -424, 280, 302, -561; 1600, 123, 1160, 390, 112, -75, -80, 150],162,41]
%
% END OF LATTICE L163.41
%
\hbox{}\par\smallskip%
%
% LATTICE L163.42
%
\leavevmode%
\llap{\spacer}%
${L_{163.42}}=\pdualwitharg{2}(\main({L_{163.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${100}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{10,9}{\infty b}{40}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{100}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{10,1}{\infty b}{40}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2600\\600\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
600\\140\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\26\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\24\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\0\\-200\\-400\\-100\\-400\\-200\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-40\\-16\\-100\\-80\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\0\\40\\-40\\-40\\-400\\-440\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-40\\-40\\40\\0\\-200\\-360\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-16\\-40\\0\\4\\0\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-100\\-400\\-200\\0\\100\\0\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-80\\-440\\-360\\-40\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-360\\-440\\-80\\-200\\-40\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2600, 600, 0; 600, 140, 0; 0, 0, -1],[200, 20, 1],[-1, -1, -7, -7, -1, -1, 1, 1; 5, 4, 26, 24, 3, 0, -6, -4; 0, -6, -60, -80, -16, -50, -20, 0],162,42]
%
% END OF LATTICE L163.42
%
\hbox{}\par\smallskip%
%
% LATTICE L163.43
%
\leavevmode%
\llap{\spacer}%
${L_{163.43}}=\pdualwitharg{2}(\main({L_{163.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${100}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{20,19}{\infty}{40}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{25}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{20,11}{\infty}{40}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-727400\\-177800\\9000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-177800\\-43460\\2200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9000\\2200\\-111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\45\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-16\\7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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91\\-366\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
125\\-504\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-101\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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37\\-150\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-94\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\44\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\0\\-200\\-400\\-100\\-200\\-200\\0
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\kern3pt%
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0\\1\\0\\-20\\-8\\-25\\-40\\-20
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-200\\0\\40\\-40\\-40\\-200\\-440\\-360
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-20\\-40\\40\\0\\-100\\-360\\-440
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-8\\-40\\0\\4\\0\\-40\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-25\\-200\\-100\\0\\25\\0\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-40\\-440\\-360\\-40\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-360\\-440\\-80\\-100\\-40\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-727400, -177800, 9000; -177800, -43460, 2200; 9000, 2200, -111],[200, 20, 1],[-11, 4, 91, 125, 25, 37, 23, -11; 45, -16, -366, -504, -101, -150, -94, 44; 0, 7, 120, 140, 24, 25, 0, -20],162,43]
%
% END OF LATTICE L163.43
%
\hbox{}\par\smallskip%
%
% LATTICE L163.44
%
\leavevmode%
\llap{\spacer}%
${L_{163.44}}=\pdualwitharg{2}({L_{163.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{200}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{10,1}{\infty}{20}\above{1pt}{1pt}{b}{2}{200}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{10,9}{\infty}{20}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110200\\-34800\\-70600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34800\\-10920\\-22240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70600\\-22240\\-45187
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-202\\-407\\516
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-627\\-1260\\1600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\-106\\135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\87\\-110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
118\\235\\-300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\-108\\136
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-170\\-344\\435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-489\\-987\\1250
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-20\\-80\\-200\\-32\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\200\\0\\-200\\-800\\-200\\-200\\-200
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\5\\-10\\-100\\-40\\-55\\-90
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-80\\-200\\-10\\20\\0\\-40\\-90\\-220
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-200\\-800\\-100\\0\\200\\0\\-100\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-200\\-40\\-40\\0\\8\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-200\\-55\\-90\\-100\\0\\5\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-200\\-90\\-220\\-400\\-40\\-10\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-110200, -34800, -70600; -34800, -10920, -22240; -70600, -22240, -45187],[200, 40, 1],[-202, -627, -53, 43, 118, -53, -170, -489; -407, -1260, -106, 87, 235, -108, -344, -987; 516, 1600, 135, -110, -300, 136, 435, 1250],162,44]
%
% END OF LATTICE L163.44
%
\hbox{}\par\smallskip%
%
% LATTICE L163.45
%
\leavevmode%
\llap{\spacer}%
${L_{163.45}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{163.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${320}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{32}\above{1pt}{1pt}{16,15}{\infty z}{8}\above{1pt}{1pt}{b}{2}{320}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{*}{2}{32}\above{1pt}{1pt}{16,7}{\infty z}{8}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-191680\\-119040\\7680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119040\\-73800\\4760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7680\\4760\\-307
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-240\\-3200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-48\\-645
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-186\\-2512
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-65\\-884
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-104\\-1440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\14\\192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-21\\-276
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
320\\0\\-320\\-320\\-1600\\-320\\-320\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\5\\0\\-20\\-160\\-50\\-80\\-20
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-320\\0\\32\\-16\\-320\\-160\\-352\\-144
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\kern3pt%
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-320\\-20\\-16\\8\\0\\-40\\-144\\-88
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1600\\-160\\-320\\0\\320\\0\\-320\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-320\\-50\\-160\\-40\\0\\20\\0\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-320\\-80\\-352\\-144\\-320\\0\\32\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-144\\-88\\-320\\-40\\-16\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-191680, -119040, 7680; -119040, -73800, 4760; 7680, 4760, -307],[320, 40, 1],[21, 4, 15, 5, 7, -1, -1, 2; -240, -48, -186, -65, -104, 10, 14, -21; -3200, -645, -2512, -884, -1440, 130, 192, -276],162,45]
%
% END OF LATTICE L163.45
%
\hbox{}\par\smallskip%
%
% LATTICE L163.46
%
\leavevmode%
\llap{\spacer}%
${L_{163.46}}=\pdualwitharg{2}({L_{163.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{400}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{40,1}{\infty}{40}\above{1pt}{1pt}{b}{2}{100}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{40,9}{\infty}{40}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\-800\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800\\1640\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-5\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-4\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-11\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-9\\-60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-40\\-80\\-100\\-32\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\400\\0\\-400\\-800\\-400\\-800\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\-40\\-200\\-160\\-440\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-400\\-40\\40\\0\\-80\\-360\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-800\\-200\\0\\100\\0\\-200\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-400\\-160\\-80\\0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-800\\-440\\-360\\-200\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-400\\-360\\-440\\-400\\-80\\-40\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[400, -800, 0; -800, 1640, 0; 0, 0, -1],[400, 40, 1],[-2, 1, 2, -3, -12, -9, -24, -19; -1, 0, 1, -1, -5, -4, -11, -9; -6, 0, 0, -20, -50, -32, -80, -60],162,46]
%
% END OF LATTICE L163.46
%
\hbox{}\par\smallskip%
%
% LATTICE L163.47
%
\leavevmode%
\llap{\spacer}%
${L_{163.47}}=\pdualwitharg{2}({L_{163.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{400}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{40,21}{\infty}{40}\above{1pt}{1pt}{s}{2}{100}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{40,29}{\infty}{40}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3047600\\-76000\\8000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76000\\40\\200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8000\\200\\-21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-1\\2654
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\-4200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\1\\-3800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-1\\4180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\-5\\17850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\-4\\13288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
92\\-11\\34920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\-9\\26940
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\0\\-40\\-80\\-100\\-32\\-40\\0
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0\\400\\0\\-400\\-800\\-400\\-800\\-400
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\40\\-40\\-200\\-160\\-440\\-360
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-800\\-200\\0\\100\\0\\-200\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-400\\-160\\-80\\0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-800\\-440\\-360\\-200\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-400\\-360\\-440\\-400\\-80\\-40\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3047600, -76000, 8000; -76000, 40, 200; 8000, 200, -21],[400, 40, 1],[7, -11, -10, 11, 47, 35, 92, 71; -1, 0, 1, -1, -5, -4, -11, -9; 2654, -4200, -3800, 4180, 17850, 13288, 34920, 26940],162,47]
%
% END OF LATTICE L163.47
%
\hbox{}\par\smallskip%
%
% LATTICE L163.48
%
\leavevmode%
\llap{\spacer}%
${L_{163.48}}=\pdualwitharg{2}({L_{163.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${400}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{40,19}{\infty}{40}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{s}{2}{100}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{40,11}{\infty}{40}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51047600\\156000\\130000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
156000\\-360\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
130000\\-400\\-331
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-100\\-4200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\62\\2674
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\631\\27120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
92\\819\\35140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\312\\13368
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\420\\17950
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\99\\4200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-89\\-3820
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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400\\0\\-400\\-800\\-400\\-800\\-400\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\4\\0\\-40\\-32\\-100\\-80\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\0\\40\\-40\\-80\\-400\\-440\\-360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800\\-40\\-40\\40\\0\\-200\\-360\\-440
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\kern3pt%
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-400\\-32\\-80\\0\\16\\0\\-80\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800\\-100\\-400\\-200\\0\\100\\0\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-80\\-440\\-360\\-80\\0\\40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-360\\-440\\-160\\-200\\-40\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-51047600, 156000, 130000; 156000, -360, -400; 130000, -400, -331],[400, 40, 1],[-11, 7, 71, 92, 35, 47, 11, -10; -100, 62, 631, 819, 312, 420, 99, -89; -4200, 2674, 27120, 35140, 13368, 17950, 4200, -3820],162,48]
%
% END OF LATTICE L163.48
%
\hbox{}\par\smallskip%
%
% LATTICE L163.49
%
\leavevmode%
\llap{\spacer}%
${L_{163.49}}=\pdualwitharg{2}({L_{163.5}})\iso \pdualwitharg{5}({L_{163.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-}{3}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1600}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{160}\above{1pt}{1pt}{80,39}{\infty z}{40}\above{1pt}{1pt}{l}{2}{64}\above{1pt}{1pt}{}{2}{25}\above{1pt}{1pt}{r}{2}{160}\above{1pt}{1pt}{80,31}{\infty z}{40}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
shares genus with 2-dual${}\iso{}$5-dual; isometric to own %
2.5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110705600\\211200\\-21212800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
211200\\-360\\40640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21212800\\40640\\-4063999
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9403\\-188080\\47200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-661\\-13222\\3318
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5849\\-117002\\29360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1781\\-35629\\8940
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-357\\-7144\\1792
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
254\\5080\\-1275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
255\\5102\\-1280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1016\\-20321\\5100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1600\\0\\-1600\\-1600\\-1600\\-800\\-1600\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-40\\-64\\-50\\-160\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1600\\0\\160\\-80\\-320\\-400\\-1760\\-720
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\kern3pt%
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-1600\\-40\\-80\\40\\0\\-100\\-720\\-440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1600\\-64\\-320\\0\\64\\0\\-320\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800\\-50\\-400\\-100\\0\\25\\0\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1600\\-160\\-1760\\-720\\-320\\0\\160\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-720\\-440\\-320\\-100\\-80\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-110705600, 211200, -21212800; 211200, -360, 40640; -21212800, 40640, -4063999],[1600, 40, 1],[-9403, -661, -5849, -1781, -357, 254, 255, -1016; -188080, -13222, -117002, -35629, -7144, 5080, 5102, -20321; 47200, 3318, 29360, 8940, 1792, -1275, -1280, 5100],162,49]
%
% END OF LATTICE L163.49
%

\medskip%
%------------------------------WEYL-GROUP-W164--------------------------------
%
\leavevmode\llap{}%
$W_{164}$%
\qquad\llap{44} lattices, $\chi=54$%
\hfill%
$42\infty2242\infty22\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L164.1
%
\leavevmode%
\llap{\spacer}%
${L_{164.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{4}{2}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{3,2}{\infty}{20}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6278220\\29700\\59400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29700\\-140\\-281
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59400\\-281\\-562
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18181\\181800\\-2017980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
91\\-911\\10101
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
172\\-1720\\19091
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-58\\982
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-102\\1111
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-200\\1480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-70\\-280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-54\\-1134
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-20\\-40\\-108\\-40\\-58\\-100\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-30\\-126\\-58\\-102\\-200\\-70\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\-20\\-180\\-100\\-200\\-420\\-180\\-180
\end{tabular}\endgroup%
\kern3pt%
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{$\left[\!\llap{\phantom{%
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-20\\-70\\-180\\-120\\-180
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-6278220, 29700, 59400; 29700, -140, -281; 59400, -281, -562],[180, 1, 1],[9, 10, 13, -3, -11, -3, 0, 7, 13, 29; -58, -102, -200, -70, -54, -2, 2, 0, -30, -126; 982, 1111, 1480, -280, -1134, -316, -1, 740, 1390, 3132],163,1]
%
% END OF LATTICE L164.1
%
\hbox{}\par\smallskip%
%
% LATTICE L164.2
%
\leavevmode%
\llap{\spacer}%
${L_{164.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-6977160, 40320, 23040; 40320, -233, -133; 23040, -133, -70],[360, 1, 1],[-44, -163, -329, -63, -55, -4, 3, 9, -17, -89; -7716, -28586, -57700, -11050, -9648, -702, 526, 1580, -2980, -15606; 175, 652, 1320, 255, 225, 17, -12, -40, 65, 351],163,2]
%
% END OF LATTICE L164.2
%
\hbox{}\par\smallskip%
%
% LATTICE L164.3
%
\leavevmode%
\llap{\spacer}%
${L_{164.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-133937640, 26822160, 347040; 26822160, -5371367, -69498; 347040, -69498, -899],[360, 1, 1],[76, -55, -53, 824, 3577, 1667, 2960, 11673, 2081, 1652; 373, -270, -260, 4045, 17559, 8183, 14530, 57300, 10215, 8109; 503, -359, -360, 5385, 23409, 10915, 19389, 76480, 13645, 10845],163,3]
%
% END OF LATTICE L164.3
%
\hbox{}\par\smallskip%
%
% LATTICE L164.4
%
\leavevmode%
\llap{\spacer}%
${L_{164.4}}=\pfillwitharg{2}({L_{164.1}})$%
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{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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-10\\-29\\-25\\-5\\0
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-29\\-102\\-100\\-35\\-27
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-25\\-100\\-105\\-45\\-45
\end{tabular}\endgroup%
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-5\\-35\\-45\\-30\\-45
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-15570, 450, 315; 450, -13, -9; 315, -9, -4],[45, 1, 1],[1, -1, -1, 6, 29, 14, 51, 51, 19, 16; 35, -36, -35, 215, 1035, 499, 1816, 1815, 675, 567; -1, 2, 0, -15, -63, -29, -102, -100, -35, -27],163,4]
%
% END OF LATTICE L164.4
%
\hbox{}\par\smallskip%
%
% LATTICE L164.5
%
\leavevmode%
\llap{\spacer}%
${L_{164.5}}=\pfillwitharg{2}({L_{164.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{6,5}{\infty}{10}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-69210\\4410\\4410
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4410\\-280\\-281
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4410\\-281\\-281
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-2701\\13500\\-56700
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
182\\-911\\3822
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172\\-860\\3611
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-3\\-1\\-46
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0\\1\\-1
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7\\0\\110
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13\\-15\\220
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-63\\522
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-20\\-29\\-50\\-10\\0
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-29\\-51\\-100\\-35\\-27
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-50\\-100\\-210\\-90\\-90
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\kern3pt%
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-10\\-35\\-90\\-60\\-90
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-27\\-90\\-90\\-180
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-69210, 4410, 4410; 4410, -280, -281; 4410, -281, -281],[90, 1, 1],[-3, 0, 7, 13, 29, 9, 10, 13, -3, -11; -1, 1, 0, -15, -63, -29, -51, -100, -35, -27; -46, -1, 110, 220, 522, 172, 211, 310, -10, -144],163,5]
%
% END OF LATTICE L164.5
%
\hbox{}\par\smallskip%
%
% LATTICE L164.6
%
\leavevmode%
\llap{\spacer}%
${L_{164.6}}=\main({L_{164.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{6,5}{\infty z}{5}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8769780\\99900\\17820
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99900\\-1138\\-203
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17820\\-203\\-35
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-342721\\-30052260\\-214200
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\kern3pt%
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3904\\342331\\2440
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624\\54717\\389
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-44\\-3858\\-29
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-163\\-14293\\-102
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-329\\-28850\\-200
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\HardButStrongLineBreak\kern3pt%
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-63\\-5525\\-35
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-55\\-4824\\-27
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-27\\-63\\-90\\0\\9
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-10\\-29\\-50\\-5\\0
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-29\\-102\\-200\\-35\\-27
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-50\\-200\\-420\\-90\\-90
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-5\\-35\\-90\\-30\\-45
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0\\-27\\-90\\-45\\-90
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-8769780, 99900, 17820; 99900, -1138, -203; 17820, -203, -35],[180, 1, 1],[-44, -163, -329, -63, -55, -4, 3, 9, -17, -89; -3858, -14293, -28850, -5525, -4824, -351, 263, 790, -1490, -7803; -29, -102, -200, -35, -27, -1, 2, 0, -15, -63],163,6]
%
% END OF LATTICE L164.6
%
\hbox{}\par\smallskip%
%
% LATTICE L164.7
%
\leavevmode%
\llap{\spacer}%
${L_{164.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{164.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{6,5}{\infty}{20}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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2375280\\90\\1157400
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\kern3pt%
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90\\2\\44
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1157400\\44\\563965
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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-981541\\-155610\\2014380
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-5740\\-911\\11780
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-153\\-24\\314
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\HardButStrongLineBreak\kern3pt%
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0\\1\\0
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-134\\-20\\275
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-1837\\-290\\3770
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7271\\-1152\\14922
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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-2\\2\\0\\-30\\-126
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-40\\-30\\-10\\20\\0
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-108\\-126\\-90\\0\\36
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-40\\-58\\-50\\-20\\0
\end{tabular}\endgroup%
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-58\\-102\\-100\\-70\\-54
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-50\\-100\\-105\\-90\\-90
\end{tabular}\endgroup%
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-20\\-70\\-90\\-120\\-180
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2375280, 90, 1157400; 90, 2, 44; 1157400, 44, 563965],[90, 2, 1],[-153, 0, -134, -1837, -7271, -3291, -5740, -5606, -3903, -3061; -24, 1, -20, -290, -1152, -522, -911, -890, -620, -486; 314, 0, 275, 3770, 14922, 6754, 11780, 11505, 8010, 6282],163,7]
%
% END OF LATTICE L164.7
%
\hbox{}\par\smallskip%
%
% LATTICE L164.8
%
\leavevmode%
\llap{\spacer}%
${L_{164.8}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{164.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{4}{10}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{3,2}{\infty}{1}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-45\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-20\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-45\\-89
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-469\\360\\360
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\kern3pt%
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-169\\129\\130
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-442\\340\\339
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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34\\-29\\-25
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
131\\-102\\-100
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27\\-20\\-21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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11\\-7\\-9
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\-27\\-45
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-5\\-5\\-10\\-135\\-50\\-145\\-25\\-5\\0
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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-5\\10\\0\\-15\\-315
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-5\\0\\1\\-1\\-45
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-10\\-15\\-1\\1\\0
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-50\\-145\\-25\\-5\\0
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-145\\-510\\-100\\-35\\-135
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-25\\-100\\-21\\-9\\-45
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-35\\-9\\-6\\-45
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-135\\-45\\-45\\-450
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-90, -45, -90; -45, -20, -45; -90, -45, -89],[45, 5, 1],[34, 131, 27, 11, 52, 5, -1, -1, 2, 65; -29, -102, -20, -7, -27, -1, 2, 0, -3, -63; -25, -100, -21, -9, -45, -5, 0, 1, -1, -45],163,8]
%
% END OF LATTICE L164.8
%
\hbox{}\par\smallskip%
%
% LATTICE L164.9
%
\leavevmode%
\llap{\spacer}%
${L_{164.9}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{164.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{5}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{4}{18}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{3,1}{\infty}{45}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
103320\\-1215\\-19890
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1215\\18\\234
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19890\\234\\3829
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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85119\\-8645\\442890
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-1280\\129\\-6660
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-16384\\1664\\-85249
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-19\\4\\-99
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
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-26\\0\\-135
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-199\\15\\-1035
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89\\8\\-463
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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9\\-9\\-45\\-90\\-27\\-90\\-261\\-225\\-45\\0
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-27\\-63\\-45\\0\\1
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-90\\-261\\-225\\-45\\0
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-261\\-918\\-900\\-315\\-27
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0\\-27\\-45\\-45\\-10
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[103320, -1215, -19890; -1215, 18, 234; -19890, 234, 3829],[45, 9, 1],[-19, 0, -26, -199, -89, -365, -1280, -1254, -441, -39; 4, 1, 0, 15, 8, 35, 129, 130, 50, 5; -99, 0, -135, -1035, -463, -1899, -6660, -6525, -2295, -203],163,9]
%
% END OF LATTICE L164.9
%
\hbox{}\par\smallskip%
%
% LATTICE L164.10
%
\leavevmode%
\llap{\spacer}%
${L_{164.10}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{164.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{4}{5}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{6,5}{\infty}{2}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-83070\\-16920\\1890
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16920\\-3440\\385
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1890\\385\\-43
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11231\\-7020\\428220
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2352\\-1471\\89670
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-256\\160\\-9761
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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5\\-1\\210
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\HardButStrongLineBreak\kern3pt%
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-1\\1\\-35
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\HardButStrongLineBreak\kern3pt%
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-1\\0\\-44
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\HardButStrongLineBreak\kern3pt%
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3\\-3\\104
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
85\\-63\\3150
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-83070, -16920, 1890; -16920, -3440, 385; 1890, 385, -43],[90, 5, 1],[5, -1, -1, 3, 85, 43, 81, 33, 13, 59; -1, 1, 0, -3, -63, -29, -51, -20, -7, -27; 210, -35, -44, 104, 3150, 1620, 3085, 1264, 506, 2340],163,10]
%
% END OF LATTICE L164.10
%
\hbox{}\par\smallskip%
%
% LATTICE L164.11
%
\leavevmode%
\llap{\spacer}%
${L_{164.11}}=\pdualwitharg{2}(\main({L_{164.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{4}{8}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{3,2}{\infty b}{20}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
814680\\436500\\204120
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
436500\\235780\\109388
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
204120\\109388\\51143
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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7291391\\342331\\-29833392
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3170170\\148840\\-12971041
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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14719\\691\\-60224
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51971\\2440\\-212644
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51117\\2400\\-209150
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18208\\855\\-74500
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14755\\693\\-60372
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{$\left.\llap{\phantom{%
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0\\0\\0
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-40\\-60\\-20\\20\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[814680, 436500, 204120; 436500, 235780, 109388; 204120, 109388, 51143],[180, 4, 1],[14719, 51971, 51117, 18208, 14755, 872, -1, 853, 7777, 32018; 691, 2440, 2400, 855, 693, 41, 0, 40, 365, 1503; -60224, -212644, -209150, -74500, -60372, -3568, 4, -3490, -31820, -131004],163,11]
%
% END OF LATTICE L164.11
%
\hbox{}\par\smallskip%
%
% LATTICE L164.12
%
\leavevmode%
\llap{\spacer}%
${L_{164.12}}=\pdualwitharg{2}({L_{164.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{4}{8}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{3,2}{\infty}{20}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
130774320\\180\\31938480
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\8\\44
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31938480\\44\\7800205
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-20569681\\-438165\\84223980
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\kern3pt%
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-42720\\-911\\174920
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-5023872\\-107016\\20570591
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-12245\\-261\\50138
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\HardButStrongLineBreak\kern3pt%
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-42720\\-911\\174920
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-20863\\-445\\85425
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-14529\\-310\\59490
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11399\\-243\\46674
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-10\\0\\5\\-10\\-90
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-50\\-200\\-105\\-90\\-90
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-20\\-140\\-90\\-120\\-180
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[130774320, 180, 31938480; 180, 8, 44; 31938480, 44, 7800205],[180, 4, 1],[-12245, -42720, -20863, -14529, -11399, -571, 0, -497, -6831, -27049; -261, -911, -445, -310, -243, -12, 1, -10, -145, -576; 50138, 174920, 85425, 59490, 46674, 2338, 0, 2035, 27970, 110754],163,12]
%
% END OF LATTICE L164.12
%
\hbox{}\par\smallskip%
%
% LATTICE L164.13
%
\leavevmode%
\llap{\spacer}%
${L_{164.13}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{164.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{}{4}{9}\above{1pt}{1pt}{}{2}{90}\above{1pt}{1pt}{6,1}{\infty}{90}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
126810\\90\\10800
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\kern3pt%
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90\\9\\9
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10800\\9\\920
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{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-11761\\-21840\\138240
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-490\\-911\\5760
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-1078\\-2002\\12671
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-13\\-24\\153
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
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-23\\-40\\270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-157\\-290\\1845
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\-128\\811
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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-9\\9\\0\\-135\\-63
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-90\\0\\90\\-90\\-90
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-180\\-135\\-90\\90\\0
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-180\\-261\\-450\\-90\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-261\\-459\\-900\\-315\\-27
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-450\\-900\\-1890\\-810\\-90
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-315\\-810\\-540\\-90
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[126810, 90, 10800; 90, 9, 9; 10800, 9, 920],[90, 9, 1],[-13, 0, -23, -157, -69, -281, -490, -957, -333, -29; -24, 1, -40, -290, -128, -522, -911, -1780, -620, -54; 153, 0, 270, 1845, 811, 3303, 5760, 11250, 3915, 341],163,13]
%
% END OF LATTICE L164.13
%
\hbox{}\par\smallskip%
%
% LATTICE L164.14
%
\leavevmode%
\llap{\spacer}%
${L_{164.14}}=\pdualwitharg{5}(\main({L_{164.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{4}{10}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{6,5}{\infty z}{1}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-206100\\-100980\\1980
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100980\\-49475\\970
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1980\\970\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13825\\31104\\144000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6756\\15200\\70375
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\-297\\-1376
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\77\\380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-123\\277\\1295
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\110\\498
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\20\\81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\81\\270
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-5\\-10\\-10\\-135\\-50\\-145\\-50\\-5\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\10\\0\\-15\\-315
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\4\\-2\\-90
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-15\\-2\\1\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-315\\-90\\0\\45
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-50\\-145\\-50\\-5\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145\\-510\\-200\\-35\\-135
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-200\\-84\\-18\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-35\\-18\\-6\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-135\\-90\\-45\\-450
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-206100, -100980, 1980; -100980, -49475, 970; 1980, 970, -19],[180, 5, 1],[-34, -123, -49, -9, -37, -2, 3, 1, -3, -71; 77, 277, 110, 20, 81, 4, -7, -2, 7, 162; 380, 1295, 498, 81, 270, -5, -45, 2, 44, 855],163,14]
%
% END OF LATTICE L164.14
%
\hbox{}\par\smallskip%
%
% LATTICE L164.15
%
\leavevmode%
\llap{\spacer}%
${L_{164.15}}=\pdualwitharg{5}({L_{164.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{4}{10}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{3,2}{\infty}{4}\above{1pt}{1pt}{*}{2}{180}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-166140\\-16920\\3780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16920\\-1720\\385
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3780\\385\\-86
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11231\\-14040\\428220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1176\\-1471\\44835
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-256\\320\\-9761
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\-58\\1620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\-102\\3085
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\-40\\1264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-14\\506
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\-54\\2340
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-10\\-20\\-40\\-540\\-200\\-290\\-100\\-20\\0
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-540\\-630\\-180\\0\\180\\0\\-270\\-180\\-180\\-1800
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\10\\0\\-30\\-630
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\4\\-4\\-180
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-40\\-30\\-4\\4\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\-630\\-180\\0\\180
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-200\\-290\\-100\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-290\\-510\\-200\\-70\\-270
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-200\\-84\\-36\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-70\\-36\\-24\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-270\\-180\\-180\\-1800
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-166140, -16920, 3780; -16920, -1720, 385; 3780, 385, -86],[180, 5, 1],[43, 81, 33, 13, 59, 5, -1, -1, 3, 85; -58, -102, -40, -14, -54, -2, 2, 0, -6, -126; 1620, 3085, 1264, 506, 2340, 210, -35, -44, 104, 3150],163,15]
%
% END OF LATTICE L164.15
%
\hbox{}\par\smallskip%
%
% LATTICE L164.16
%
\leavevmode%
\llap{\spacer}%
${L_{164.16}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{164.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{4}{10}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{6,5}{\infty}{4}\above{1pt}{1pt}{s}{2}{180}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4287780\\2970\\2093850
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2970\\10\\1450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2093850\\1450\\1022489
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8510471\\-764694\\-17426700
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16360\\-1471\\-33500
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4155440\\-373380\\-8509001
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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713\\-62\\-1460
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881\\-80\\-1804
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19339\\-1746\\-39600
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4287780, 2970, 2093850; 2970, 10, 1450; 2093850, 1450, 1022489],[90, 10, 1],[713, 0, 21, 881, 19339, 9103, 16360, 3251, 2391, 10109; -62, 1, -2, -80, -1746, -820, -1471, -292, -214, -900; -1460, 0, -43, -1804, -39600, -18640, -33500, -6657, -4896, -20700],163,16]
%
% END OF LATTICE L164.16
%
\hbox{}\par\smallskip%
%
% LATTICE L164.17
%
\leavevmode%
\llap{\spacer}%
${L_{164.17}}=\pdualwitharg{3}(\main({L_{164.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{4}{18}\above{1pt}{1pt}{b}{2}{180}\above{1pt}{1pt}{6,1}{\infty z}{45}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16020\\2340\\180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2340\\-315\\-27
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\-27\\-2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-1361\\-3400\\-82620
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156\\389\\9477
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\40\\971
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\-1\\9
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1\\2\\63
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-1\\0\\-90
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\HardButStrongLineBreak\kern3pt%
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-7\\-15\\-450
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-7\\-187
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-90\\-261\\-450\\-45\\0
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-261\\-918\\-1800\\-315\\-27
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-450\\-1800\\-3780\\-810\\-90
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-45\\-315\\-810\\-270\\-45
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0\\-27\\-90\\-45\\-10
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16020, 2340, 180; 2340, -315, -27; 180, -27, -2],[180, 9, 1],[0, 1, -1, -7, -3, -12, -41, -79, -13, -1; -1, 2, 0, -15, -7, -29, -102, -200, -35, -3; 9, 63, -90, -450, -187, -738, -2493, -4770, -765, -56],163,17]
%
% END OF LATTICE L164.17
%
\hbox{}\par\smallskip%
%
% LATTICE L164.18
%
\leavevmode%
\llap{\spacer}%
${L_{164.18}}=\pdualwitharg{3}({L_{164.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{*}{4}{18}\above{1pt}{1pt}{l}{2}{180}\above{1pt}{1pt}{3,1}{\infty}{180}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-163980\\14400\\21420
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14400\\-1260\\-1881
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\kern3pt%
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21420\\-1881\\-2798
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-11761\\9800\\-97020
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1092\\-911\\9009
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1536\\-1280\\12671
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-19\\-2\\-144
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-1\\2\\-9
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47\\0\\360
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103\\-30\\810
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29\\-14\\232
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-180\\0\\180\\-180\\-180
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-360\\-522\\-900\\-180\\0
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-900\\-1800\\-3780\\-1620\\-180
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-180\\-630\\-1620\\-1080\\-180
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0\\-54\\-180\\-180\\-40
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-163980, 14400, 21420; 14400, -1260, -1881; 21420, -1881, -2798],[180, 9, 1],[-19, -1, 47, 103, 29, 91, 121, 193, 17, -5; -2, 2, 0, -30, -14, -58, -102, -200, -70, -6; -144, -9, 360, 810, 232, 738, 999, 1620, 180, -34],163,18]
%
% END OF LATTICE L164.18
%
\hbox{}\par\smallskip%
%
% LATTICE L164.19
%
\leavevmode%
\llap{\spacer}%
${L_{164.19}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{164.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{}{4}{18}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{6,1}{\infty}{180}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7757730\\-198090\\-3805200
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198090\\-5040\\-97164
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3805200\\-97164\\-1866467
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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4225979\\-33700\\-8613720
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114114\\-911\\-232596
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2072862\\-16530\\-4225069
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-53\\1\\108
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\HardButStrongLineBreak\kern3pt%
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1214\\0\\-2475
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5342\\-15\\-10890
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1508\\-7\\-3074
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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-90\\0\\45\\-90\\-90
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-360\\-522\\-450\\-180\\0
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-522\\-918\\-900\\-630\\-54
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-180\\-630\\-810\\-1080\\-180
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7757730, -198090, -3805200; -198090, -5040, -97164; -3805200, -97164, -1866467],[90, 18, 1],[-980, -53, 1214, 5342, 1508, 4742, 6323, 5056, 928, -254; -1, 1, 0, -15, -7, -29, -51, -50, -35, -3; 1998, 108, -2475, -10890, -3074, -9666, -12888, -10305, -1890, 518],163,19]
%
% END OF LATTICE L164.19
%
\hbox{}\par\smallskip%
%
% LATTICE L164.20
%
\leavevmode%
\llap{\spacer}%
${L_{164.20}}=\pdualwitharg{5}({L_{164.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{*}{4}{20}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{12,5}{\infty z}{2}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-412200\\-110880\\3960
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110880\\-29825\\1065
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3960\\1065\\-38
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-13825\\62208\\299520
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3708\\16685\\80340
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
132\\-594\\-2861
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2\\8\\15
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\HardButStrongLineBreak\kern3pt%
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3\\-14\\-80
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\HardButStrongLineBreak\kern3pt%
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1\\-4\\-8
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\HardButStrongLineBreak\kern3pt%
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-3\\14\\79
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71\\324\\1665
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-10\\20\\0\\-30\\-630
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-20\\0\\8\\-4\\-180
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-20\\-30\\-4\\2\\0
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-270\\-630\\-180\\0\\90
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-100\\-290\\-100\\-10\\0
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\kern3pt%
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-290\\-1020\\-400\\-70\\-270
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-100\\-400\\-168\\-36\\-180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-70\\-36\\-12\\-90
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-270\\-180\\-90\\-900
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-412200, -110880, 3960; -110880, -29825, 1065; 3960, 1065, -38],[360, 5, 1],[-2, 3, 1, -3, -71, -34, -123, -49, -9, -37; 8, -14, -4, 14, 324, 154, 554, 220, 40, 162; 15, -80, -8, 79, 1665, 765, 2680, 1048, 181, 675],163,20]
%
% END OF LATTICE L164.20
%
\hbox{}\par\smallskip%
%
% LATTICE L164.21
%
\leavevmode%
\llap{\spacer}%
${L_{164.21}}=\pdualwitharg{5}({L_{164.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{4}{5}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{12,11}{\infty z}{2}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13661640\\49320\\24480
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49320\\-175\\-90
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24480\\-90\\-43
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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113723\\16288776\\30618000
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-403\\-57723\\-108500
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-208\\-29792\\-56001
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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36\\5155\\9695
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\HardButStrongLineBreak\kern3pt%
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66\\9453\\17770
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53\\7592\\14268
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10\\1433\\2691
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\6165\\11565
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-270\\-315\\-180\\0\\90\\0\\-135\\-180\\-90\\-900
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-5\\5\\0\\-15\\-315
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-20\\0\\8\\-4\\-180
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-20\\-15\\-4\\2\\0
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-270\\-315\\-180\\0\\90
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-100\\-145\\-100\\-10\\0
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-145\\-255\\-200\\-35\\-135
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-100\\-200\\-168\\-36\\-180
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-10\\-35\\-36\\-12\\-90
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-135\\-180\\-90\\-900
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13661640, 49320, 24480; 49320, -175, -90; 24480, -90, -43],[360, 5, 1],[36, 66, 53, 10, 43, 3, -1, -1, 3, 74; 5155, 9453, 7592, 1433, 6165, 431, -143, -144, 429, 10593; 9695, 17770, 14268, 2691, 11565, 805, -270, -268, 809, 19935],163,21]
%
% END OF LATTICE L164.21
%
\hbox{}\par\smallskip%
%
% LATTICE L164.22
%
\leavevmode%
\llap{\spacer}%
${L_{164.22}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{164.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{}{4}{90}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{3,1}{\infty}{9}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\405\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
405\\24120\\-5310
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-5310\\1169
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
129\\3760\\17100
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[90, 405, -90; 405, 24120, -5310; -90, -5310, 1169],[45, 45, 1],[35, 129, 26, 10, 5, 4, 1, 0, 3, 8; 1039, 3760, 750, 279, 133, 89, 0, 2, 97, 243; 4725, 17100, 3411, 1269, 605, 405, 0, 9, 441, 1105],163,22]
%
% END OF LATTICE L164.22
%
\hbox{}\par\smallskip%
%
% LATTICE L164.23
%
\leavevmode%
\llap{\spacer}%
${L_{164.23}}=\pdualwitharg{2}({L_{164.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{4}{8}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{6,5}{\infty a}{80}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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0\\8\\0
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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60420\\-5035\\237651
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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419\\-29\\1648
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600\\-51\\2360
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511\\-50\\2010
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183\\-35\\720
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-55\\-27\\-216
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1109880, 0, 282240; 0, 8, 0; 282240, 0, -71773],[360, 8, 1],[419, 600, 511, 183, -55, -59, 0, 89, 417, 1135; -29, -51, -50, -35, -27, -1, 1, 0, -15, -63; 1648, 2360, 2010, 720, -216, -232, 0, 350, 1640, 4464],163,23]
%
% END OF LATTICE L164.23
%
\hbox{}\par\smallskip%
%
% LATTICE L164.24
%
\leavevmode%
\llap{\spacer}%
${L_{164.24}}=\pdualwitharg{2}({L_{164.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{4}{8}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{6,5}{\infty b}{80}\above{1pt}{1pt}{s}{2}{144}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71480520\\153360\\261000
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\kern3pt%
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153360\\-328\\-560
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{$\left.\llap{\phantom{%
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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43\\387\\11552
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71\\650\\19068
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41\\395\\11000
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25\\261\\6696
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-71480520, 153360, 261000; 153360, -328, -560; 261000, -560, -953],[360, 8, 1],[43, 71, 67, 41, 25, -1, -1, 3, 29, 101; 387, 650, 620, 395, 261, -3, -10, 20, 245, 891; 11552, 19068, 17990, 11000, 6696, -272, -268, 810, 7800, 27144],163,24]
%
% END OF LATTICE L164.24
%
\hbox{}\par\smallskip%
%
% LATTICE L164.25
%
\leavevmode%
\llap{\spacer}%
${L_{164.25}}=\pdualwitharg{3}({L_{164.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{*}{4}{36}\above{1pt}{1pt}{s}{2}{360}\above{1pt}{1pt}{12,1}{\infty z}{90}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-471240\\17280\\720
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{$\left.\llap{\phantom{%
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-13\\-295\\-1530
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-23\\-112
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-471240, 17280, 720; 17280, -630, -27; 720, -27, -1],[360, 9, 1],[-12, -41, -79, -13, -1, 0, 1, -1, -7, -3; -269, -922, -1780, -295, -23, -1, 22, -20, -155, -67; -1476, -4986, -9540, -1530, -112, 18, 126, -180, -900, -374],163,25]
%
% END OF LATTICE L164.25
%
\hbox{}\par\smallskip%
%
% LATTICE L164.26
%
\leavevmode%
\llap{\spacer}%
${L_{164.26}}=\pdualwitharg{3}({L_{164.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{}{4}{9}\above{1pt}{1pt}{r}{2}{360}\above{1pt}{1pt}{12,7}{\infty z}{90}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-458297640\\-1925640\\-256129200
\end{tabular}\endgroup%
\kern3pt%
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-1925640\\-8091\\-1076184
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-256129200\\-1076184\\-143143147
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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60025\\1635549\\-119700
\end{tabular}\endgroup%
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7978866\\217406652\\-15911209
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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4\\245\\-9
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45\\1261\\-90
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-89\\-2760\\180
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-608\\-16905\\1215
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-259\\-7125\\517
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-458297640, -1925640, -256129200; -1925640, -8091, -1076184; -256129200, -1076184, -143143147],[360, 9, 1],[4, 45, -89, -608, -259, -1033, -1760, -6771, -1107, -84; 245, 1261, -2760, -16905, -7125, -28283, -47991, -184160, -29825, -2221; -9, -90, 180, 1215, 517, 2061, 3510, 13500, 2205, 167],163,26]
%
% END OF LATTICE L164.26
%
\hbox{}\par\smallskip%
%
% LATTICE L164.27
%
\leavevmode%
\llap{\spacer}%
${L_{164.27}}=\pdualwitharg{2.5}(\main({L_{164.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{4}{40}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{3,2}{\infty b}{4}\above{1pt}{1pt}{}{2}{180}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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-31320\\500\\-7700
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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2106620\\-876591\\-8625780
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-36531\\15200\\149580
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518199\\-215629\\-2121821
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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3678\\-1531\\-15060
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13403\\-5577\\-54880
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2685\\-1117\\-10994
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4484\\-1863\\-18360
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-540\\-1260\\-180\\0\\180\\0\\-540\\-180\\-180\\-1800
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{$\left[\!\llap{\phantom{%
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-20\\0\\4\\-4\\-180
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2304900, -31320, 566100; -31320, 500, -7700; 566100, -7700, 139039],[180, 20, 1],[3678, 13403, 2685, 1015, 4484, 381, 127, 21, 338, 7693; -1531, -5577, -1117, -422, -1863, -158, -53, -9, -141, -3204; -15060, -54880, -10994, -4156, -18360, -1560, -520, -86, -1384, -31500],163,27]
%
% END OF LATTICE L164.27
%
\hbox{}\par\smallskip%
%
% LATTICE L164.28
%
\leavevmode%
\llap{\spacer}%
${L_{164.28}}=\pdualwitharg{2.5}({L_{164.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{4}{40}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{3,2}{\infty}{4}\above{1pt}{1pt}{b}{2}{180}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3867120\\5580\\943020
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\kern3pt%
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-40\\-60\\-2\\4\\0
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-580\\-2040\\-200\\-140\\-540
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-50\\-200\\-21\\-18\\-90
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-20\\-140\\-18\\-24\\-180
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3867120, 5580, 943020; 5580, 40, 1360; 943020, 1360, 229961],[180, 20, 1],[4329, 15560, 1546, 1137, 4807, 339, 0, 10, 419, 9197; -410, -1471, -146, -107, -450, -31, 1, -1, -40, -873; -17750, -63800, -6339, -4662, -19710, -1390, 0, -41, -1718, -37710],163,28]
%
% END OF LATTICE L164.28
%
\hbox{}\par\smallskip%
%
% LATTICE L164.29
%
\leavevmode%
\llap{\spacer}%
${L_{164.29}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{164.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${90}\above{1pt}{1pt}{}{4}{45}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{6,1}{\infty}{18}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
637830\\-2250\\-62280
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2250\\45\\225
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{$\left.\llap{\phantom{%
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247151\\-360240\\2544480
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-2710\\3949\\-27900
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-24390\\35550\\-251101
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-118\\174\\-1215
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0\\1\\0
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-7\\10\\-72
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-146\\212\\-1503
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-356\\518\\-3665
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-90\\0\\18\\-18\\-90
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-180\\-135\\-18\\18\\0
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-900\\-1305\\-450\\-90\\0
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-1305\\-2295\\-900\\-315\\-135
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-90\\-315\\-162\\-108\\-90
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[637830, -2250, -62280; -2250, 45, 225; -62280, 225, 6082],[90, 45, 1],[-118, 0, -7, -146, -356, -1508, -2710, -1077, -396, -186; 174, 1, 10, 212, 518, 2196, 3949, 1570, 578, 272; -1215, 0, -72, -1503, -3665, -15525, -27900, -11088, -4077, -1915],163,29]
%
% END OF LATTICE L164.29
%
\hbox{}\par\smallskip%
%
% LATTICE L164.30
%
\leavevmode%
\llap{\spacer}%
${L_{164.30}}=\pdualwitharg{2.3}(\main({L_{164.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{}{4}{72}\above{1pt}{1pt}{*}{2}{180}\above{1pt}{1pt}{3,1}{\infty a}{180}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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1895220\\-10620\\462600
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-10620\\72\\-2592
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\kern3pt%
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462600\\-2592\\112915
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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7120\\389\\-29160
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-197224\\-10803\\807731
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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167\\8\\-684
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0\\1\\0
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22\\5\\-90
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923\\55\\-3780
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461\\26\\-1888
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-180\\0\\180\\-180\\-180
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-360\\-540\\-180\\180\\0
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-360\\-1044\\-900\\-180\\0
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-1044\\-3672\\-3600\\-1260\\-108
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-900\\-3600\\-3780\\-1620\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-1260\\-1620\\-1080\\-180
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-108\\-180\\-180\\-40
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1895220, -10620, 462600; -10620, 72, -2592; 462600, -2592, 112915],[180, 36, 1],[167, 0, 22, 923, 461, 1969, 7120, 7098, 2637, 251; 8, 1, 5, 55, 26, 109, 389, 385, 140, 13; -684, 0, -90, -3780, -1888, -8064, -29160, -29070, -10800, -1028],163,30]
%
% END OF LATTICE L164.30
%
\hbox{}\par\smallskip%
%
% LATTICE L164.31
%
\leavevmode%
\llap{\spacer}%
${L_{164.31}}=\pdualwitharg{2.3}({L_{164.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{*}{4}{72}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{3,1}{\infty}{180}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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852480\\900\\203220
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\kern3pt%
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900\\72\\216
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
203220\\216\\48445
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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0\\1\\0
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-118\\-10\\495
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\HardButStrongLineBreak\kern3pt%
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-1609\\-145\\6750
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-707\\-64\\2966
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\72\\0\\-540\\-252
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\45\\-90\\-90
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-360\\-540\\-90\\180\\0
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-108\\-252\\-90\\0\\4
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-360\\-1044\\-450\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1044\\-3672\\-1800\\-1260\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-450\\-1800\\-945\\-810\\-90
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-1260\\-810\\-1080\\-180
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0\\-108\\-90\\-180\\-40
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[852480, 900, 203220; 900, 72, 216; 203220, 216, 48445],[180, 36, 1],[-133, 0, -118, -1609, -707, -2879, -10040, -4902, -3411, -297; -12, 1, -10, -145, -64, -261, -911, -445, -310, -27; 558, 0, 495, 6750, 2966, 12078, 42120, 20565, 14310, 1246],163,31]
%
% END OF LATTICE L164.31
%
\hbox{}\par\smallskip%
%
% LATTICE L164.32
%
\leavevmode%
\llap{\spacer}%
${L_{164.32}}=\pdualwitharg{3.5}(\main({L_{164.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{}{4}{90}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{6,1}{\infty z}{9}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26820\\-26460\\2700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26460\\-26010\\2655
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2700\\2655\\-271
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5772\\56160
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5401\\52560
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-555\\-5401
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\90\\855
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\371\\3600
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\158\\1548
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\35\\351
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\205
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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-45\\90\\0\\-135\\-315
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\36\\-18\\-90
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-90\\-135\\-18\\9\\0
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-135\\-315\\-90\\0\\5
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-450\\-1305\\-450\\-45\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1305\\-4590\\-1800\\-315\\-135
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-450\\-1800\\-756\\-162\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-315\\-162\\-54\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-135\\-90\\-45\\-50
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26820, -26460, 2700; -26460, -26010, 2655; 2700, 2655, -271],[180, 45, 1],[-2, -1, 1, 1, 1, 2, 1, -1, -1, -1; 90, 371, 158, 35, 20, 21, -1, -10, 2, 17; 855, 3600, 1548, 351, 205, 225, 0, -108, 9, 155],163,32]
%
% END OF LATTICE L164.32
%
\hbox{}\par\smallskip%
%
% LATTICE L164.33
%
\leavevmode%
\llap{\spacer}%
${L_{164.33}}=\pdualwitharg{3.5}({L_{164.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{*}{4}{90}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{3,1}{\infty}{36}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-444060\\527040\\195300
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
527040\\-625500\\-231795
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
195300\\-231795\\-85894
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-429369\\7640\\-997020
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511982\\-9111\\1188855
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188832\\-3360\\438479
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2939\\-58\\6840
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5677\\-102\\13185
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2343\\-40\\5436
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
957\\-14\\2214
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
503\\-6\\1160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\-90\\-180\\-360\\-540\\-1800\\-2610\\-900\\-180\\0
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-540\\-630\\-180\\0\\20\\0\\-270\\-180\\-180\\-200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\-90\\-180\\-360\\-540
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\90\\0\\-270\\-630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\36\\-36\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-270\\-36\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\-630\\-180\\0\\20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-2610\\-900\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2610\\-4590\\-1800\\-630\\-270
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\-1800\\-756\\-324\\-180
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-180\\-630\\-324\\-216\\-180
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-270\\-180\\-180\\-200
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-444060, 527040, 195300; 527040, -625500, -231795; 195300, -231795, -85894],[180, 45, 1],[2939, 5677, 2343, 957, 503, 433, -57, -95, 167, 621; -58, -102, -40, -14, -6, -2, 2, 0, -6, -14; 6840, 13185, 5436, 2214, 1160, 990, -135, -216, 396, 1450],163,33]
%
% END OF LATTICE L164.33
%
\hbox{}\par\smallskip%
%
% LATTICE L164.34
%
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${L_{164.34}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{164.2}}))$%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-483660, -2724750, -1324800; -2724750, -15346710, -7461720; -1324800, -7461720, -3627961],[90, 90, 1],[-1, 1, 0, -3, -7, -29, -51, -10, -7, -3; -5776, 875, 617, -2590, -9052, -42266, -80945, -16631, -13424, -6962; 11880, -1800, -1269, 5328, 18620, 86940, 166500, 34209, 27612, 14320],163,34]
%
% END OF LATTICE L164.34
%
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%
% LATTICE L164.35
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${L_{164.35}}=\pdualwitharg{2.5}({L_{164.3}})$%
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% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L164.35
%
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%
% LATTICE L164.36
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${L_{164.36}}=\pdualwitharg{2.5}({L_{164.2}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-136943280, -16925400, 1099080; -16925400, -2091880, 135840; 1099080, 135840, -8821],[360, 40, 1],[-77, -169, -37, -35, -191, -25, -1, 3, 1, -115; 280, 571, 121, 106, 540, 62, -1, -7, 8, 486; -5280, -12260, -2746, -2728, -15480, -2160, -140, 266, 248, -6840],163,36]
%
% END OF LATTICE L164.36
%
\hbox{}\par\smallskip%
%
% LATTICE L164.37
%
\leavevmode%
\llap{\spacer}%
${L_{164.37}}=\pdualwitharg{3.5}({L_{164.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-363240, -124560, 12600; -124560, -42705, 4320; 12600, 4320, -437],[360, 45, 1],[-2, -1, 1, 1, 1, 2, 1, -1, -1, -1; 175, 652, 264, 51, 25, 17, -12, -8, 13, 39; 1665, 6390, 2628, 531, 275, 225, -90, -108, 99, 355],163,37]
%
% END OF LATTICE L164.37
%
\hbox{}\par\smallskip%
%
% LATTICE L164.38
%
\leavevmode%
\llap{\spacer}%
${L_{164.38}}=\pdualwitharg{3.5}({L_{164.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${90}\above{1pt}{1pt}{}{4}{45}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{12,7}{\infty z}{18}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-101170440, -67313880, 3740400; -67313880, -44787375, 2488680; 3740400, 2488680, -138287],[360, 45, 1],[506, -169, -169, 506, 1387, 6073, 11134, 8941, 1687, 806; -743, 249, 248, -745, -2041, -8935, -16379, -13152, -2481, -1185; 315, -90, -108, 279, 785, 3465, 6390, 5148, 981, 475],163,38]
%
% END OF LATTICE L164.38
%
\hbox{}\par\smallskip%
%
% LATTICE L164.39
%
\leavevmode%
\llap{\spacer}%
${L_{164.39}}=\pdualwitharg{2.3}({L_{164.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3169080, 3538800, -12960; 3538800, -3951576, 14472; -12960, 14472, -53],[360, 72, 1],[-7, 2, 7, 1, -5, -29, -62, -67, -61, -7; -1, 1, 0, -15, -7, -29, -51, -50, -35, -3; 1440, -216, -1710, -4320, -680, -792, 1296, 2790, 5400, 896],163,39]
%
% END OF LATTICE L164.39
%
\hbox{}\par\smallskip%
%
% LATTICE L164.40
%
\leavevmode%
\llap{\spacer}%
${L_{164.40}}=\pdualwitharg{2.3}({L_{164.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14447880, 208080, 129960; 208080, -2952, -1872; 129960, -1872, -1169],[360, 72, 1],[-17, -1, 27, 141, 45, 155, 231, 203, 89, 1; -11, -2, 20, 125, 43, 155, 242, 220, 115, 5; -1872, -108, 2970, 15480, 4936, 16992, 25308, 22230, 9720, 104],163,40]
%
% END OF LATTICE L164.40
%
\hbox{}\par\smallskip%
%
% LATTICE L164.41
%
\leavevmode%
\llap{\spacer}%
${L_{164.41}}=\pdualwitharg{2.3.5}(\main({L_{164.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[21780, -226260, -55260; -226260, 2354040, 574920; -55260, 574920, 140411],[180, 180, 1],[-399, -1450, -290, -109, -53, -39, -10, -2, -37, -93; 2863, 10367, 2069, 772, 371, 260, 43, 13, 269, 670; -11880, -43020, -8586, -3204, -1540, -1080, -180, -54, -1116, -2780],163,41]
%
% END OF LATTICE L164.41
%
\hbox{}\par\smallskip%
%
% LATTICE L164.42
%
\leavevmode%
\llap{\spacer}%
${L_{164.42}}=\pdualwitharg{2.3.5}({L_{164.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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0\\0\\0\\0\\0
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}}\right.$}%
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-180\\360\\0\\-540\\-1260
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-90\\0\\9\\-18\\-90
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-5220\\-18360\\-1800\\-1260\\-540
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[360, -16740, -4320; -16740, 6537600, 1682820; -4320, 1682820, 433169],[180, 180, 1],[-2536, -9111, -905, -665, -312, -197, 1, -6, -246, -599; 18689, 67160, 6672, 4905, 2303, 1459, 0, 44, 1811, 4413; -72630, -261000, -25929, -19062, -8950, -5670, 0, -171, -7038, -17150],163,42]
%
% END OF LATTICE L164.42
%
\hbox{}\par\smallskip%
%
% LATTICE L164.43
%
\leavevmode%
\llap{\spacer}%
${L_{164.43}}=\pdualwitharg{2.3.5}({L_{164.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${720}\above{1pt}{1pt}{}{4}{360}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{6,1}{\infty a}{144}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-4838040\\-4838040\\48240
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-10\\121\\11142
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-7\\106\\9936
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-3\\60\\5720
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-360\\0\\36\\-72\\-360
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-1440\\-1080\\-72\\144\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4837680, -4838040, 48240; -4838040, -4838040, 48240; 48240, 48240, -481],[360, 360, 1],[-29, -51, -10, -7, -3, -1, 1, 0, -3, -7; 280, 571, 121, 106, 60, 62, -1, -7, 8, 54; 25200, 52200, 11142, 9936, 5720, 6120, 0, -702, 504, 4720],163,43]
%
% END OF LATTICE L164.43
%
\hbox{}\par\smallskip%
%
% LATTICE L164.44
%
\leavevmode%
\llap{\spacer}%
${L_{164.44}}=\pdualwitharg{2.3.5}({L_{164.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${720}\above{1pt}{1pt}{*}{4}{360}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{6,1}{\infty b}{144}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-21754800\\2606684040\\-18096480
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\kern3pt%
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2606684040\\-312282032040\\2167966440
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18096480\\2167966440\\-15050749
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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34283\\137412\\19752120
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71475\\286471\\41178420
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15301\\61325\\8815086
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13741\\55070\\7915968
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7961\\31904\\4586000
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21754800, 2606684040, -18096480; 2606684040, -312282032040, 2167966440; -18096480, 2167966440, -15050749],[360, 360, 1],[34283, 71475, 15301, 13741, 7961, 8623, 35, -999, 561, 6341; 137412, 286471, 61325, 55070, 31904, 34554, 139, -4003, 2252, 25418; 19752120, 41178420, 8815086, 7915968, 4586000, 4966920, 19980, -575406, 323712, 3653680],163,44]
%
% END OF LATTICE L164.44
%

\medskip%
%------------------------------WEYL-GROUP-W165--------------------------------
%
\leavevmode\llap{}%
$W_{165}$%
\qquad\llap{12} lattices, $\chi=6$%
\hfill%
$22222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L165.1
%
\leavevmode%
\llap{\spacer}%
${L_{165.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
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{${90}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-400860\\-197640\\1080
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-197640\\-97442\\531
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1080\\531\\-2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-44\\90\\135
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43\\-88\\-144
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\-45\\-70
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-299\\612\\1008
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\88\\143
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\0\\-45\\-180\\0
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0\\4\\0\\-36\\-4
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-45\\0\\10\\0\\-5
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-180\\-36\\0\\36\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-5\\0\\2
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-400860, -197640, 1080; -197640, -97442, 531; 1080, 531, -2],[180, 1, 1],[-44, 43, 22, -299, -43; 90, -88, -45, 612, 88; 135, -144, -70, 1008, 143],164,1]
%
% END OF LATTICE L165.1
%
\hbox{}\par\smallskip%
%
% LATTICE L165.2
%
\leavevmode%
\llap{\spacer}%
${L_{165.2}}=\pfillwitharg{2}({L_{165.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${90}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-990\\450\\45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
450\\-71\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\-18\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-9\\432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\107
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\0\\-45\\-90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-9\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\10\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-9\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-5\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-990, 450, 45; 450, -71, -18; 45, -18, -2],[45, 1, 1],[2, -2, -1, 16, 4; 0, 1, 0, -9, -2; 45, -54, -25, 432, 107],164,2]
%
% END OF LATTICE L165.2
%
\hbox{}\par\smallskip%
%
% LATTICE L165.3
%
\leavevmode%
\llap{\spacer}%
${L_{165.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{165.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7515\\225\\315
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
225\\-5\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
315\\-10\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\36\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-9\\-90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-45\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\2\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-45\\0\\45\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-5\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7515, 225, 315; 225, -5, -10; 315, -10, -13],[45, 5, 1],[-1, -1, 0, 5, 1; -9, -6, 1, 36, 5; -18, -20, -1, 90, 20],164,3]
%
% END OF LATTICE L165.3
%
\hbox{}\par\smallskip%
%
% LATTICE L165.4
%
\leavevmode%
\llap{\spacer}%
${L_{165.4}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{165.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82035\\-46575\\-18135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46575\\26442\\10296
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18135\\10296\\4009
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\0\\95
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\5\\-135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\2\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\2\\99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-45\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-9\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\90\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-9\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-45\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[82035, -46575, -18135; -46575, 26442, 10296; -18135, 10296, 4009],[45, 9, 1],[21, 2, -27, 4, 23; 0, 0, 5, 2, 2; 95, 9, -135, 13, 99],164,4]
%
% END OF LATTICE L165.4
%
\hbox{}\par\smallskip%
%
% LATTICE L165.5
%
\leavevmode%
\llap{\spacer}%
${L_{165.5}}=\pdualwitharg{2}({L_{165.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${360}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
210909240\\-1307340\\52194420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1307340\\8104\\-323532
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52194420\\-323532\\12916729
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
968\\-1935\\-3960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-22\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
171\\-360\\-700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
596\\-1224\\-2439
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
218\\-443\\-892
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\-180\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-9\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\40\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-9\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-20\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[210909240, -1307340, 52194420; -1307340, 8104, -323532; 52194420, -323532, 12916729],[180, 4, 1],[968, 11, 171, 596, 218; -1935, -22, -360, -1224, -443; -3960, -45, -700, -2439, -892],164,5]
%
% END OF LATTICE L165.5
%
\hbox{}\par\smallskip%
%
% LATTICE L165.6
%
\leavevmode%
\llap{\spacer}%
${L_{165.6}}=\pdualwitharg{5}({L_{165.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{180}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13140\\-360\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\10\\5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\5\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-27\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-28\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\36\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-9\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-180\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\2\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-180\\0\\180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-5\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13140, -360, -180; -360, 10, 5; -180, 5, 2],[180, 5, 1],[-1, -1, 0, 1, 0; -27, -28, 0, 36, 1; -18, -20, -1, 0, 0],164,6]
%
% END OF LATTICE L165.6
%
\hbox{}\par\smallskip%
%
% LATTICE L165.7
%
\leavevmode%
\llap{\spacer}%
${L_{165.7}}=\pdualwitharg{3}({L_{165.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71460\\28080\\1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28080\\11034\\495
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1260\\495\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-10\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-28\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\8\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-45\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\0\\90\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-36\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-45\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[71460, 28080, 1260; 28080, 11034, 495; 1260, 495, 22],[180, 9, 1],[4, 11, 2, -3, -3; -10, -28, -5, 8, 8; -5, 0, 0, -8, -9],164,7]
%
% END OF LATTICE L165.7
%
\hbox{}\par\smallskip%
%
% LATTICE L165.8
%
\leavevmode%
\llap{\spacer}%
${L_{165.8}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{165.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1530\\2250\\-405
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2250\\3735\\-675
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-405\\-675\\122
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-17\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-12\\-63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-15\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-9\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\45\\0\\-45\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\18\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-45\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-45\\0\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1530, 2250, -405; 2250, 3735, -675; -405, -675, 122],[45, 45, 1],[0, 1, 1, 1, 0; -2, -17, -12, -15, -8; -11, -90, -63, -80, -45],164,8]
%
% END OF LATTICE L165.8
%
\hbox{}\par\smallskip%
%
% LATTICE L165.9
%
\leavevmode%
\llap{\spacer}%
${L_{165.9}}=\pdualwitharg{2.5}({L_{165.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15480\\-180\\-3780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\680\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3780\\-40\\-923
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\0\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-1\\135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-1\\172
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\2\\-340
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\0\\-36\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-45\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\8\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-45\\0\\45\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-20\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15480, -180, -3780; -180, 680, -40; -3780, -40, -923],[180, 20, 1],[35, -33, -42, -11, 83; 0, -1, -1, 0, 2; -144, 135, 172, 45, -340],164,9]
%
% END OF LATTICE L165.9
%
\hbox{}\par\smallskip%
%
% LATTICE L165.10
%
\leavevmode%
\llap{\spacer}%
${L_{165.10}}=\pdualwitharg{2.3}({L_{165.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{360}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36360\\-15660\\9540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15660\\3096\\-3960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9540\\-3960\\2497
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
328\\-55\\-1340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-2\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-661\\110\\2700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\2\\49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
282\\-47\\-1152
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-180\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-9\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\360\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-9\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-180\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[36360, -15660, 9540; -15660, 3096, -3960; 9540, -3960, 2497],[180, 36, 1],[328, 11, -661, -12, 282; -55, -2, 110, 2, -47; -1340, -45, 2700, 49, -1152],164,10]
%
% END OF LATTICE L165.10
%
\hbox{}\par\smallskip%
%
% LATTICE L165.11
%
\leavevmode%
\llap{\spacer}%
${L_{165.11}}=\pdualwitharg{3.5}({L_{165.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{180}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\-180\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\90\\45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\45\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\12\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\28\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-9\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\180\\0\\-180\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\18\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-180\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-45\\0\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[180, -180, -180; -180, 90, 45; -180, 45, -22],[180, 45, 1],[-1, -1, 4, 9, 0; -3, -4, 12, 28, 1; 2, 0, -9, -20, 0],164,11]
%
% END OF LATTICE L165.11
%
\hbox{}\par\smallskip%
%
% LATTICE L165.12
%
\leavevmode%
\llap{\spacer}%
${L_{165.12}}=\pdualwitharg{2.3.5}({L_{165.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{360}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36360\\72540\\18000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72540\\145080\\36000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18000\\36000\\8933
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-33\\135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-62\\252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-31\\125
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-36\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\45\\0\\-45\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\72\\0\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-45\\0\\5\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-180\\0\\360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[36360, 72540, 18000; 72540, 145080, 36000; 18000, 36000, 8933],[180, 180, 1],[0, -1, -1, 0, 2; -1, -33, -62, -31, -1; 4, 135, 252, 125, 0],164,12]
%
% END OF LATTICE L165.12
%

\medskip%
%------------------------------WEYL-GROUP-W166--------------------------------
%
\leavevmode\llap{}%
$W_{166}$%
\qquad\llap{12} lattices, $\chi=24$%
\hfill%
$22|22|22|22|\rtimes D_{4}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L166.1
%
\leavevmode%
\llap{\spacer}%
${L_{166.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2340\\-1080\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\498\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\24\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\9\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-15\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-13\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-24\\-68\\-72\\-20\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-18\\-72\\-99\\-45\\-9
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-18
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\10\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-18\\0\\2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68\\-72\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-99\\-45\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-45\\-35\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-15\\-11
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[2340, -1080, 0; -1080, 498, 1; 0, 1, -2],[180, 1, 1],[11, 4, -7, -6, -11, -4, 7, 6; 24, 9, -15, -13, -24, -9, 15, 13; 8, 0, -10, -8, -16, -9, 5, 5],165,1]
%
% END OF LATTICE L166.1
%
\hbox{}\par\smallskip%
%
% LATTICE L166.2
%
\leavevmode%
\llap{\spacer}%
${L_{166.2}}=\pfillwitharg{2}({L_{166.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3465\\90\\90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\-1\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\-3\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\-12\\-29
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-13\\-27
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\27
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\0\\-10\\-36\\-17\\-12\\-10\\0
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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1\\0\\-10\\-36
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-18
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-10\\0\\10\\0
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-36\\-18\\0\\18
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-17\\-12\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-11\\-15\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-15\\-35\\-45
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-45\\-99
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3465, 90, 90; 90, -1, -3; 90, -3, -2],[45, 1, 1],[-1, -1, -1, 1, 1, 1, 1, -1; -12, -13, -15, 9, 12, 13, 15, -9; -29, -27, -25, 27, 25, 24, 20, -36],165,2]
%
% END OF LATTICE L166.2
%
\hbox{}\par\smallskip%
%
% LATTICE L166.3
%
\leavevmode%
\llap{\spacer}%
${L_{166.3}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{166.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\0\\45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\-5\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\18\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-7\\15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\4\\-11
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\1\\-10
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-1\\-4\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\2
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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5\\0\\-10\\-180\\-85\\-60\\-10\\0
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\kern3pt%
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0\\10\\0\\-90\\-60\\-55\\-15\\-45
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\kern3pt%
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-10\\0\\2\\0\\-10\\-15\\-7\\-45
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-10\\-180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-90
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\2\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-90\\0\\90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85\\-60\\-10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-55\\-15\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-15\\-7\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-45\\-495
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[90, 0, 45; 0, -5, -5; 45, -5, 18],[45, 5, 1],[3, -1, -1, 1, 5, 7, 3, 17; 1, -4, -2, 0, 7, 10, 4, 18; -10, 0, 2, 0, -10, -15, -7, -45],165,3]
%
% END OF LATTICE L166.3
%
\hbox{}\par\smallskip%
%
% LATTICE L166.4
%
\leavevmode%
\llap{\spacer}%
${L_{166.4}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{166.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}{90}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
585\\675\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
675\\927\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-99\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-27\\4\\-144
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-19\\3\\-99
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\5\\-135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\1\\-25
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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9\\0\\-90\\-36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-18
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-90\\0\\90\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-18\\0\\2
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-153\\-108\\-90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[585, 675, -90; 675, 927, -99; -90, -99, 14],[45, 9, 1],[-27, -19, -27, -5, -21, -17, -33, -7; 4, 3, 5, 1, 4, 3, 5, 1; -144, -99, -135, -25, -108, -90, -180, -38],165,4]
%
% END OF LATTICE L166.4
%
\hbox{}\par\smallskip%
%
% LATTICE L166.5
%
\leavevmode%
\llap{\spacer}%
${L_{166.5}}=\pdualwitharg{2}({L_{166.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
377640\\6300\\95220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6300\\104\\1588
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
95220\\1588\\24009
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
89\\45\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\11\\-89
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\20\\-37
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79\\162\\-324
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\80\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-2\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-20\\-24\\-17\\-72\\-20\\0
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\kern3pt%
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0\\72\\0\\-72\\-72\\-396\\-180\\-36
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\kern3pt%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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1\\0\\-20\\-24
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0\\72\\0\\-72
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-24\\-72\\0\\8
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-17\\-72\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-396\\-180\\-36
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-20\\-180\\-140\\-60
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0\\-36\\-60\\-44
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[377640, 6300, 95220; 6300, 104, 1588; 95220, 1588, 24009],[180, 4, 1],[9, 79, 39, -3, -13, -97, -49, -3; 20, 162, 80, -2, -22, -171, -85, -1; -37, -324, -160, 12, 53, 396, 200, 12],165,5]
%
% END OF LATTICE L166.5
%
\hbox{}\par\smallskip%
%
% LATTICE L166.6
%
\leavevmode%
\llap{\spacer}%
${L_{166.6}}=\pdualwitharg{5}({L_{166.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260\\-540\\180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\-230\\85
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\85\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-649\\1440\\-360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-297\\659\\-165
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\40\\-11
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\56\\-20
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-13\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-11\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\9\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-360\\-340\\-120\\-20\\0
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\kern3pt%
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0\\10\\0\\-90\\-120\\-55\\-15\\-45
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-360
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\0\\-90
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-20\\0\\2\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-90\\0\\90
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-340\\-120\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-55\\-15\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-15\\-7\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-45\\-495
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1260, -540, 180; -540, -230, 85; 180, 85, 18],[180, 5, 1],[-25, 6, 5, -4, -47, -33, -14, -77; 56, -13, -11, 9, 104, 73, 31, 171; -20, 0, 2, 0, -20, -15, -7, -45],165,6]
%
% END OF LATTICE L166.6
%
\hbox{}\par\smallskip%
%
% LATTICE L166.7
%
\leavevmode%
\llap{\spacer}%
${L_{166.7}}=\pdualwitharg{3}({L_{166.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13860\\7920\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7920\\4482\\-99
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-99\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
559\\-1260\\-11340
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
284\\-640\\-5751
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\9\\80
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\12\\144
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\HardButStrongLineBreak\kern3pt%
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0\\0\\-1
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-30\\-315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13860, 7920, -180; 7920, 4482, -99; -180, -99, 2],[180, 9, 1],[-5, 0, 13, 14, 37, 4, 7, -2; 12, 0, -30, -32, -84, -9, -15, 5; 144, -1, -315, -315, -792, -80, -90, 72],165,7]
%
% END OF LATTICE L166.7
%
\hbox{}\par\smallskip%
%
% LATTICE L166.8
%
\leavevmode%
\llap{\spacer}%
${L_{166.8}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{166.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
405\\-2475\\540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2475\\13815\\-3015
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
540\\-3015\\658
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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9\\-136\\-630
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4\\-68\\-315
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\9
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\55
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[405, -2475, 540; -2475, 13815, -3015; 540, -3015, 658],[45, 45, 1],[9, 4, 0, 0, 3, 5, 3, 3; -136, -68, 2, 12, 20, -19, -31, -41; -630, -315, 9, 55, 90, -90, -144, -190],165,8]
%
% END OF LATTICE L166.8
%
\hbox{}\par\smallskip%
%
% LATTICE L166.9
%
\leavevmode%
\llap{\spacer}%
${L_{166.9}}=\pdualwitharg{2.5}({L_{166.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{360}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1328760\\-90180\\333000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90180\\6120\\-22600
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
333000\\-22600\\83453
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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27341\\-9207\\-111600
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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157\\-50\\-640
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\4
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-89\\18\\360
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-85\\-120\\-20\\0
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-120\\-220\\-60\\-180
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1328760, -90180, 333000; -90180, 6120, -22600; 333000, -22600, 83453],[180, 20, 1],[168, 157, -1, -89, 28, 137, 99, 971; -54, -50, 0, 18, -12, -49, -33, -315; -685, -640, 4, 360, -115, -560, -404, -3960],165,9]
%
% END OF LATTICE L166.9
%
\hbox{}\par\smallskip%
%
% LATTICE L166.10
%
\leavevmode%
\llap{\spacer}%
${L_{166.10}}=\pdualwitharg{2.3}({L_{166.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{*}{2}{360}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4003560\\-61380\\981540
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61380\\936\\-15048
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
981540\\-15048\\240641
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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642\\-151\\-2628
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{$\left[\!\llap{\phantom{%
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-180\\-180\\-1260\\-540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-540\\-396
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4003560, -61380, 981540; -61380, 936, -15048; 981540, -15048, 240641],[180, 36, 1],[11, -42, 176, 642, 587, 341, 1319, 255; -2, 11, -35, -151, -140, -82, -320, -62; -45, 172, -720, -2628, -2403, -1396, -5400, -1044],165,10]
%
% END OF LATTICE L166.10
%
\hbox{}\par\smallskip%
%
% LATTICE L166.11
%
\leavevmode%
\llap{\spacer}%
${L_{166.11}}=\pdualwitharg{3.5}({L_{166.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13140\\-1080\\-540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\90\\45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-540\\45\\22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\84\\1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-10\\-135
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-3\\-46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\0\\-180\\-360\\-3060\\-1080\\-180\\0
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\kern3pt%
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0\\90\\0\\-90\\-1080\\-495\\-135\\-45
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\0\\-180\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\90\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-90\\0\\10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3060\\-1080\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\-495\\-135\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-135\\-63\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-45\\-55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[13140, -1080, -540; -1080, 90, 45; -540, 45, 22],[180, 45, 1],[1, 0, -1, -2, -17, -6, -1, 0; 12, 1, -3, -5, -36, -10, 0, 2; 0, 0, -18, -40, -360, -135, -27, -5],165,11]
%
% END OF LATTICE L166.11
%
\hbox{}\par\smallskip%
%
% LATTICE L166.12
%
\leavevmode%
\llap{\spacer}%
${L_{166.12}}=\pdualwitharg{2.3.5}({L_{166.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{r}{2}{360}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2520\\-5940\\-1440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5940\\-15480\\-3780
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1440\\-3780\\-923
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-789\\3240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
111\\9730\\-39960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\2367\\-9721
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\219\\-900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\35\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-151\\620
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-180\\-360\\-765\\-1080\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\360\\0\\-360\\-1080\\-1980\\-540\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\72\\0\\-180\\-540\\-252\\-180
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-360\\0\\40\\0\\-180\\-180\\-220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-180\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\360\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-360\\0\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-765\\-1080\\-180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\-1980\\-540\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-540\\-252\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-180\\-180\\-220
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2520, -5940, -1440; -5940, -15480, -3780; -1440, -3780, -923],[180, 180, 1],[0, 2, 0, -2, -6, -11, -3, -1; -11, 219, 35, -151, -515, -1008, -298, -112; 45, -900, -144, 620, 2115, 4140, 1224, 460],165,12]
%
% END OF LATTICE L166.12
%

\medskip%
%------------------------------WEYL-GROUP-W167--------------------------------
%
\leavevmode\llap{}%
$W_{167}$%
\qquad\llap{4} lattices, $\chi=6$%
\hfill%
$4|42|2\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L167.1
%
\leavevmode%
\llap{\spacer}%
${L_{167.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2352\\-672\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672\\-191\\67
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\67\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\18\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-14\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-40\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-2\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-2\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2352, -672, 240; -672, -191, 67; 240, 67, -22],[48, 1, 1],[-1, -4, 3, 9; 5, 18, -14, -40; 4, 11, -10, -24],166,1]
%
% END OF LATTICE L167.1
%
\hbox{}\par\smallskip%
%
% LATTICE L167.2
%
\leavevmode%
\llap{\spacer}%
${L_{167.2}}=\pdualwitharg{3}({L_{167.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{4}{6}\above{1pt}{1pt}{*}{4}{12}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7824\\-720\\-816
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-66\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-816\\-75\\-85
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-3\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\6\\-6\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-6\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7824, -720, -816; -720, -66, -75; -816, -75, -85],[48, 3, 1],[1, 0, -1, 1; -1, -7, 4, 16; -9, 6, 6, -24],166,2]
%
% END OF LATTICE L167.2
%
\hbox{}\par\smallskip%
%
% LATTICE L167.3
%
\leavevmode%
\llap{\spacer}%
${L_{167.3}}=\pdualwitharg{2}({L_{167.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{4}{32}\above{1pt}{1pt}{}{4}{16}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-480, -48, 192; -48, 16, 0; 192, 0, -59],[48, 16, 1],[7, -5, 0, 3; 23, -14, -1, 9; 24, -16, 0, 10],166,3]
%
% END OF LATTICE L167.3
%
\hbox{}\par\smallskip%
%
% LATTICE L167.4
%
\leavevmode%
\llap{\spacer}%
${L_{167.4}}=\pdualwitharg{2.3}({L_{167.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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96\\48\\0
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\kern3pt%
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48\\48\\0
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\kern3pt%
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0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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1\\-1\\0
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-1\\2\\0
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-2\\-1\\-24
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0\\-1\\-6
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[96, 48, 0; 48, 48, 0; 0, 0, -1],[48, 48, 1],[1, -1, -2, 0; -1, 2, -1, -1; 0, 0, -24, -6],166,4]
%
% END OF LATTICE L167.4
%

\medskip%
%------------------------------WEYL-GROUP-W168--------------------------------
%
\leavevmode\llap{}%
$W_{168}$%
\qquad\llap{6} lattices, $\chi=8$%
\hfill%
$6262\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L168.1
%
\leavevmode%
\llap{\spacer}%
${L_{168.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{3}%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-48528\\1296\\720
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1296\\-30\\-21
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\kern3pt%
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720\\-21\\-10
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{$\left.\llap{\phantom{%
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0\\0\\0
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2\\31\\78
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0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-48528, 1296, 720; 1296, -30, -21; 720, -21, -10],[144, 3, 1],[2, -2, -1, 1; 31, -30, -16, 15; 78, -81, -39, 40],167,1]
%
% END OF LATTICE L168.1
%
\hbox{}\par\smallskip%
%
% LATTICE L168.2
%
\leavevmode%
\llap{\spacer}%
${L_{168.2}}=\pfillwitharg{3}({L_{168.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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384\\-10\\-5
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\kern3pt%
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192\\-5\\-2
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0\\0\\0
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0\\-1\\5
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0\\0\\1
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-14736, 384, 192; 384, -10, -5; 192, -5, -2],[48, 1, 1],[-1, 0, 1, 0; -39, 1, 39, -1; 0, -3, -3, 2],167,2]
%
% END OF LATTICE L168.2
%
\hbox{}\par\smallskip%
%
% LATTICE L168.3
%
\leavevmode%
\llap{\spacer}%
${L_{168.3}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{168.1}}))$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\kern3pt%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21936, 960, 480; 960, -42, -21; 480, -21, -10],[48, 3, 1],[0, -1, 0, 1; 2, -23, -1, 24; -5, 0, 2, -3],167,3]
%
% END OF LATTICE L168.3
%
\hbox{}\par\smallskip%
%
% LATTICE L168.4
%
\leavevmode%
\llap{\spacer}%
${L_{168.4}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{168.1}}))$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29856\\-1104\\864
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1104\\-32\\32
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\kern3pt%
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864\\32\\-25
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0\\1\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\-48\\-240\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
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96\\-48
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-48\\32
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-29856, -1104, 864; -1104, -32, 32; 864, 32, -25],[48, 16, 1],[-7, 1, 7, -1; 0, -3, -3, 2; -240, 32, 240, -32],167,4]
%
% END OF LATTICE L168.4
%
\hbox{}\par\smallskip%
%
% LATTICE L168.5
%
\leavevmode%
\llap{\spacer}%
${L_{168.5}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{168.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${32}\above{1pt}{1pt}{}{6}{96}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672\\-10320\\624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10320\\-156576\\9456
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624\\9456\\-571
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\21\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-115\\344\\5520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-21\\-337
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\16\\240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-48\\-80\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\96\\0\\-144
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
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32\\-48
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\96
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-672, -10320, 624; -10320, -156576, 9456; 624, 9456, -571],[48, 48, 1],[2, -23, -1, 24; -5, 16, 2, -19; -80, 240, 32, -288],167,5]
%
% END OF LATTICE L168.5
%
\hbox{}\par\smallskip%
%
% LATTICE L168.6
%
\leavevmode%
\llap{\spacer}%
${L_{168.6}}=\pdualwitharg{2}({L_{168.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${96}\above{1pt}{1pt}{}{6}{32}\above{1pt}{1pt}{*}{2}{96}\above{1pt}{1pt}{}{6}{288}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20448\\24912\\-720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24912\\-29856\\864
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\864\\-25
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\7\\272
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\13\\480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-15\\-576
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\-48\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\32\\0\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\0\\96\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-144\\288
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-20448, 24912, -720; 24912, -29856, 864; -720, 864, -25],[144, 48, 1],[0, -1, -1, 2; -7, 7, 13, -15; -240, 272, 480, -576],167,6]
%
% END OF LATTICE L168.6
%

\medskip%
%------------------------------WEYL-GROUP-W169--------------------------------
%
\leavevmode\llap{}%
$W_{169}$%
\qquad\llap{36} lattices, $\chi=24$%
\hfill%
$2|2\slashinfty2|2\slashinfty\rtimes D_{4}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L169.1
%
\leavevmode%
\llap{\spacer}%
${L_{169.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3}%
\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{24,1}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-305712\\-9360\\-9936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9360\\-285\\-303
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9936\\-303\\-322
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-449\\-32256\\44352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-937\\1287
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-1008\\1385
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\85\\-111
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\120\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-82\\108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-12\\-72\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-72\\-720\\-144
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\-6\\-144\\-48
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-72\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-720\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-144\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-305712, -9360, -9936; -9360, -285, -303; -9936, -303, -322],[144, 3, 1],[1, 1, -1, 0, 7, 3; 85, 120, -82, -13, 456, 226; -111, -144, 108, 12, -648, -306],168,1]
%
% END OF LATTICE L169.1
%
\hbox{}\par\smallskip%
%
% LATTICE L169.2
%
\leavevmode%
\llap{\spacer}%
${L_{169.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{12,1}{\infty z}{48}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{24,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30096\\4032\\-6336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4032\\-492\\891
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6336\\891\\-1297
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
57\\152\\-174
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\174\\-198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165\\-440\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-527\\-1408\\1608
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-239\\-639\\729
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62\\-166\\189
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-96\\-36\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-144\\-90\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\-48\\-72\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-144\\-48\\48\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-90\\-72\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-36\\-48\\-24\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-30096, 4032, -6336; 4032, -492, 891; -6336, 891, -1297],[144, 3, 1],[57, 65, -165, -527, -239, -62; 152, 174, -440, -1408, -639, -166; -174, -198, 504, 1608, 729, 189],168,2]
%
% END OF LATTICE L169.2
%
\hbox{}\par\smallskip%
%
% LATTICE L169.3
%
\leavevmode%
\llap{\spacer}%
${L_{169.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{6,1}{\infty}{48}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{24,19}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100944\\-20592\\-11808
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20592\\-4128\\-2487
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11808\\-2487\\-1297
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\304\\282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-106\\339\\315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
275\\-880\\-816
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1293\\-4136\\-3840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1315\\-4206\\-3906
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
207\\-662\\-615
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-96\\-72\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-72\\-90\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\-48\\-144\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-72\\-48\\48\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-90\\-144\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-18\\-48\\-24\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-100944, -20592, -11808; -20592, -4128, -2487; -11808, -2487, -1297],[144, 3, 1],[-95, -106, 275, 1293, 1315, 207; 304, 339, -880, -4136, -4206, -662; 282, 315, -816, -3840, -3906, -615],168,3]
%
% END OF LATTICE L169.3
%
\hbox{}\par\smallskip%
%
% LATTICE L169.4
%
\leavevmode%
\llap{\spacer}%
${L_{169.4}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{144}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{24,13}{\infty}{12}\above{1pt}{1pt}{}{2}{144}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{12,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-227952\\-35280\\9072
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35280\\-5460\\1404
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9072\\1404\\-361
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-8\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\13\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\35\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\120\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\27
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-24\\-48\\-144\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-144\\-720\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\12\\-12\\-144\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-144\\-12\\12\\0\\-12
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-720\\-144\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-72\\-24\\-12\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-227952, -35280, 9072; -35280, -5460, 1404; 9072, 1404, -361],[144, 12, 1],[1, -1, -2, -3, -7, 0; -8, -12, 13, 35, 120, 7; -6, -72, 0, 60, 288, 27],168,4]
%
% END OF LATTICE L169.4
%
\hbox{}\par\smallskip%
%
% LATTICE L169.5
%
\leavevmode%
\llap{\spacer}%
${L_{169.5}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{12,1}{\infty}{48}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{12,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30096\\15984\\-6336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15984\\-7980\\3228
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6336\\3228\\-1297
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\76\\282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\87\\324
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\-220\\-816
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
177\\-704\\-2616
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
161\\-639\\-2376
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-83\\-309
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-96\\-72\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-144\\-180\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\-48\\-144\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-144\\-48\\48\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-180\\-144\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-36\\-48\\-24\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-30096, 15984, -6336; 15984, -7980, 3228; -6336, 3228, -1297],[144, 12, 1],[-19, -22, 55, 177, 161, 21; 76, 87, -220, -704, -639, -83; 282, 324, -816, -2616, -2376, -309],168,5]
%
% END OF LATTICE L169.5
%
\hbox{}\par\smallskip%
%
% LATTICE L169.6
%
\leavevmode%
\llap{\spacer}%
${L_{169.6}}=\pfillwitharg{3}({L_{169.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{8,5}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45744\\240\\1152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-1\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1152\\-6\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1296\\-432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-33\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-12\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\-42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12\\-48\\-48\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-48\\-80\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3\\-6\\-24\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-48\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-80\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-24\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-45744, 240, 1152; 240, -1, -6; 1152, -6, -29],[48, 1, 1],[1, -1, -1, -1, 1, 1; -12, 0, 9, 30, 24, 0; 42, -40, -42, -48, 32, 39],168,6]
%
% END OF LATTICE L169.6
%
\hbox{}\par\smallskip%
%
% LATTICE L169.7
%
\leavevmode%
\llap{\spacer}%
${L_{169.7}}=\pfillwitharg{3}({L_{169.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{8,3}{\infty z}{3}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{4,3}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31632\\10896\\1056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10896\\-3753\\-364
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1056\\-364\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\48\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-24\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-18\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\69\\66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\66\\67
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
137\\360\\384
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-48\\-24\\-48
\end{tabular}\endgroup%
\kern3pt%
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0\\4\\0\\-12\\-10\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\12\\-6\\-12\\-96
\end{tabular}\endgroup%
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-48\\-12\\-6\\3\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-10\\-12\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-48\\-96\\-24\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-31632, 10896, 1056; 10896, -3753, -364; 1056, -364, -35],[48, 1, 1],[19, -9, -7, 26, 25, 137; 48, -24, -18, 69, 66, 360; 72, -22, -24, 66, 67, 384],168,7]
%
% END OF LATTICE L169.7
%
\hbox{}\par\smallskip%
%
% LATTICE L169.8
%
\leavevmode%
\llap{\spacer}%
${L_{169.8}}=\pfillwitharg{3}({L_{169.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{8,7}{\infty z}{3}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{1,0}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1104\\432\\432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\-169\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\-168\\-143
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\48\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-24\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-18\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\141\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
95\\252\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
245\\648\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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48\\0\\-48\\-48\\-48\\-48
\end{tabular}\endgroup%
\kern3pt%
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0\\1\\0\\-6\\-10\\-24
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\kern3pt%
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-48\\0\\12\\-6\\-24\\-96
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-6\\-6\\3\\0\\-24
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-10\\-24\\0\\4\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-24\\-96\\-24\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1104, 432, 432; 432, -169, -168; 432, -168, -143],[48, 1, 1],[19, -9, -7, 53, 95, 245; 48, -24, -18, 141, 252, 648; 0, 1, 0, -6, -10, -24],168,8]
%
% END OF LATTICE L169.8
%
\hbox{}\par\smallskip%
%
% LATTICE L169.9
%
\leavevmode%
\llap{\spacer}%
${L_{169.9}}=\pfillwitharg{3}({L_{169.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{4,1}{\infty z}{3}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{8,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45744\\480\\1152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
480\\-4\\-12
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1152\\-12\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\9\\-84
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\42
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\15\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-24\\-24\\-48\\-12
\end{tabular}\endgroup%
\kern3pt%
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0\\16\\0\\-24\\-80\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\12\\-6\\-48\\-48
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\-6\\3\\0\\-12
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-80\\-48\\0\\16\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-48\\-48\\-12\\0\\12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-45744, 480, 1152; 480, -4, -12; 1152, -12, -29],[48, 4, 1],[-2, -1, 1, 1, 1, -1; 9, 0, -6, 0, 12, 15; -84, -40, 42, 39, 32, -48],168,9]
%
% END OF LATTICE L169.9
%
\hbox{}\par\smallskip%
%
% LATTICE L169.10
%
\leavevmode%
\llap{\spacer}%
${L_{169.10}}=\pfillwitharg{3}({L_{169.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{4,3}{\infty z}{3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{2,1}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1104\\432\\432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\-140\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\-168\\-169
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\0\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\1\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\-3\\69
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\-5\\132
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
137\\-12\\360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-48\\-48\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-12\\-20\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\12\\-6\\-24\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-12\\-6\\3\\0\\-24
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-20\\-24\\0\\4\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-48\\-96\\-24\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1104, 432, 432; 432, -140, -168; 432, -168, -169],[48, 4, 1],[19, -9, -7, 26, 50, 137; 0, 1, 0, -3, -5, -12; 48, -24, -18, 69, 132, 360],168,10]
%
% END OF LATTICE L169.10
%
\hbox{}\par\smallskip%
%
% LATTICE L169.11
%
\leavevmode%
\llap{\spacer}%
${L_{169.11}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{169.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{8,1}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68112\\480\\2544
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
480\\-3\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2544\\-18\\-95
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-353\\-352\\-9504
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\2\\81
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\13\\350
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\-4\\-24\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-24\\-240\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\4\\-2\\-48\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\4
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-24\\-2
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-240\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-48\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-68112, 480, 2544; 480, -3, -18; 2544, -18, -95],[48, 3, 1],[-1, -1, 1, 0, -7, -3; -5, -16, 4, 4, 8, -6; -26, -24, 26, -1, -192, -80],168,11]
%
% END OF LATTICE L169.11
%
\hbox{}\par\smallskip%
%
% LATTICE L169.12
%
\leavevmode%
\llap{\spacer}%
${L_{169.12}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{169.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{8,3}{\infty z}{1}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{4,3}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10032\\336\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-9\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-12\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\24\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\27\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\64\\200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\-16\\-24\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12\\-30\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\4\\-2\\-12\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-12\\-2\\1\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-30\\-12\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-48\\-32\\-8\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10032, 336, 336; 336, -9, -12; 336, -12, -11],[48, 3, 1],[3, -1, -1, 1, 4, 9; 24, -6, -8, 6, 27, 64; 64, -24, -22, 23, 90, 200],168,12]
%
% END OF LATTICE L169.12
%
\hbox{}\par\smallskip%
%
% LATTICE L169.13
%
\leavevmode%
\llap{\spacer}%
${L_{169.13}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{169.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{8,7}{\infty z}{1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{1,0}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5232\\528\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\-45\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-24\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\47
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-10\\300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-8\\296
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\-16\\-48\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-6\\-30\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\4\\-2\\-24\\-32
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-6\\-2\\1\\0\\-8
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-30\\-24\\0\\12\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-24\\-32\\-8\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5232, 528, 240; 528, -45, -24; 240, -24, -11],[48, 3, 1],[3, -1, -1, 2, 13, 13; 0, 1, 0, -2, -10, -8; 64, -24, -22, 47, 300, 296],168,13]
%
% END OF LATTICE L169.13
%
\hbox{}\par\smallskip%
%
% LATTICE L169.14
%
\leavevmode%
\llap{\spacer}%
${L_{169.14}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{169.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{4,3}{\infty}{48}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{8,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6288\\2448\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2448\\-948\\184
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\184\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\75\\132
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\39\\67
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\228\\384
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\48\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-12\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-15\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-96\\-24\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-24\\-10\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\-48\\-48\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-24\\-48\\48\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-10\\-48\\0\\4\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\-96\\-48\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6288, 2448, -480; 2448, -948, 184; -480, 184, -35],[48, 4, 1],[19, 10, 59, 13, -3, -4; 75, 39, 228, 48, -12, -15; 132, 67, 384, 72, -22, -24],168,14]
%
% END OF LATTICE L169.14
%
\hbox{}\par\smallskip%
%
% LATTICE L169.15
%
\leavevmode%
\llap{\spacer}%
${L_{169.15}}=\pdualwitharg{3}({L_{169.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{24,17}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9949104\\663984\\-16992
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
663984\\-44313\\1134
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16992\\1134\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6335\\95040\\6336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-423\\-6346\\-423
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\165\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-104\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-166\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12\\-12\\-24\\-6
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\kern3pt%
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0\\16\\0\\-24\\-80\\-48
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\-6\\-48\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-12
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-80\\-48
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-48\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-9949104, 663984, -16992; 663984, -44313, 1134; -16992, 1134, -29],[144, 3, 1],[1, -7, -11, -4, -1, 5; 16, -104, -166, -61, -16, 76; 39, 32, -48, -42, -40, 42],168,15]
%
% END OF LATTICE L169.15
%
\hbox{}\par\smallskip%
%
% LATTICE L169.16
%
\leavevmode%
\llap{\spacer}%
${L_{169.16}}=\pdualwitharg{3}({L_{169.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{12,5}{\infty z}{48}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{24,23}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-311184\\3312\\3312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3312\\-33\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3312\\-36\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\64\\216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\392\\1200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\70\\211
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\71\\210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-96\\-12\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-48\\-10\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\-48\\-24\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-48\\-48\\48\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-10\\-24\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-12\\-48\\-24\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-311184, 3312, 3312; 3312, -33, -36; 3312, -36, -35],[144, 3, 1],[-1, -1, 3, 17, 3, 3; -22, -24, 64, 392, 70, 71; -72, -70, 216, 1200, 211, 210],168,16]
%
% END OF LATTICE L169.16
%
\hbox{}\par\smallskip%
%
% LATTICE L169.17
%
\leavevmode%
\llap{\spacer}%
${L_{169.17}}=\pdualwitharg{3}({L_{169.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{6,5}{\infty}{48}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{24,11}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15696\\720\\1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-33\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1584\\-72\\-143
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\64\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\680\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\260\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\143\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-96\\-24\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-24\\-10\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\-48\\-48\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-24\\-48\\48\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-10\\-48\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-6\\-48\\-24\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15696, 720, 1584; 720, -33, -72; 1584, -72, -143],[144, 3, 1],[-1, -1, 3, 29, 11, 6; -22, -24, 64, 680, 260, 143; 0, 1, 0, -24, -10, -6],168,17]
%
% END OF LATTICE L169.17
%
\hbox{}\par\smallskip%
%
% LATTICE L169.18
%
\leavevmode%
\llap{\spacer}%
${L_{169.18}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{169.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{4,1}{\infty z}{1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{8,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60624\\912\\2400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
912\\-12\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2400\\-36\\-95
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\3\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\32\\-192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\9\\-80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-8\\-8\\-48\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-24\\-240\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\4\\-2\\-48\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-24\\-2\\1\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-240\\-48\\0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-48\\-16\\-4\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-60624, 912, 2400; 912, -12, -36; 2400, -36, -95],[48, 12, 1],[-2, -1, 1, 0, -7, -3; 3, -4, -2, 2, 32, 9; -52, -24, 26, -1, -192, -80],168,18]
%
% END OF LATTICE L169.18
%
\hbox{}\par\smallskip%
%
% LATTICE L169.19
%
\leavevmode%
\llap{\spacer}%
${L_{169.19}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{169.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{4,3}{\infty z}{1}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{2,1}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5232\\528\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\-36\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-24\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-5\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-4\\200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\-16\\-48\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-12\\-60\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\4\\-2\\-24\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-12\\-2\\1\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-60\\-24\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-48\\-32\\-8\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5232, 528, 240; 528, -36, -24; 240, -24, -11],[48, 12, 1],[3, -1, -1, 1, 8, 9; 0, 1, 0, -1, -5, -4; 64, -24, -22, 23, 180, 200],168,19]
%
% END OF LATTICE L169.19
%
\hbox{}\par\smallskip%
%
% LATTICE L169.20
%
\leavevmode%
\llap{\spacer}%
${L_{169.20}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{169.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{8,7}{\infty}{4}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{2,1}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-624\\192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\-1116\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
192\\336\\-101
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-72\\-224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-30\\-93
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-13\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-28\\-88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\-32\\-48\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-12\\-30\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\4\\-4\\-24\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-12\\-4\\4\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-30\\-24\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-24\\-32\\-16\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-240, -624, 192; -624, -1116, 336; 192, 336, -101],[48, 12, 1],[9, 4, 2, -1, -1, 3; -72, -30, -13, 9, 6, -28; -224, -93, -40, 28, 18, -88],168,20]
%
% END OF LATTICE L169.20
%
\hbox{}\par\smallskip%
%
% LATTICE L169.21
%
\leavevmode%
\llap{\spacer}%
${L_{169.21}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{169.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{2,1}{\infty z}{3}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{4,1}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45744\\-8256\\1152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8256\\-1488\\208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1152\\208\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\9\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-2\\15\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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48\\0\\-48\\-48\\-96\\-48
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-45744, -8256, 1152; -8256, -1488, 208; 1152, 208, -29],[48, 16, 1],[-4, -1, 1, 1, 1, -2; 9, 0, -3, 0, 6, 15; -96, -40, 18, 39, 80, 24],168,21]
%
% END OF LATTICE L169.21
%
\hbox{}\par\smallskip%
%
% LATTICE L169.22
%
\leavevmode%
\llap{\spacer}%
${L_{169.22}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{169.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1104\\288\\144
\end{tabular}\endgroup%
\kern3pt%
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288\\-48\\-64
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144\\-64\\7
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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19\\48\\48
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16\\39\\42
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40\\99\\104
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65\\162\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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48\\0\\-48\\-96\\-96\\-48
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0\\16\\0\\-48\\-80\\-96
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-48\\0\\12\\-12\\-48\\-96
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-96\\-48\\-12\\12\\0\\-48
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-96\\-80\\-48\\0\\16\\0
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-48\\-96\\-96\\-48\\0\\48
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1104, 288, 144; 288, -48, -64; 144, -64, 7],[48, 16, 1],[19, -6, -7, 16, 40, 65; 48, -15, -18, 39, 99, 162; 48, -16, -18, 42, 104, 168],168,22]
%
% END OF LATTICE L169.22
%
\hbox{}\par\smallskip%
%
% LATTICE L169.23
%
\leavevmode%
\llap{\spacer}%
${L_{169.23}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{169.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{4,3}{\infty b}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-298320\\2208\\12960
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2208\\-16\\-96
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12960\\-96\\-563
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{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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1106\\9963\\23616
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-9\\-82\\-192
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-48\\-432\\-1025
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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11\\72\\240
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4\\33\\86
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\87\\168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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48\\0\\-96\\-192\\-48\\-48
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-298320, 2208, 12960; 2208, -16, -96; 12960, -96, -563],[48, 16, 1],[11, 4, 8, -2, -1, 1; 72, 33, 87, 9, -6, -6; 240, 86, 168, -48, -22, 24],168,23]
%
% END OF LATTICE L169.23
%
\hbox{}\par\smallskip%
%
% LATTICE L169.24
%
\leavevmode%
\llap{\spacer}%
${L_{169.24}}=\pdualwitharg{3}({L_{169.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{24,17}{\infty}{12}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{12,5}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-377712\\127296\\-3312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127296\\-42900\\1116
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3312\\1116\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\39
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-3\\-8\\32
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-6\\-19\\-48
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\10\\42
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\right.$}%
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3\\0\\-12\\-24\\-24\\-6
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0\\16\\0\\-48\\-80\\-48
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-12\\0\\12\\-12\\-48\\-48
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-24\\-48\\-12\\12\\0\\-24
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-24\\-80\\-48\\0\\16\\0
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-6\\-48\\-48\\-24\\0\\12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-377712, 127296, -3312; 127296, -42900, 1116; -3312, 1116, -29],[144, 12, 1],[1, -3, -6, -5, -1, 3; 4, -8, -19, -17, -4, 10; 39, 32, -48, -84, -40, 42],168,24]
%
% END OF LATTICE L169.24
%
\hbox{}\par\smallskip%
%
% LATTICE L169.25
%
\leavevmode%
\llap{\spacer}%
${L_{169.25}}=\pdualwitharg{3}({L_{169.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{6,5}{\infty a}{48}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{12,11}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15696\\-7632\\2304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7632\\-3516\\1056
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2304\\1056\\-317
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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3\\-74\\-225
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6\\-145\\-440
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17\\-404\\-1224
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3\\-64\\-192
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\22\\66
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3\\0\\-24\\-48\\-12\\-6
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0\\4\\0\\-48\\-20\\-24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\48\\-48\\-48\\-96
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-48\\-48\\-48\\48\\0\\-48
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-12\\-20\\-48\\0\\4\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-24\\-96\\-48\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15696, -7632, 2304; -7632, -3516, 1056; 2304, 1056, -317],[144, 12, 1],[3, 6, 17, 3, -1, -1; -74, -145, -404, -64, 25, 22; -225, -440, -1224, -192, 76, 66],168,25]
%
% END OF LATTICE L169.25
%
\hbox{}\par\smallskip%
%
% LATTICE L169.26
%
\leavevmode%
\llap{\spacer}%
${L_{169.26}}=\pdualwitharg{2}({L_{169.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{6,1}{\infty b}{48}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{24,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\432\\288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\-132\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\-168\\-115
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\1\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-4\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\-8\\192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-3\\81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-1\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-96\\-36\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-144\\-90\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\-48\\-72\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-144\\-48\\48\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-90\\-72\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-72\\-96\\-48\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-720, 432, 288; 432, -132, -168; 288, -168, -115],[144, 12, 1],[-9, -7, 27, 73, 31, 14; 1, 0, -4, -8, -3, -1; -24, -18, 72, 192, 81, 36],168,26]
%
% END OF LATTICE L169.26
%
\hbox{}\par\smallskip%
%
% LATTICE L169.27
%
\leavevmode%
\llap{\spacer}%
${L_{169.27}}=\pdualwitharg{2.3}({L_{169.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{24,11}{\infty}{12}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{12,11}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-171216\\5040\\2448
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\-132\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2448\\-72\\-35
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-4\\216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-1\\420
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1\\211
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\-8\\1200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-96\\-24\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-24\\-10\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\12\\-12\\-12\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-24\\-12\\12\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-10\\-12\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-48\\-96\\-48\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-171216, 5040, 2448; 5040, -132, -72; 2448, -72, -35],[144, 12, 1],[3, -1, -1, 6, 3, 17; -4, 0, 1, -1, -1, -8; 216, -70, -72, 420, 211, 1200],168,27]
%
% END OF LATTICE L169.27
%
\hbox{}\par\smallskip%
%
% LATTICE L169.28
%
\leavevmode%
\llap{\spacer}%
${L_{169.28}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{169.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{2,1}{\infty z}{1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{4,1}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51312\\-55776\\2208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55776\\-60624\\2400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2208\\2400\\-95
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-4\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-7\\192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-6\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\-16\\-96\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-24\\-240\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\4\\-2\\-48\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-24\\-2\\1\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-240\\-48\\0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-96\\-32\\-8\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-51312, -55776, 2208; -55776, -60624, 2400; 2208, 2400, -95],[48, 48, 1],[3, -2, -1, 1, 16, 9; -4, -1, 1, 0, -7, -6; -32, -72, 2, 23, 192, 56],168,28]
%
% END OF LATTICE L169.28
%
\hbox{}\par\smallskip%
%
% LATTICE L169.29
%
\leavevmode%
\llap{\spacer}%
${L_{169.29}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{169.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{2,1}{\infty a}{4}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{4,1}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5232\\-5232\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5232\\-5184\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\240\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-5\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-2\\104
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-16\\-32\\-96\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-48\\-240\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\4\\-4\\-48\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-48\\-4\\4\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-240\\-48\\0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-96\\-32\\-16\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5232, -5232, 240; -5232, -5184, 240; 240, 240, -11],[48, 48, 1],[3, -1, -1, 1, 11, 7; 0, 1, 0, -1, -5, -2; 64, 0, -22, -2, 120, 104],168,29]
%
% END OF LATTICE L169.29
%
\hbox{}\par\smallskip%
%
% LATTICE L169.30
%
\leavevmode%
\llap{\spacer}%
${L_{169.30}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{169.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{8,3}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\672\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
672\\-9072\\1296
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\1296\\-185
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
385\\34\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\-5\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\0\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\2\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\3\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-32\\-64\\-48\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-48\\-60\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\0\\16\\-16\\-48\\-64
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\0\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\-48\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-60\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-48\\-64
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-48, 672, -96; 672, -9072, 1296; -96, 1296, -185],[48, 48, 1],[13, 13, 8, -2, -2, 3; 0, 2, 3, 1, -1, -2; -8, 6, 16, 8, -6, -16],168,30]
%
% END OF LATTICE L169.30
%
\hbox{}\par\smallskip%
%
% LATTICE L169.31
%
\leavevmode%
\llap{\spacer}%
${L_{169.31}}=\pdualwitharg{2}({L_{169.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{144}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{24,13}{\infty}{48}\above{1pt}{1pt}{l}{2}{144}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{6,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1152\\5328\\-1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5328\\-24432\\7248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1584\\7248\\-2149
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-27\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-17\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\11\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\57\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\8\\21
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-96\\-144\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-288\\-720\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\-48\\-288\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-288\\-48\\48\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-720\\-288\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-72\\-48\\-24\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1152, 5328, -1584; 5328, -24432, 7248; -1584, 7248, -2149],[144, 48, 1],[1, -26, -13, 17, 64, 8; 2, -27, -17, 11, 57, 8; 6, -72, -48, 24, 144, 21],168,31]
%
% END OF LATTICE L169.31
%
\hbox{}\par\smallskip%
%
% LATTICE L169.32
%
\leavevmode%
\llap{\spacer}%
${L_{169.32}}=\pdualwitharg{2.3}({L_{169.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{12,5}{\infty b}{48}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{6,5}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140112\\-80064\\2016
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80064\\-45744\\1152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2016\\1152\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-2\\264
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-4\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-24\\-48\\-24\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-96\\-80\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\48\\-48\\-96\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-96\\-48\\48\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-80\\-96\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-48\\-96\\-48\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-140112, -80064, 2016; -80064, -45744, 1152; 2016, 1152, -29],[144, 48, 1],[0, 2, 5, 3, 0, -1; 1, 1, -2, -4, -1, 1; 39, 176, 264, 48, -40, -30],168,32]
%
% END OF LATTICE L169.32
%
\hbox{}\par\smallskip%
%
% LATTICE L169.33
%
\leavevmode%
\llap{\spacer}%
${L_{169.33}}=\pdualwitharg{2}({L_{169.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{l}{2}{144}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{24,1}{\infty}{48}\above{1pt}{1pt}{b}{2}{144}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{6,1}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-960\\432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\432\\-169
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-7\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\19\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\29\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\30\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-96\\-144\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-288\\-720\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\-48\\-288\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-288\\-48\\48\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-720\\-288\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-144\\-96\\-48\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[144, -144, 0; -144, -960, 432; 0, 432, -169],[144, 48, 1],[-7, -1, 19, 31, 35, -1; -7, 0, 19, 29, 30, -2; -18, 0, 48, 72, 72, -6],168,33]
%
% END OF LATTICE L169.33
%
\hbox{}\par\smallskip%
%
% LATTICE L169.34
%
\leavevmode%
\llap{\spacer}%
${L_{169.34}}=\pdualwitharg{2.3}({L_{169.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{12,5}{\infty a}{48}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{6,5}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15696\\-1008\\576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1008\\192\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
576\\-48\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\61\\186
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\145\\440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\230\\696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\64\\192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-21\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\-66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-96\\-48\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-96\\-80\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\-48\\-96\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-96\\-48\\48\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-80\\-96\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-48\\-96\\-48\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15696, -1008, 576; -1008, 192, -48; 576, -48, 7],[144, 48, 1],[3, 7, 11, 3, -1, -1; 61, 145, 230, 64, -21, -22; 186, 440, 696, 192, -64, -66],168,34]
%
% END OF LATTICE L169.34
%
\hbox{}\par\smallskip%
%
% LATTICE L169.35
%
\leavevmode%
\llap{\spacer}%
${L_{169.35}}=\pdualwitharg{2}({L_{169.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{24,19}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31392\\77616\\-4896
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77616\\-191760\\12096
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4896\\12096\\-763
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-1368\\-21888
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\3419\\54720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-216\\-3457
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\-11\\-168
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\19\\288
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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0\\36\\0\\-144\\-180\\-144
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-96\\0\\48\\-48\\-144\\-192
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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0\\36\\0
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-96\\0\\48
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\-144\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-180\\-144
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-31392, 77616, -4896; 77616, -191760, 12096; -4896, 12096, -763],[144, 48, 1],[-1, 1, 2, 0, -2, -3; -11, -3, 19, 47, 39, 17; -168, -54, 288, 744, 630, 288],168,35]
%
% END OF LATTICE L169.35
%
\hbox{}\par\smallskip%
%
% LATTICE L169.36
%
\leavevmode%
\llap{\spacer}%
${L_{169.36}}=\pdualwitharg{2.3}({L_{169.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{12,11}{\infty b}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\6624\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6624\\-298320\\12960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\12960\\-563
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82\\-27\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3321\\1106\\23616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-48\\-1025
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-96\\-192\\-48\\-48
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0\\4\\0\\-48\\-20\\-48
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\0\\48\\-48\\-48\\-192
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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48\\0\\-96
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0\\4\\0
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-96\\0\\48
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\-48\\-48
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-20\\-48
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-48\\-192
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-144, 6624, -288; 6624, -298320, 12960; -288, 12960, -563],[144, 48, 1],[3, -2, -2, 24, 11, 29; -2, -1, 1, 11, 4, 8; -48, -22, 24, 240, 86, 168],168,36]
%
% END OF LATTICE L169.36
%

\medskip%
%------------------------------WEYL-GROUP-W170--------------------------------
%
\leavevmode\llap{}%
$W_{170}$%
\qquad\llap{8} lattices, $\chi=12$%
\hfill%
$22|22\slashinfty\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L170.1
%
\leavevmode%
\llap{\spacer}%
${L_{170.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{144}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{24,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1105776\\18288\\7200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18288\\-285\\-123
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7200\\-123\\-46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-154\\-684
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-240\\-1080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\66\\293
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
103\\2256\\10080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\131\\588
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-144\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-720\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-720\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-72\\-3\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1105776, 18288, 7200; 18288, -285, -123; 7200, -123, -46],[144, 3, 1],[-7, -11, 3, 103, 6; -154, -240, 66, 2256, 131; -684, -1080, 293, 10080, 588],169,1]
%
% END OF LATTICE L170.1
%
\hbox{}\par\smallskip%
%
% LATTICE L170.2
%
\leavevmode%
\llap{\spacer}%
${L_{170.2}}=\pfillwitharg{3}({L_{170.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{8,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-116592\\2352\\1872
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2352\\-46\\-39
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1872\\-39\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-180\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\24\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\77\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\288\\320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\-33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-6\\-48\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-80\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-24\\-3\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-116592, 2352, 1872; 2352, -46, -39; 1872, -39, -29],[48, 1, 1],[-7, 1, 3, 11, -1; -180, 24, 77, 288, -24; -210, 32, 90, 320, -33],169,2]
%
% END OF LATTICE L170.2
%
\hbox{}\par\smallskip%
%
% LATTICE L170.3
%
\leavevmode%
\llap{\spacer}%
${L_{170.3}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{170.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{8,5}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49104\\-960\\2160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-960\\-15\\42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2160\\42\\-95
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\4\\47
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\64\\912
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\5\\93
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-70
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-3\\-24\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-240\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\6\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-240\\0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-48\\-6\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-49104, -960, 2160; -960, -15, 42; 2160, 42, -95],[48, 3, 1],[2, 39, 4, -3, -3; 4, 64, 5, -8, -4; 47, 912, 93, -72, -70],169,3]
%
% END OF LATTICE L170.3
%
\hbox{}\par\smallskip%
%
% LATTICE L170.4
%
\leavevmode%
\llap{\spacer}%
${L_{170.4}}=\pdualwitharg{3}({L_{170.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{24,5}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1124208\\9648\\5760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9648\\-78\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5760\\-51\\-29
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-40\\-129
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\128\\368
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\165\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\40\\128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-124\\-378
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-9\\-24\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\18\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-80\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-48\\-18\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1124208, 9648, 5760; 9648, -78, -51; 5760, -51, -29],[144, 3, 1],[-1, 3, 4, 1, -3; -40, 128, 165, 40, -124; -129, 368, 504, 128, -378],169,4]
%
% END OF LATTICE L170.4
%
\hbox{}\par\smallskip%
%
% LATTICE L170.5
%
\leavevmode%
\llap{\spacer}%
${L_{170.5}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{170.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{32}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{8,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-353616\\-120096\\15312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120096\\-40784\\5200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15312\\5200\\-663
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\19\\195
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\84\\912
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-24\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-10\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\32\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-10\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-48\\-48\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-353616, -120096, 15312; -120096, -40784, 5200; 15312, 5200, -663],[48, 16, 1],[1, -1, -1, 2, 11; -6, -7, 5, 19, 84; -24, -78, 16, 195, 912],169,5]
%
% END OF LATTICE L170.5
%
\hbox{}\par\smallskip%
%
% LATTICE L170.6
%
\leavevmode%
\llap{\spacer}%
${L_{170.6}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{170.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{96}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{8,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-163056\\-49824\\6912
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49824\\-15216\\2112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6912\\2112\\-293
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\-78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-3\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\1\\219
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\3\\304
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-48\\-24\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-30\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\96\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-30\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-48\\-48\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-163056, -49824, 6912; -49824, -15216, 2112; 6912, 2112, -293],[48, 48, 1],[-2, -3, 5, 9, 12; 1, -1, -3, 1, 3; -40, -78, 96, 219, 304],169,6]
%
% END OF LATTICE L170.6
%
\hbox{}\par\smallskip%
%
% LATTICE L170.7
%
\leavevmode%
\llap{\spacer}%
${L_{170.7}}=\pdualwitharg{2}({L_{170.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{*}{2}{32}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{24,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61200\\-146304\\6336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-146304\\-349008\\15120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6336\\15120\\-655
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-11\\-312
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-15\\-414
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\9\\256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\45\\1251
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\59\\1632
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-48\\-72\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-90\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\32\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-90\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-144\\-48\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-61200, -146304, 6336; -146304, -349008, 15120; 6336, 15120, -655],[144, 48, 1],[-6, -7, 5, 22, 28; -11, -15, 9, 45, 59; -312, -414, 256, 1251, 1632],169,7]
%
% END OF LATTICE L170.7
%
\hbox{}\par\smallskip%
%
% LATTICE L170.8
%
\leavevmode%
\llap{\spacer}%
${L_{170.8}}=\pdualwitharg{2.3}({L_{170.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{288}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{24,23}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4546512\\-1474272\\63504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1474272\\-478032\\20592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63504\\20592\\-887
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-1\\-238
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-3\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\2\\619
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\11\\2832
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-144\\-24\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-10\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\0\\288\\0\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-10\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-48\\-144\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4546512, -1474272, 63504; -1474272, -478032, 20592; 63504, 20592, -887],[144, 48, 1],[-2, -3, 5, 8, 36; 1, -1, -3, 2, 11; -120, -238, 288, 619, 2832],169,8]
%
% END OF LATTICE L170.8
%

\medskip%
%------------------------------WEYL-GROUP-W171--------------------------------
%
\leavevmode\llap{}%
$W_{171}$%
\qquad\llap{8} lattices, $\chi=48$%
\hfill%
$\infty2|2\infty|\infty2|2\infty|\rtimes D_{4}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L171.1
%
\leavevmode%
\llap{\spacer}%
${L_{171.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{3}%
\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{3,2}{\infty a}{48}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{6,1}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8784\\1152\\2304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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2304\\-327\\-572
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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263\\795\\603
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
575\\1736\\1320
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\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-240\\-336\\-72\\-48
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-8784, 1152, 2304; 1152, -132, -327; 2304, -327, -572],[144, 3, 1],[179, 629, 263, 575, 125, -21, -35, 33; 544, 1904, 795, 1736, 376, -64, -105, 104; 408, 1440, 603, 1320, 288, -48, -81, 72],170,1]
%
% END OF LATTICE L171.1
%
\hbox{}\par\smallskip%
%
% LATTICE L171.2
%
\leavevmode%
\llap{\spacer}%
${L_{171.2}}=\pfillwitharg{3}({L_{171.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{1,0}{\infty z}{48}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{1,0}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\131
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\131\\-286
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-17\\960\\432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\1259\\567
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\-2760\\-1243
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\264\\120
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\HardButStrongLineBreak\kern3pt%
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-11\\1008\\456
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\HardButStrongLineBreak\kern3pt%
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-2\\144\\65
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\960\\432
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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48\\-48\\-24\\-336
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-336\\-24\\-48
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-816\\-96\\-528
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-48\\-528\\-96\\-816
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[48, 0, 0; 0, -60, 131; 0, 131, -286],[48, 1, 1],[-1, -11, -2, -17, -7, -5, 0, 1; 264, 1008, 144, 960, 216, -48, -24, 0; 120, 456, 65, 432, 96, -24, -11, 0],170,2]
%
% END OF LATTICE L171.2
%
\hbox{}\par\smallskip%
%
% LATTICE L171.3
%
\leavevmode%
\llap{\spacer}%
${L_{171.3}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{171.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{1,0}{\infty a}{16}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{2,1}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2268144\\444768\\141744
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
444768\\-87216\\-27795
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
141744\\-27795\\-8858
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41183\\247104\\-116688
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8082\\-48493\\22899
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2580\\-15480\\7309
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-33\\-232\\200
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\16
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\80\\-75
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\176\\-248
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-16\\-24\\-112\\-80\\-112\\-24\\-16
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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16\\-16\\-24\\-112
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\16\\0\\-80
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-24\\0\\6\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-80\\0\\16
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-112\\-24\\-16
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-272\\-96\\-176
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-96\\-42\\-96
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-176\\-96\\-272
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2268144, 444768, 141744; 444768, -87216, -27795; 141744, -27795, -8858],[48, 3, 1],[-33, 1, 11, 19, -15, -97, -47, -115; -232, 0, 80, 176, -56, -576, -296, -752; 200, 16, -75, -248, -64, 256, 177, 520],170,3]
%
% END OF LATTICE L171.3
%
\hbox{}\par\smallskip%
%
% LATTICE L171.4
%
\leavevmode%
\llap{\spacer}%
${L_{171.4}}=\pdualwitharg{3}({L_{171.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{3,1}{\infty z}{48}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{3,2}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1400400\\437616\\20016
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
437616\\-136752\\-6255
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20016\\-6255\\-286
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81599\\248064\\283968
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25525\\-77597\\-88827
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1150\\-3496\\-4003
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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93\\280\\384
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
391\\1184\\1464
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\176\\209
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
405\\1232\\1392
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-48\\-24\\-336\\-240\\-336\\-24\\-48
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\48\\0\\-240\\-336\\-816\\-96\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\2\\0\\-24\\-96\\-14\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-240\\0\\48\\-48\\-528\\-96\\-816
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-48\\-24\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\48\\0\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-240\\0\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-336\\-24\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-816\\-96\\-528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-96\\-14\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-528\\-96\\-816
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1400400, 437616, 20016; 437616, -136752, -6255; 20016, -6255, -286],[144, 3, 1],[93, 391, 58, 405, 107, 9, -8, -5; 280, 1184, 176, 1232, 328, 32, -24, -16; 384, 1464, 209, 1392, 312, -72, -35, 0],170,4]
%
% END OF LATTICE L171.4
%
\hbox{}\par\smallskip%
%
% LATTICE L171.5
%
\leavevmode%
\llap{\spacer}%
${L_{171.5}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{171.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{1,0}{\infty z}{3}\above{1pt}{1pt}{r}{2}{32}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{1,0}{\infty b}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1536\\1296\\-624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1296\\-1088\\528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-624\\528\\-253
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1526\\-675\\2400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1281\\566\\-2016
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-610\\-270\\959
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\6\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\21\\-81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
131\\57\\-208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127\\57\\-198
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-48\\-84\\-60\\-42\\-48\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-30\\-42\\-51\\-96\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\32\\0\\-48\\-96\\-224\\-192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-30\\0\\12\\-12\\-66\\-192\\-204
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-48\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\32\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-30\\0\\12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-42\\-48\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-51\\-96\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-96\\-224\\-192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-66\\-192\\-204
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1536, 1296, -624; 1296, -1088, 528; -624, 528, -253],[48, 16, 1],[17, 50, 131, 127, 44, 11, -9, -5; 6, 21, 57, 57, 21, 6, -3, -3; -30, -81, -208, -198, -66, -15, 16, 6],170,5]
%
% END OF LATTICE L171.5
%
\hbox{}\par\smallskip%
%
% LATTICE L171.6
%
\leavevmode%
\llap{\spacer}%
${L_{171.6}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{171.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{1,0}{\infty z}{1}\above{1pt}{1pt}{r}{2}{96}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{1,0}{\infty b}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\1104\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1104\\-12096\\528
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\528\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85\\24\\768
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
819\\-235\\-7488
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\10\\319
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-1\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-48\\-28\\-20\\-14\\-48\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-10\\-14\\-17\\-96\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\96\\0\\-48\\-96\\-672\\-192
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-28\\-10\\0\\4\\-4\\-22\\-192\\-68
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-2\\-48\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0\\-10
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\96\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-10\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-14\\-48\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-17\\-96\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-96\\-672\\-192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-22\\-192\\-68
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-96, 1104, -48; 1104, -12096, 528; -48, 528, -23],[48, 48, 1],[6, 2, -1, -1, 1, 5, 43, 15; -1, 0, 2, 0, -1, -2, -14, -4; -38, -5, 48, 2, -26, -59, -432, -130],170,6]
%
% END OF LATTICE L171.6
%
\hbox{}\par\smallskip%
%
% LATTICE L171.7
%
\leavevmode%
\llap{\spacer}%
${L_{171.7}}=\pdualwitharg{2}({L_{171.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{3,1}{\infty z}{3}\above{1pt}{1pt}{r}{2}{288}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{3,1}{\infty a}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\-144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-3264\\-2064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2064\\-1277
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
437\\888\\-1440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2701\\-5477\\8880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1533\\-3108\\5039
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-11\\18
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-98\\159
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-437\\-888\\1440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-159\\-322\\522
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-144\\-84\\-60\\-42\\-144\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-30\\-42\\-51\\-288\\-66
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-30\\0\\12\\-12\\-66\\-576\\-204
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-6\\-144\\-84
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\3\\0\\-30
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\0\\288\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-30\\0\\12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-42\\-144\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-51\\-288\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-288\\-2016\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-66\\-576\\-204
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[288, -144, 0; -144, -3264, -2064; 0, -2064, -1277],[144, 48, 1],[-5, -48, -437, -159, -68, -25, -1, 13; -11, -98, -888, -322, -137, -50, 0, 26; 18, 159, 1440, 522, 222, 81, 0, -42],170,7]
%
% END OF LATTICE L171.7
%
\hbox{}\par\smallskip%
%
% LATTICE L171.8
%
\leavevmode%
\llap{\spacer}%
${L_{171.8}}=\pdualwitharg{2.3}({L_{171.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{3,2}{\infty a}{12}\above{1pt}{1pt}{*}{2}{32}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{3,2}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\720\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-1536\\624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\624\\-253
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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-109\\408\\1152
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\kern3pt%
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225\\-851\\-2400
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-90\\340\\959
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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7\\-23\\-66
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19\\-70\\-198
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19\\-74\\-208
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-29\\-81
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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12\\-12\\-48\\-42
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\12\\0\\-30
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-48\\0\\32\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-30\\0\\3
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-84\\-48\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-204\\-192\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-192\\-224\\-96
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-66\\-96\\-51
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-288, 720, -288; 720, -1536, 624; -288, 624, -253],[144, 48, 1],[7, 19, 19, 7, 2, -1, -1, 2; -23, -70, -74, -29, -11, 2, 6, -5; -66, -198, -208, -81, -30, 6, 16, -15],170,8]
%
% END OF LATTICE L171.8
%

\medskip%
%------------------------------WEYL-GROUP-W172--------------------------------
%
\leavevmode\llap{}%
$W_{172}$%
\qquad\llap{16} lattices, $\chi=6$%
\hfill%
$2|22\slashtwo2\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L172.1
%
\leavevmode%
\llap{\spacer}%
${L_{172.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2,m}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5976\\216\\216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\-6\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\-9\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-10\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\60\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\11\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-72\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-6\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\72\\0\\-72
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-6\\0\\2\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-72\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-5976, 216, 216; 216, -6, -9; 216, -9, -7],[72, 3, 1],[-1, -1, 5, 1, 1; -12, -10, 60, 11, 9; -16, -18, 72, 16, 18],171,1]
%
% END OF LATTICE L172.1
%
\hbox{}\par\smallskip%
%
% LATTICE L172.2
%
\leavevmode%
\llap{\spacer}%
${L_{172.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3,m}%
\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{144}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
shares genus with 3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13968\\-288\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\6\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\3\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-40\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\48\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-42\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-144\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-3\\-18
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\0\\144\\0\\-144
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-3\\0\\1\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-144\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[13968, -288, -144; -288, 6, 3; -144, 3, 1],[144, 3, 1],[-1, 0, 1, 0, -1; -40, 1, 48, 0, -42; -16, 0, 0, -1, -18],171,2]
%
% END OF LATTICE L172.2
%
\hbox{}\par\smallskip%
%
% LATTICE L172.3
%
\leavevmode%
\llap{\spacer}%
${L_{172.3}}=\pfillwitharg{3}({L_{172.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\528\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
528\\166\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-15\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-24\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
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\HardButStrongLineBreak\kern3pt%
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-5\\16\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-10\\-6
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-48\\-8\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-3\\-6
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\16\\0\\-16
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-3\\0\\1\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-16\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[1680, 528, -48; 528, 166, -15; -48, -15, 1],[48, 1, 1],[7, -1, -5, 0, 3; -24, 3, 16, 0, -10; -16, 0, 0, -1, -6],171,3]
%
% END OF LATTICE L172.3
%
\hbox{}\par\smallskip%
%
% LATTICE L172.4
%
\leavevmode%
\llap{\spacer}%
${L_{172.4}}=\pfillwitharg{2}({L_{172.1}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\12\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-3
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-4
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\HardButStrongLineBreak\kern3pt%
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1\\3\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-18\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-9\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\2\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-9\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-4\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[18, -18, 0; -18, 12, 3; 0, 3, -1],[18, 3, 1],[1, -1, -1, 1, 1; 0, -1, 0, 3, 1; 0, -3, -4, 0, 2],171,4]
%
% END OF LATTICE L172.4
%
\hbox{}\par\smallskip%
%
% LATTICE L172.5
%
\leavevmode%
\llap{\spacer}%
${L_{172.5}}=\main({L_{172.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-828\\72\\108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-3\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108\\-9\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-36\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-9\\-3
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\4\\0\\-4
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-9\\0\\9\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-4\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-828, 72, 108; 72, -3, -9; 108, -9, -14],[36, 3, 1],[5, -1, -1, 1, 1; 0, 2, 0, -3, -1; 36, -9, -8, 9, 8],171,5]
%
% END OF LATTICE L172.5
%
\hbox{}\par\smallskip%
%
% LATTICE L172.6
%
\leavevmode%
\llap{\spacer}%
${L_{172.6}}=\main({L_{172.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-21\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\24\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-24\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-7\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-72\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-3\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\72\\0\\-72
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-3\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-72\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[72, 0, -72; 0, -6, -21; -72, -21, -1],[72, 3, 1],[-7, -2, 7, 2, -1; 24, 7, -24, -7, 3; -8, -3, 0, 2, 0],171,6]
%
% END OF LATTICE L172.6
%
\hbox{}\par\smallskip%
%
% LATTICE L172.7
%
\leavevmode%
\llap{\spacer}%
${L_{172.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{172.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-18\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\6\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-6\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-3\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\1\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\6\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-9\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-18\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\1\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-18\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-4\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-72, -18, -36; -18, 6, -6; -36, -6, -17],[18, 6, 1],[-4, 0, 2, 7, -1; -3, -1, 1, 6, 0; 9, 0, -5, -18, 2],171,7]
%
% END OF LATTICE L172.7
%
\hbox{}\par\smallskip%
%
% LATTICE L172.8
%
\leavevmode%
\llap{\spacer}%
${L_{172.8}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{172.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-96\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-21\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\18\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\7\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\40\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-144\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\0\\48\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-6\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-24\\0\\3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[48, -96, 48; -96, -21, 18; 48, 18, -13],[48, 3, 1],[1, 1, 5, -1, -1; 0, 7, 40, -6, -8; 0, 13, 72, -12, -15],171,8]
%
% END OF LATTICE L172.8
%
\hbox{}\par\smallskip%
%
% LATTICE L172.9
%
\leavevmode%
\llap{\spacer}%
${L_{172.9}}=\pdualwitharg{2}(\main({L_{172.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1332\\-108\\-324
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\24\\36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-324\\36\\85
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\18\\-27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\21\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\10\\-16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-9\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-36\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\1\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-36\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-4\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1332, -108, -324; -108, 24, 36; -324, 36, 85],[36, 12, 1],[-5, 0, -1, -7, -3; 18, 1, 3, 21, 10; -27, 0, -5, -36, -16],171,9]
%
% END OF LATTICE L172.9
%
\hbox{}\par\smallskip%
%
% LATTICE L172.10
%
\leavevmode%
\llap{\spacer}%
${L_{172.10}}=\pdualwitharg{3}({L_{172.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}{144}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
shares genus with 3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10224\\288\\144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\33\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-3\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\74
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-144\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-6\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\0\\144\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-6\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-72\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10224, 288, 144; 288, 33, -3; 144, -3, -2],[144, 3, 1],[-1, -1, 1, 1, 1; 0, 2, 0, -2, -3; -80, -75, 72, 74, 72],171,10]
%
% END OF LATTICE L172.10
%
\hbox{}\par\smallskip%
%
% LATTICE L172.11
%
\leavevmode%
\llap{\spacer}%
${L_{172.11}}=\pdualwitharg{2}(\pfillwitharg{3}({L_{172.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{96}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-304\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\-7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\5\\32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\96\\0\\-48\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-48\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-16\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[96, -48, 0; -48, -304, 48; 0, 48, -7],[48, 16, 1],[1, -1, -1, 0, 3; 2, 0, -2, -1, 5; 13, 0, -14, -8, 32],171,11]
%
% END OF LATTICE L172.11
%
\hbox{}\par\smallskip%
%
% LATTICE L172.12
%
\leavevmode%
\llap{\spacer}%
${L_{172.12}}=\pdualwitharg{2}(\main({L_{172.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-4872\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\336\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\5\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-9\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-72\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\1\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-72\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\-8\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[144, -72, 0; -72, -4872, 336; 0, 336, -23],[72, 24, 1],[-1, 3, 1, -1, -3; -3, 5, 2, 0, -6; -45, 72, 29, 0, -88],171,12]
%
% END OF LATTICE L172.12
%
\hbox{}\par\smallskip%
%
% LATTICE L172.13
%
\leavevmode%
\llap{\spacer}%
${L_{172.13}}=\pdualwitharg{2}({L_{172.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-648\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-648\\2472\\1104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\1104\\493
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\24\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\14\\-32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-36\\-144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0\\-72\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\4\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-72\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\-16\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[144, -648, -288; -648, 2472, 1104; -288, 1104, 493],[72, 24, 1],[-1, 1, 1, 1, -1; 24, -5, -6, 0, 14; -54, 12, 14, 0, -32],171,13]
%
% END OF LATTICE L172.13
%
\hbox{}\par\smallskip%
%
% LATTICE L172.14
%
\leavevmode%
\llap{\spacer}%
${L_{172.14}}=\pdualwitharg{2.3}(\pfillwitharg{3}({L_{172.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{32}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-960\\384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-960\\-336\\144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
384\\144\\-61
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\14\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-11\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\51\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-18\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\32\\0\\-48\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\12\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-48\\0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-48\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1680, -960, 384; -960, -336, 144; 384, 144, -61],[48, 48, 1],[1, -1, -1, 1, 4; 14, -11, -15, 8, 51; 39, -32, -42, 24, 144],171,14]
%
% END OF LATTICE L172.14
%
\hbox{}\par\smallskip%
%
% LATTICE L172.15
%
\leavevmode%
\llap{\spacer}%
${L_{172.15}}=\pdualwitharg{2}({L_{172.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{96}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{144}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
shares genus with 3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\3312\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3312\\-37488\\1632
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\1632\\-71
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\-32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-18\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\96\\0\\-144\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\4\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-144\\0\\144\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-48\\-16\\0\\16
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-288, 3312, -144; 3312, -37488, 1632; -144, 1632, -71],[144, 48, 1],[4, -1, -1, 1, 4; 0, 2, 0, -3, -1; -9, 48, 2, -72, -32],171,15]
%
% END OF LATTICE L172.15
%
\hbox{}\par\smallskip%
%
% LATTICE L172.16
%
\leavevmode%
\llap{\spacer}%
${L_{172.16}}=\pdualwitharg{2.3}({L_{172.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{*}{2}{96}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{144}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
shares genus with 3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-144\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-48\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-144\\-287
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\0\\-18
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\1
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-1\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-18\\-144\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\96\\0\\-144\\-48
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-18\\0\\1\\0\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-144\\0\\144\\0
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0\\-48\\-8\\0\\16
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-288, -144, -288; -144, -48, -144; -288, -144, -287],[144, 48, 1],[17, -1, -1, 1, 8; 0, 2, 0, -3, -1; -18, 0, 1, 0, -8],171,16]
%
% END OF LATTICE L172.16
%

\medskip%
%------------------------------WEYL-GROUP-W173--------------------------------
%
\leavevmode\llap{}%
$W_{173}$%
\qquad\llap{8} lattices, $\chi=40$%
\hfill%
$3\infty\infty3\infty\infty\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L173.1
%
\leavevmode%
\llap{\spacer}%
${L_{173.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{-}{3}{2}\above{1pt}{1pt}{20,17}{\infty b}{8}\above{1pt}{1pt}{10,7}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45400\\600\\800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
600\\-6\\-11
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
800\\-11\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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799\\7200\\39200
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-64\\-343
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-135\\-736
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\16\\99
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\71\\393
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\48\\244
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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2\\-1\\-8\\-11\\-23\\-4
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\kern3pt%
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-1\\2\\-4\\-23\\-66\\-28
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\-4
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-4\\8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-23\\-4
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-66\\-28
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-28\\-24
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-45400, 600, 800; 600, -6, -11; 800, -11, -14],[200, 1, 1],[2, 8, 5, 1, 0, -1; 16, 71, 48, 11, 1, -12; 99, 393, 244, 48, -1, -48],172,1]
%
% END OF LATTICE L173.1
%
\hbox{}\par\smallskip%
%
% LATTICE L173.2
%
\leavevmode%
\llap{\spacer}%
${L_{173.2}}=\pfillwitharg{2}({L_{173.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{+}{3}{2}\above{1pt}{1pt}{10,7}{\infty b}{2}\above{1pt}{1pt}{5,2}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2350\\450\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
450\\-86\\19
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\19\\-4
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-1\\150\\900
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-29\\-168
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\29
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{$\left.\llap{\phantom{%
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0\\0\\0
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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1\\6\\3
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-4\\-2\\2
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-11\\-23\\-2
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-23\\-66\\-14
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-2\\-14\\-6
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-2350, 450, -100; 450, -86, 19; -100, 19, -4],[50, 1, 1],[1, 0, -1, -1, 0, 1; 6, 1, -6, -9, -9, 4; 3, 4, -4, -21, -52, -8],172,2]
%
% END OF LATTICE L173.2
%
\hbox{}\par\smallskip%
%
% LATTICE L173.3
%
\leavevmode%
\llap{\spacer}%
${L_{173.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{173.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{-}{3}{4}\above{1pt}{1pt}{20,7}{\infty z}{1}\above{1pt}{1pt}{5,2}{\infty}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6700\\-150\\3200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-150\\24\\-68
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3200\\-68\\1529
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6700, -150, 3200; -150, 24, -68; 3200, -68, 1529],[50, 2, 1],[-26, -27, -13, -205, -589, -64; -9, -10, -5, -75, -214, -23; 54, 56, 27, 426, 1224, 133],172,3]
%
% END OF LATTICE L173.3
%
\hbox{}\par\smallskip%
%
% LATTICE L173.4
%
\leavevmode%
\llap{\spacer}%
${L_{173.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{173.1}}))$%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12450, -900, 5900; -900, 200, -425; 5900, -425, 2796],[50, 25, 1],[83, 95, 95, 700, 1993, 427; 2, 3, 2, 13, 37, 8; -175, -200, -200, -1475, -4200, -900],172,4]
%
% END OF LATTICE L173.4
%
\hbox{}\par\smallskip%
%
% LATTICE L173.5
%
\leavevmode%
\llap{\spacer}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[101200, -600, 24800; -600, -16, -144; 24800, -144, 6077],[200, 8, 1],[-41, 2, 21, 41, -2, -21; 24, -7, -14, -27, -1, 13; 168, -8, -86, -168, 8, 86],172,5]
%
% END OF LATTICE L173.5
%
\hbox{}\par\smallskip%
%
% LATTICE L173.6
%
\leavevmode%
\llap{\spacer}%
${L_{173.6}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{173.1}}))$%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-200, -36550, -17900; -36550, -6384300, -3126650; -17900, -3126650, -1531247],[50, 50, 1],[3, 4, -2, -21, -52, -4; 906, -49, -453, -759, 441, 502; -1850, 100, 925, 1550, -900, -1025],172,6]
%
% END OF LATTICE L173.6
%
\hbox{}\par\smallskip%
%
% LATTICE L173.7
%
\leavevmode%
\llap{\spacer}%
${L_{173.7}}=\pdualwitharg{5}({L_{173.1}})$%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-100\\200\\-100\\-700\\-600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\-25\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\50\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-100\\200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-275\\-575\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-575\\-1650\\-700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-700\\-600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21400, 23200, 10400; 23200, -25150, -11275; 10400, -11275, -5054],[200, 25, 1],[12, -11, -13, 54, 187, 101; 11, 1, -12, 16, 71, 48; 0, -25, 0, 75, 225, 100],172,7]
%
% END OF LATTICE L173.7
%
\hbox{}\par\smallskip%
%
% LATTICE L173.8
%
\leavevmode%
\llap{\spacer}%
${L_{173.8}}=\pdualwitharg{2.5}({L_{173.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${400}\above{1pt}{1pt}{+}{3}{400}\above{1pt}{1pt}{40,3}{\infty z}{100}\above{1pt}{1pt}{5,3}{\infty a}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
591600\\591400\\-5600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
591400\\590800\\-5600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5600\\-5600\\53
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-7\\1800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\5\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\5\\-950
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\-200\\-400\\-2200\\-4600\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\400\\-200\\-4600\\-13200\\-1400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-200\\100\\-200\\-1400\\-300
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\-200\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\400\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-200\\100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2200\\-4600\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4600\\-13200\\-1400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-1400\\-300
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[591600, 591400, -5600; 591400, 590800, -5600; -5600, -5600, 53],[200, 200, 1],[24, -7, -14, -27, -1, 13; -7, 5, 5, 10, 3, -4; 1800, -200, -950, -1800, 200, 950],172,8]
%
% END OF LATTICE L173.8
%

\medskip%
%------------------------------WEYL-GROUP-W174--------------------------------
%
\leavevmode\llap{}%
$W_{174}$%
\qquad\llap{8} lattices, $\chi=30$%
\hfill%
$2\infty2\infty2\infty\rtimes C_{3}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L174.1
%
\leavevmode%
\llap{\spacer}%
${L_{174.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${50}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{10,1}{\infty z}{2}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{10,9}{\infty z}{2}\above{1pt}{1pt}{b}{2}{200}\above{1pt}{1pt}{2,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12200\\-5000\\200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5000\\2042\\-55
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\-55\\-98
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-152\\-375\\-100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73\\-180\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75\\185\\49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
459\\1132\\300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
176\\434\\115
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1379\\3400\\900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\0\\-25\\-100\\-25\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\-4\\-40\\-20\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-4\\2\\0\\-5\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-40\\0\\8\\-4\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-20\\-5\\-4\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-200\\-100\\-200\\0\\200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[12200, -5000, 200; -5000, 2042, -55; 200, -55, -98],[200, 1, 1],[-152, -73, 75, 459, 176, 1379; -375, -180, 185, 1132, 434, 3400; -100, -48, 49, 300, 115, 900],173,1]
%
% END OF LATTICE L174.1
%
\hbox{}\par\smallskip%
%
% LATTICE L174.2
%
\leavevmode%
\llap{\spacer}%
${L_{174.2}}=\pfillwitharg{2}({L_{174.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{5,4}{\infty}{2}\above{1pt}{1pt}{l}{2}{50}\above{1pt}{1pt}{1,0}{\infty}{50}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{5,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29950\\12100\\1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12100\\-4888\\-485
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\-485\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\5\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\36\\110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\39\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
101\\200\\500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-5\\-50\\-25\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-2\\-50\\-50\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-2\\2\\0\\-25\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-50\\0\\50\\-50\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-50\\-25\\-50\\50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-10\\-10\\-50\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-29950, 12100, 1200; 12100, -4888, -485; 1200, -485, -48],[50, 1, 1],[3, 19, 20, 101, -1, -5; 5, 36, 39, 200, 0, -10; 24, 110, 105, 500, -25, -24],173,2]
%
% END OF LATTICE L174.2
%
\hbox{}\par\smallskip%
%
% LATTICE L174.3
%
\leavevmode%
\llap{\spacer}%
${L_{174.3}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{174.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{5,4}{\infty}{4}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{1,0}{\infty}{100}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{5,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
130100\\-3350\\60750
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3350\\96\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60750\\-1560\\28369
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
241\\220\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
318\\291\\-665
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
483\\443\\-1010
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1088\\1000\\-2275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\25\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\10\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-10\\-50\\-50\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2\\-25\\-50\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-2\\4\\0\\-50\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-25\\0\\25\\-50\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-50\\-50\\-50\\100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-10\\-25\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[130100, -3350, 60750; -3350, 96, -1560; 60750, -1560, 28369],[50, 2, 1],[241, 318, 483, 1088, 24, 11; 220, 291, 443, 1000, 25, 10; -504, -665, -1010, -2275, -50, -23],173,3]
%
% END OF LATTICE L174.3
%
\hbox{}\par\smallskip%
%
% LATTICE L174.4
%
\leavevmode%
\llap{\spacer}%
${L_{174.4}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{174.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${50}\above{1pt}{1pt}{l}{2}{50}\above{1pt}{1pt}{5,1}{\infty}{50}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{1,0}{\infty}{2}\above{1pt}{1pt}{l}{2}{50}\above{1pt}{1pt}{5,4}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
636050\\-66750\\304200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-66750\\7200\\-31925
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
304200\\-31925\\145488
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2365\\23\\4950
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6259\\60\\13100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4766\\45\\9975
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-861\\8\\1802
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\0\\23
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-215\\2\\450
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\0\\-125\\-50\\-25\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\50\\-50\\-50\\-50\\-250
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-125\\-50\\50\\0\\-25\\-250
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-50\\0\\2\\-2\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-50\\-25\\-2\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-250\\-250\\-50\\0\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[636050, -66750, 304200; -66750, 7200, -31925; 304200, -31925, 145488],[50, 25, 1],[-2365, -6259, -4766, -861, -11, -215; 23, 60, 45, 8, 0, 2; 4950, 13100, 9975, 1802, 23, 450],173,4]
%
% END OF LATTICE L174.4
%
\hbox{}\par\smallskip%
%
% LATTICE L174.5
%
\leavevmode%
\llap{\spacer}%
${L_{174.5}}=\pdualwitharg{2}({L_{174.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${400}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{5,1}{\infty a}{16}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{5,4}{\infty b}{16}\above{1pt}{1pt}{*}{2}{100}\above{1pt}{1pt}{1,0}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41112400\\-165000\\293200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-165000\\-560\\1176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
293200\\1176\\-2091
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\25\\3800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\23\\3238
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\-46\\-6616
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111\\-110\\-15626
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-215\\-30408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-526\\-525\\-74050
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\0\\-200\\-200\\-200\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-8\\-20\\-40\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-8\\16\\0\\-40\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-20\\0\\4\\-8\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-40\\-40\\-8\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-100\\-200\\-100\\0\\100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-41112400, -165000, 293200; -165000, -560, 1176; 293200, 1176, -2091],[200, 8, 1],[27, 23, -47, -111, -216, -526; 25, 23, -46, -110, -215, -525; 3800, 3238, -6616, -15626, -30408, -74050],173,5]
%
% END OF LATTICE L174.5
%
\hbox{}\par\smallskip%
%
% LATTICE L174.6
%
\leavevmode%
\llap{\spacer}%
${L_{174.6}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{174.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${100}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{5,1}{\infty}{100}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{1,0}{\infty}{4}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{5,4}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2400\\1068250\\523200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1068250\\-474214400\\-232257400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
523200\\-232257400\\-113753399
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\3355\\-6850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\10518\\-21475
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
105\\22089\\-45100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\2228\\-4549
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-48\\98
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-2755\\5625
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
100\\0\\-250\\-50\\-50\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\25\\-50\\-25\\-50\\-125
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-250\\-50\\100\\0\\-50\\-250
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-25\\0\\1\\-2\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-50\\-50\\-2\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-125\\-250\\-25\\0\\25
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2400, 1068250, 523200; 1068250, -474214400, -232257400; 523200, -232257400, -113753399],[50, 50, 1],[24, 55, 105, 10, -1, -12; 3355, 10518, 22089, 2228, -48, -2755; -6850, -21475, -45100, -4549, 98, 5625],173,6]
%
% END OF LATTICE L174.6
%
\hbox{}\par\smallskip%
%
% LATTICE L174.7
%
\leavevmode%
\llap{\spacer}%
${L_{174.7}}=\pdualwitharg{5}({L_{174.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${50}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{2,1}{\infty z}{2}\above{1pt}{1pt}{b}{2}{200}\above{1pt}{1pt}{10,9}{\infty z}{50}\above{1pt}{1pt}{b}{2}{200}\above{1pt}{1pt}{10,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7000\\-1800\\1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-150\\125
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\125\\-98
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-59\\-100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-28\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\29\\49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\884\\1500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\338\\575
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\528\\900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\0\\-25\\-500\\-125\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\-4\\-200\\-100\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-4\\2\\0\\-25\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-500\\-200\\0\\200\\-100\\-1000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-125\\-100\\-25\\-100\\50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-200\\-100\\-1000\\0\\200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7000, -1800, 1200; -1800, -150, 125; 1200, 125, -98],[200, 25, 1],[-2, -1, 1, 31, 12, 19; -59, -28, 29, 884, 338, 528; -100, -48, 49, 1500, 575, 900],173,7]
%
% END OF LATTICE L174.7
%
\hbox{}\par\smallskip%
%
% LATTICE L174.8
%
\leavevmode%
\llap{\spacer}%
${L_{174.8}}=\pdualwitharg{2.5}({L_{174.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{100}\above{1pt}{1pt}{5,4}{\infty b}{400}\above{1pt}{1pt}{*}{2}{100}\above{1pt}{1pt}{5,1}{\infty a}{400}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{1,0}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-854800\\-3979400\\-946600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3979400\\-18436400\\-4385400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-946600\\-4385400\\-1043139
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\1346\\-5616
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-555\\15926\\-66450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\31013\\-129400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-526\\15111\\-63050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\-767\\3200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-661\\2758
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-200\\-200\\-200\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\100\\-200\\-500\\-1000\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-200\\400\\0\\-1000\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-854800, -3979400, -946600; -3979400, -18436400, -4385400; -946600, -4385400, -1043139],[200, 200, 1],[-47, -555, -1080, -526, 27, 23; 1346, 15926, 31013, 15111, -767, -661; -5616, -66450, -129400, -63050, 3200, 2758],173,8]
%
% END OF LATTICE L174.8
%

\medskip%
%------------------------------WEYL-GROUP-W175--------------------------------
%
\leavevmode\llap{}%
$W_{175}$%
\qquad\llap{27} lattices, $\chi=12$%
\hfill%
$\slashtwo2|2\slashtwo2|2\rtimes D_{4}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L175.1
%
\leavevmode%
\llap{\spacer}%
${L_{175.1}}$%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1800, 288, -72; 288, -15, -6; -72, -6, 7],[72, 3, 1],[-1, 1, 2, 2, -1, -3; -12, 11, 22, 21, -12, -34; -18, 20, 39, 36, -22, -60],174,1]
%
% END OF LATTICE L175.1
%
\hbox{}\par\smallskip%
%
% LATTICE L175.2
%
\leavevmode%
\llap{\spacer}%
${L_{175.2}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[8136, -576, 288; -576, 39, -18; 288, -18, 7],[72, 3, 1],[-1, 1, 1, 1, -1, -1; -24, 22, 22, 18, -24, -23; -18, 16, 15, 0, -22, -18],174,2]
%
% END OF LATTICE L175.2
%
\hbox{}\par\smallskip%
%
% LATTICE L175.3
%
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\llap{\spacer}%
${L_{175.3}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1368, 144, 360; 144, -15, -36; 360, -36, -71],[72, 3, 1],[1, -1, -1, 8, 4, 7; 9, -12, -10, 96, 47, 80; 0, 1, 0, -9, -4, -6],174,3]
%
% END OF LATTICE L175.3
%
\hbox{}\par\smallskip%
%
% LATTICE L175.4
%
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${L_{175.4}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-5904, 864, 288; 864, -78, -30; 288, -30, -11],[144, 6, 1],[-1, -5, 0, 1, 1, -1; 20, 108, 1, -20, -24, 16; -82, -432, -3, 80, 90, -72],174,4]
%
% END OF LATTICE L175.4
%
\hbox{}\par\smallskip%
%
% LATTICE L175.5
%
\leavevmode%
\llap{\spacer}%
${L_{175.5}}$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90576\\-6048\\2448
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6048\\-402\\162
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2448\\162\\-65
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-29\\-35
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\HardButStrongLineBreak\kern3pt%
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-1\\32\\42
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-1\\28\\32
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\HardButStrongLineBreak\kern3pt%
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1\\-33\\-45
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-152\\-192
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\0\\-6\\-16\\-9\\0
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\kern3pt%
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0\\144\\0\\-144\\-144\\-144
\end{tabular}\endgroup%
\kern3pt%
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-6\\0\\12\\0\\-18\\-48
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\kern3pt%
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-16\\-144\\0\\16\\0\\-48
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\kern3pt%
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-9\\-144\\-18\\0\\9\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-48\\-48\\0\\48
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-90576, -6048, 2448; -6048, -402, 162; 2448, 162, -65],[144, 6, 1],[1, 5, -1, -1, 1, 5; -29, -132, 32, 28, -33, -152; -35, -144, 42, 32, -45, -192],174,5]
%
% END OF LATTICE L175.5
%
\hbox{}\par\smallskip%
%
% LATTICE L175.6
%
\leavevmode%
\llap{\spacer}%
${L_{175.6}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{144}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8784\\432\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\-6\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-6\\7
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-58
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\-72
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\54
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\16\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-52\\-168
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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4\\0\\-12\\-32\\-36\\0
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\kern3pt%
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0\\144\\0\\-144\\-288\\-144
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\kern3pt%
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-12\\0\\12\\0\\-36\\-48
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-32\\-144\\0\\16\\0\\-48
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-36\\-288\\-36\\0\\36\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[8784, 432, -288; 432, -6, -6; -288, -6, 7],[144, 6, 1],[-1, -1, 1, 1, -1, -3; -18, -24, 16, 16, -18, -52; -58, -72, 54, 56, -54, -168],174,6]
%
% END OF LATTICE L175.6
%
\hbox{}\par\smallskip%
%
% LATTICE L175.7
%
\leavevmode%
\llap{\spacer}%
${L_{175.7}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-}{5}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{3,2}%
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{${64}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{*}{2}{192}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{576}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
shares genus with 2-dual${}\iso{}$3-dual; isometric to own %
2.3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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318528\\576\\-576
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
576\\-24\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-576\\0\\1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-1\\-16\\-544
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-558
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\-576
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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1\\24\\576
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\6
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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64\\0\\-192\\-32\\-576\\0
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\kern3pt%
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0\\36\\0\\-18\\-576\\-36
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\0\\192\\0\\-576\\-96
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\kern3pt%
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-32\\-18\\0\\1\\0\\-6
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-576\\-576\\-576\\0\\576\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-96\\-6\\0\\12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[318528, 576, -576; 576, -24, 0; -576, 0, 1],[576, 24, 1],[-1, -1, -1, 0, 1, 0; -16, -18, -20, 0, 24, 1; -544, -558, -576, -1, 576, 6],174,7]
%
% END OF LATTICE L175.7
%
\hbox{}\par\smallskip%
%
% LATTICE L175.8
%
\leavevmode%
\llap{\spacer}%
${L_{175.8}}=\pfillwitharg{2}({L_{175.1}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-18\\0\\-18
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\kern3pt%
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0\\3\\6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\6\\-5
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-1\\18\\54
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0\\-8\\-21
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0\\3\\8
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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1\\3\\0
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-4\\-9\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-6
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-18, 0, -18; 0, 3, 6; -18, 6, -5],[18, 3, 1],[-1, 1, 2, 1, -1, -2; -2, 3, 4, 1, -6, -5; 1, 0, -3, -4, -9, 0],174,8]
%
% END OF LATTICE L175.8
%
\hbox{}\par\smallskip%
%
% LATTICE L175.9
%
\leavevmode%
\llap{\spacer}%
${L_{175.9}}=\main({L_{175.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-252\\0\\108
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\33\\30
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108\\30\\-19
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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-49\\96\\-144
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\kern3pt%
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-16\\31\\-48
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-252, 0, 108; 0, 33, 30; 108, 30, -19],[36, 3, 1],[-4, -8, -11, -2, 6, 9; 9, 17, 23, 3, -13, -19; -9, -19, -27, -9, 13, 21],174,9]
%
% END OF LATTICE L175.9
%
\hbox{}\par\smallskip%
%
% LATTICE L175.10
%
\leavevmode%
\llap{\spacer}%
${L_{175.10}}=\pfillwitharg{2}({L_{175.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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108\\-54\\7
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\3\\9
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\8\\16
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\HardButStrongLineBreak\kern3pt%
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3\\8\\15
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\HardButStrongLineBreak\kern3pt%
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13\\36\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\7
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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9\\0\\-9\\-72\\-9\\0
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\kern3pt%
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0\\4\\0\\-36\\-8\\-12
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\kern3pt%
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-9\\0\\3\\0\\-3\\-12
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\kern3pt%
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-72\\-36\\0\\36\\0\\-36
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-9\\-8\\-3\\0\\1\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-12\\-36\\0\\12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[1764, -864, 108; -864, 426, -54; 108, -54, 7],[36, 6, 1],[1, 3, 3, 13, 1, -1; 3, 8, 8, 36, 3, -2; 9, 16, 15, 72, 7, 0],174,10]
%
% END OF LATTICE L175.10
%
\hbox{}\par\smallskip%
%
% LATTICE L175.11
%
\leavevmode%
\llap{\spacer}%
${L_{175.11}}=\main({L_{175.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{6}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{24}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
504\\144\\0
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\kern3pt%
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144\\42\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\-3\\0
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\HardButStrongLineBreak\kern3pt%
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1\\-4\\-4
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\HardButStrongLineBreak\kern3pt%
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0\\-1\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\12\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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18\\0\\-18\\-144\\-18\\0
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\kern3pt%
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0\\8\\0\\-72\\-16\\-24
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\kern3pt%
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-18\\0\\6\\0\\-6\\-24
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-144\\-72\\0\\72\\0\\-72
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-18\\-16\\-6\\0\\2\\0
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0\\-24\\-24\\-72\\0\\24
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[504, 144, 0; 144, 42, 0; 0, 0, -1],[72, 6, 1],[1, 1, 0, -5, -1, -1; -3, -4, -1, 12, 3, 4; 0, -4, -6, -36, -4, 0],174,11]
%
% END OF LATTICE L175.11
%
\hbox{}\par\smallskip%
%
% LATTICE L175.12
%
\leavevmode%
\llap{\spacer}%
${L_{175.12}}=\main({L_{175.6}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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1440\\474\\48
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\kern3pt%
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144\\48\\5
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\-3\\0
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-3\\11\\-18
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\HardButStrongLineBreak\kern3pt%
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-23\\84\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-3\\11\\-20
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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18\\0\\-18\\-144\\-18\\0
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0\\8\\0\\-72\\-16\\-24
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-18\\0\\6\\0\\-6\\-24
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-144\\-72\\0\\72\\0\\-72
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-16\\-6\\0\\2\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-24\\-24\\-72\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[4392, 1440, 144; 1440, 474, 48; 144, 48, 5],[72, 6, 1],[1, -1, -3, -23, -3, -1; -3, 4, 11, 84, 11, 4; 0, -8, -18, -144, -20, -12],174,12]
%
% END OF LATTICE L175.12
%
\hbox{}\par\smallskip%
%
% LATTICE L175.13
%
\leavevmode%
\llap{\spacer}%
${L_{175.13}}=\pfillwitharg{3}({L_{175.7}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-}{5}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${64}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{192}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{64}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31296\\10560\\-192
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\kern3pt%
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10560\\3560\\-64
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-5\\16\\64
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0\\0\\-1
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\HardButStrongLineBreak\kern3pt%
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11\\-36\\-192
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\HardButStrongLineBreak\kern3pt%
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3\\-10\\-58
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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7\\-24\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\0\\-192\\-64\\-192\\0
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0\\1\\0\\-6\\-32\\-6
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\0\\192\\0\\-192\\-96
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\-6\\0\\4\\0\\-12
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\-32\\-192\\0\\64\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-96\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[31296, 10560, -192; 10560, 3560, -64; -192, -64, 1],[192, 8, 1],[-5, 0, 11, 3, 7, -1; 16, 0, -36, -10, -24, 3; 64, -1, -192, -58, -160, 6],174,13]
%
% END OF LATTICE L175.13
%
\hbox{}\par\smallskip%
%
% LATTICE L175.14
%
\leavevmode%
\llap{\spacer}%
${L_{175.14}}=\pfillwitharg{2}({L_{175.7}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{144}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3312, 144, 144; 144, 120, 24; 144, 24, 1],[144, 24, 1],[-1, 1, 5, 1, 5, -1; 8, -6, -34, -7, -36, 7; -32, 27, 144, 29, 144, -30],174,14]
%
% END OF LATTICE L175.14
%
\hbox{}\par\smallskip%
%
% LATTICE L175.15
%
\leavevmode%
\llap{\spacer}%
${L_{175.15}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{175.1}}))$%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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324\\78\\156
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684\\156\\329
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1422, 324, 684; 324, 78, 156; 684, 156, 329],[18, 6, 1],[-1, 8, -11, -32, -107, -22; 0, 3, -2, -7, -24, -5; 2, -18, 24, 70, 234, 48],174,15]
%
% END OF LATTICE L175.15
%
\hbox{}\par\smallskip%
%
% LATTICE L175.16
%
\leavevmode%
\llap{\spacer}%
${L_{175.16}}=\pdualwitharg{2}(\main({L_{175.2}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[65736, 12276, -8352; 12276, 2292, -1560; -8352, -1560, 1061],[36, 12, 1],[17, -1, -17, -59, -13, 3; -42, 0, 34, 120, 26, -8; 72, -8, -84, -288, -64, 12],174,16]
%
% END OF LATTICE L175.16
%
\hbox{}\par\smallskip%
%
% LATTICE L175.17
%
\leavevmode%
\llap{\spacer}%
${L_{175.17}}=\pdualwitharg{2}(\main({L_{175.5}}))$%
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{$1\above{1pt}{1pt}{-}{3}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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-6\\0\\3\\0\\-18\\-12
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2952, 720, -144; 720, 156, -36; -144, -36, 7],[72, 12, 1],[-1, 1, 1, 1, -1, -2; 0, -3, -1, -1, 0, 1; -22, 0, 15, 16, -18, -36],174,17]
%
% END OF LATTICE L175.17
%
\hbox{}\par\smallskip%
%
% LATTICE L175.18
%
\leavevmode%
\llap{\spacer}%
${L_{175.18}}=\pdualwitharg{2}(\main({L_{175.6}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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2\\-25\\-77
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\HardButStrongLineBreak\kern3pt%
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7\\-93\\-288
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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-1\\13\\40
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-9\\-27
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-47\\-144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\0\\-6\\-8\\-9\\0
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\end{tabular}\endgroup%
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0\\-36\\-24\\-12\\0\\12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1368, -1656, 504; -1656, -1716, 516; 504, 516, -155],[72, 12, 1],[2, 7, -1, -1, 1, 4; -25, -93, 10, 13, -9, -47; -77, -288, 30, 40, -27, -144],174,18]
%
% END OF LATTICE L175.18
%
\hbox{}\par\smallskip%
%
% LATTICE L175.19
%
\leavevmode%
\llap{\spacer}%
${L_{175.19}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{175.7}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{144}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{48}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38160\\4608\\-432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4608\\570\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-432\\-66\\19
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-8\\73\\71
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-113\\1032\\1008
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\HardButStrongLineBreak\kern3pt%
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-7\\64\\63
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\HardButStrongLineBreak\kern3pt%
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-7\\64\\64
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\HardButStrongLineBreak\kern3pt%
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-1\\9\\9
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\100\\96
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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1\\0\\-3\\-16\\-9\\0
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0\\144\\0\\-144\\-144\\-144
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-3\\0\\3\\0\\-9\\-24
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-16\\-144\\0\\16\\0\\-48
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-9\\-144\\-9\\0\\9\\0
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0\\-144\\-24\\-48\\0\\48
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[38160, 4608, -432; 4608, 570, -66; -432, -66, 19],[144, 6, 1],[-8, -113, -7, -7, -1, -11; 73, 1032, 64, 64, 9, 100; 71, 1008, 63, 64, 9, 96],174,19]
%
% END OF LATTICE L175.19
%
\hbox{}\par\smallskip%
%
% LATTICE L175.20
%
\leavevmode%
\llap{\spacer}%
${L_{175.20}}=\pdualwitharg{2}({L_{175.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\0\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\24\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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1\\0\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-4
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-1\\-2\\-12
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-3\\-36
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-8
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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72\\0\\-72\\-288\\-72\\0
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0\\8\\0\\-72\\-32\\-24
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-72\\0\\24\\0\\-24\\-48
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-288\\-72\\0\\72\\0\\-72
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-72\\-32\\-24\\0\\8\\0
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0\\-24\\-48\\-72\\0\\24
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[72, 0, 0; 0, 24, 0; 0, 0, -1],[72, 24, 1],[1, 0, -1, -4, -1, 0; 0, -1, -2, -3, 0, 1; 0, -4, -12, -36, -8, 0],174,20]
%
% END OF LATTICE L175.20
%
\hbox{}\par\smallskip%
%
% LATTICE L175.21
%
\leavevmode%
\llap{\spacer}%
${L_{175.21}}=\pdualwitharg{2}({L_{175.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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72\\0\\0
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0\\-8760\\408
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\408\\-19
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\3\\64
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0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[72, 0, 0; 0, -8760, 408; 0, 408, -19],[72, 24, 1],[1, 0, -1, -4, -1, 0; 0, 3, 4, -3, -4, -5; 0, 64, 84, -72, -88, -108],174,21]
%
% END OF LATTICE L175.21
%
\hbox{}\par\smallskip%
%
% LATTICE L175.22
%
\leavevmode%
\llap{\spacer}%
${L_{175.22}}=\pdualwitharg{2}({L_{175.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[24696, 5472, -648; 5472, 1176, -144; -648, -144, 17],[72, 24, 1],[-1, 2, 3, -2, -3, -4; 0, -1, -2, -3, 0, 1; -36, 68, 96, -108, -116, -144],174,22]
%
% END OF LATTICE L175.22
%
\hbox{}\par\smallskip%
%
% LATTICE L175.23
%
\leavevmode%
\llap{\spacer}%
${L_{175.23}}=\pdualwitharg{2}({L_{175.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{144}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\-576\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-576\\264\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\72\\19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\30\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\-24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-6\\-32\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-144\\-288\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\3\\0\\-18\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-144\\0\\16\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-288\\-18\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-24\\-48\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1008, -576, -144; -576, 264, 72; -144, 72, 19],[144, 24, 1],[-1, -5, 0, 1, 1, -1; 5, 30, 1, -2, -3, 4; -26, -144, -3, 16, 18, -24],174,23]
%
% END OF LATTICE L175.23
%
\hbox{}\par\smallskip%
%
% LATTICE L175.24
%
\leavevmode%
\llap{\spacer}%
${L_{175.24}}=\pdualwitharg{2}({L_{175.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{144}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92304\\-6624\\1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6624\\-456\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\72\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-106
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-6\\-504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-48\\-32\\-144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\0\\-36\\-288\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\48\\0\\-144\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-36\\0\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-288\\-144\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-48\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-92304, -6624, 1008; -6624, -456, 72; 1008, 72, -11],[144, 24, 1],[1, 1, -1, -1, -5, 0; 2, 0, -4, -2, -6, 1; 104, 90, -120, -106, -504, 6],174,24]
%
% END OF LATTICE L175.24
%
\hbox{}\par\smallskip%
%
% LATTICE L175.25
%
\leavevmode%
\llap{\spacer}%
${L_{175.25}}=\pdualwitharg{2}({L_{175.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{144}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2736\\-1872\\144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1872\\1272\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\-96\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-32\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\0\\-144\\-288\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-36\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-144\\0\\16\\0\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-288\\-36\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-48\\-48\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2736, -1872, 144; -1872, 1272, -96; 144, -96, 7],[144, 24, 1],[-1, -1, 1, 1, -1, -3; -3, -6, 2, 2, -3, -8; -22, -72, 6, 8, -18, -48],174,25]
%
% END OF LATTICE L175.25
%
\hbox{}\par\smallskip%
%
% LATTICE L175.26
%
\leavevmode%
\llap{\spacer}%
${L_{175.26}}=\pdualwitharg{3}(\pfillwitharg{3}({L_{175.7}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${192}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{*}{2}{64}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{192}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27456\\13056\\1536
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13056\\-5448\\-648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1536\\-648\\-77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-32\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\24\\-222
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\28\\-256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-23\\213
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-312\\2880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-25\\230
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
192\\0\\-192\\-96\\-576\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-18\\-192\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\0\\64\\0\\-192\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-18\\0\\3\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-576\\-192\\-192\\0\\192\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-12\\-32\\-6\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-27456, 13056, 1536; 13056, -5448, -648; 1536, -648, -77],[192, 24, 1],[1, -1, -1, 1, 13, 1; -32, 24, 28, -23, -312, -25; 288, -222, -256, 213, 2880, 230],174,26]
%
% END OF LATTICE L175.26
%
\hbox{}\par\smallskip%
%
% LATTICE L175.27
%
\leavevmode%
\llap{\spacer}%
${L_{175.27}}=\pdualwitharg{2}({L_{175.7}})\iso \pdualwitharg{3}({L_{175.7}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-}{5}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${64}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{192}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{576}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 2-dual${}\iso{}$3-dual; isometric to own %
2.3-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40896\\9792\\-2304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9792\\552\\-168
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2304\\-168\\49
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-80\\-320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\78\\315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\548\\2208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\78\\314
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\72\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-79\\-318
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\0\\-192\\-64\\-576\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\9\\0\\-18\\-288\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\0\\192\\0\\-576\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\-18\\0\\4\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-576\\-288\\-576\\0\\576\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-96\\-12\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-40896, 9792, -2304; 9792, 552, -168; -2304, -168, 49],[576, 24, 1],[-1, 1, 7, 1, 1, -1; -80, 78, 548, 78, 72, -79; -320, 315, 2208, 314, 288, -318],174,27]
%
% END OF LATTICE L175.27
%

\medskip%
%------------------------------WEYL-GROUP-W176--------------------------------
%
\leavevmode\llap{}%
$W_{176}$%
\qquad\llap{34} lattices, $\chi=24$%
\hfill%
$222|222\slashinfty\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L176.1
%
\leavevmode%
\llap{\spacer}%
${L_{176.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{1,0}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-740040\\12600\\12992
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12600\\-211\\-225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12992\\-225\\-224
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
211\\6328\\5880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\1469\\1366
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
181\\5425\\5047
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\1228\\1144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\686\\644
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-450\\-418
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-448\\-420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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56\\0\\-56\\-56\\-280\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
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0\\1\\0\\-4\\-42\\-12\\-28
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\7\\0\\-112\\-42\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-4\\0\\8\\0\\-8\\-56
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-42\\-112\\0\\28\\0\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-12\\-42\\-8\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-28\\-140\\-56\\-112\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-740040, 12600, 12992; 12600, -211, -225; 12992, -225, -224],[56, 1, 1],[211, 49, 181, 41, 23, -15, -15; 6328, 1469, 5425, 1228, 686, -450, -448; 5880, 1366, 5047, 1144, 644, -418, -420],175,1]
%
% END OF LATTICE L176.1
%
\hbox{}\par\smallskip%
%
% LATTICE L176.2
%
\leavevmode%
\llap{\spacer}%
${L_{176.2}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2,m}%
\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{8,5}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-345296\\7840\\8624
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7840\\-178\\-196
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8624\\-196\\-211
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-133\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\204\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\3024\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\209\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\966\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\44\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-630\\-28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-112\\-14\\-140\\-56\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-4\\-84\\-48\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\112\\0\\-448\\-336\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-4\\0\\2\\0\\-8\\-28
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-84\\-448\\0\\28\\0\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-48\\-336\\-8\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-56\\-560\\-28\\-112\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-345296, 7840, 8624; 7840, -178, -196; 8624, -196, -211],[112, 2, 1],[-3, 5, 73, 5, 23, 1, -15; -133, 204, 3024, 209, 966, 44, -630; 0, 14, 168, 10, 42, 0, -28],175,2]
%
% END OF LATTICE L176.2
%
\hbox{}\par\smallskip%
%
% LATTICE L176.3
%
\leavevmode%
\llap{\spacer}%
${L_{176.3}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{8,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-655984\\-185808\\11312
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-185808\\-52630\\3204
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11312\\3204\\-195
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-14\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-28\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\28\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\62\\148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74\\301\\651
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-83\\336\\704
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\189\\378
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-224\\-56\\-140\\-112\\-28
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\kern3pt%
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0\\4\\0\\-8\\-42\\-48\\-28
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-224\\0\\112\\0\\-224\\-336\\-280
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-8\\0\\8\\0\\-16\\-28
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-42\\-224\\0\\7\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-48\\-336\\-16\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-28\\-280\\-28\\-28\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-655984, -185808, 11312; -185808, -52630, 3204; 11312, 3204, -195],[112, 2, 1],[3, 7, -5, -15, -74, -83, -47; -14, -28, 28, 62, 301, 336, 189; -56, -54, 168, 148, 651, 704, 378],175,3]
%
% END OF LATTICE L176.3
%
\hbox{}\par\smallskip%
%
% LATTICE L176.4
%
\leavevmode%
\llap{\spacer}%
${L_{176.4}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{8,5}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\-112\\784
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-22\\168
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
784\\168\\-783
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3\\-14\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-28\\1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\28\\0
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\HardButStrongLineBreak\kern3pt%
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-29\\118\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-295\\1190\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-167\\672\\-24
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\385\\-14
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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56\\0\\-224\\-56\\-280\\-112\\-28
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\kern3pt%
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0\\1\\0\\-4\\-42\\-24\\-14
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-224\\0\\112\\0\\-448\\-336\\-280
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-4\\0\\8\\0\\-16\\-28
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-42\\-448\\0\\28\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-24\\-336\\-16\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-14\\-280\\-28\\-56\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-560, -112, 784; -112, -22, 168; 784, 168, -783],[112, 2, 1],[3, 7, -5, -29, -295, -167, -96; -14, -28, 28, 118, 1190, 672, 385; 0, 1, 0, -4, -42, -24, -14],175,4]
%
% END OF LATTICE L176.4
%
\hbox{}\par\smallskip%
%
% LATTICE L176.5
%
\leavevmode%
\llap{\spacer}%
${L_{176.5}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{1}{1}64\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{b}{2}{448}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{*}{2}{64}\above{1pt}{1pt}{s}{2}{224}\above{1pt}{1pt}{16,1}{\infty z}$}%
\nopagebreak\par%
shares genus with {$ {L_{176.6}}$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-189504\\-9856\\448
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9856\\-504\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
448\\24\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-35\\476
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-11\\170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-112\\2016
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\14\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\12\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-14\\112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-448\\-56\\-280\\-224\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-84\\-96\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-448\\0\\448\\0\\-896\\-1344\\-2240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-8\\0\\8\\0\\-32\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-84\\-896\\0\\28\\0\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224\\-96\\-1344\\-32\\0\\64\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-112\\-2240\\-112\\-224\\0\\224
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-189504, -9856, 448; -9856, -504, 24; 448, 24, -1],[448, 8, 1],[3, 1, 11, 0, -1, -1, 1; -35, -11, -112, 1, 14, 12, -14; 476, 170, 2016, 16, -126, -160, 112],175,5]
%
% END OF LATTICE L176.5
%
\hbox{}\par\smallskip%
%
% LATTICE L176.6
%
\leavevmode%
\llap{\spacer}%
${L_{176.6}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{1}{1}64\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{l}{2}{448}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{64}\above{1pt}{1pt}{*}{2}{224}\above{1pt}{1pt}{16,9}{\infty z}$}%
\nopagebreak\par%
shares genus with {$ {L_{176.5}}$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3777088\\165312\\-54656
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
165312\\-7224\\2384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54656\\2384\\-785
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-133\\-196
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\249\\338
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
157\\6552\\8960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\419\\576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\462\\637
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\44\\64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-1218\\-1680
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-448\\-56\\-140\\-224\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-42\\-96\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-448\\0\\448\\0\\-448\\-1344\\-2240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-8\\0\\8\\0\\-32\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-42\\-448\\0\\7\\0\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224\\-96\\-1344\\-32\\0\\64\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-112\\-2240\\-112\\-112\\0\\224
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-3777088, 165312, -54656; 165312, -7224, 2384; -54656, 2384, -785],[448, 8, 1],[-3, 6, 157, 10, 11, 1, -29; -133, 249, 6552, 419, 462, 44, -1218; -196, 338, 8960, 576, 637, 64, -1680],175,6]
%
% END OF LATTICE L176.6
%
\hbox{}\par\smallskip%
%
% LATTICE L176.7
%
\leavevmode%
\llap{\spacer}%
${L_{176.7}}=\pfillwitharg{2}({L_{176.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{4,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
700\\-112\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\14\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-49\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-25\\43
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-196\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-13\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-14\\21
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\7\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-56\\-14\\-70\\-28\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-2\\-21\\-12\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\28\\0\\-112\\-84\\-140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-2\\0\\2\\0\\-4\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-21\\-112\\0\\7\\0\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-12\\-84\\-4\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-14\\-140\\-14\\-28\\0\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[700, -112, 0; -112, 14, -2; 0, -2, -1],[28, 2, 1],[-8, -4, -31, -2, -2, 1, 1; -49, -25, -196, -13, -14, 6, 7; 84, 43, 336, 22, 21, -12, -14],175,7]
%
% END OF LATTICE L176.7
%
\hbox{}\par\smallskip%
%
% LATTICE L176.8
%
\leavevmode%
\llap{\spacer}%
${L_{176.8}}=\main({L_{176.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{4,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12712\\168\\1064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
168\\-2\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1064\\-14\\-89
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\7\\22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\84\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\21\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-14\\14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-56\\-7\\-70\\-28\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-2\\-42\\-24\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\0\\56\\0\\-224\\-168\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-2\\0\\1\\0\\-4\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-42\\-224\\0\\14\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-24\\-168\\-4\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-28\\-280\\-14\\-56\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-12712, 168, 1064; 168, -2, -14; 1064, -14, -89],[56, 2, 1],[3, 2, 11, 0, -2, -1, 1; 0, 7, 84, 5, 21, 0, -14; 35, 22, 112, -1, -28, -12, 14],175,8]
%
% END OF LATTICE L176.8
%
\hbox{}\par\smallskip%
%
% LATTICE L176.9
%
\leavevmode%
\llap{\spacer}%
${L_{176.9}}=\main({L_{176.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{4,3}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-616\\560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-616\\-1122\\992
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
560\\992\\-873
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-61\\238\\231
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\HardButStrongLineBreak\kern3pt%
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-107\\420\\408
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190\\749\\728
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\76\\74
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\28\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-35\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-14\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-28\\-14\\-140\\-14\\-14
\end{tabular}\endgroup%
\kern3pt%
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0\\8\\0\\-8\\-168\\-24\\-56
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-28\\0\\14\\0\\-224\\-42\\-140
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-8\\0\\4\\0\\-4\\-28
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-168\\-224\\0\\56\\0\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-24\\-42\\-4\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-56\\-140\\-28\\-112\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-280, -616, 560; -616, -1122, 992; 560, 992, -873],[56, 2, 1],[-61, -107, -190, -19, -5, 9, 3; 238, 420, 749, 76, 28, -35, -14; 231, 408, 728, 74, 28, -34, -14],175,9]
%
% END OF LATTICE L176.9
%
\hbox{}\par\smallskip%
%
% LATTICE L176.10
%
\leavevmode%
\llap{\spacer}%
${L_{176.10}}=\pfillwitharg{2}({L_{176.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{112}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{8,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2393104\\378448\\-41328
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
378448\\-59848\\6536
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41328\\6536\\-713
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-183\\-1099\\532
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-89\\-535\\254
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-675\\-4060\\1904
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\-247\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-91\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\162\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\133\\-56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-224\\-56\\-140\\-112\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-42\\-48\\-56
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224\\0\\112\\0\\-224\\-336\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-8\\0\\8\\0\\-16\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-42\\-224\\0\\7\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-48\\-336\\-16\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-56\\-560\\-56\\-56\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-2393104, 378448, -41328; 378448, -59848, 6536; -41328, 6536, -713],[112, 8, 1],[-183, -89, -675, -41, -15, 27, 22; -1099, -535, -4060, -247, -91, 162, 133; 532, 254, 1904, 112, 35, -80, -56],175,10]
%
% END OF LATTICE L176.10
%
\hbox{}\par\smallskip%
%
% LATTICE L176.11
%
\leavevmode%
\llap{\spacer}%
${L_{176.11}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{176.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{112}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{8,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47376\\-2912\\-3136
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2912\\-178\\-192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3136\\-192\\-207
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-133\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-52\\63
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-672\\784
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-11\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\21\\-35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\44\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\0\\-112\\-14\\-70\\-56\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-2\\-21\\-24\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\112\\0\\-224\\-336\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-2\\0\\2\\0\\-8\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-21\\-224\\0\\7\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-24\\-336\\-8\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-28\\-560\\-28\\-56\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-47376, -2912, -3136; -2912, -178, -192; -3136, -192, -207],[112, 2, 1],[-3, -1, -11, 0, 1, 1, -1; -133, -52, -672, -11, 21, 44, -14; 168, 63, 784, 10, -35, -56, 28],175,11]
%
% END OF LATTICE L176.11
%
\hbox{}\par\smallskip%
%
% LATTICE L176.12
%
\leavevmode%
\llap{\spacer}%
${L_{176.12}}=\pdualwitharg{2}(\main({L_{176.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{4,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-780\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-28\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-14\\364
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\154
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-21\\532
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-112\\-56\\-140\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-42\\-24\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\28\\0\\-112\\-84\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-8\\0\\8\\0\\-8\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-42\\-112\\0\\7\\0\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-24\\-84\\-8\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-56\\-280\\-56\\-56\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[56, 0, 0; 0, -780, -28; 0, -28, -1],[56, 4, 1],[-1, -1, -5, -1, -1, 0, 1; -14, -6, -21, -2, 0, 1, 0; 364, 154, 532, 48, -7, -28, 0],175,12]
%
% END OF LATTICE L176.12
%
\hbox{}\par\smallskip%
%
% LATTICE L176.13
%
\leavevmode%
\llap{\spacer}%
${L_{176.13}}=\pdualwitharg{2}(\main({L_{176.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{2,1}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-728\\784\\392
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
784\\-780\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
392\\-420\\-211
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
197\\-14\\392
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\-3\\91
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
341\\-21\\672
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\-2\\76
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\0\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\1\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\0\\-28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-112\\-56\\-280\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-4\\-42\\-12\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\28\\0\\-224\\-84\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-4\\0\\8\\0\\-8\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-42\\-224\\0\\28\\0\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-12\\-84\\-8\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-28\\-280\\-56\\-112\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-728, 784, 392; 784, -780, -420; 392, -420, -211],[56, 4, 1],[197, 46, 341, 39, 23, -14, -15; -14, -3, -21, -2, 0, 1, 0; 392, 91, 672, 76, 42, -28, -28],175,13]
%
% END OF LATTICE L176.13
%
\hbox{}\par\smallskip%
%
% LATTICE L176.14
%
\leavevmode%
\llap{\spacer}%
${L_{176.14}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{176.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{4,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2702\\406
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\406\\-61
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\14\\91
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\22\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\17\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\33
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-8\\-14\\-10\\-28\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-14\\-21\\-84\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\4\\0\\-16\\-84\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-14\\0\\14\\0\\-28\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-21\\-16\\0\\1\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-84\\-84\\-28\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-14\\-20\\-14\\-4\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28, 0, 0; 0, -2702, 406; 0, 406, -61],[28, 14, 1],[1, 3, 3, 1, 0, -1, 0; 1, 14, 22, 17, 5, 0, -3; 6, 91, 144, 112, 33, 0, -20],175,14]
%
% END OF LATTICE L176.14
%
\hbox{}\par\smallskip%
%
% LATTICE L176.15
%
\leavevmode%
\llap{\spacer}%
${L_{176.15}}=\pdualwitharg{7}({L_{176.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{1,0}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18424\\1064\\1064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1064\\-56\\-63
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1064\\-63\\-61
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\24\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\58\\217
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\37\\135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\88\\308
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\20\\66
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-12\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-8\\-56\\-40\\-56\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\0\\-28\\-42\\-84\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\1\\0\\-16\\-42\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-28\\0\\56\\0\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-42\\-16\\0\\4\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-84\\-42\\-56\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-28\\-20\\-56\\-16\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18424, 1064, 1064; 1064, -56, -63; 1064, -63, -61],[56, 7, 1],[7, 16, 10, 23, 5, -3, -3; 24, 58, 37, 88, 20, -10, -12; 96, 217, 135, 308, 66, -42, -40],175,15]
%
% END OF LATTICE L176.15
%
\hbox{}\par\smallskip%
%
% LATTICE L176.16
%
\leavevmode%
\llap{\spacer}%
${L_{176.16}}=\pdualwitharg{2}({L_{176.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{2,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12712\\-3920\\1064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3920\\-1208\\328
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1064\\328\\-89
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\7\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\42\\280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\21\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-7\\-14
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-56\\-14\\-140\\-28\\-14
\end{tabular}\endgroup%
\kern3pt%
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0\\8\\0\\-8\\-168\\-48\\-56
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-56\\0\\56\\0\\-448\\-168\\-280
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-14\\-8\\0\\4\\0\\-8\\-28
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-168\\-448\\0\\56\\0\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-48\\-168\\-8\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-56\\-280\\-28\\-112\\0\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12712, -3920, 1064; -3920, -1208, 328; 1064, 328, -89],[56, 8, 1],[3, 4, 11, 0, -4, -1, 1; 0, 7, 42, 5, 21, 0, -7; 35, 72, 280, 18, 28, -12, -14],175,16]
%
% END OF LATTICE L176.16
%
\hbox{}\par\smallskip%
%
% LATTICE L176.17
%
\leavevmode%
\llap{\spacer}%
${L_{176.17}}=\pdualwitharg{7}(\main({L_{176.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{4,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95032\\1232\\3248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1232\\-14\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3248\\-42\\-111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\19\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\24\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\7\\-91
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\3\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-8\\-7\\-10\\-28\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\14\\0\\-14\\-42\\-168\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\8\\0\\-32\\-168\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-14\\0\\7\\0\\-28\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-42\\-32\\0\\2\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-168\\-168\\-28\\0\\56\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-28\\-40\\-14\\-8\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-95032, 1232, 3248; 1232, -14, -42; 3248, -42, -111],[56, 14, 1],[0, -4, -7, -3, -2, -1, 1; 2, 19, 24, 7, 3, -4, -2; -1, -126, -216, -91, -60, -28, 30],175,17]
%
% END OF LATTICE L176.17
%
\hbox{}\par\smallskip%
%
% LATTICE L176.18
%
\leavevmode%
\llap{\spacer}%
${L_{176.18}}=\pdualwitharg{7}(\main({L_{176.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{4,3}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8680\\728\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
728\\-42\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336\\-28\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-12\\504
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\158
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-2\\182
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\76
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\-14\\-20\\-14\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\56\\0\\-56\\-168\\-168\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\0\\2\\0\\-32\\-42\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-56\\0\\28\\0\\-28\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-168\\-32\\0\\8\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-168\\-42\\-28\\0\\14\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-56\\-20\\-28\\-16\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8680, 728, 336; 728, -42, -28; 336, -28, -13],[56, 14, 1],[1, 19, 6, 7, 3, -1, -1; -1, -12, -3, -2, 0, 1, 0; 27, 504, 158, 182, 76, -28, -26],175,18]
%
% END OF LATTICE L176.18
%
\hbox{}\par\smallskip%
%
% LATTICE L176.19
%
\leavevmode%
\llap{\spacer}%
${L_{176.19}}=\pdualwitharg{2}({L_{176.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{112}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{8,5}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8630608\\1395744\\-43680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1395744\\-225720\\7064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43680\\7064\\-221
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-106\\-637\\588
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53\\-319\\278
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-409\\-2464\\2072
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-157\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-140\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\90\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\91\\-56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-224\\-56\\-280\\-112\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-8\\-84\\-48\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224\\0\\112\\0\\-448\\-336\\-560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-8\\0\\8\\0\\-16\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-280\\-84\\-448\\0\\28\\0\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-48\\-336\\-16\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-56\\-560\\-56\\-112\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8630608, 1395744, -43680; 1395744, -225720, 7064; -43680, 7064, -221],[112, 8, 1],[-106, -53, -409, -26, -23, 15, 15; -637, -319, -2464, -157, -140, 90, 91; 588, 278, 2072, 120, 70, -88, -56],175,19]
%
% END OF LATTICE L176.19
%
\hbox{}\par\smallskip%
%
% LATTICE L176.20
%
\leavevmode%
\llap{\spacer}%
${L_{176.20}}=\pdualwitharg{2}({L_{176.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{112}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{8,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1483664\\207200\\-34944
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
207200\\-28936\\4880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34944\\4880\\-823
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-7\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\7\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\25\\148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\476\\2352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\33\\153
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\147\\616
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-112\\-56\\-560\\-28\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-16\\-336\\-24\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\28\\0\\-448\\-42\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-16\\0\\8\\0\\-4\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\-336\\-448\\0\\112\\0\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-24\\-42\\-4\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-112\\-280\\-56\\-224\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1483664, 207200, -34944; 207200, -28936, 4880; -34944, 4880, -823],[112, 8, 1],[1, -1, -2, 0, 11, 1, 6; -7, -22, 7, 25, 476, 33, 147; -84, -88, 126, 148, 2352, 153, 616],175,20]
%
% END OF LATTICE L176.20
%
\hbox{}\par\smallskip%
%
% LATTICE L176.21
%
\leavevmode%
\llap{\spacer}%
${L_{176.21}}=\pdualwitharg{2}({L_{176.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${56}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}{112}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{8,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-102256\\16016\\-3808
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16016\\-2504\\600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3808\\600\\-139
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\63\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\64\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-91\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-107\\140
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-347\\-1694\\2184
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-220\\282
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-441\\560
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56\\0\\-112\\-56\\-560\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-16\\-336\\-48\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\0\\28\\0\\-448\\-84\\-280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-16\\0\\8\\0\\-8\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\-336\\-448\\0\\112\\0\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-48\\-84\\-8\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-112\\-280\\-56\\-224\\0\\56
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-102256, 16016, -3808; 16016, -2504, 600; -3808, 600, -139],[112, 8, 1],[13, 13, -19, -22, -347, -45, -90; 63, 64, -91, -107, -1694, -220, -441; -84, -80, 126, 140, 2184, 282, 560],175,21]
%
% END OF LATTICE L176.21
%
\hbox{}\par\smallskip%
%
% LATTICE L176.22
%
\leavevmode%
\llap{\spacer}%
${L_{176.22}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{176.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{s}{2}{112}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{8,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-190064\\-52864\\7728
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52864\\-14658\\2142
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7728\\2142\\-313
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\20\\112
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-1\\33\\201
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-3\\105\\644
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-7\\264\\1632
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-2\\82\\511
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\HardButStrongLineBreak\kern3pt%
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0\\5\\34
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8\\0\\-8\\-28\\-80\\-28\\-4
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-8\\0\\1\\0\\-32\\-21\\-10
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-80\\-336\\-32\\0\\16\\0\\-16
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-28\\-168\\-21\\-14\\0\\7\\0
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-4\\-56\\-10\\-14\\-16\\0\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-190064, -52864, 7728; -52864, -14658, 2142; 7728, 2142, -313],[112, 14, 1],[1, -1, -1, -3, -7, -2, 0; -34, 20, 33, 105, 264, 82, 5; -208, 112, 201, 644, 1632, 511, 34],175,22]
%
% END OF LATTICE L176.22
%
\hbox{}\par\smallskip%
%
% LATTICE L176.23
%
\leavevmode%
\llap{\spacer}%
${L_{176.23}}=\pdualwitharg{7}({L_{176.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{14}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{8,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-118832\\-21952\\2800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21952\\-3906\\490
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2800\\490\\-61
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3\\54\\296
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-3\\52\\280
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\HardButStrongLineBreak\kern3pt%
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5\\-90\\-494
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\HardButStrongLineBreak\kern3pt%
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10\\-179\\-980
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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31\\-552\\-3016
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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23\\-408\\-2226
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-35\\-190
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8\\0\\-16\\-28\\-80\\-56\\-4
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0\\112\\0\\-56\\-336\\-336\\-56
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-16\\0\\4\\0\\-64\\-84\\-20
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-28\\-56\\0\\14\\0\\-28\\-14
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-80\\-336\\-64\\0\\16\\0\\-16
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-56\\-336\\-84\\-28\\0\\28\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-56\\-20\\-14\\-16\\0\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-118832, -21952, 2800; -21952, -3906, 490; 2800, 490, -61],[112, 14, 1],[-3, -3, 5, 10, 31, 23, 2; 54, 52, -90, -179, -552, -408, -35; 296, 280, -494, -980, -3016, -2226, -190],175,23]
%
% END OF LATTICE L176.23
%
\hbox{}\par\smallskip%
%
% LATTICE L176.24
%
\leavevmode%
\llap{\spacer}%
${L_{176.24}}=\pdualwitharg{7}({L_{176.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{2}{112}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{28}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{8,3}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30352\\-11648\\1792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11648\\-4382\\672
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1792\\672\\-103
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\-21\\-120
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-376\\-2128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-58\\-327
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-130\\-728
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-52\\-288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\22\\126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\18\\100
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2\\0\\-4\\-28\\-40\\-28\\-4
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0\\112\\0\\-112\\-336\\-336\\-112
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-4\\0\\1\\0\\-32\\-42\\-20
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-28\\-112\\0\\56\\0\\-56\\-56
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-40\\-336\\-32\\0\\16\\0\\-32
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-28\\-336\\-42\\-56\\0\\28\\0
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-4\\-112\\-20\\-56\\-32\\0\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30352, -11648, 1792; -11648, -4382, 672; 1792, 672, -103],[112, 14, 1],[1, 19, 3, 7, 3, -1, -1; -21, -376, -58, -130, -52, 22, 18; -120, -2128, -327, -728, -288, 126, 100],175,24]
%
% END OF LATTICE L176.24
%
\hbox{}\par\smallskip%
%
% LATTICE L176.25
%
\leavevmode%
\llap{\spacer}%
${L_{176.25}}=\pdualwitharg{7}({L_{176.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{8,7}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17360\\-13440\\2128
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13440\\-9758\\1540
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2128\\1540\\-243
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-57\\-344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-864\\-5208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-248\\-1494
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-242\\-1456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-76\\-456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\29\\175
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\26\\156
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2\\0\\-8\\-28\\-40\\-14\\-4
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0\\112\\0\\-112\\-336\\-168\\-112
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-8\\0\\4\\0\\-64\\-42\\-40
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-28\\-112\\0\\56\\0\\-28\\-56
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-40\\-336\\-64\\0\\16\\0\\-32
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-14\\-168\\-42\\-28\\0\\7\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-112\\-40\\-56\\-32\\0\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-17360, -13440, 2128; -13440, -9758, 1540; 2128, 1540, -243],[112, 14, 1],[2, 31, 9, 9, 3, -1, -1; -57, -864, -248, -242, -76, 29, 26; -344, -5208, -1494, -1456, -456, 175, 156],175,25]
%
% END OF LATTICE L176.25
%
\hbox{}\par\smallskip%
%
% LATTICE L176.26
%
\leavevmode%
\llap{\spacer}%
${L_{176.26}}=\pdualwitharg{2.7}(\main({L_{176.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{28}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{2,1}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3640\\2968\\-1064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2968\\-2380\\868
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1064\\868\\-311
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\0\\-24
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\1\\-28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-2\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\-3\\148
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-6\\238
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-2\\52
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8\\0\\-8\\-56\\-40\\-56\\-8
\end{tabular}\endgroup%
\kern3pt%
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0\\28\\0\\-56\\-84\\-168\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\1\\0\\-16\\-42\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-56\\0\\56\\0\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-84\\-16\\0\\4\\0\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-168\\-42\\-56\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-56\\-20\\-56\\-16\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3640, 2968, -1064; 2968, -2380, 868; -1064, 868, -311],[56, 28, 1],[7, 9, -5, -51, -46, -75, -17; 0, 1, 0, -2, -3, -6, -2; -24, -28, 17, 168, 148, 238, 52],175,26]
%
% END OF LATTICE L176.26
%
\hbox{}\par\smallskip%
%
% LATTICE L176.27
%
\leavevmode%
\llap{\spacer}%
${L_{176.27}}=\pdualwitharg{2.7}(\main({L_{176.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{4,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104104\\3472\\2520
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3472\\-84\\-84
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2520\\-84\\-61
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3\\0\\-124
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-2\\1\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\206
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\HardButStrongLineBreak\kern3pt%
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21\\-2\\868
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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17\\-3\\704
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\HardButStrongLineBreak\kern3pt%
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13\\-3\\539
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-2\\208
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8\\0\\-16\\-56\\-40\\-28\\-8
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0\\28\\0\\-56\\-84\\-84\\-56
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\kern3pt%
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-16\\0\\4\\0\\-32\\-42\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-56\\0\\56\\0\\-28\\-56
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-84\\-32\\0\\4\\0\\-16
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-84\\-42\\-28\\0\\7\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-56\\-40\\-56\\-16\\0\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-104104, 3472, 2520; 3472, -84, -84; 2520, -84, -61],[56, 28, 1],[-3, -2, 5, 21, 17, 13, 5; 0, 1, 0, -2, -3, -3, -2; -124, -84, 206, 868, 704, 539, 208],175,27]
%
% END OF LATTICE L176.27
%
\hbox{}\par\smallskip%
%
% LATTICE L176.28
%
\leavevmode%
\llap{\spacer}%
${L_{176.28}}=\pdualwitharg{2.7}({L_{176.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{2,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82376\\-88480\\3024
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-88480\\-95032\\3248
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3024\\3248\\-111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\0\\27
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-8\\280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-7\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-6\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-4\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\0\\-8\\-14\\-20\\-28\\-2
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0\\56\\0\\-56\\-168\\-336\\-56
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-8\\0\\8\\0\\-64\\-168\\-40
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-14\\-56\\0\\28\\0\\-56\\-28
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-20\\-168\\-64\\0\\8\\0\\-16
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-28\\-336\\-168\\-56\\0\\56\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-56\\-40\\-28\\-16\\0\\4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-82376, -88480, 3024; -88480, -95032, 3248; 3024, 3248, -111],[56, 56, 1],[1, 19, 12, 7, 3, -2, -1; 0, -8, -7, -6, -4, -1, 1; 27, 280, 120, 14, -36, -84, 2],175,28]
%
% END OF LATTICE L176.28
%
\hbox{}\par\smallskip%
%
% LATTICE L176.29
%
\leavevmode%
\llap{\spacer}%
${L_{176.29}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{176.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{}{2}{112}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{4,3}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
112\\-112\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-3528\\1064
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1064\\-311
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-7\\-24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\5\\17
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\33\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\38\\128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\21\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8\\0\\-8\\-56\\-80\\-56\\-8
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0\\112\\0\\-112\\-336\\-336\\-112
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-8\\0\\1\\0\\-32\\-42\\-20
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-56\\-112\\0\\56\\0\\-56\\-56
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-80\\-336\\-32\\0\\16\\0\\-32
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-56\\-336\\-42\\-56\\0\\28\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-112\\-20\\-56\\-32\\0\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[112, -112, 0; -112, -3528, 1064; 0, 1064, -311],[112, 56, 1],[-7, -1, 5, 34, 41, 24, 0; -7, 0, 5, 33, 38, 21, -1; -24, 0, 17, 112, 128, 70, -4],175,29]
%
% END OF LATTICE L176.29
%
\hbox{}\par\smallskip%
%
% LATTICE L176.30
%
\leavevmode%
\llap{\spacer}%
${L_{176.30}}=\pdualwitharg{2.7}({L_{176.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{4,3}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5712\\6048\\1792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6048\\-952\\-336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1792\\-336\\-115
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\55\\-192
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-3\\80\\-280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-82\\286
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-353\\1232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-566\\1976
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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16\\-429\\1498
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-79\\276
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-16\\-56\\-80\\-56\\-8
\end{tabular}\endgroup%
\kern3pt%
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0\\112\\0\\-112\\-336\\-336\\-112
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-16\\0\\4\\0\\-64\\-84\\-40
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-56\\-112\\0\\56\\0\\-56\\-56
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-336\\-64\\0\\16\\0\\-32
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-336\\-84\\-56\\0\\28\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-112\\-40\\-56\\-32\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5712, 6048, 1792; 6048, -952, -336; 1792, -336, -115],[112, 56, 1],[-2, -3, 3, 13, 21, 16, 3; 55, 80, -82, -353, -566, -429, -79; -192, -280, 286, 1232, 1976, 1498, 276],175,30]
%
% END OF LATTICE L176.30
%
\hbox{}\par\smallskip%
%
% LATTICE L176.31
%
\leavevmode%
\llap{\spacer}%
${L_{176.31}}=\pdualwitharg{2.7}({L_{176.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{b}{2}{112}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{7}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{8,7}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\3360\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3360\\-32872\\2184
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224\\2184\\-145
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\1\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\7\\84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\14\\176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\6\\77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\3\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-16\\-56\\-80\\-28\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\112\\0\\-112\\-336\\-168\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\4\\0\\-64\\-42\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-112\\0\\56\\0\\-28\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-336\\-64\\0\\16\\0\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-168\\-42\\-28\\0\\7\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-112\\-40\\-56\\-32\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-336, 3360, -224; 3360, -32872, 2184; -224, 2184, -145],[112, 56, 1],[-2, -3, 3, 13, 21, 8, 3; -1, -4, 1, 7, 14, 6, 3; -12, -56, 10, 84, 176, 77, 40],175,31]
%
% END OF LATTICE L176.31
%
\hbox{}\par\smallskip%
%
% LATTICE L176.32
%
\leavevmode%
\llap{\spacer}%
${L_{176.32}}=\pdualwitharg{2.7}({L_{176.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{112}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{b}{2}{56}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{s}{2}{28}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{8,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-416304\\624512\\-5040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624512\\-936824\\7560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5040\\7560\\-61
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-3\\-124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\5\\206
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\19\\700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\28\\904
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\20\\574
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\3\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-16\\-56\\-80\\-56\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\112\\0\\-112\\-336\\-336\\-112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\4\\0\\-64\\-84\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-112\\0\\56\\0\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-336\\-64\\0\\16\\0\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56\\-336\\-84\\-56\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-112\\-40\\-56\\-32\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-416304, 624512, -5040; 624512, -936824, 7560; -5040, 7560, -61],[112, 56, 1],[-3, -3, 5, 20, 31, 23, 4; -3, -2, 5, 19, 28, 20, 3; -124, 0, 206, 700, 904, 574, 40],175,32]
%
% END OF LATTICE L176.32
%
\hbox{}\par\smallskip%
%
% LATTICE L176.33
%
\leavevmode%
\llap{\spacer}%
${L_{176.33}}=\pdualwitharg{7}({L_{176.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${32}\above{1pt}{1pt}{s}{2}{448}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{r}{2}{64}\above{1pt}{1pt}{b}{2}{28}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{8,7}{\infty b}$}%
\nopagebreak\par%
shares genus with {$ \pdualwitharg{7}({L_{176.6}})$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6455232\\-501312\\144256
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-501312\\-38920\\11200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144256\\11200\\-3223
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\70\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-116\\-448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-74\\-302
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-203\\-840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-416\\-1760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-99\\-434
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\0\\-32\\-112\\-320\\-112\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\448\\0\\-224\\-1344\\-672\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\0\\4\\0\\-128\\-84\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-224\\0\\56\\0\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-320\\-1344\\-128\\0\\64\\0\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-672\\-84\\-56\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-224\\-40\\-56\\-64\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6455232, -501312, 144256; -501312, -38920, 11200; 144256, 11200, -3223],[448, 56, 1],[1, -1, -1, -3, -7, -2, 0; 70, -116, -74, -203, -416, -99, 15; 288, -448, -302, -840, -1760, -434, 52],175,33]
%
% END OF LATTICE L176.33
%
\hbox{}\par\smallskip%
%
% LATTICE L176.34
%
\leavevmode%
\llap{\spacer}%
${L_{176.34}}=\pdualwitharg{7}({L_{176.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${32}\above{1pt}{1pt}{*}{2}{448}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{}{2}{64}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{8,7}{\infty a}$}%
\nopagebreak\par%
shares genus with {$ \pdualwitharg{7}({L_{176.5}})$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6078912\\-672000\\186368
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-672000\\-73864\\20496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
186368\\20496\\-5687
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-434\\-1728
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-620\\-2464
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\173\\689
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\1477\\5880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\4792\\19072
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\1833\\7294
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\351\\1396
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\0\\-16\\-112\\-320\\-112\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\448\\0\\-224\\-1344\\-672\\-224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\1\\0\\-64\\-42\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-224\\0\\56\\0\\-56\\-56
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-320\\-1344\\-64\\0\\64\\0\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112\\-672\\-42\\-56\\0\\28\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-224\\-20\\-56\\-64\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6078912, -672000, 186368; -672000, -73864, 20496; 186368, 20496, -5687],[448, 56, 1],[-5, -7, 2, 17, 55, 21, 4; -434, -620, 173, 1477, 4792, 1833, 351; -1728, -2464, 689, 5880, 19072, 7294, 1396],175,34]
%
% END OF LATTICE L176.34
%

\medskip%
%------------------------------WEYL-GROUP-W177--------------------------------
%
\leavevmode\llap{}%
$W_{177}$%
\qquad\llap{44} lattices, $\chi=24$%
\hfill%
$2|22|22|22|2\rtimes D_{4}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L177.1
%
\leavevmode%
\llap{\spacer}%
${L_{177.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96720\\720\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-4\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-5\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-120\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-118\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-120\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\115\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\816\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\117\\49
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\234\\99
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\350\\150
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-96\\-120\\-336\\-24\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-10\\-48\\-6\\-12\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\0\\12\\0\\-48\\-12\\-42\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-10\\0\\5\\0\\-5\\-30\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-48\\-48\\0\\48\\0\\-48\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-6\\-12\\-5\\0\\1\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\-42\\-30\\-48\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-120\\-110\\-240\\-10\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-96720, 720, 240; 720, -4, -5; 240, -5, 7],[240, 1, 1],[-1, -1, -1, 1, 7, 1, 2, 3; -120, -118, -120, 115, 816, 117, 234, 350; -48, -50, -54, 45, 336, 49, 99, 150],176,1]
%
% END OF LATTICE L177.1
%
\hbox{}\par\smallskip%
%
% LATTICE L177.2
%
\leavevmode%
\llap{\spacer}%
${L_{177.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
205680\\1680\\-1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1680\\-19\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1200\\-10\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-174
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-10\\1180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-24\\3216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-6\\848
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-6\\849
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-5\\680
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-96\\-240\\-336\\-48\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-10\\-24\\-6\\-6\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\0\\12\\0\\-48\\-24\\-42\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-10\\0\\20\\0\\-20\\-60\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-24\\-48\\0\\48\\0\\-48\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-6\\-24\\-20\\0\\4\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-6\\-42\\-60\\-48\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-60\\-110\\-120\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[205680, 1680, -1200; 1680, -19, -10; -1200, -10, 7],[240, 1, 1],[-1, -1, -1, 7, 19, 5, 5, 4; 0, 1, 0, -10, -24, -6, -6, -5; -168, -170, -174, 1180, 3216, 848, 849, 680],176,2]
%
% END OF LATTICE L177.2
%
\hbox{}\par\smallskip%
%
% LATTICE L177.3
%
\leavevmode%
\llap{\spacer}%
${L_{177.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
205680\\4080\\-1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4080\\52\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1200\\-24\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-174
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-5\\320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-12\\1152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\332
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\333
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-5\\500
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-96\\-240\\-336\\-48\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20\\-48\\-12\\-12\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\0\\12\\0\\-48\\-24\\-42\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-20\\0\\20\\0\\-20\\-60\\-220
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-48\\-48\\0\\48\\0\\-48\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-12\\-24\\-20\\0\\4\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-12\\-42\\-60\\-48\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-120\\-220\\-240\\-20\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[205680, 4080, -1200; 4080, 52, -24; -1200, -24, 7],[240, 4, 1],[-1, -1, -1, 2, 7, 2, 2, 3; 0, 1, 0, -5, -12, -3, -3, -5; -168, -168, -174, 320, 1152, 332, 333, 500],176,3]
%
% END OF LATTICE L177.3
%
\hbox{}\par\smallskip%
%
% LATTICE L177.4
%
\leavevmode%
\llap{\spacer}%
${L_{177.4}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30480\\1440\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1440\\-4\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\21\\168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-108\\-864
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-65\\-515
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-87\\-684
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\-170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\-168
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\170
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-96\\-60\\-84\\-24\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-5\\-12\\-6\\-24\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\0\\48\\0\\-48\\-48\\-336\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-5\\0\\5\\0\\-10\\-120\\-110
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-12\\-48\\0\\12\\0\\-96\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-6\\-48\\-10\\0\\4\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-24\\-336\\-120\\-96\\0\\48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-240\\-110\\-120\\-20\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-30480, 1440, 0; 1440, -4, -8; 0, -8, 1],[240, 4, 1],[1, 0, -5, -3, -4, -1, -1, 1; 21, 0, -108, -65, -87, -22, -24, 20; 168, -1, -864, -515, -684, -170, -168, 170],176,4]
%
% END OF LATTICE L177.4
%
\hbox{}\par\smallskip%
%
% LATTICE L177.5
%
\leavevmode%
\llap{\spacer}%
${L_{177.5}}=\pfillwitharg{2}({L_{177.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4620\\180\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
180\\7\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-5\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-96\\7
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\49\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\168\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\45\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\0\\-24\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-5
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\12\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-5\\0\\5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-6\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-3\\-12\\-5
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-12\\-84\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-60\\-55
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[4620, 180, -120; 180, 7, -5; -120, -5, -4],[60, 1, 1],[-4, -2, -7, -2, 1, 1, 1, -3; 99, 49, 168, 45, -27, -25, -24, 75; -6, -3, -12, -5, 0, 1, 0, -5],176,5]
%
% END OF LATTICE L177.5
%
\hbox{}\par\smallskip%
%
% LATTICE L177.6
%
\leavevmode%
\llap{\spacer}%
${L_{177.6}}=\main({L_{177.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{24}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\360\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-154\\-49
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-49\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
449\\1080\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-185\\-445\\148
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\12\\-5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-39\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-17\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\12\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\75\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-48\\-60\\-42\\-12\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-10\\-12\\-6\\-24\\-10
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-48\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-10
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\24\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-10\\0\\10
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-12\\-24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-6\\-24\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-24\\-168\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-120\\-110
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-840, 360, 120; 360, -154, -49; 120, -49, 4],[120, 1, 1],[-16, -7, 5, 31, 31, 12, 25, -6; -39, -17, 12, 75, 75, 29, 60, -15; 0, 2, 0, -10, -12, -6, -24, -10],176,6]
%
% END OF LATTICE L177.6
%
\hbox{}\par\smallskip%
%
% LATTICE L177.7
%
\leavevmode%
\llap{\spacer}%
${L_{177.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{177.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1740\\300\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
300\\-51\\21
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\21\\-8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-120\\180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\19\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-8\\11
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-8
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-8\\-30\\-7\\-6\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-15\\-6\\-9\\-12\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-8\\-30
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-15
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\4\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-15\\0\\15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-6\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-9\\-12\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-12\\-28\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-60\\-165
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1740, 300, -120; 300, -51, 21; -120, 21, -8],[60, 3, 1],[1, 0, -1, -2, 0, 1, 3, 7; 3, -1, -4, -5, 1, 5, 12, 25; -8, -3, 4, 15, 2, -3, -16, -45],176,7]
%
% END OF LATTICE L177.7
%
\hbox{}\par\smallskip%
%
% LATTICE L177.8
%
\leavevmode%
\llap{\spacer}%
${L_{177.8}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{177.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6420\\-3780\\-1560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3780\\2192\\920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1560\\920\\379
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\-30\\-420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
96\\23\\336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\7\\97
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\3\\96
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-2\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-6\\-117
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109\\-20\\-400
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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12\\0\\-24\\-120\\-84\\-24\\-12\\0
\end{tabular}\endgroup%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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12\\0\\-24\\-120
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\0\\3\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-20\\0\\20
\end{tabular}\endgroup%
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-84\\-24\\-12\\0
\end{tabular}\endgroup%
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-24\\-12\\-12\\-20
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6420, -3780, -1560; -3780, 2192, 920; -1560, 920, 379],[60, 4, 1],[25, -9, -32, -109, -49, 1, 20, 69; 3, -2, -6, -20, -9, 0, 3, 10; 96, -32, -117, -400, -180, 4, 75, 260],176,8]
%
% END OF LATTICE L177.8
%
\hbox{}\par\smallskip%
%
% LATTICE L177.9
%
\leavevmode%
\llap{\spacer}%
${L_{177.9}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{177.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${15}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-24\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-13\\-30
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\8\\19
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2\\-15\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\0\\-120\\-30\\-105\\-30\\-60\\0
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\kern3pt%
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0\\5\\0\\-5\\-30\\-15\\-60\\-5
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{$\left.\llap{\phantom{%
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\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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15\\0\\-120\\-30
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-5
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\60\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-5\\0\\1
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-105\\-30\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-15\\-60\\-5
\end{tabular}\endgroup%
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-60\\-60\\-420\\-60
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0\\-5\\-60\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[60, 0, 0; 0, -5, 5; 0, 5, -4],[60, 5, 1],[-2, 0, 1, 0, -1, -1, -7, -1; -15, -3, 0, 1, 3, -1, -24, -5; -30, -5, 0, 1, 0, -5, -60, -11],176,9]
%
% END OF LATTICE L177.9
%
\hbox{}\par\smallskip%
%
% LATTICE L177.10
%
\leavevmode%
\llap{\spacer}%
${L_{177.10}}=\pdualwitharg{3}(\main({L_{177.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13080\\-2760\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2760\\-582\\-51
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-51\\-4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71\\280\\840
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\59\\180
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\11
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3\\-13\\-16
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2\\-9\\-6
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\35\\30
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2\\0\\-16\\-60\\-14\\-12\\-8\\0
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\kern3pt%
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0\\6\\0\\-30\\-12\\-18\\-24\\-30
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-16\\-60
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\0\\-30
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-16\\0\\8\\0
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-60\\-30\\0\\30
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-14\\-12\\-8\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-18\\-24\\-30
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-24\\-56\\-120
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0\\-30\\-120\\-330
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13080, -2760, -240; -2760, -582, -51; -240, -51, -4],[120, 3, 1],[3, 2, -1, -8, -2, -1, 3, 13; -13, -9, 4, 35, 9, 5, -12, -55; -16, -6, 8, 30, 4, -6, -32, -90],176,10]
%
% END OF LATTICE L177.10
%
\hbox{}\par\smallskip%
%
% LATTICE L177.11
%
\leavevmode%
\llap{\spacer}%
${L_{177.11}}=\pdualwitharg{5}(\main({L_{177.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${30}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{120}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73560\\-5880\\1440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5880\\470\\-115
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1440\\-115\\28
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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113\\912\\-2280
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2\\16\\-41
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\-15\\-10
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\12\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\13\\2
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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30\\0\\-240\\-60\\-210\\-60\\-120\\0
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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30\\0\\-240\\-60
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-210\\-60\\-120\\0
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-60\\-30\\-120\\-10
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[73560, -5880, 1440; -5880, 470, -115; 1440, -115, 28],[120, 5, 1],[-2, -1, 1, 1, 5, 2, 5, 0; -39, -15, 12, 13, 63, 23, 36, -5; -60, -10, 0, 2, 0, -10, -120, -22],176,11]
%
% END OF LATTICE L177.11
%
\hbox{}\par\smallskip%
%
% LATTICE L177.12
%
\leavevmode%
\llap{\spacer}%
${L_{177.12}}=\pdualwitharg{3}({L_{177.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-741360, 3600, 2880; 3600, -12, -15; 2880, -15, -11],[240, 3, 1],[7, 2, 1, 3, -1, -1, 1, 6; 272, 77, 38, 110, -40, -38, 40, 235; 1456, 417, 209, 630, -208, -210, 206, 1245],176,12]
%
% END OF LATTICE L177.12
%
\hbox{}\par\smallskip%
%
% LATTICE L177.13
%
\leavevmode%
\llap{\spacer}%
${L_{177.13}}=\pdualwitharg{3}({L_{177.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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7\\-6\\624
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4\\-5\\360
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-83760, 2640, 960; 2640, -57, -30; 960, -30, -11],[240, 3, 1],[11, 7, 2, 4, -1, -1, 1, 17; -8, -6, -2, -5, 0, 1, 0, -10; 976, 624, 179, 360, -88, -90, 86, 1500],176,13]
%
% END OF LATTICE L177.13
%
\hbox{}\par\smallskip%
%
% LATTICE L177.14
%
\leavevmode%
\llap{\spacer}%
${L_{177.14}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{177.1}}))$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[16140, 1020, -4200; 1020, 96, -264; -4200, -264, 1093],[60, 12, 1],[39, 19, 5, 63, 21, 41, 25, 237; -7, -4, -1, -10, -3, -6, -4, -40; 148, 72, 19, 240, 80, 156, 95, 900],176,14]
%
% END OF LATTICE L177.14
%
\hbox{}\par\smallskip%
%
% LATTICE L177.15
%
\leavevmode%
\llap{\spacer}%
${L_{177.15}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{177.1}}))$%
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{$1\above{1pt}{1pt}{2}{0}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-40\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-15\\0\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35\\-30\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-45\\-60\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-60\\-140\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-15\\-60\\-33
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4020, 300, 180; 300, -15, 15; 180, 15, 8],[60, 15, 1],[-1, 2, 13, 8, 8, 5, 1, -1; 1, -1, -8, -5, -5, -3, 0, 1; 20, -45, -280, -171, -170, -105, -20, 21],176,15]
%
% END OF LATTICE L177.15
%
\hbox{}\par\smallskip%
%
% LATTICE L177.16
%
\leavevmode%
\llap{\spacer}%
${L_{177.16}}=\pdualwitharg{2}({L_{177.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40080\\1920\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1920\\92\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\21\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\20\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-24\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-200\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-87\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-64\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-21\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-20\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-48\\-240\\-84\\-48\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-48\\-48\\-24\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\0\\12\\0\\-24\\-48\\-42\\-240
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\kern3pt%
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-240\\-80\\0\\80\\0\\-80\\-120\\-880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-48\\-24\\0\\12\\0\\-24\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-48\\-48\\-80\\0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\-42\\-120\\-24\\0\\3\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-240\\-880\\-240\\-80\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[40080, 1920, 0; 1920, 92, 0; 0, 0, -1],[240, 4, 1],[-1, -1, 1, 9, 4, 3, 1, 1; 21, 20, -24, -200, -87, -64, -21, -20; 0, -8, -30, -120, -36, -16, -3, 0],176,16]
%
% END OF LATTICE L177.16
%
\hbox{}\par\smallskip%
%
% LATTICE L177.17
%
\leavevmode%
\llap{\spacer}%
${L_{177.17}}=\pdualwitharg{2}(\main({L_{177.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
480\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\32\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\60\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-17\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-9\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-96\\-480\\-336\\-96\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\16\\0\\-80\\-96\\-48\\-48\\-80
\end{tabular}\endgroup%
\kern3pt%
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\kern3pt%
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-480\\-80\\0\\80\\0\\-80\\-240\\-880
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\0\\-96\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\0\\12\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-80\\0\\80
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-96\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-48\\-48\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-48\\-84\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-240\\-880
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[480, -120, 0; -120, 32, 0; 0, 0, -1],[120, 8, 1],[1, 1, 1, 1, -1, -1, -1, -1; -9, 1, 3, 5, -3, -5, -9, -25; -72, -16, -6, 0, 0, -8, -30, -120],176,17]
%
% END OF LATTICE L177.17
%
\hbox{}\par\smallskip%
%
% LATTICE L177.18
%
\leavevmode%
\llap{\spacer}%
${L_{177.18}}=\pdualwitharg{5}({L_{177.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
138480\\8880\\-2880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8880\\545\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2880\\-180\\59
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-432\\-2160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\42\\225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\22\\116
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-70\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-312\\-1590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-49\\-248
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\0\\-240\\-240\\-1680\\-240\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0\\-10\\-120\\-30\\-60\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\0\\15\\0\\-120\\-60\\-210\\-30
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\kern3pt%
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-240\\-10\\0\\4\\0\\-20\\-120\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-120\\-120\\0\\240\\0\\-480\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-30\\-60\\-20\\0\\20\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-60\\-210\\-120\\-480\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-30\\-22\\-120\\-10\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[138480, 8880, -2880; 8880, 545, -180; -2880, -180, 59],[240, 5, 1],[-17, 0, 2, 1, -1, -3, -13, -2; -432, -3, 42, 22, -24, -70, -312, -49; -2160, -10, 225, 116, -120, -360, -1590, -248],176,18]
%
% END OF LATTICE L177.18
%
\hbox{}\par\smallskip%
%
% LATTICE L177.19
%
\leavevmode%
\llap{\spacer}%
${L_{177.19}}=\pdualwitharg{5}({L_{177.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{l}{2}{15}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
878640\\29280\\-7200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29280\\965\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7200\\-240\\59
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-24\\-720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-6\\340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-6\\585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\118
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\-1590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-2\\-376
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\0\\-240\\-120\\-1680\\-120\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-10\\-240\\-30\\-120\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\0\\15\\0\\-120\\-30\\-210\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-10\\0\\1\\0\\-5\\-60\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-240\\-120\\0\\240\\0\\-480\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-30\\-30\\-5\\0\\5\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-120\\-210\\-60\\-480\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-60\\-22\\-240\\-10\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[878640, 29280, -7200; 29280, 965, -240; -7200, -240, 59],[240, 5, 1],[-5, 3, 5, 1, -1, -2, -13, -3; -24, -6, -6, -1, 0, 1, 0, -2; -720, 340, 585, 118, -120, -240, -1590, -376],176,19]
%
% END OF LATTICE L177.19
%
\hbox{}\par\smallskip%
%
% LATTICE L177.20
%
\leavevmode%
\llap{\spacer}%
${L_{177.20}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{177.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3660\\540\\540
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
540\\80\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
540\\80\\79
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\90\\300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\11\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\15\\49
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\24\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\13\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\12\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-120\\-120\\-420\\-120\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20\\-120\\-60\\-60\\-20
\end{tabular}\endgroup%
\kern3pt%
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-120\\-20\\0\\4\\0\\-20\\-60\\-44
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-120\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\15\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-20\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-120\\-60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-60\\-60\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\-105\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-60\\-44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3660, 540, 540; 540, 80, 80; 540, 80, 79],[60, 20, 1],[-13, -5, -4, -1, 1, 1, -2, -3; 24, 13, 12, 3, -6, -7, -3, 3; 60, 20, 15, 4, 0, 0, 15, 16],176,20]
%
% END OF LATTICE L177.20
%
\hbox{}\par\smallskip%
%
% LATTICE L177.21
%
\leavevmode%
\llap{\spacer}%
${L_{177.21}}=\pdualwitharg{3.5}(\main({L_{177.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{b}{2}{30}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2760\\-1200\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1200\\480\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\-45\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-48\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\7\\60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-13\\-85
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-7\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\27
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-80\\-60\\-70\\-60\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-30\\-60\\-90\\-120\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\40\\0\\-40\\-120\\-280\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-30\\0\\6\\0\\-30\\-120\\-66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-80\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\30\\0\\-30
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-30\\0\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70\\-60\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-90\\-120\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-120\\-280\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-120\\-66
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2760, -1200, 120; -1200, 480, -45; 120, -45, 4],[120, 15, 1],[-2, -1, 1, 1, 1, 0, -3, -2; -13, -7, 4, 5, 5, -1, -20, -13; -85, -45, 20, 27, 25, -15, -140, -87],176,21]
%
% END OF LATTICE L177.21
%
\hbox{}\par\smallskip%
%
% LATTICE L177.22
%
\leavevmode%
\llap{\spacer}%
${L_{177.22}}=\pdualwitharg{3}({L_{177.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-83760\\-3120\\960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3120\\-84\\36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
960\\36\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\83
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-3\\336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-4\\592
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-5\\1020
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\86
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-84
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-5\\240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-16\\-60\\-14\\-12\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-60\\-24\\-36\\-48\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\16\\0\\-16\\-48\\-112\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-60\\0\\60\\0\\-60\\-240\\-660
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-24\\-16\\0\\4\\0\\-32\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-36\\-48\\-60\\0\\12\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-48\\-112\\-240\\-32\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-240\\-660\\-120\\-60\\0\\60
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-83760, -3120, 960; -3120, -84, 36; 960, 36, -11],[240, 12, 1],[1, 4, 7, 12, 1, -1, -1, 3; -1, -3, -4, -5, 0, 1, 0, -5; 83, 336, 592, 1020, 86, -84, -88, 240],176,22]
%
% END OF LATTICE L177.22
%
\hbox{}\par\smallskip%
%
% LATTICE L177.23
%
\leavevmode%
\llap{\spacer}%
${L_{177.23}}=\pdualwitharg{3}({L_{177.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{12}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28560\\1200\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1200\\-12\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-12\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\10\\90
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\11\\88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\11\\87
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\32\\256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\20\\165
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-32\\-120\\-28\\-12\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\12\\0\\-60\\-24\\-18\\-48\\-30
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-32\\0\\16\\0\\-16\\-24\\-112\\-120
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\kern3pt%
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-120\\-60\\0\\60\\0\\-30\\-240\\-330
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-24\\-16\\0\\4\\0\\-32\\-60
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-18\\-24\\-30\\0\\3\\0\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-48\\-112\\-240\\-32\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-120\\-330\\-60\\-15\\0\\15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[28560, 1200, -480; 1200, -12, -12; -480, -12, 7],[240, 12, 1],[0, -1, -1, 1, 1, 1, 3, 2; -1, -12, -12, 10, 11, 11, 32, 20; -4, -90, -88, 90, 88, 87, 256, 165],176,23]
%
% END OF LATTICE L177.23
%
\hbox{}\par\smallskip%
%
% LATTICE L177.24
%
\leavevmode%
\llap{\spacer}%
${L_{177.24}}=\pdualwitharg{2.3}({L_{177.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{240}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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240\\0\\0
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\kern3pt%
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0\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\60\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\8\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\0\\-8\\-120\\-14\\-24\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-240\\-48\\-144\\-48\\-240
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\kern3pt%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-240\\-240\\-2640\\-240\\-240\\0\\240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240, 0, 0; 0, -12, 0; 0, 0, 1],[240, 12, 1],[0, -1, -1, -11, -1, -1, 0, 1; 0, 4, 5, 60, 6, 8, 1, 0; -1, 0, 8, 120, 14, 24, 4, 0],176,24]
%
% END OF LATTICE L177.24
%
\hbox{}\par\smallskip%
%
% LATTICE L177.25
%
\leavevmode%
\llap{\spacer}%
${L_{177.25}}=\pdualwitharg{2.3}(\main({L_{177.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{s}{2}{240}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\360\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\384\\120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
120\\120\\37
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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19\\-60\\120
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\kern3pt%
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12\\-37\\72
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-9\\17
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-20\\48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-18\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-60\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-32\\-240
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-240
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-32\\0\\16\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-240\\0\\240
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-48\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-144\\-96\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-96\\-112\\-480
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-240\\-480\\-2640
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240, 360, 120; 360, 384, 120; 120, 120, 37],[120, 24, 1],[1, 5, 5, 19, 1, -1, -1, 1; -4, -20, -18, -60, -2, 8, 6, 0; 10, 48, 40, 120, 2, -24, -16, 0],176,25]
%
% END OF LATTICE L177.25
%
\hbox{}\par\smallskip%
%
% LATTICE L177.26
%
\leavevmode%
\llap{\spacer}%
${L_{177.26}}=\pdualwitharg{3.5}({L_{177.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{60}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2640\\960\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
960\\195\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-60\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-24\\-130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-22\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\6\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-9\\-48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-160\\-120\\-70\\-60\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-60\\-60\\-90\\-240\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\0\\80\\0\\-40\\-120\\-560\\-120
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\kern3pt%
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-120\\-60\\0\\12\\0\\-30\\-240\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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-60\\-90\\-120\\-30\\0\\15\\0\\-15
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2640, 960, -240; 960, 195, -60; -240, -60, 17],[240, 15, 1],[-3, -3, -1, 1, 1, 1, -1, -1; -24, -22, -8, 6, 6, 5, -16, -9; -130, -120, -40, 36, 35, 30, -80, -48],176,26]
%
% END OF LATTICE L177.26
%
\hbox{}\par\smallskip%
%
% LATTICE L177.27
%
\leavevmode%
\llap{\spacer}%
${L_{177.27}}=\pdualwitharg{3.5}({L_{177.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30480, 6960, -720; 6960, 1560, -165; -720, -165, 17],[240, 15, 1],[-3, -2, -1, 1, 2, 5, 3, -1; 0, 1, 0, -1, -2, -6, -8, -2; -130, -75, -40, 33, 65, 150, 40, -66],176,27]
%
% END OF LATTICE L177.27
%
\hbox{}\par\smallskip%
%
% LATTICE L177.28
%
\leavevmode%
\llap{\spacer}%
${L_{177.28}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{177.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\kern3pt%
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-7500\\213420\\-56340
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
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-1\\-37\\-140
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-140\\-120\\-20\\0
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-20\\-60\\-35\\-60
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0\\-60\\-60\\-132
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240, -7500, 1980; -7500, 213420, -56340; 1980, -56340, 14873],[60, 60, 1],[-1, -4, -3, -6, -5, -2, 0, 0; -37, -111, -41, -35, 21, 95, 33, 19; -140, -420, -155, -132, 80, 360, 125, 72],176,28]
%
% END OF LATTICE L177.28
%
\hbox{}\par\smallskip%
%
% LATTICE L177.29
%
\leavevmode%
\llap{\spacer}%
${L_{177.29}}=\pdualwitharg{2}({L_{177.1}})$%
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{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${48}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{b}{2}{80}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1890480\\331680\\-4560
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
331680\\58192\\-800
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-4560\\-800\\11
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-6\\25\\-680
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-5\\24\\-336
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1890480, 331680, -4560; 331680, 58192, -800; -4560, -800, 11],[240, 16, 1],[-1, 0, -1, -6, -5, -2, -1, -4; 6, -1, 3, 25, 24, 11, 6, 25; 24, -72, -198, -680, -336, -32, 21, 160],176,29]
%
% END OF LATTICE L177.29
%
\hbox{}\par\smallskip%
%
% LATTICE L177.30
%
\leavevmode%
\llap{\spacer}%
${L_{177.30}}=\pdualwitharg{2}({L_{177.2}})$%
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{$1\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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0\\0\\-1
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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1\\-5\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-96\\-48\\-48\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\0\\12\\0\\-48\\-48\\-84\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-80\\0\\80\\0\\-80\\-240\\-880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-336\\-96\\-48\\0\\48\\0\\-96\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-48\\-48\\-80\\0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-48\\-84\\-240\\-96\\0\\12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-240\\-880\\-480\\-80\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4080, 720, 0; 720, 128, 0; 0, 0, -1],[240, 16, 1],[-1, -1, -1, -1, 1, 1, 1, 1; 6, 5, 3, -5, -12, -7, -6, -5; 0, -8, -30, -120, -72, -16, -6, 0],176,30]
%
% END OF LATTICE L177.30
%
\hbox{}\par\smallskip%
%
% LATTICE L177.31
%
\leavevmode%
\llap{\spacer}%
${L_{177.31}}=\pdualwitharg{5}({L_{177.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
878640\\43920\\-7200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43920\\2180\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7200\\-360\\59
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-12\\-2160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\116
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\1\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\-1590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-1\\-496
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\0\\-240\\-240\\-1680\\-240\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20\\-240\\-60\\-120\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\0\\15\\0\\-120\\-60\\-210\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-20\\0\\4\\0\\-20\\-120\\-44
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-240\\-120\\0\\240\\0\\-480\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-60\\-60\\-20\\0\\20\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-120\\-210\\-120\\-480\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-60\\-44\\-240\\-20\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[878640, 43920, -7200; 43920, 2180, -360; -7200, -360, 59],[240, 20, 1],[-17, 0, 2, 1, -1, -3, -13, -4; -12, -3, -3, -1, 0, 1, 0, -1; -2160, -20, 225, 116, -120, -360, -1590, -496],176,31]
%
% END OF LATTICE L177.31
%
\hbox{}\par\smallskip%
%
% LATTICE L177.32
%
\leavevmode%
\llap{\spacer}%
${L_{177.32}}=\pdualwitharg{2.5}({L_{177.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21840\\7200\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7200\\2380\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-160\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\105\\300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\56\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\15\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\12\\40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\72\\210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\64\\184
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-120\\-240\\-420\\-240\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-80\\-240\\-240\\-240\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\0\\15\\0\\-60\\-120\\-210\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-80\\0\\16\\0\\-80\\-240\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-240\\-60\\0\\60\\0\\-240\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-240\\-120\\-80\\0\\80\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-240\\-210\\-240\\-240\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-120\\-176\\-240\\-80\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[21840, 7200, -480; 7200, 2380, -160; -480, -160, 11],[240, 20, 1],[-28, -15, -4, -1, 1, -3, -19, -17; 105, 56, 15, 4, -3, 12, 72, 64; 300, 160, 45, 16, 0, 40, 210, 184],176,32]
%
% END OF LATTICE L177.32
%
\hbox{}\par\smallskip%
%
% LATTICE L177.33
%
\leavevmode%
\llap{\spacer}%
${L_{177.33}}=\pdualwitharg{2.5}(\main({L_{177.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-120\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\640\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-84\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\7\\49
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\12\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\22\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\84\\600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\26\\184
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-480\\-240\\-420\\-240\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-80\\-240\\-240\\-480\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\0\\240\\0\\-240\\-480\\-1680\\-480
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-80\\0\\16\\0\\-80\\-480\\-176
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-480\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\0\\240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-80\\0\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-240\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-240\\-480\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-480\\-1680\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-480\\-176
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240, -120, 0; -120, 640, -80; 0, -80, 11],[120, 40, 1],[7, 13, 49, 15, 17, 3, -1, 1; 12, 22, 84, 26, 30, 6, 0, 2; 90, 160, 600, 184, 210, 40, 0, 16],176,33]
%
% END OF LATTICE L177.33
%
\hbox{}\par\smallskip%
%
% LATTICE L177.34
%
\leavevmode%
\llap{\spacer}%
${L_{177.34}}=\pdualwitharg{5}({L_{177.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${60}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{s}{2}{240}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{240}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
677040\\47040\\-5760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47040\\3260\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5760\\-400\\49
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-15\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\134
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\9\\660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\125
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-12\\480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60\\0\\-480\\-120\\-420\\-60\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20\\-120\\-30\\-240\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\0\\240\\0\\-240\\-120\\-1680\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-20\\0\\4\\0\\-10\\-240\\-22
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-420\\-120\\-240\\0\\60\\0\\-480\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-30\\-120\\-10\\0\\5\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-240\\-1680\\-240\\-480\\0\\240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-120\\-22\\-60\\-5\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[677040, 47040, -5760; 47040, 3260, -400; -5760, -400, 49],[240, 20, 1],[-2, -1, 1, 1, 5, 1, 5, 0; -15, -4, 0, 2, 9, 1, -12, -2; -360, -150, 120, 134, 660, 125, 480, -17],176,34]
%
% END OF LATTICE L177.34
%
\hbox{}\par\smallskip%
%
% LATTICE L177.35
%
\leavevmode%
\llap{\spacer}%
${L_{177.35}}=\pdualwitharg{2.3}({L_{177.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{240}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56640\\240\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-48\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-464
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-25\\-2520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-226
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-16\\-120\\-14\\-24\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-240\\-48\\-144\\-96\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\0\\16\\0\\-16\\-96\\-112\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-240\\0\\240\\0\\-240\\-480\\-2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-48\\-16\\0\\4\\0\\-32\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-144\\-96\\-240\\0\\48\\0\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-96\\-112\\-480\\-32\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-240\\-480\\-2640\\-240\\-240\\0\\240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[56640, 240, -240; 240, -48, 0; -240, 0, 1],[240, 48, 1],[0, -1, -2, -11, -1, -1, 0, 1; 0, -3, -5, -25, -2, -1, 1, 5; -1, -240, -464, -2520, -226, -216, 8, 240],176,35]
%
% END OF LATTICE L177.35
%
\hbox{}\par\smallskip%
%
% LATTICE L177.36
%
\leavevmode%
\llap{\spacer}%
${L_{177.36}}=\pdualwitharg{2.3}({L_{177.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{b}{2}{240}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{240}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2832\\-192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-192\\13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\17\\248
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\90\\1320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\8\\118
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\8\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-32\\-240\\-28\\-48\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\48\\0\\-240\\-48\\-144\\-96\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\0\\16\\0\\-16\\-96\\-112\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-240\\0\\240\\0\\-240\\-480\\-2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-48\\-16\\0\\4\\0\\-32\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-144\\-96\\-240\\0\\48\\0\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-96\\-112\\-480\\-32\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-240\\-480\\-2640\\-240\\-240\\0\\240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240, 0, 0; 0, 2832, -192; 0, -192, 13],[240, 48, 1],[0, -1, -2, -11, -1, -1, 0, 1; 1, 10, 17, 90, 8, 8, 1, 0; 14, 144, 248, 1320, 118, 120, 16, 0],176,36]
%
% END OF LATTICE L177.36
%
\hbox{}\par\smallskip%
%
% LATTICE L177.37
%
\leavevmode%
\llap{\spacer}%
${L_{177.37}}=\pdualwitharg{3.5}({L_{177.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{l}{2}{60}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{12}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{}{2}{60}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{12}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30480\\5040\\-720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5040\\780\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-120\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\1\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-3\\60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-160\\-240\\-560\\-240\\-40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\60\\0\\-60\\-240\\-180\\-60\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\0\\20\\0\\-80\\-120\\-70\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-60\\0\\12\\0\\-60\\-60\\-132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\-240\\-80\\0\\80\\0\\-80\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-180\\-120\\-60\\0\\60\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-60\\-70\\-60\\-80\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-60\\-120\\-132\\-240\\-60\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30480, 5040, -720; 5040, 780, -120; -720, -120, 17],[240, 60, 1],[-1, -3, -3, -2, -1, 2, 1, 1; 0, 1, 0, -1, -4, -3, -1, -1; -40, -120, -130, -96, -80, 60, 35, 36],176,37]
%
% END OF LATTICE L177.37
%
\hbox{}\par\smallskip%
%
% LATTICE L177.38
%
\leavevmode%
\llap{\spacer}%
${L_{177.38}}=\pdualwitharg{2.3.5}({L_{177.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{}{2}{240}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6960\\-9600\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9600\\22020\\1080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\1080\\53
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-40\\840
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-74\\1550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-196\\4104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-107\\2240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-172\\3600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-16\\335
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-16\\336
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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20\\0\\-80\\-240\\-140\\-240\\-20\\0
\end{tabular}\endgroup%
\kern3pt%
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0\\240\\0\\-240\\-240\\-720\\-120\\-240
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-80\\0\\20\\0\\-40\\-240\\-70\\-240
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-240\\-240\\0\\48\\0\\-240\\-120\\-528
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-240\\-40\\0\\20\\0\\-40\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-720\\-240\\-240\\0\\240\\0\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-120\\-70\\-120\\-40\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-240\\-240\\-528\\-240\\-240\\0\\48
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6960, -9600, -480; -9600, 22020, 1080; -480, 1080, 53],[240, 60, 1],[0, 3, 5, 13, 7, 11, 1, 1; 1, -40, -74, -196, -107, -172, -16, -16; -20, 840, 1550, 4104, 2240, 3600, 335, 336],176,38]
%
% END OF LATTICE L177.38
%
\hbox{}\par\smallskip%
%
% LATTICE L177.39
%
\leavevmode%
\llap{\spacer}%
${L_{177.39}}=\pdualwitharg{2.3.5}(\main({L_{177.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{*}{2}{240}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{48}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30480\\76200\\-720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76200\\190560\\-1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-1800\\17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\44\\6600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108\\107\\16200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-2\\-432
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-160\\-480\\-560\\-480\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\240\\0\\-240\\-480\\-720\\-240\\-240
\end{tabular}\endgroup%
\kern3pt%
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-160\\0\\20\\0\\-80\\-240\\-140\\-240
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-240\\0\\48\\0\\-240\\-240\\-528
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-160\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\240\\0\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-240\\0\\48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-560\\-480\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-720\\-240\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-240\\-140\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-240\\-240\\-528
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[30480, 76200, -720; 76200, 190560, -1800; -720, -1800, 17],[120, 120, 1],[-1, -5, -3, -5, -3, 1, 1, 1; 0, 2, 0, -2, -4, -6, -2, -2; -40, 0, -130, -432, -560, -600, -170, -168],176,39]
%
% END OF LATTICE L177.39
%
\hbox{}\par\smallskip%
%
% LATTICE L177.40
%
\leavevmode%
\llap{\spacer}%
${L_{177.40}}=\pdualwitharg{3.5}({L_{177.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{s}{2}{12}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-535920\\-121920\\2640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121920\\-27660\\600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2640\\600\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-297
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-390
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\2\\294
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-160\\-60\\-140\\-120\\-80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\15\\0\\-15\\-60\\-90\\-120\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\0\\80\\0\\-80\\-240\\-560\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-15\\0\\3\\0\\-30\\-120\\-66
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-140\\-60\\-80\\0\\20\\0\\-160\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-90\\-240\\-30\\0\\60\\0\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-120\\-560\\-120\\-160\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-30\\-240\\-66\\-120\\-60\\0\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-535920, -121920, 2640; -121920, -27660, 600; 2640, 600, -13],[240, 60, 1],[1, 0, -3, -1, -2, -1, 1, 1; 3, 1, -4, -2, -5, -4, 0, 2; 340, 45, -800, -297, -640, -390, 200, 294],176,40]
%
% END OF LATTICE L177.40
%
\hbox{}\par\smallskip%
%
% LATTICE L177.41
%
\leavevmode%
\llap{\spacer}%
${L_{177.41}}=\pdualwitharg{2.5}({L_{177.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{}{2}{15}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{b}{2}{80}\above{1pt}{1pt}{b}{2}{60}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26160\\3600\\-1680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3600\\320\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-160\\79
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-102\\-240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\31\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\42\\105
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\45\\112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\48\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-49\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-111\\-270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-63\\-152
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\0\\-240\\-480\\-1680\\-480\\-240\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-80\\-480\\-240\\-240\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\0\\15\\0\\-120\\-120\\-210\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-80\\0\\16\\0\\-80\\-240\\-176
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1680\\-480\\-120\\0\\240\\0\\-480\\-480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-240\\-120\\-80\\0\\80\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-240\\-210\\-240\\-480\\0\\60\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-120\\-176\\-480\\-80\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26160, 3600, -1680; 3600, 320, -160; -1680, -160, 79],[240, 80, 1],[-1, 1, 1, 1, 1, -1, -2, -1; -102, 31, 42, 45, 48, -49, -111, -63; -240, 80, 105, 112, 120, -120, -270, -152],176,41]
%
% END OF LATTICE L177.41
%
\hbox{}\par\smallskip%
%
% LATTICE L177.42
%
\leavevmode%
\llap{\spacer}%
${L_{177.42}}=\pdualwitharg{2.5}({L_{177.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${240}\above{1pt}{1pt}{l}{2}{80}\above{1pt}{1pt}{r}{2}{60}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{240}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{s}{2}{60}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2320\\-160
\end{tabular}\endgroup%
\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[240, 0, 0; 0, 2320, -160; 0, -160, 11],[240, 80, 1],[-7, -2, -1, 0, 1, 0, -2, -2; 42, 11, 6, 1, 0, 3, 15, 13; 600, 160, 90, 16, 0, 40, 210, 184],176,42]
%
% END OF LATTICE L177.42
%
\hbox{}\par\smallskip%
%
% LATTICE L177.43
%
\leavevmode%
\llap{\spacer}%
${L_{177.43}}=\pdualwitharg{2.3.5}({L_{177.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{b}{2}{240}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{b}{2}{48}\above{1pt}{1pt}{l}{2}{80}\above{1pt}{1pt}{}{2}{240}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{48}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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480\\2640\\1037
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\47\\-120
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-5\\161\\-408
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-240, 1200, 480; 1200, 6720, 2640; 480, 2640, 1037],[240, 240, 1],[1, -1, -2, -5, -5, -3, 0, 1; -16, 47, 67, 161, 158, 95, 2, -19; 40, -120, -170, -408, -400, -240, -5, 48],176,43]
%
% END OF LATTICE L177.43
%
\hbox{}\par\smallskip%
%
% LATTICE L177.44
%
\leavevmode%
\llap{\spacer}%
${L_{177.44}}=\pdualwitharg{2.3.5}({L_{177.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{r}{2}{240}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}{48}\above{1pt}{1pt}{b}{2}{80}\above{1pt}{1pt}{l}{2}{240}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{48}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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14160\\92160\\-960
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\kern3pt%
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92160\\1032240\\-10800
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\-16\\-1520
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8\\-125\\-11880
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8\\-123\\-11690
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\HardButStrongLineBreak\kern3pt%
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18\\-275\\-26136
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17\\-258\\-24520
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10\\-149\\-14160
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1\\-14\\-1330
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0\\1\\96
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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0\\-240\\-240\\-528\\-480\\-240\\0\\48
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14160, 92160, -960; 92160, 1032240, -10800; -960, -10800, 113],[240, 240, 1],[1, 8, 8, 18, 17, 10, 1, 0; -16, -125, -123, -275, -258, -149, -14, 1; -1520, -11880, -11690, -26136, -24520, -14160, -1330, 96],176,44]
%
% END OF LATTICE L177.44
%

\medskip%
%------------------------------WEYL-GROUP-W178--------------------------------
%
\leavevmode\llap{}%
$W_{178}$%
\qquad\llap{12} lattices, $\chi=18$%
\hfill%
$422422\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L178.1
%
\leavevmode%
\llap{\spacer}%
${L_{178.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-506772\\2268\\4032
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2268\\-10\\-19
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4032\\-19\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5039\\876960\\138600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-3829\\-605
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44\\-7656\\-1211
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\173\\28
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\3654\\578
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\3483\\549
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\right.$}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\18\\0\\-90\\-99
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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2\\-2\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-104\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-506772, 2268, 4032; 2268, -10, -19; 4032, -19, -26],[252, 1, 1],[1, 21, 20, 1, -1, -2; 173, 3654, 3483, 175, -174, -351; 28, 578, 549, 27, -28, -54],177,1]
%
% END OF LATTICE L178.1
%
\hbox{}\par\smallskip%
%
% LATTICE L178.2
%
\leavevmode%
\llap{\spacer}%
${L_{178.2}}=\pfillwitharg{2}({L_{178.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{3}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{}{4}{1}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1888173, 68292, -17388; 68292, -2470, 629; -17388, 629, -158],[63, 1, 1],[-16, 7, 28, -18, -177, -334; -437, 191, 765, -491, -4831, -9117; 21, -10, -36, 26, 245, 459],177,2]
%
% END OF LATTICE L178.2
%
\hbox{}\par\smallskip%
%
% LATTICE L178.3
%
\leavevmode%
\llap{\spacer}%
${L_{178.3}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{178.1}}))$%
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{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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-7245\\2835\\-567
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7245, 2835, -567; 2835, -1106, 224; -567, 224, -43],[63, 7, 1],[13, 125, 235, 11, -5, -19; 26, 245, 459, 21, -10, -36; -35, -364, -693, -35, 14, 63],177,3]
%
% END OF LATTICE L178.3
%
\hbox{}\par\smallskip%
%
% LATTICE L178.4
%
\leavevmode%
\llap{\spacer}%
${L_{178.4}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{178.1}}))$%
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7009939\\235116\\16654050
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[12618081, -133749, -5309262; -133749, 1422, 56277; -5309262, 56277, 2233958],[63, 9, 1],[572, 125, 16, 1038, 7925, 1594; 17, 3, 1, 37, 267, 53; 1359, 297, 38, 2466, 18828, 3787],177,4]
%
% END OF LATTICE L178.4
%
\hbox{}\par\smallskip%
%
% LATTICE L178.5
%
\leavevmode%
\llap{\spacer}%
${L_{178.5}}=\pdualwitharg{2}({L_{178.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3499272, -413532, -891072; -413532, 48904, 105304; -891072, 105304, 226907],[252, 4, 1],[-367, -2995, -5507, -251, -95, -55; 6, 53, 99, 5, 2, 0; -1444, -11786, -21672, -988, -374, -216],177,5]
%
% END OF LATTICE L178.5
%
\hbox{}\par\smallskip%
%
% LATTICE L178.6
%
\leavevmode%
\llap{\spacer}%
${L_{178.6}}=\pdualwitharg{7}({L_{178.1}})$%
\nopagebreak\par%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-108770508, -6986196, -506016; -6986196, -448714, -32501; -506016, -32501, -2354],[252, 7, 1],[-21, -431, -409, -20, 21, 40; 257, 5298, 5031, 247, -258, -495; 966, 19502, 18459, 889, -952, -1764],177,6]
%
% END OF LATTICE L178.6
%
\hbox{}\par\smallskip%
%
% LATTICE L178.7
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2772, 504, 252; 504, -90, -45; 252, -45, -22],[252, 9, 1],[0, -1, 0, 1, 11, 1; 4, -6, -1, 5, 96, 10; -9, 0, 2, 0, -90, -11],177,7]
%
% END OF LATTICE L178.7
%
\hbox{}\par\smallskip%
%
% LATTICE L178.8
%
\leavevmode%
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${L_{178.8}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{178.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[9954, -61362, 27027; -61362, 379197, -167013; 27027, -167013, 73559],[63, 63, 1],[37, 267, 53, 17, 3, 1; -438, -3370, -679, -247, -55, -6; -1008, -7749, -1561, -567, -126, -14],177,8]
%
% END OF LATTICE L178.8
%
\hbox{}\par\smallskip%
%
% LATTICE L178.9
%
\leavevmode%
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${L_{178.9}}=\pdualwitharg{2.7}({L_{178.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[905776200, -25407900, -229743864; -25407900, 712936, 6444536; -229743864, 6444536, 58272941],[252, 28, 1],[-21505, -142427, -249209, -7365, -1923, -10621; -764, -5057, -8847, -261, -68, -378; -84700, -560966, -981540, -29008, -7574, -41832],177,9]
%
% END OF LATTICE L178.9
%
\hbox{}\par\smallskip%
%
% LATTICE L178.10
%
\leavevmode%
\llap{\spacer}%
${L_{178.10}}=\pdualwitharg{2.3}({L_{178.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}7\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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36792\\3276\\-9324
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3276\\360\\-828
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[36792, 3276, -9324; 3276, 360, -828; -9324, -828, 2363],[252, 36, 1],[-46, -23, 1, -37, -392, -84; 5, 2, 0, 6, 53, 11; -180, -90, 4, -144, -1530, -328],177,10]
%
% END OF LATTICE L178.10
%
\hbox{}\par\smallskip%
%
% LATTICE L178.11
%
\leavevmode%
\llap{\spacer}%
${L_{178.11}}=\pdualwitharg{3.7}({L_{178.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-49644, 55944, -1512; 55944, -62874, 1701; -1512, 1701, -46],[252, 63, 1],[-2, -71, -8, -5, 1, 1; 5, 96, 10, 4, -6, -1; 252, 5922, 637, 315, -252, -70],177,11]
%
% END OF LATTICE L178.11
%
\hbox{}\par\smallskip%
%
% LATTICE L178.12
%
\leavevmode%
\llap{\spacer}%
${L_{178.12}}=\pdualwitharg{2.3.7}({L_{178.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}7\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${504}\above{1pt}{1pt}{*}{4}{252}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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2520\\-148932\\37800
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-148932\\9195480\\-2333772
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37800\\-2333772\\592301
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2520, -148932, 37800; -148932, 9195480, -2333772; 37800, -2333772, 592301],[252, 252, 1],[38, 519, 115, 73, 36, -4; -575, -8338, -1860, -1214, -607, 71; -2268, -32886, -7336, -4788, -2394, 280],177,12]
%
% END OF LATTICE L178.12
%

\medskip%
%------------------------------WEYL-GROUP-W179--------------------------------
%
\leavevmode\llap{}%
$W_{179}$%
\qquad\llap{44} lattices, $\chi=72$%
\hfill%
$222\infty222222\infty222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L179.1
%
\leavevmode%
\llap{\spacer}%
${L_{179.1}}$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-642852, 2772, 1512; 2772, -8, -9; 1512, -9, -2],[252, 1, 1],[1, 1, -1, -1, 15, 34, 54, 32, 2078, 309, 309, 139, 65, 23; 124, 126, -124, -126, 1848, 4194, 6664, 3950, 256536, 38148, 38150, 17164, 8028, 2842; 197, 189, -198, -196, 2968, 6723, 10675, 6325, 410697, 61070, 61068, 27468, 12843, 4543],178,1]
%
% END OF LATTICE L179.1
%
\hbox{}\par\smallskip%
%
% LATTICE L179.2
%
\leavevmode%
\llap{\spacer}%
${L_{179.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{252}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{b}{2}{14}\above{1pt}{1pt}{12,5}{\infty a}{56}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{28}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-5026392\\-29232\\-504
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29232\\-170\\-3
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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266111\\-45671472\\-2794176
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-24\\4119\\251
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{$\left.\llap{\phantom{%
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0\\0\\0
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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449\\-77060\\-4720
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222\\-38101\\-2338
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193\\-33124\\-2044
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43\\-7380\\-459
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{$\left.\llap{\phantom{%
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\GramMatrix{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-5026392, -29232, -504; -29232, -170, -3; -504, -3, 1],[504, 1, 1],[24, 3035, 449, 222, 193, 43, 27, 0, -11, -1, 2, 31, 29, 85; -4119, -520884, -77060, -38101, -33124, -7380, -4634, 0, 1890, 172, -343, -5320, -4977, -14588; -251, -31878, -4720, -2338, -2044, -459, -294, -1, 126, 16, -14, -308, -297, -882],178,2]
%
% END OF LATTICE L179.2
%
\hbox{}\par\smallskip%
%
% LATTICE L179.3
%
\leavevmode%
\llap{\spacer}%
${L_{179.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{63}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{14}\above{1pt}{1pt}{12,5}{\infty b}{56}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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289800\\4536\\0
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\kern3pt%
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4536\\71\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
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{$\left.\llap{\phantom{%
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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225\\-14568\\-1736
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113\\-7315\\-861
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[289800, 4536, 0; 4536, 71, 0; 0, 0, -1],[504, 1, 1],[23, 755, 225, 113, 103, 49, 9, 1, 1, -1, -1, 9, 23, 19; -1490, -48888, -14568, -7315, -6664, -3168, -581, -64, -63, 64, 63, -588, -1494, -1232; -184, -5859, -1736, -861, -756, -342, -56, -2, 0, 0, -7, -112, -216, -161],178,3]
%
% END OF LATTICE L179.3
%
\hbox{}\par\smallskip%
%
% LATTICE L179.4
%
\leavevmode%
\llap{\spacer}%
${L_{179.4}}=\pfillwitharg{2}({L_{179.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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126\\0\\63
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0\\-1\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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1\\1\\-2
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[126, 0, 63; 0, -1, -1; 63, -1, 31],[63, 1, 1],[-1, 1, 1, 3, 4, 7, 5, 1, -1, -1, -3, -4, -7, -5; -2, 0, 1, 0, -21, -90, -140, -82, -5292, -393, -392, -175, -162, -56; 2, 0, -2, -7, -14, -36, -42, -20, -1134, -82, -77, -28, -18, 0],178,4]
%
% END OF LATTICE L179.4
%
\hbox{}\par\smallskip%
%
% LATTICE L179.5
%
\leavevmode%
\llap{\spacer}%
${L_{179.5}}=\pfillwitharg{2}({L_{179.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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0\\0\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-148303\\-9154278\\417942
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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1\\62\\-1
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\HardButStrongLineBreak\kern3pt%
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1\\63\\0
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-1\\-62\\0
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15\\924\\-56
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54\\3332\\-161
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[250110, -4032, 0; -4032, 65, 0; 0, 0, -1],[126, 1, 1],[1, 1, -1, -1, 15, 34, 54, 32, 2078, 309, 309, 139, 65, 23; 62, 63, -62, -63, 924, 2097, 3332, 1975, 128268, 19074, 19075, 8582, 4014, 1421; -1, 0, 0, -7, -56, -108, -161, -92, -5859, -868, -861, -378, -171, -56],178,5]
%
% END OF LATTICE L179.5
%
\hbox{}\par\smallskip%
%
% LATTICE L179.6
%
\leavevmode%
\llap{\spacer}%
${L_{179.6}}=\main({L_{179.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-15372\\-7560\\252
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-15372, -7560, 252; -7560, -3718, 125; 252, 125, 31],[252, 1, 1],[32, -31, -63, -93, -227, -172, -85, 10, 2677, 455, 289, 423, 298, 183; -65, 63, 128, 189, 462, 351, 175, -19, -5355, -912, -581, -854, -603, -371; 2, 0, -4, -7, -28, -36, -42, -20, -1134, -164, -77, -56, -18, 0],178,6]
%
% END OF LATTICE L179.6
%
\hbox{}\par\smallskip%
%
% LATTICE L179.7
%
\leavevmode%
\llap{\spacer}%
${L_{179.7}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{179.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8064\\3970\\4032
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4284\\4032\\2173
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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0\\0\\0
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11023910\\358918\\-22401777
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{$\left.\llap{\phantom{%
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0\\0\\0
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\HardButStrongLineBreak\kern3pt%
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1922\\63\\-3906
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\HardButStrongLineBreak\kern3pt%
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31\\1\\-63
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\HardButStrongLineBreak\kern3pt%
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-1929\\-63\\3920
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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-24040\\-783\\48852
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35253\\-1148\\71638
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8442, 8064, 4284; 8064, 3970, 4032; 4284, 4032, 2173],[126, 2, 1],[-63, 1922, 31, -1929, -6445, -24040, -35253, -19932, -1261607, -93341, -184691, -40210, -35945, -11402; -2, 63, 1, -63, -210, -783, -1148, -649, -41076, -3039, -6013, -1309, -1170, -371; 128, -3906, -63, 3920, 13097, 48852, 71638, 40504, 2563722, 189679, 375312, 81711, 73044, 23170],178,7]
%
% END OF LATTICE L179.7
%
\hbox{}\par\smallskip%
%
% LATTICE L179.8
%
\leavevmode%
\llap{\spacer}%
${L_{179.8}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{179.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{l}{2}{7}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{6,5}{\infty}{1}\above{1pt}{1pt}{r}{2}{126}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\63\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\-7\\-7
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-811\\-1260\\-7560
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306\\475\\2856
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\56\\335
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-2\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\7
\end{tabular}\endgroup%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-63, 63, 0; 63, -7, -7; 0, -7, 1],[63, 7, 1],[-1, 1, 1, 0, -2, -59, -13, -53, -487, -253, -36, -16, -103, -5; -2, 0, 1, 0, -3, -90, -20, -82, -756, -393, -56, -25, -162, -8; -7, 9, 7, -1, -20, -567, -123, -497, -4545, -2359, -335, -148, -945, -45],178,8]
%
% END OF LATTICE L179.8
%
\hbox{}\par\smallskip%
%
% LATTICE L179.9
%
\leavevmode%
\llap{\spacer}%
${L_{179.9}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{179.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{s}{2}{14}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{63}\above{1pt}{1pt}{6,1}{\infty}{63}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{126}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
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\kern3pt%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[10647, 441, -4788; 441, 18, -198; -4788, -198, 2153],[63, 9, 1],[0, 6, 4, 1, -22, -10, -136, -78, -552, -368, -365, -160, -16, -46; 1, 7, 0, -28, -161, -65, -847, -475, -3325, -2212, -2184, -945, -93, -259; 0, 14, 9, 0, -63, -28, -378, -216, -1526, -1017, -1008, -441, -44, -126],178,9]
%
% END OF LATTICE L179.9
%
\hbox{}\par\smallskip%
%
% LATTICE L179.10
%
\leavevmode%
\llap{\spacer}%
${L_{179.10}}=\pdualwitharg{7}(\pfillwitharg{2}({L_{179.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${7}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{14}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{6,5}{\infty a}{2}\above{1pt}{1pt}{}{2}{63}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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882\\-112\\-63
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{$\left.\llap{\phantom{%
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6930, 882, 504; 882, -112, -63; 504, -63, -31],[126, 7, 1],[4, 4, -1, -1, 3, 61, 15, 65, 617, 645, 93, 43, 146, 8; 35, 36, -8, -9, 24, 504, 125, 544, 5175, 5412, 781, 362, 1233, 68; -7, -9, 0, 2, -2, -63, -17, -77, -747, -784, -114, -54, -189, -11],178,10]
%
% END OF LATTICE L179.10
%
\hbox{}\par\smallskip%
%
% LATTICE L179.11
%
\leavevmode%
\llap{\spacer}%
${L_{179.11}}=\pdualwitharg{2}(\main({L_{179.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[127897560, -1047060, -31966452; -1047060, 8572, 261700; -31966452, 261700, 7989629],[252, 4, 1],[1, 127, -16, -284, -1473, -5321, -7695, -4315, -271909, -10050, -19848, -8593, -7621, -2371; 0, 126, -31, -497, -2548, -9198, -13300, -7458, -469980, -17371, -34307, -14854, -13176, -4102; 4, 504, -63, -1120, -5810, -20988, -30352, -17020, -1072512, -39641, -78288, -33894, -30060, -9352],178,11]
%
% END OF LATTICE L179.11
%
\hbox{}\par\smallskip%
%
% LATTICE L179.12
%
\leavevmode%
\llap{\spacer}%
${L_{179.12}}=\pdualwitharg{2}({L_{179.1}})$%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[153681696, -1562148, -39042864; -1562148, 15880, 396864; -39042864, 396864, 9918847],[252, 4, 1],[-318, -1984, -32, -345, -2036, -16552, -25278, -14664, -941882, -69884, -69571, -15443, -28394, -9680; 7, 63, 1, 0, 21, 189, 301, 179, 11655, 867, 868, 196, 369, 133; -1252, -7812, -126, -1358, -8015, -65160, -99512, -57728, -3707928, -275114, -273882, -60795, -111780, -38108],178,12]
%
% END OF LATTICE L179.12
%
\hbox{}\par\smallskip%
%
% LATTICE L179.13
%
\leavevmode%
\llap{\spacer}%
${L_{179.13}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{179.2}}))$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[71946, 36288, 16002; 36288, 17865, 8064; 16002, 8064, 3559],[126, 9, 1],[-29, 93, 27, -851, -5701, -1182, -15604, -8824, -62064, -82655, -81777, -35613, -1769, -5053; -2, 7, 2, -63, -420, -87, -1148, -649, -4564, -6078, -6013, -2618, -130, -371; 135, -434, -126, 3969, 26586, 5512, 72765, 41148, 289415, 385434, 381339, 166068, 8249, 23562],178,13]
%
% END OF LATTICE L179.13
%
\hbox{}\par\smallskip%
%
% LATTICE L179.14
%
\leavevmode%
\llap{\spacer}%
${L_{179.14}}=\pdualwitharg{7}(\main({L_{179.3}}))$%
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{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${14}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}{28}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{12,5}{\infty}{4}\above{1pt}{1pt}{b}{2}{126}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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\kern3pt%
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-10332\\-4606\\-7
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-22932, -10332, 0; -10332, -4606, -7; 0, -7, 1],[252, 7, 1],[5, -4, -9, 0, 19, 281, 62, 253, 2326, 2417, 172, 153, 493, 24; -11, 9, 20, 0, -42, -621, -137, -559, -5139, -5340, -380, -338, -1089, -53; -70, 72, 140, -1, -292, -4284, -942, -3836, -35226, -36596, -2603, -2312, -7434, -360],178,14]
%
% END OF LATTICE L179.14
%
\hbox{}\par\smallskip%
%
% LATTICE L179.15
%
\leavevmode%
\llap{\spacer}%
${L_{179.15}}=\pdualwitharg{7}({L_{179.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1252188, 8820, 8820; 8820, -56, -63; 8820, -63, -62],[252, 7, 1],[4, 4, -1, -1, 3, 61, 15, 65, 617, 645, 93, 43, 146, 8; 70, 72, -16, -18, 48, 1008, 250, 1088, 10350, 10824, 1562, 724, 2466, 136; 497, 495, -126, -124, 376, 7623, 1873, 8113, 76995, 80486, 11604, 5364, 18207, 997],178,15]
%
% END OF LATTICE L179.15
%
\hbox{}\par\smallskip%
%
% LATTICE L179.16
%
\leavevmode%
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${L_{179.16}}=\pdualwitharg{2.7}(\pfillwitharg{2}({L_{179.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[65520, 16758, 32508; 16758, 4354, 8316; 32508, 8316, 16129],[126, 14, 1],[158, 221, 31, -1, 124, 3953, 904, 3775, 35176, 18323, 5245, 1187, 7846, 407; 7, 9, 1, 0, 6, 189, 43, 179, 1665, 867, 248, 56, 369, 19; -322, -450, -63, 2, -253, -8064, -1844, -7700, -71748, -37373, -10698, -2421, -16002, -830],178,16]
%
% END OF LATTICE L179.16
%
\hbox{}\par\smallskip%
%
% LATTICE L179.17
%
\leavevmode%
\llap{\spacer}%
${L_{179.17}}=\pdualwitharg{3}({L_{179.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[185724, 2016, -1260; 2016, -18, -9; -1260, -9, 8],[252, 9, 1],[1, 1, -1, -9, -33, -6, -74, -40, -274, -363, -355, -149, -7, -17; 21, 21, -22, -196, -728, -133, -1645, -891, -6111, -8098, -7924, -3332, -157, -385; 180, 182, -180, -1638, -6048, -1102, -13608, -7362, -50456, -66852, -65394, -27468, -1292, -3150],178,17]
%
% END OF LATTICE L179.17
%
\hbox{}\par\smallskip%
%
% LATTICE L179.18
%
\leavevmode%
\llap{\spacer}%
${L_{179.18}}=\pdualwitharg{3}(\main({L_{179.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5796, 756, 0; 756, 99, 0; 0, 0, -1],[252, 9, 1],[0, -1, -1, 1, 19, 4, 53, 30, 211, 281, 139, 121, 6, 17; -1, 7, 8, -7, -154, -33, -441, -251, -1771, -2360, -1169, -1022, -51, -147; -9, -7, 0, 0, -126, -31, -441, -261, -1883, -2520, -1260, -1134, -59, -189],178,18]
%
% END OF LATTICE L179.18
%
\hbox{}\par\smallskip%
%
% LATTICE L179.19
%
\leavevmode%
\llap{\spacer}%
${L_{179.19}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{179.2}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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-36\\-154\\-90\\-126\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
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-378\\-2016\\-1260\\-2142\\-252\\0\\126
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\kern3pt%
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\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[450048942, 725760, 222524190; 725760, 1170, 358848; 222524190, 358848, 110025845],[126, 18, 1],[-899, -90, 405, 187, -8567, -4128, -58069, -34108, -244959, -163771, -326919, -73116, -7541, -23614; 62, 7, -31, -63, 462, 233, 3332, 1975, 14252, 9537, 19075, 4291, 446, 1421; 1818, 182, -819, -378, 17325, 8348, 117432, 68976, 495376, 331191, 661122, 147861, 15250, 47754],178,19]
%
% END OF LATTICE L179.19
%
\hbox{}\par\smallskip%
%
% LATTICE L179.20
%
\leavevmode%
\llap{\spacer}%
${L_{179.20}}=\pdualwitharg{7}({L_{179.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2520\\504\\-504
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504\\98\\-119
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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21599\\-94176\\24192
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5050\\-22019\\5656
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2520, 504, -504; 504, 98, -119; -504, -119, -17],[504, 7, 1],[-66, -1325, -1399, -103, -101, -184, -23, -9, -25, -1, 3, 1, -41, -27; 288, 5778, 6100, 449, 440, 801, 100, 39, 108, 4, -13, -4, 180, 118; -77, -1494, -1568, -114, -108, -189, -22, -7, -18, 0, 2, -4, -63, -34],178,20]
%
% END OF LATTICE L179.20
%
\hbox{}\par\smallskip%
%
% LATTICE L179.21
%
\leavevmode%
\llap{\spacer}%
${L_{179.21}}=\pdualwitharg{7}({L_{179.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${28}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{56}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{12,5}{\infty b}{8}\above{1pt}{1pt}{s}{2}{252}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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\kern3pt%
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504\\-70\\-63
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\kern3pt%
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504\\-63\\-31
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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3455\\29592\\-6048
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{$\left.\llap{\phantom{%
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428\\3663\\-747
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897\\7680\\-1568
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6\\52\\-11
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3528, 504, 504; 504, -70, -63; 504, -63, -31],[504, 7, 1],[89, 428, 897, 65, 61, 211, 6, 7, 4, -1, -1, 3, 77, 10; 760, 3663, 7680, 557, 524, 1818, 52, 62, 36, -8, -9, 24, 648, 85; -154, -747, -1568, -114, -108, -378, -11, -14, -9, 0, 2, -4, -126, -17],178,21]
%
% END OF LATTICE L179.21
%
\hbox{}\par\smallskip%
%
% LATTICE L179.22
%
\leavevmode%
\llap{\spacer}%
${L_{179.22}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{179.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[63, -126, 63; -126, 126, -63; 63, -63, 32],[63, 63, 1],[0, 0, 1, 1, 5, 14, 26, 102, 102, 475, 67, 29, 20, 8; 1, 1, 0, -4, -23, -65, -121, -475, -475, -2212, -312, -135, -93, -37; 0, 2, 0, -9, -54, -154, -288, -1134, -1136, -5292, -747, -324, -224, -90],178,22]
%
% END OF LATTICE L179.22
%
\hbox{}\par\smallskip%
%
% LATTICE L179.23
%
\leavevmode%
\llap{\spacer}%
${L_{179.23}}=\pdualwitharg{2}({L_{179.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{l}{2}{504}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{112}\above{1pt}{1pt}{24,23}{\infty z}{28}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{l}{2}{56}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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504\\0\\0
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0\\-8\\8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\-7
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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-21\\-176\\-393
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[504, 0, 0; 0, -8, 8; 0, 8, -7],[504, 8, 1],[-9, -568, -21, -83, -18, -16, -5, 0, 1, 0, -1, -3, -11, -16; -72, -4725, -176, -707, -161, -153, -56, -3, 0, 1, 7, -14, -72, -119; -164, -10584, -393, -1568, -350, -324, -112, -4, 0, 1, 0, -42, -180, -280],178,23]
%
% END OF LATTICE L179.23
%
\hbox{}\par\smallskip%
%
% LATTICE L179.24
%
\leavevmode%
\llap{\spacer}%
${L_{179.24}}=\pdualwitharg{2}({L_{179.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{504}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{112}\above{1pt}{1pt}{24,11}{\infty z}{28}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-63000\\0\\-31248
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0\\8\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31248\\0\\-15499
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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953378\\-6993\\-1920744
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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77426\\-567\\-155988
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5725\\-41\\-11534
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-63000, 0, -31248; 0, 8, 0; -31248, 0, -15499],[504, 8, 1],[1227, 77426, 5725, 11313, 2453, 2180, 681, 0, -125, 1, 139, 410, 1501, 2182; -10, -567, -41, -77, -14, -9, 0, 1, 0, -1, -7, -7, -18, -21; -2472, -155988, -11534, -22792, -4942, -4392, -1372, 0, 252, -2, -280, -826, -3024, -4396],178,24]
%
% END OF LATTICE L179.24
%
\hbox{}\par\smallskip%
%
% LATTICE L179.25
%
\leavevmode%
\llap{\spacer}%
${L_{179.25}}=\pdualwitharg{3}({L_{179.2}})$%
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{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-115416, 1512, 504; 1512, 9, -9; 504, -9, -2],[504, 9, 1],[-15, -211, -281, -139, -121, -3, -17, 0, 1, 1, -1, -19, -2, -53; -251, -3542, -4720, -2338, -2044, -51, -294, -1, 14, 16, -14, -308, -33, -882; -2781, -39116, -52092, -25767, -22428, -556, -3150, 0, 182, 180, -189, -3528, -371, -9828],178,25]
%
% END OF LATTICE L179.25
%
\hbox{}\par\smallskip%
%
% LATTICE L179.26
%
\leavevmode%
\llap{\spacer}%
${L_{179.26}}=\pdualwitharg{3}({L_{179.3}})$%
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{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-121464, 15120, -504; 15120, -1881, 63; -504, 63, -2],[504, 9, 1],[-19, -64, -169, -82, -67, -3, -3, 1, 1, 1, -2, -17, -3, -18; -174, -595, -1576, -770, -644, -30, -35, 6, 7, 8, -14, -140, -26, -161; -954, -3556, -9576, -4851, -4536, -248, -441, -72, -35, 0, 63, -252, -94, -756],178,26]
%
% END OF LATTICE L179.26
%
\hbox{}\par\smallskip%
%
% LATTICE L179.27
%
\leavevmode%
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${L_{179.27}}=\pdualwitharg{2.7}(\main({L_{179.3}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
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%
% END OF LATTICE L179.27
%
\hbox{}\par\smallskip%
%
% LATTICE L179.28
%
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${L_{179.28}}=\pdualwitharg{2.7}({L_{179.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[64655136, -1060164, -16428888; -1060164, 17416, 269388; -16428888, 269388, 4174585],[252, 28, 1],[-1928, -1930, -32, -31, -1041, -63646, -14296, -59050, -546872, -284532, -40621, -9122, -119288, -6030; 7, 9, 1, 0, 3, 189, 43, 179, 1665, 867, 124, 28, 369, 19; -7588, -7596, -126, -122, -4097, -250488, -56264, -232400, -2152296, -1119818, -159870, -35901, -469476, -23732],178,28]
%
% END OF LATTICE L179.28
%
\hbox{}\par\smallskip%
%
% LATTICE L179.29
%
\leavevmode%
\llap{\spacer}%
${L_{179.29}}=\pdualwitharg{3.7}(\pfillwitharg{2}({L_{179.2}}))$%
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{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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3654\\-8442\\-4536
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3654, -8442, -4536; -8442, 19593, 10521; -4536, 10521, 5650],[126, 63, 1],[-25, -6, -27, 2, -11, -27, -62, -275, -294, -2773, -402, -189, -73, -38; 57, 13, 56, -4, 34, 75, 167, 729, 773, 7280, 1052, 490, 187, 95; -126, -29, -126, 9, -72, -161, -360, -1575, -1672, -15750, -2277, -1062, -406, -207],178,29]
%
% END OF LATTICE L179.29
%
\hbox{}\par\smallskip%
%
% LATTICE L179.30
%
\leavevmode%
\llap{\spacer}%
${L_{179.30}}=\pdualwitharg{2.3}({L_{179.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{s}{2}{56}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{b}{2}{252}\above{1pt}{1pt}{6,1}{\infty z}{63}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{504}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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49590072\\-849492\\-12598740
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-849492\\14544\\215820
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-12598740\\215820\\3200807
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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311\\2\\1224
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1457\\28\\5733
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1477\\24\\5812
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21069\\322\\82908
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[49590072, -849492, -12598740; -849492, 14544, 215820; -12598740, 215820, 3200807],[252, 36, 1],[311, -313, -407, -352, 1457, 1477, 21069, 12451, 89647, 59963, 59908, 13432, 2777, 8709; 2, 0, 0, 7, 28, 24, 322, 184, 1302, 868, 861, 189, 38, 112; 1224, -1232, -1602, -1386, 5733, 5812, 82908, 48996, 352772, 235962, 235746, 52857, 10928, 34272],178,30]
%
% END OF LATTICE L179.30
%
\hbox{}\par\smallskip%
%
% LATTICE L179.31
%
\leavevmode%
\llap{\spacer}%
${L_{179.31}}=\pdualwitharg{2.3}(\main({L_{179.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{*}{2}{56}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{252}\above{1pt}{1pt}{12,7}{\infty}{252}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{504}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4788, -3024, 1260; -3024, 828, 756; 1260, 756, -331],[252, 36, 1],[-183, -121, 16, 128, -737, -417, -6215, -3771, -27557, -9242, -18580, -8489, -901, -3011; -11, -7, 1, 7, -49, -27, -399, -241, -1757, -589, -1183, -539, -57, -189; -720, -476, 63, 504, -2898, -1640, -24444, -14832, -108388, -36351, -73080, -33390, -3544, -11844],178,31]
%
% END OF LATTICE L179.31
%
\hbox{}\par\smallskip%
%
% LATTICE L179.32
%
\leavevmode%
\llap{\spacer}%
${L_{179.32}}=\pdualwitharg{3.7}({L_{179.1}})$%
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{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-95004, -13608, -3780; -13608, -1260, -315; -3780, -315, -76],[252, 63, 1],[7, 1, -1, -3, -3, 0, 4, 26, 32, 309, 47, 25, 11, 7; -311, -45, 42, 136, 152, 27, -127, -955, -1221, -11858, -1824, -996, -449, -295; 945, 137, -126, -414, -468, -91, 369, 2835, 3643, 35406, 5454, 2988, 1351, 891],178,32]
%
% END OF LATTICE L179.32
%
\hbox{}\par\smallskip%
%
% LATTICE L179.33
%
\leavevmode%
\llap{\spacer}%
${L_{179.33}}=\pdualwitharg{3.7}(\main({L_{179.3}}))$%
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{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[252, 0, 0; 0, -1827, 504; 0, 504, -139],[252, 63, 1],[-1, 0, 1, 0, -3, -5, -10, -41, -42, -393, -28, -25, -9, -4; 17, 3, 0, -5, -26, -27, -41, -137, -123, -1120, -75, -54, -13, 1; 63, 11, 0, -18, -90, -91, -135, -441, -389, -3528, -234, -162, -35, 9],178,33]
%
% END OF LATTICE L179.33
%
\hbox{}\par\smallskip%
%
% LATTICE L179.34
%
\leavevmode%
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${L_{179.34}}=\pdualwitharg{2.3.7}(\pfillwitharg{2}({L_{179.2}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-69804, -98910, -48258; -98910, -139230, -67914; -48258, -67914, -33127],[126, 126, 1],[35, 4, -4, -9, 12, 56, 125, 544, 575, 2706, 781, 181, 137, 68; -701, -77, 95, 182, -250, -1153, -2565, -11143, -11767, -55367, -15974, -3698, -2795, -1383; 1386, 152, -189, -360, 495, 2282, 5076, 22050, 23284, 109557, 31608, 7317, 5530, 2736],178,34]
%
% END OF LATTICE L179.34
%
\hbox{}\par\smallskip%
%
% LATTICE L179.35
%
\leavevmode%
\llap{\spacer}%
${L_{179.35}}=\pdualwitharg{2.7}({L_{179.3}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3717000, 0, 17136; 0, -56, 0; 17136, 0, 79],[504, 56, 1],[101, 944, 246, 141, 32, 212, 11, 4, 1, -1, -1, 3, 103, 24; 92, 837, 217, 123, 27, 171, 8, 1, 0, 0, 1, 4, 108, 23; -21980, -205416, -53529, -30680, -6962, -46116, -2392, -868, -216, 217, 216, -654, -22428, -5224],178,35]
%
% END OF LATTICE L179.35
%
\hbox{}\par\smallskip%
%
% LATTICE L179.36
%
\leavevmode%
\llap{\spacer}%
${L_{179.36}}=\pdualwitharg{2.7}({L_{179.2}})$%
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{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5544, 5544, -2016; 5544, -5488, 2016; -2016, 2016, -733],[504, 56, 1],[1319, 11893, 6156, 1738, 377, 2347, 105, 1, -13, 4, 22, 63, 1613, 335; -10, -81, -41, -11, -2, -9, 0, 1, 0, -1, -1, -1, -18, -3; -3696, -33300, -17234, -4864, -1054, -6552, -292, 0, 36, -14, -64, -178, -4536, -940],178,36]
%
% END OF LATTICE L179.36
%
\hbox{}\par\smallskip%
%
% LATTICE L179.37
%
\leavevmode%
\llap{\spacer}%
${L_{179.37}}=\pdualwitharg{3.7}({L_{179.2}})$%
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{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[365400, -87696, 31248; -87696, 21042, -7497; 31248, -7497, 2671],[504, 63, 1],[18, 41, 391, 29, 29, 6, 7, 3, 1, 1, -1, -1, 1, 7; -80, -102, -844, -43, 8, 13, 36, 35, 12, 4, -17, -68, -28, -66; -441, -778, -7056, -468, -324, -35, 18, 63, 22, 0, -36, -180, -91, -270],178,37]
%
% END OF LATTICE L179.37
%
\hbox{}\par\smallskip%
%
% LATTICE L179.38
%
\leavevmode%
\llap{\spacer}%
${L_{179.38}}=\pdualwitharg{3.7}({L_{179.3}})$%
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{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[504, 0, 0; 0, -3654, 1323; 0, 1323, -479],[504, 63, 1],[-41, -21, -393, -28, -25, -9, -2, -1, 0, 1, 0, -3, -5, -5; -304, -133, -2408, -159, -108, -22, 4, 46, 4, 0, -13, -64, -64, -47; -882, -389, -7056, -468, -324, -70, 9, 126, 11, 0, -36, -180, -182, -135],178,38]
%
% END OF LATTICE L179.38
%
\hbox{}\par\smallskip%
%
% LATTICE L179.39
%
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\llap{\spacer}%
${L_{179.39}}=\pdualwitharg{2.3}({L_{179.2}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[457128, 65520, -1512; 65520, 9000, -216; -1512, -216, 5],[504, 72, 1],[2, 1, -1, -9, -6, -1, -4, 1, 20, 15, 37, 13, 2, 11; 1, 0, -1, -7, -7, -2, -21, -10, -63, -41, -77, -14, -1, 0; 648, 308, -342, -3024, -2142, -400, -2268, -216, 2716, 2358, 7056, 3150, 544, 3276],178,39]
%
% END OF LATTICE L179.39
%
\hbox{}\par\smallskip%
%
% LATTICE L179.40
%
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${L_{179.40}}=\pdualwitharg{2.3}({L_{179.3}})$%
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{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}7\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[504, 0, 0; 0, -2952, 144; 0, 144, -7],[504, 72, 1],[-3, -16, -5, -17, -2, 0, 1, 0, -5, -2, -11, -5, -1, -8; -4, -7, -1, 7, 7, 1, 0, -5, -56, -20, -91, -28, -4, -21; -108, -280, -63, 0, 126, 20, 0, -108, -1232, -441, -2016, -630, -92, -504],178,40]
%
% END OF LATTICE L179.40
%
\hbox{}\par\smallskip%
%
% LATTICE L179.41
%
\leavevmode%
\llap{\spacer}%
${L_{179.41}}=\pdualwitharg{2.3.7}({L_{179.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${504}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{252}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{6,1}{\infty z}{9}\above{1pt}{1pt}{r}{2}{56}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7068096, 20886012, -5306868; 20886012, 61717824, -15681708; -5306868, -15681708, 3984521],[252, 252, 1],[-17, -1, 3, -1, -13, -85, -169, -691, -707, -3307, -471, -105, -151, -67; -1595, -145, 31, -114, -1155, -7541, -14999, -61363, -62813, -293835, -41858, -9338, -13445, -5987; -6300, -572, 126, -450, -4563, -29792, -59256, -242424, -248152, -1160838, -165366, -36891, -53116, -23652],178,41]
%
% END OF LATTICE L179.41
%
\hbox{}\par\smallskip%
%
% LATTICE L179.42
%
\leavevmode%
\llap{\spacer}%
${L_{179.42}}=\pdualwitharg{2.3.7}(\main({L_{179.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}7\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\16\\63
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[35028, -127008, 32256; -127008, 461412, -117180; 32256, -117180, 29759],[252, 252, 1],[11, 1, 0, 1, 17, 55, 109, 445, 455, 1064, 303, 135, 97, 43; -253, -17, 16, -18, -393, -1279, -2539, -10373, -10609, -24810, -7066, -3149, -2263, -1003; -1008, -68, 63, -72, -1566, -5096, -10116, -41328, -42268, -98847, -28152, -12546, -9016, -3996],178,42]
%
% END OF LATTICE L179.42
%
\hbox{}\par\smallskip%
%
% LATTICE L179.43
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-194544, -21672, 9576; -21672, 8568, -3528; 9576, -3528, 1451],[504, 504, 1],[67, 74, 351, 103, 25, 20, 11, 8, 1, -1, -3, 0, 5, 14; 10111, 11217, 53246, 15650, 3815, 3067, 1701, 1265, 159, -158, -482, -23, 725, 2091; 24192, 26836, 127386, 37440, 9126, 7336, 4068, 3024, 380, -378, -1152, -54, 1736, 5004],178,43]
%
% END OF LATTICE L179.43
%
\hbox{}\par\smallskip%
%
% LATTICE L179.44
%
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${L_{179.44}}=\pdualwitharg{2.3.7}({L_{179.3}})$%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1512\\-1152\\-2520\\-1224\\-144\\0\\72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5040\\-4536\\-10332\\-5544\\-1008\\-504\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4536\\-4544\\-10584\\-5976\\-1296\\-896\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10332\\-10584\\-24759\\-14112\\-3150\\-2268\\-1008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5544\\-5976\\-14112\\-8208\\-1944\\-1512\\-792
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1008\\-1296\\-3150\\-1944\\-540\\-504\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\-896\\-2268\\-1512\\-504\\-560\\-504
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-1008\\-792\\-360\\-504\\-576
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-20664, 217728, 1008; 217728, 1104264, 5040; 1008, 5040, 23],[504, 504, 1],[-5, -8, -20, -13, -4, -4, -3, -4, -1, -1, 1, 1, 1, 0; 108, 171, 427, 277, 85, 85, 64, 87, 24, 28, -17, -20, -20, 1; -23436, -37112, -92673, -60120, -18450, -18452, -13896, -18900, -5224, -6111, 3672, 4338, 4340, -216],178,44]
%
% END OF LATTICE L179.44
%

\medskip%
%------------------------------WEYL-GROUP-W180--------------------------------
%
\leavevmode\llap{}%
$W_{180}$%
\qquad\llap{2} lattices, $\chi=24$%
\hfill%
$\slashtwo\infty4|4\infty\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L180.1
%
\leavevmode%
\llap{\spacer}%
${L_{180.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}64\above{1pt}{1pt}{1}{7}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{16,15}{\infty z}{1}\above{1pt}{1pt}{}{4}{2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{16,7}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16960\\-896\\320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-896\\-47\\17
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
320\\17\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-3
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-10\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-7\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\10\\-26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-4\\-3\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-2\\-6\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-2\\1\\-1\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-1\\2\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-16\\-16\\-2\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-16960, -896, 320; -896, -47, 17; 320, 17, -6],[64, 1, 1],[0, 1, 1, 0, -1; -1, -10, -7, 2, 10; -3, 24, 32, 5, -26],179,1]
%
% END OF LATTICE L180.1
%
\hbox{}\par\smallskip%
%
% LATTICE L180.2
%
\leavevmode%
\llap{\spacer}%
${L_{180.2}}=\pdualwitharg{2}({L_{180.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{2}{2}$}\EasyButWeakLineBreak%
{${64}\above{1pt}{1pt}{l}{2}{64}\above{1pt}{1pt}{16,9}{\infty}{64}\above{1pt}{1pt}{*}{4}{128}\above{1pt}{1pt}{}{4}{64}\above{1pt}{1pt}{16,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
640\\-3520\\64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3520\\21056\\-384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\-384\\7
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-3\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-192
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-9\\-448
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\0\\-256\\-192\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\64\\-64\\-192\\-256
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-256\\-64\\64\\-64\\-512
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\-192\\-64\\128\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\-256\\-512\\-64\\64
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[640, -3520, 64; -3520, 21056, -384; 64, -384, 7],[64, 64, 1],[0, 1, -1, -3, -4; -3, -1, -3, -4, -9; -160, -64, -160, -192, -448],179,2]
%
% END OF LATTICE L180.2
%

\medskip%
%------------------------------WEYL-GROUP-W181--------------------------------
%
\leavevmode\llap{}%
$W_{181}$%
\qquad\llap{14} lattices, $\chi=12$%
\hfill%
$2|22\slashinfty2\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L181.1
%
\leavevmode%
\llap{\spacer}%
${L_{181.1}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}32\above{1pt}{1pt}{-}{5}$}\EasyButWeakLineBreak%
{${32}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{32}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{8,5}{\infty}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13664\\-416\\928
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-416\\-10\\28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
928\\28\\-63
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\6\\76
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\40\\384
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\4\\31
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-46
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\0\\-96\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-4\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\0\\32\\0\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-4\\0\\1\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-32\\-2\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-13664, -416, 928; -416, -10, 28; 928, 28, -63],[32, 2, 1],[-3, 5, 25, 2, -3; -8, 6, 40, 4, -4; -48, 76, 384, 31, -46],180,1]
%
% END OF LATTICE L181.1
%
\hbox{}\par\smallskip%
%
% LATTICE L181.2
%
\leavevmode%
\llap{\spacer}%
${L_{181.2}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}64\above{1pt}{1pt}{1}{1}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${64}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{64}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{16,9}{\infty}{8}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
shares genus with {$ {L_{181.3}}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59840\\-2752\\-3008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2752\\-126\\-138
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3008\\-138\\-151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\48\\-64
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\24\\-42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\64\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-22\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64\\0\\-192\\-32\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-4\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-192\\0\\64\\0\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-4\\0\\2\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-64\\-4\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-59840, -2752, -3008; -2752, -126, -138; -3008, -138, -151],[64, 2, 1],[1, 1, 5, 0, -1; 48, 24, 64, -9, -22; -64, -42, -160, 8, 40],180,2]
%
% END OF LATTICE L181.2
%
\hbox{}\par\smallskip%
%
% LATTICE L181.3
%
\leavevmode%
\llap{\spacer}%
${L_{181.3}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}64\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${64}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{64}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{16,1}{\infty}{8}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
shares genus with {$ {L_{181.2}}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-342464\\6656\\2880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6656\\-126\\-58
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2880\\-58\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-208\\-352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\60\\99
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\1344\\2240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\119\\200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-150\\-248
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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64\\0\\-192\\-32\\0
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0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-342464, 6656, 2880; 6656, -126, -58; 2880, -58, -23],[64, 2, 1],[-7, 2, 45, 4, -5; -208, 60, 1344, 119, -150; -352, 99, 2240, 200, -248],180,3]
%
% END OF LATTICE L181.3
%
\hbox{}\par\smallskip%
%
% LATTICE L181.4
%
\leavevmode%
\llap{\spacer}%
${L_{181.4}}$%
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480\\-4\\-32
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3712\\-32\\-247
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-55776, 480, 3712; 480, -4, -32; 3712, -32, -247],[32, 4, 1],[-1, -1, 1, 7, 1; -8, 6, 14, 64, -8; -14, -16, 13, 96, 16],180,4]
%
% END OF LATTICE L181.4
%
\hbox{}\par\smallskip%
%
% LATTICE L181.5
%
\leavevmode%
\llap{\spacer}%
${L_{181.5}}$%
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-46112\\2112\\2144
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-46112, 2112, 2144; 2112, -92, -100; 2144, -100, -99],[32, 4, 1],[-5, 2, 9, 41, -5; -30, 11, 53, 244, -28; -78, 32, 141, 640, -80],180,5]
%
% END OF LATTICE L181.5
%
\hbox{}\par\smallskip%
%
% LATTICE L181.6
%
\leavevmode%
\llap{\spacer}%
${L_{181.6}}$%
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64\\0\\0
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0\\-56\\-8
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[64, 0, 0; 0, -56, -8; 0, -8, -1],[64, 8, 1],[-1, 0, 3, 1, 0; 0, -1, -8, -1, 1; 0, 6, 32, 0, -8],180,6]
%
% END OF LATTICE L181.6
%
\hbox{}\par\smallskip%
%
% LATTICE L181.7
%
\leavevmode%
\llap{\spacer}%
${L_{181.7}}=\main({L_{181.2}})$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-22240, 224, 1184; 224, -2, -12; 1184, -12, -63],[32, 2, 1],[-1, -1, -5, 0, 1; -16, -5, 0, 4, 4; -16, -18, -96, -1, 18],180,7]
%
% END OF LATTICE L181.7
%
\hbox{}\par\smallskip%
%
% LATTICE L181.8
%
\leavevmode%
\llap{\spacer}%
${L_{181.8}}=\pdualwitharg{2}({L_{181.4}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18784, 416, 1088; 416, -8, -24; 1088, -24, -63],[32, 8, 1],[-1, -2, -5, 0, 1; -4, 3, 20, 2, -2; -16, -36, -96, -1, 18],180,8]
%
% END OF LATTICE L181.8
%
\hbox{}\par\smallskip%
%
% LATTICE L181.9
%
\leavevmode%
\llap{\spacer}%
${L_{181.9}}=\pdualwitharg{2}({L_{181.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{1}{1}32\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
{${32}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{32}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{4,1}{\infty a}{4}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
320\\-1048\\208
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\208\\-41
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-16
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\1\\0
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17\\12\\32
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\HardButStrongLineBreak\kern3pt%
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3\\3\\10
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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32\\0\\-96\\-32\\0
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0\\8\\0\\-8\\-8
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-96\\0\\32\\0\\-32
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-32\\-8\\0\\4\\-4
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0\\-8\\-32\\-4\\4
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-96, 320, -64; 320, -1048, 208; -64, 208, -41],[32, 8, 1],[-3, 3, 17, 3, -2; -4, 1, 12, 3, -1; -16, 0, 32, 10, -2],180,9]
%
% END OF LATTICE L181.9
%
\hbox{}\par\smallskip%
%
% LATTICE L181.10
%
\leavevmode%
\llap{\spacer}%
${L_{181.10}}=\pdualwitharg{2}({L_{181.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{1}64\above{1pt}{1pt}{1}{7}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{64}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{16,7}{\infty}{8}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13376\\960\\448
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
960\\-56\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
448\\-32\\-15
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\61
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-4\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-8\\0
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0\\64\\0\\-32\\-32
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-6\\0\\4\\0\\-16
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-8\\-32\\0\\8\\-8
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0\\-32\\-16\\-8\\8
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-13376, 960, 448; 960, -56, -32; 448, -32, -15],[64, 8, 1],[2, 3, -1, -1, 4; -1, -4, 0, 1, -1; 61, 96, -30, -32, 120],180,10]
%
% END OF LATTICE L181.10
%
\hbox{}\par\smallskip%
%
% LATTICE L181.11
%
\leavevmode%
\llap{\spacer}%
${L_{181.11}}=\pdualwitharg{2}(\main({L_{181.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[16\above{1pt}{1pt}{1}{}32\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{32}\above{1pt}{1pt}{4,3}{\infty b}{32}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55776\\960\\3712
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
960\\-16\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3712\\-64\\-247
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\7\\13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\32\\96
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-16\\0
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0\\16\\0\\-32\\-32
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-6\\0\\4\\0\\-32
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-16\\-32\\0\\32\\-32
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-32\\-32\\-32\\32
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-55776, 960, 3712; 960, -16, -64; 3712, -64, -247],[32, 16, 1],[1, -1, -1, 1, 7; 7, 3, -4, -4, 32; 13, -16, -14, 16, 96],180,11]
%
% END OF LATTICE L181.11
%
\hbox{}\par\smallskip%
%
% LATTICE L181.12
%
\leavevmode%
\llap{\spacer}%
${L_{181.12}}=\pdualwitharg{2}({L_{181.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[16\above{1pt}{1pt}{-}{}32\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{b}{2}{32}\above{1pt}{1pt}{4,3}{\infty a}{32}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-224\\1216\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1216\\-5712\\752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\752\\-99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\17\\130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\40\\304
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\-32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-32\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-32\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-32
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-32\\0\\32\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-32\\-32\\-32\\32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-224, 1216, -160; 1216, -5712, 752; -160, 752, -99],[32, 16, 1],[0, -1, -1, -1, 1; -5, 3, 17, 40, -4; -38, 24, 130, 304, -32],180,12]
%
% END OF LATTICE L181.12
%
\hbox{}\par\smallskip%
%
% LATTICE L181.13
%
\leavevmode%
\llap{\spacer}%
${L_{181.13}}=\pdualwitharg{2}({L_{181.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[32\above{1pt}{1pt}{1}{}64\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{64}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{l}{2}{32}\above{1pt}{1pt}{16,7}{\infty}{32}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
shares genus with {$ \pdualwitharg{2}({L_{181.3}})$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\1920\\-128
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1920\\-55776\\3712
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128\\3712\\-247
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-14
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\2\\26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\7\\96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-12\\-32\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\64\\0\\-64\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-64\\0\\32\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-64\\-32\\-32\\32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-64, 1920, -128; 1920, -55776, 3712; -128, 3712, -247],[64, 32, 1],[-2, 3, 7, 16, -2; -1, -2, 2, 7, 1; -14, -32, 26, 96, 16],180,13]
%
% END OF LATTICE L181.13
%
\hbox{}\par\smallskip%
%
% LATTICE L181.14
%
\leavevmode%
\llap{\spacer}%
${L_{181.14}}=\pdualwitharg{2}({L_{181.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[32\above{1pt}{1pt}{1}{}64\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{64}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{32}\above{1pt}{1pt}{16,15}{\infty}{32}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
shares genus with {$ \pdualwitharg{2}({L_{181.2}})$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-390208\\-188032\\12352
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-188032\\-90592\\5952
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12352\\5952\\-391
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-390208, -188032, 12352; -188032, -90592, 5952; 12352, 5952, -391],[64, 32, 1],[-3, 3, 5, 24, -2; -1, -2, 1, 7, 1; -110, 64, 173, 864, -48],180,14]
%
% END OF LATTICE L181.14
%

\medskip%
%------------------------------WEYL-GROUP-W182--------------------------------
%
\leavevmode\llap{}%
$W_{182}$%
\qquad\llap{6} lattices, $\chi=48$%
\hfill%
$\slashinfty\slashtwo\slashinfty\slashtwo\slashinfty\slashtwo\slashinfty\slashtwo\rtimes D_{8}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L182.1
%
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\llap{\spacer}%
${L_{182.1}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-181184, 448, 4800; 448, -1, -12; 4800, -12, -127],[64, 1, 1],[-15, -1, 1, 0, -9, -23, -7, -3; -608, -64, 40, 8, -256, -800, -256, -116; -512, -32, 34, -1, -320, -800, -242, -103],181,1]
%
% END OF LATTICE L182.1
%
\hbox{}\par\smallskip%
%
% LATTICE L182.2
%
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\llap{\spacer}%
${L_{182.2}}$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-153536, 832, 4416; 832, -4, -24; 4416, -24, -127],[64, 4, 1],[-23, -9, 0, 1, -1, -15, -3, -7; -32, 16, 4, 4, -16, -64, -10, -16; -800, -320, -1, 34, -32, -512, -103, -242],181,2]
%
% END OF LATTICE L182.2
%
\hbox{}\par\smallskip%
%
% LATTICE L182.3
%
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\llap{\spacer}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-120000, -1728, 3904; -1728, -20, 56; 3904, 56, -127],[64, 4, 1],[99, 53, 2, -3, -3, 43, 10, 27; 152, 88, 4, -4, -8, 56, 14, 40; 3104, 1664, 63, -94, -96, 1344, 313, 846],181,3]
%
% END OF LATTICE L182.3
%
\hbox{}\par\smallskip%
%
% LATTICE L182.4
%
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\llap{\spacer}%
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0\\0\\0
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0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-140480, 1600, 4224; 1600, -16, -48; 4224, -48, -127],[64, 16, 1],[-15, -1, 1, 0, -9, -23, -7, -3; 28, -4, -2, 2, 44, 76, 20, 7; -512, -32, 34, -1, -320, -800, -242, -103],181,4]
%
% END OF LATTICE L182.4
%
\hbox{}\par\smallskip%
%
% LATTICE L182.5
%
\leavevmode%
\llap{\spacer}%
${L_{182.5}}=\pdualwitharg{2}({L_{182.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{-}{3}64\above{1pt}{1pt}{1}{1}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-192\\1152\\-128
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1152\\-6800\\752
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128\\752\\-83
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-129\\-80\\-512
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720\\449\\2880
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-192, 1152, -128; 1152, -6800, 752; -128, 752, -83],[64, 16, 1],[3, -2, -3, 27, 13, 18, 67, 37; 3, -1, -4, 12, 7, 11, 44, 28; 22, -6, -32, 64, 42, 70, 288, 192],181,5]
%
% END OF LATTICE L182.5
%
\hbox{}\par\smallskip%
%
% LATTICE L182.6
%
\leavevmode%
\llap{\spacer}%
${L_{182.6}}=\pdualwitharg{2}({L_{182.1}})$%
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{$1\above{1pt}{1pt}{1}{1}64\above{1pt}{1pt}{2}{0}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-123968\\-131968\\3968
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-131968\\-140480\\4224
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3968\\4224\\-127
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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14\\-15\\-64
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-1\\1\\2
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1\\0\\31
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22\\-9\\384
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7\\-6\\18
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\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-123968, -131968, 3968; -131968, -140480, 4224; 3968, 4224, -127],[64, 64, 1],[14, -2, -1, 1, 22, 38, 10, 7; -15, -1, 1, 0, -9, -23, -7, -6; -64, -96, 2, 31, 384, 416, 78, 18],181,6]
%
% END OF LATTICE L182.6
%

\medskip%
%------------------------------WEYL-GROUP-W183--------------------------------
%
\leavevmode\llap{}%
$W_{183}$%
\qquad\llap{34} lattices, $\chi=18$%
\hfill%
$\slashinfty22\slashtwo22\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L183.1
%
\leavevmode%
\llap{\spacer}%
${L_{183.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{6,1}{\infty}{8}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-444600\\4680\\4392
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4680\\-47\\-49
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4392\\-49\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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15\\808\\656
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-376\\-308
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-378\\-306
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\161\\132
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17\\914\\746
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46\\2475\\2016
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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8\\-8\\-72\\-8\\-8\\0
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-8\\8\\0\\-4\\-16\\-36
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-72\\0\\36\\0\\-36\\-144
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-8\\-4\\0\\1\\0\\-9
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-8\\-16\\-36\\0\\4\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-144\\-9\\0\\9
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-444600, 4680, 4392; 4680, -47, -49; 4392, -49, -40],[72, 1, 1],[15, -7, -7, 3, 17, 46; 808, -376, -378, 161, 914, 2475; 656, -308, -306, 132, 746, 2016],182,1]
%
% END OF LATTICE L183.1
%
\hbox{}\par\smallskip%
%
% LATTICE L183.2
%
\leavevmode%
\llap{\spacer}%
${L_{183.2}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{24,1}{\infty z}{2}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-463824\\-11520\\-12240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11520\\-286\\-304
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12240\\-304\\-323
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\2\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-13\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-108\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-28\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\36\\-72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-4\\-72\\-32\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\2\\0\\-8\\-8\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\36\\0\\-36\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-8\\0\\16\\0\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-8\\-36\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-576\\-144\\0\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-463824, -11520, -12240; -11520, -286, -304; -12240, -304, -323],[144, 2, 1],[-1, 0, 5, 3, 1, 1; 2, -13, -108, -28, 0, 36; 36, 12, -90, -88, -38, -72],182,2]
%
% END OF LATTICE L183.2
%
\hbox{}\par\smallskip%
%
% LATTICE L183.3
%
\leavevmode%
\llap{\spacer}%
${L_{183.3}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{24,13}{\infty z}{2}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-123408\\6048\\2448
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6048\\-286\\-124
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2448\\-124\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-70\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\59\\156
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\396\\1035
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\308\\800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\30\\77
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-108\\-288
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-4\\-36\\-32\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\2\\0\\-8\\-4\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\9\\0\\-9\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-8\\0\\16\\0\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-4\\-9\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-288\\-144\\0\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-123408, 6048, 2448; 6048, -286, -124; 2448, -124, -47],[144, 2, 1],[-7, 6, 40, 31, 3, -11; -70, 59, 396, 308, 30, -108; -180, 156, 1035, 800, 77, -288],182,3]
%
% END OF LATTICE L183.3
%
\hbox{}\par\smallskip%
%
% LATTICE L183.4
%
\leavevmode%
\llap{\spacer}%
${L_{183.4}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{24,13}{\infty}{8}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{144}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60336\\17712\\2880
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17712\\-5198\\-848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2880\\-848\\-133
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-87\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\78\\44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\90\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-41\\-128\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\-190\\-110
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-671\\-2088\\-1224
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-36\\-16\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-16\\-16\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\36\\0\\-36\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-16\\0\\16\\0\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-16\\-36\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-576\\-144\\0\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-60336, 17712, 2880; 17712, -5198, -848; 2880, -848, -133],[144, 2, 1],[-28, 25, 29, -41, -61, -671; -87, 78, 90, -128, -190, -2088; -52, 44, 54, -72, -110, -1224],182,4]
%
% END OF LATTICE L183.4
%
\hbox{}\par\smallskip%
%
% LATTICE L183.5
%
\leavevmode%
\llap{\spacer}%
${L_{183.5}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{24,1}{\infty b}{32}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{s}{2}{64}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{576}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
shares genus with {$ {L_{183.6}}$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-607680\\9216\\95040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9216\\-136\\-1488
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
95040\\-1488\\-14287
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62\\2231\\180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-1186\\-96
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-31\\-1116\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\1976\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\1906\\154
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1289\\46368\\3744
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-16\\-72\\-64\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\32\\0\\-64\\-32\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\36\\0\\-36\\-1152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\-64\\0\\64\\0\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-32\\-36\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-576\\-1152\\-576\\0\\576
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-607680, 9216, 95040; 9216, -136, -1488; 95040, -1488, -14287],[576, 8, 1],[62, -33, -31, 55, 53, 1289; 2231, -1186, -1116, 1976, 1906, 46368; 180, -96, -90, 160, 154, 3744],182,5]
%
% END OF LATTICE L183.5
%
\hbox{}\par\smallskip%
%
% LATTICE L183.6
%
\leavevmode%
\llap{\spacer}%
${L_{183.6}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{24,1}{\infty a}{32}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{64}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{576}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
shares genus with {$ {L_{183.5}}$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42328512\\78336\\-13117824
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78336\\-136\\24384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13117824\\24384\\-4063999
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1016\\-36577\\3060
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
255\\9182\\-768
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
254\\9144\\-765
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-425\\-15304\\1280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-339\\-12205\\1021
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18551\\-667872\\55872
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-16\\-36\\-64\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\32\\0\\-64\\-16\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\9\\0\\-9\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\-64\\0\\64\\0\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-16\\-9\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-576\\-576\\-576\\0\\576
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-42328512, 78336, -13117824; 78336, -136, 24384; -13117824, 24384, -4063999],[576, 8, 1],[-1016, 255, 254, -425, -339, -18551; -36577, 9182, 9144, -15304, -12205, -667872; 3060, -768, -765, 1280, 1021, 55872],182,6]
%
% END OF LATTICE L183.6
%
\hbox{}\par\smallskip%
%
% LATTICE L183.7
%
\leavevmode%
\llap{\spacer}%
${L_{183.7}}=\pfillwitharg{2}({L_{183.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{12,7}{\infty}{2}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-34\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-20\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-36\\27
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-36\\-4\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\2\\0\\-2\\-8\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\36\\0\\-36\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-2\\0\\1\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-8\\-36\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-144\\-9\\0\\9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[36, 0, 0; 0, -34, -40; 0, -40, -47],[36, 2, 1],[-1, 0, 1, 0, -1, -4; -1, 7, 0, -6, -20, -36; 0, -6, 0, 5, 16, 27],182,7]
%
% END OF LATTICE L183.7
%
\hbox{}\par\smallskip%
%
% LATTICE L183.8
%
\leavevmode%
\llap{\spacer}%
${L_{183.8}}=\main({L_{183.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{12,7}{\infty}{4}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-231912\\720\\5760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-2\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5760\\-18\\-143
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\18\\38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-36\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-19\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-44\\-116
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-45\\-198
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-36\\-4\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-4\\-16\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\72\\0\\-72\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-4\\0\\2\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-16\\-72\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-288\\-18\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-231912, 720, 5760; 720, -2, -18; 5760, -18, -143],[72, 2, 1],[0, 1, -1, -1, -3, -5; 6, 18, -36, -19, -44, -45; -1, 38, -36, -38, -116, -198],182,8]
%
% END OF LATTICE L183.8
%
\hbox{}\par\smallskip%
%
% LATTICE L183.9
%
\leavevmode%
\llap{\spacer}%
${L_{183.9}}=\main({L_{183.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{12,1}{\infty}{4}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30168\\-9576\\720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9576\\-3038\\228
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\228\\-17
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\9\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-12\\-34
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-9\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\20\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\25\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59\\252\\864
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-18\\-8\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
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-2\\4\\0\\-8\\-8\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\18\\0\\-18\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-8\\0\\8\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-8\\-18\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-288\\-72\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-30168, -9576, 720; -9576, -3038, 228; 720, 228, -17],[72, 2, 1],[-2, 3, 2, -5, -6, -59; 9, -12, -9, 20, 25, 252; 35, -34, -36, 56, 80, 864],182,9]
%
% END OF LATTICE L183.9
%
\hbox{}\par\smallskip%
%
% LATTICE L183.10
%
\leavevmode%
\llap{\spacer}%
${L_{183.10}}=\pfillwitharg{2}({L_{183.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{12,1}{\infty a}{8}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{144}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-136\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-7\\-12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\18\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\5
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-8\\-36\\-32\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-16\\-8\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\9\\0\\-9\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-16\\0\\16\\0\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-8\\-9\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-288\\-144\\0\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[144, 0, 0; 0, -136, 80; 0, 80, -47],[144, 8, 1],[0, 1, 2, 1, 0, -1; -7, 1, 18, 20, 3, 0; -12, 0, 27, 32, 5, 0],182,10]
%
% END OF LATTICE L183.10
%
\hbox{}\par\smallskip%
%
% LATTICE L183.11
%
\leavevmode%
\llap{\spacer}%
${L_{183.11}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{183.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{24,1}{\infty}{8}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{144}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-151920\\-56016\\6912
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-56016\\-20654\\2548
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6912\\2548\\-313
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\-123\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\54\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\63\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-88\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34\\-97\\-39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
857\\-2448\\-1008
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-4\\-18\\-16\\-2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\8\\0\\-16\\-8\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\9\\0\\-9\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-16\\0\\16\\0\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-8\\-9\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-288\\-144\\0\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-151920, -56016, 6912; -56016, -20654, 2548; 6912, 2548, -313],[144, 2, 1],[43, -19, -22, 31, 34, 857; -123, 54, 63, -88, -97, -2448; -52, 20, 27, -32, -39, -1008],182,11]
%
% END OF LATTICE L183.11
%
\hbox{}\par\smallskip%
%
% LATTICE L183.12
%
\leavevmode%
\llap{\spacer}%
${L_{183.12}}=\pdualwitharg{2}(\main({L_{183.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{12,1}{\infty}{8}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2232\\216\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\-20\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\8\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-26\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-144\\18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-25\\8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\9\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-8\\-72\\-16\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-8\\-16\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\36\\0\\-36\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-8\\0\\4\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-16\\-36\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-288\\-36\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2232, 216, -72; 216, -20, 8; -72, 8, -1],[72, 4, 1],[1, -3, -17, -3, -1, 1; 8, -26, -144, -25, -8, 9; -8, -4, 18, 8, 6, 0],182,12]
%
% END OF LATTICE L183.12
%
\hbox{}\par\smallskip%
%
% LATTICE L183.13
%
\leavevmode%
\llap{\spacer}%
${L_{183.13}}=\pdualwitharg{2}(\main({L_{183.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{6,1}{\infty a}{8}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-150840\\51840\\2664
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51840\\-17812\\-916
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2664\\-916\\-47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-14\\-124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\20\\232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\72\\747
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\29\\284
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\6\\53
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-9\\-108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-8\\-36\\-16\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-8\\-8\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\9\\0\\-9\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-8\\0\\4\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-8\\-9\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-144\\-36\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-150840, 51840, 2664; 51840, -17812, -916; 2664, -916, -47],[72, 4, 1],[-7, 11, 38, 15, 3, -5; -14, 20, 72, 29, 6, -9; -124, 232, 747, 284, 53, -108],182,13]
%
% END OF LATTICE L183.13
%
\hbox{}\par\smallskip%
%
% LATTICE L183.14
%
\leavevmode%
\llap{\spacer}%
${L_{183.14}}=\pdualwitharg{2}({L_{183.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{6,1}{\infty}{4}\above{1pt}{1pt}{b}{2}{72}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-576\\1080\\-288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1080\\-2008\\536
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\536\\-143
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-9\\-36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\5\\12
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\45\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\-36\\-8\\-4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-8\\-16\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\72\\0\\-72\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-8\\0\\8\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-16\\-72\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-576\\-72\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-576, 1080, -288; 1080, -2008, 536; -288, 536, -143],[72, 8, 1],[0, -1, 1, 2, 3, 10; 3, 0, -9, -1, 5, 45; 11, 2, -36, -8, 12, 144],182,14]
%
% END OF LATTICE L183.14
%
\hbox{}\par\smallskip%
%
% LATTICE L183.15
%
\leavevmode%
\llap{\spacer}%
${L_{183.15}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{183.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{12,5}{\infty}{18}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24012\\2808\\-10620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2808\\-306\\1242
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10620\\1242\\-4697
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\1\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-207\\-1\\468
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-104\\-1\\235
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-271\\-4\\612
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-1\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\0\\-52
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-18\\-18\\-72\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\18\\0\\-36\\-36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\1\\0\\-9\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-36\\0\\36\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-36\\-9\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-16\\-36\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24012, 2808, -10620; 2808, -306, 1242; -10620, 1242, -4697],[36, 18, 1],[24, -207, -104, -271, -20, 23; 1, -1, -1, -4, -1, 0; -54, 468, 235, 612, 45, -52],182,15]
%
% END OF LATTICE L183.15
%
\hbox{}\par\smallskip%
%
% LATTICE L183.16
%
\leavevmode%
\llap{\spacer}%
${L_{183.16}}=\pdualwitharg{3}({L_{183.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{3,2}{\infty b}{72}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-765432\\18288\\18360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18288\\-423\\-441
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18360\\-441\\-440
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-40\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127\\760\\4536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\179\\1072
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\434\\2610
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\17\\108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-42\\-250
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-72\\-36\\-144\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\72\\0\\-72\\-72\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\1\\0\\-9\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-72\\0\\36\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-72\\-9\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-16\\-36\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-765432, 18288, 18360; 18288, -423, -441; 18360, -441, -440],[72, 9, 1],[-7, 127, 30, 73, 3, -7; -40, 760, 179, 434, 17, -42; -252, 4536, 1072, 2610, 108, -250],182,16]
%
% END OF LATTICE L183.16
%
\hbox{}\par\smallskip%
%
% LATTICE L183.17
%
\leavevmode%
\llap{\spacer}%
${L_{183.17}}=\pdualwitharg{2}({L_{183.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{12,7}{\infty b}{8}\above{1pt}{1pt}{}{2}{144}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
144\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-56\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\9\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-8\\-144\\-16\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-8\\-32\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\0\\144\\0\\-144\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-8\\0\\4\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-32\\-144\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-576\\-36\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[144, 0, 0; 0, -56, 8; 0, 8, -1],[144, 8, 1],[1, 0, -1, 0, 1, 4; 1, -1, 0, 1, 4, 9; -4, -8, 0, 6, 16, 18],182,17]
%
% END OF LATTICE L183.17
%
\hbox{}\par\smallskip%
%
% LATTICE L183.18
%
\leavevmode%
\llap{\spacer}%
${L_{183.18}}=\pdualwitharg{2}({L_{183.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{12,7}{\infty a}{8}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-229680\\4320\\1728
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4320\\328\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1728\\-32\\-13
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-1\\932
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\1\\-800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\-936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-1\\666
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-4\\3064
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\-9\\7326
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-8\\-144\\-16\\-16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-8\\-32\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\0\\144\\0\\-144\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-8\\0\\4\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-32\\-144\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-576\\-36\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-229680, 4320, 1728; 4320, 328, -32; 1728, -32, -13],[144, 8, 1],[7, -6, -7, 5, 23, 55; -1, 1, 0, -1, -4, -9; 932, -800, -936, 666, 3064, 7326],182,18]
%
% END OF LATTICE L183.18
%
\hbox{}\par\smallskip%
%
% LATTICE L183.19
%
\leavevmode%
\llap{\spacer}%
${L_{183.19}}=\pdualwitharg{2}({L_{183.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{12,1}{\infty b}{8}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{144}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2552112\\21024\\18288
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21024\\-136\\-152
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18288\\-152\\-131
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-29\\-804
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\35\\936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55\\270\\7362
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\112\\3080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\24\\670
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-36\\-936
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-8\\-72\\-32\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\8\\0\\-16\\-16\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\36\\0\\-36\\-576
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-16\\0\\16\\0\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-16\\-36\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-576\\-144\\0\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2552112, 21024, 18288; 21024, -136, -152; 18288, -152, -131],[144, 8, 1],[-6, 7, 55, 23, 5, -7; -29, 35, 270, 112, 24, -36; -804, 936, 7362, 3080, 670, -936],182,19]
%
% END OF LATTICE L183.19
%
\hbox{}\par\smallskip%
%
% LATTICE L183.20
%
\leavevmode%
\llap{\spacer}%
${L_{183.20}}=\pdualwitharg{3}(\main({L_{183.3}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{12,11}{\infty}{36}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-407880\\2808\\7848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2808\\-18\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7848\\-54\\-151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\207
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-6\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\16\\252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\3\\154
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-18\\-36\\-36\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\36\\0\\-36\\-144\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\8\\0\\-72\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-36\\0\\18\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-144\\-72\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-32\\-18\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-407880, 2808, 7848; 2808, -18, -54; 7848, -54, -151],[72, 18, 1],[4, 1, -1, -1, 5, 3; 0, -6, 0, 5, 16, 3; 207, 54, -52, -54, 252, 154],182,20]
%
% END OF LATTICE L183.20
%
\hbox{}\par\smallskip%
%
% LATTICE L183.21
%
\leavevmode%
\llap{\spacer}%
${L_{183.21}}=\pdualwitharg{3}(\main({L_{183.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{12,5}{\infty}{36}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1656\\-13752\\1584
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13752\\-108774\\12546
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1584\\12546\\-1447
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\116\\1035
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-13\\-116
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\16\\144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\129\\1152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\216\\1928
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-18\\-18\\-72\\-18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\36\\0\\-72\\-72\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\0\\2\\0\\-18\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-72\\0\\72\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-72\\-18\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-32\\-72\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1656, -13752, 1584; -13752, -108774, 12546; 1584, 12546, -1447],[72, 18, 1],[27, -3, -3, 5, 31, 51; 116, -10, -13, 16, 129, 216; 1035, -90, -116, 144, 1152, 1928],182,21]
%
% END OF LATTICE L183.21
%
\hbox{}\par\smallskip%
%
% LATTICE L183.22
%
\leavevmode%
\llap{\spacer}%
${L_{183.22}}=\pdualwitharg{3}({L_{183.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{24,11}{\infty}{72}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-861264\\-168192\\19008
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168192\\-32814\\3708
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19008\\3708\\-419
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-175\\-1368
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-58\\-468
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\44\\344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\56\\450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-180\\-1368
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-124\\-962
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-36\\-72\\-72\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\72\\0\\-72\\-288\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\16\\0\\-144\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-72\\0\\36\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-288\\-144\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-64\\-36\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-861264, -168192, 19008; -168192, -32814, 3708; 19008, 3708, -419],[144, 18, 1],[4, 1, -1, -1, 5, 3; -175, -58, 44, 56, -180, -124; -1368, -468, 344, 450, -1368, -962],182,22]
%
% END OF LATTICE L183.22
%
\hbox{}\par\smallskip%
%
% LATTICE L183.23
%
\leavevmode%
\llap{\spacer}%
${L_{183.23}}=\pdualwitharg{3}({L_{183.3}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{24,23}{\infty}{72}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121104\\-112032\\13248
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-112032\\-103374\\12222
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13248\\12222\\-1445
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\81\\648
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\114\\900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-20\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-49\\-387
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-148\\-1152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\7\\59
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-36\\-72\\-36\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\72\\0\\-36\\-288\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\16\\0\\-144\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-36\\0\\9\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-288\\-144\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-32\\-9\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-121104, -112032, 13248; -112032, -103374, 12222; 13248, 12222, -1445],[144, 18, 1],[-4, -7, 1, 3, 11, 0; 81, 114, -20, -49, -148, 7; 648, 900, -160, -387, -1152, 59],182,23]
%
% END OF LATTICE L183.23
%
\hbox{}\par\smallskip%
%
% LATTICE L183.24
%
\leavevmode%
\llap{\spacer}%
${L_{183.24}}=\pdualwitharg{3}({L_{183.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{24,11}{\infty z}{18}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{s}{2}{144}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1833264\\24336\\24336
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24336\\-306\\-324
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24336\\-324\\-323
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\107\\1926
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\200\\3640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\120\\2214
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\16\\360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-12\\-214
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-36\\-144\\-144\\-144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\18\\0\\-36\\-144\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\0\\16\\0\\-144\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-36\\0\\36\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-144\\-144\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-64\\-36\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1833264, 24336, 24336; 24336, -306, -324; 24336, -324, -323],[144, 18, 1],[-3, 27, 51, 31, 5, -3; -10, 107, 200, 120, 16, -12; -216, 1926, 3640, 2214, 360, -214],182,24]
%
% END OF LATTICE L183.24
%
\hbox{}\par\smallskip%
%
% LATTICE L183.25
%
\leavevmode%
\llap{\spacer}%
${L_{183.25}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{183.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{24,23}{\infty z}{18}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{144}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3312\\1008\\2160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1008\\-306\\-648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2160\\-648\\-1295
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\215\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99\\392\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\114\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\16\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-12\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-36\\-144\\-72\\-144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\18\\0\\-18\\-144\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\0\\16\\0\\-144\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-18\\0\\9\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-144\\-144\\0\\144\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-18\\-32\\-9\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3312, 1008, 2160; 1008, -306, -648; 2160, -648, -1295],[144, 18, 1],[-3, 54, 99, 29, 5, -3; -10, 215, 392, 114, 16, -12; 0, -18, -32, -9, 0, 1],182,25]
%
% END OF LATTICE L183.25
%
\hbox{}\par\smallskip%
%
% LATTICE L183.26
%
\leavevmode%
\llap{\spacer}%
${L_{183.26}}=\pdualwitharg{2.3}(\main({L_{183.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{6,5}{\infty b}{72}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-504\\8352\\-936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8352\\-120780\\13536
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-936\\13536\\-1517
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\230\\2052
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-8\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-13\\-116
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\6\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\125\\1116
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\106\\946
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-72\\-72\\-144\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\72\\0\\-72\\-144\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\4\\0\\-36\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-72\\0\\36\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-144\\-36\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-32\\-36\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-504, 8352, -936; 8352, -120780, 13536; -936, 13536, -1517],[72, 36, 1],[-1, 1, 0, -1, -2, -1; 230, -8, -13, 6, 125, 106; 2052, -72, -116, 54, 1116, 946],182,26]
%
% END OF LATTICE L183.26
%
\hbox{}\par\smallskip%
%
% LATTICE L183.27
%
\leavevmode%
\llap{\spacer}%
${L_{183.27}}=\pdualwitharg{2.3}(\main({L_{183.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{12,11}{\infty}{72}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1224\\-12168\\-5328
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12168\\-112500\\-49212
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5328\\-49212\\-21527
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
127\\-990\\2232
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\48\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\55\\-124
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-20\\45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73\\-559\\1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30\\-232\\523
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-72\\-72\\-72\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\72\\0\\-36\\-144\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\4\\0\\-36\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-36\\0\\9\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-144\\-36\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-16\\-9\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1224, -12168, -5328; -12168, -112500, -49212; -5328, -49212, -21527],[72, 36, 1],[127, -7, -7, 3, 73, 30; -990, 48, 55, -20, -559, -232; 2232, -108, -124, 45, 1260, 523],182,27]
%
% END OF LATTICE L183.27
%
\hbox{}\par\smallskip%
%
% LATTICE L183.28
%
\leavevmode%
\llap{\spacer}%
${L_{183.28}}=\pdualwitharg{2.3}({L_{183.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{6,5}{\infty}{36}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187992\\-276912\\5328
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-276912\\-407880\\7848
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5328\\7848\\-151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\207
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\1\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-2\\72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\5\\540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\6\\416
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-18\\-36\\-72\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\36\\0\\-72\\-144\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\8\\0\\-72\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-72\\0\\72\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-144\\-72\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-64\\-72\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-187992, -276912, 5328; -276912, -407880, 7848; 5328, 7848, -151],[72, 72, 1],[0, -3, 0, 5, 8, 3; 4, 1, -1, -2, 5, 6; 207, -54, -52, 72, 540, 416],182,28]
%
% END OF LATTICE L183.28
%
\hbox{}\par\smallskip%
%
% LATTICE L183.29
%
\leavevmode%
\llap{\spacer}%
${L_{183.29}}=\pdualwitharg{2.3}({L_{183.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{24,17}{\infty}{72}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{l}{2}{144}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-909936\\1182528\\-236016
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1182528\\-1536696\\306720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-236016\\306720\\-61217
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
76\\1\\-288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-739\\-1\\2844
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-366\\-1\\1406
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-939\\-4\\3600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-62\\-1\\234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
83\\0\\-320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-72\\-72\\-288\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\72\\0\\-144\\-144\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\4\\0\\-36\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-144\\0\\144\\0\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-144\\-36\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-64\\-144\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-909936, 1182528, -236016; 1182528, -1536696, 306720; -236016, 306720, -61217],[144, 72, 1],[76, -739, -366, -939, -62, 83; 1, -1, -1, -4, -1, 0; -288, 2844, 1406, 3600, 234, -320],182,29]
%
% END OF LATTICE L183.29
%
\hbox{}\par\smallskip%
%
% LATTICE L183.30
%
\leavevmode%
\llap{\spacer}%
${L_{183.30}}=\pdualwitharg{2.3}({L_{183.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{24,5}{\infty}{72}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{b}{2}{144}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1260720\\-804384\\23472
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-804384\\-513144\\14976
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23472\\14976\\-437
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\1\\-288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\-1\\3348
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\-1\\1630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79\\-4\\4104
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-1\\234
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\0\\-376
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-72\\-72\\-288\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\72\\0\\-144\\-144\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\4\\0\\-36\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-144\\0\\144\\0\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-144\\-36\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-64\\-144\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1260720, -804384, 23472; -804384, -513144, 14976; 23472, 14976, -437],[144, 72, 1],[-6, 63, 31, 79, 5, -7; 1, -1, -1, -4, -1, 0; -288, 3348, 1630, 4104, 234, -376],182,30]
%
% END OF LATTICE L183.30
%
\hbox{}\par\smallskip%
%
% LATTICE L183.31
%
\leavevmode%
\llap{\spacer}%
${L_{183.31}}=\pdualwitharg{2.3}({L_{183.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{12,5}{\infty a}{72}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{144}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4494384\\74880\\82080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
74880\\-1224\\-1368
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
82080\\-1368\\-1499
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-5\\-324
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
63\\71\\3384
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\34\\1666
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79\\84\\4248
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\4\\270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-8\\-376
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-72\\-72\\-288\\-72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\72\\0\\-144\\-144\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\0\\4\\0\\-36\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-144\\0\\144\\0\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-144\\-36\\0\\36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-64\\-144\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4494384, 74880, 82080; 74880, -1224, -1368; 82080, -1368, -1499],[144, 72, 1],[-6, 63, 31, 79, 5, -7; -5, 71, 34, 84, 4, -8; -324, 3384, 1666, 4248, 270, -376],182,31]
%
% END OF LATTICE L183.31
%
\hbox{}\par\smallskip%
%
% LATTICE L183.32
%
\leavevmode%
\llap{\spacer}%
${L_{183.32}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{183.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{12,5}{\infty b}{72}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{144}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-909936\\-875376\\218736
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-875376\\-842040\\210384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
218736\\210384\\-52559
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\-161\\-324
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-692\\1439\\2880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-173\\360\\721
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-897\\1868\\3744
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\67\\135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
77\\-160\\-320
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-72\\-36\\-288\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\72\\0\\-144\\-72\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\1\\0\\-9\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-144\\0\\144\\0\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-72\\-9\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-32\\-144\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-909936, -875376, 218736; -875376, -842040, 210384; 218736, 210384, -52559],[144, 72, 1],[77, -692, -173, -897, -32, 77; -161, 1439, 360, 1868, 67, -160; -324, 2880, 721, 3744, 135, -320],182,32]
%
% END OF LATTICE L183.32
%
\hbox{}\par\smallskip%
%
% LATTICE L183.33
%
\leavevmode%
\llap{\spacer}%
${L_{183.33}}=\pdualwitharg{3}({L_{183.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${288}\above{1pt}{1pt}{48,23}{\infty z}{72}\above{1pt}{1pt}{b}{2}{64}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{s}{2}{576}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}$}%
\nopagebreak\par%
shares genus with {$ \pdualwitharg{3}({L_{183.6}})$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-109914048\\377280\\377280
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
377280\\-1224\\-1296
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
377280\\-1296\\-1295
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-864
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\215\\15516
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
99\\392\\28448
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\114\\8334
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\16\\1440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-12\\-862
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\-144\\-576\\-288\\-576\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\72\\0\\-72\\-576\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-576\\0\\64\\0\\-576\\-128
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-288\\-72\\0\\36\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-576\\-576\\-576\\0\\576\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-128\\-36\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-109914048, 377280, 377280; 377280, -1224, -1296; 377280, -1296, -1295],[576, 72, 1],[-3, 54, 99, 29, 5, -3; -10, 215, 392, 114, 16, -12; -864, 15516, 28448, 8334, 1440, -862],182,33]
%
% END OF LATTICE L183.33
%
\hbox{}\par\smallskip%
%
% LATTICE L183.34
%
\leavevmode%
\llap{\spacer}%
${L_{183.34}}=\pdualwitharg{3}({L_{183.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${288}\above{1pt}{1pt}{48,47}{\infty z}{72}\above{1pt}{1pt}{l}{2}{64}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{576}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}$}%
\nopagebreak\par%
shares genus with {$ \pdualwitharg{3}({L_{183.5}})$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1729561536\\1497024\\2994048
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1497024\\-1224\\-2592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2994048\\-2592\\-5183
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-10\\-1728
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108\\431\\62172
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
195\\776\\112256
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\111\\16119
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\16\\2880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-12\\-1727
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288\\-144\\-576\\-144\\-576\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\72\\0\\-36\\-576\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-576\\0\\64\\0\\-576\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-144\\-36\\0\\9\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-576\\-576\\-576\\0\\576\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-64\\-9\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1729561536, 1497024, 2994048; 1497024, -1224, -2592; 2994048, -2592, -5183],[576, 72, 1],[-3, 108, 195, 28, 5, -3; -10, 431, 776, 111, 16, -12; -1728, 62172, 112256, 16119, 2880, -1727],182,34]
%
% END OF LATTICE L183.34
%

\medskip%
%------------------------------WEYL-GROUP-W184--------------------------------
%
\leavevmode\llap{}%
$W_{184}$%
\qquad\llap{34} lattices, $\chi=72$%
\hfill%
$\slashinfty2|2\slashinfty2|2\slashinfty2|2\slashinfty2|2\rtimes D_{8}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L184.1
%
\leavevmode%
\llap{\spacer}%
${L_{184.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-639288, 7128, 7200; 7128, -79, -81; 7200, -81, -80],[72, 1, 1],[121, 23, -3, -3, 19, 15, 221, 319, 79, 117, 95, 23; 6480, 1224, -162, -160, 1024, 807, 11880, 17136, 4242, 6280, 5096, 1233; 4320, 828, -106, -108, 672, 532, 7848, 11340, 2810, 4164, 3384, 820],183,1]
%
% END OF LATTICE L184.1
%
\hbox{}\par\smallskip%
%
% LATTICE L184.2
%
\leavevmode%
\llap{\spacer}%
${L_{184.2}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{8,1}{\infty z}{18}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{24,11}{\infty}{8}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-32976\\-8208\\-3456
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8208\\-2042\\-856
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3456\\-856\\-345
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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\HardButStrongLineBreak\kern3pt%
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202\\-909\\234
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35\\-158\\42
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\HardButStrongLineBreak\kern3pt%
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\HardButStrongLineBreak\kern3pt%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-32976, -8208, -3456; -8208, -2042, -856; -3456, -856, -345],[144, 2, 1],[757, 202, 35, -3, -29, 1, 197, 275, 177, 162, 347, 423; -3402, -909, -158, 13, 130, -4, -882, -1233, -794, -727, -1558, -1900; 864, 234, 42, -2, -32, 0, 216, 306, 198, 182, 392, 480],183,2]
%
% END OF LATTICE L184.2
%
\hbox{}\par\smallskip%
%
% LATTICE L184.3
%
\leavevmode%
\llap{\spacer}%
${L_{184.3}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{8,5}{\infty z}{18}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{24,23}{\infty}{8}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1440\\-30\\-52
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2304\\-52\\-79
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-120\\-2064\\-2113
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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67\\1134\\1188
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\261\\288
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\14\\19
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-21\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-54\\-52
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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}}\!\right]$}%
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}}{}{\hbox{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-65808, 1440, 2304; 1440, -30, -52; 2304, -52, -79],[144, 2, 1],[67, 16, 1, -1, -3, 1, 23, 29, 9, 16, 33, 39; 1134, 261, 14, -21, -54, 20, 414, 513, 158, 279, 570, 668; 1188, 288, 19, -16, -52, 16, 396, 504, 157, 280, 580, 688],183,3]
%
% END OF LATTICE L184.3
%
\hbox{}\par\smallskip%
%
% LATTICE L184.4
%
\leavevmode%
\llap{\spacer}%
${L_{184.4}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-396720\\3744\\3744
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\kern3pt%
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3744\\-34\\-36
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3744\\-36\\-35
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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-18\\-108\\-36\\-68\\-38\\-96
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-396720, 3744, 3744; 3744, -34, -36; 3744, -36, -35],[144, 2, 1],[19, 7, -1, -1, 3, 19, 35, 101, 25, 37, 15, 29; 657, 234, -36, -34, 107, 672, 1233, 3546, 876, 1294, 523, 1008; 1350, 504, -70, -72, 210, 1336, 2466, 7128, 1766, 2616, 1062, 2056],183,4]
%
% END OF LATTICE L184.4
%
\hbox{}\par\smallskip%
%
% LATTICE L184.5
%
\leavevmode%
\llap{\spacer}%
${L_{184.5}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
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{${72}\above{1pt}{1pt}{8,1}{\infty b}{288}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{32}\above{1pt}{1pt}{48,23}{\infty z}{8}\above{1pt}{1pt}{b}{2}{64}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with {$ {L_{184.6}}$}%
; isometric to own %
2-dual\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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46080\\-144\\-143
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-14837184, 46080, 46080; 46080, -136, -144; 46080, -144, -143],[576, 8, 1],[62, 173, 21, 61, 24, 45, 28, 7, -1, -1, 6, 35; 2205, 6138, 744, 2158, 847, 1584, 981, 234, -36, -34, 215, 1248; 17748, 49536, 6014, 17472, 6876, 12896, 8028, 2016, -286, -288, 1716, 10016],183,5]
%
% END OF LATTICE L184.5
%
\hbox{}\par\smallskip%
%
% LATTICE L184.6
%
\leavevmode%
\llap{\spacer}%
${L_{184.6}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{8,1}{\infty a}{288}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{32}\above{1pt}{1pt}{48,47}{\infty z}{8}\above{1pt}{1pt}{l}{2}{64}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with {$ {L_{184.5}}$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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170496\\-136\\-288
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-202207680, 170496, 340992; 170496, -136, -288; 340992, -288, -575],[576, 8, 1],[116, 317, 19, 109, 42, 77, 46, 7, -1, -1, 12, 67; 4149, 11322, 678, 3886, 1495, 2736, 1629, 234, -36, -34, 431, 2400; 66708, 182304, 10927, 62688, 24156, 44288, 26460, 4032, -575, -576, 6900, 38528],183,6]
%
% END OF LATTICE L184.6
%
\hbox{}\par\smallskip%
%
% LATTICE L184.7
%
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${L_{184.7}}=\pfillwitharg{2}({L_{184.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1980, 288, -864; 288, -34, 126; -864, 126, -377],[36, 2, 1],[-223, -56, 7, 8, -21, -40, -317, -484, -247, -188, -159, -80; -27, -9, 0, 1, -1, -3, -27, -45, -24, -19, -17, -9; 504, 126, -16, -18, 48, 91, 720, 1098, 560, 426, 360, 181],183,7]
%
% END OF LATTICE L184.7
%
\hbox{}\par\smallskip%
%
% LATTICE L184.8
%
\leavevmode%
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${L_{184.8}}=\main({L_{184.3}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-6984, 720, -360; 720, -74, 38; -360, 38, -15],[72, 2, 1],[29, 23, 1, -3, -1, 2, 16, 67, 39, 33, 16, 18; 252, 198, 8, -26, -8, 19, 144, 594, 344, 290, 140, 157; -63, -54, -4, 6, 3, -2, -27, -126, -76, -66, -33, -38],183,8]
%
% END OF LATTICE L184.8
%
\hbox{}\par\smallskip%
%
% LATTICE L184.9
%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-19512, 1224, 576; 1224, -70, -36; 576, -36, -17],[72, 2, 1],[19, 7, -1, -1, 3, 19, 35, 101, 25, 37, 15, 29; -9, 0, 1, 0, -3, -16, -27, -72, -17, -24, -9, -16; 657, 234, -36, -34, 107, 672, 1233, 3546, 876, 1294, 523, 1008],183,9]
%
% END OF LATTICE L184.9
%
\hbox{}\par\smallskip%
%
% LATTICE L184.10
%
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${L_{184.10}}=\pfillwitharg{2}({L_{184.5}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-7920, 5184, 864; 5184, -3336, -536; 864, -536, -79],[144, 8, 1],[12, -25, -26, -425, -664, -343, -264, -227, -58, -331, -92, 7; 27, -57, -59, -963, -1503, -776, -597, -513, -131, -747, -207, 16; -52, 112, 115, 1872, 2916, 1504, 1156, 992, 253, 1440, 396, -32],183,10]
%
% END OF LATTICE L184.10
%
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%
% LATTICE L184.11
%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
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-36\\72\\0\\-72\\-108\\-432
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\kern3pt%
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-108\\-72\\0\\8\\-4\\-48
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-90\\-108\\-6\\-4\\2\\0
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\kern3pt%
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-612\\-1368\\-144\\-360\\-108\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-144\\-17\\-48\\-18\\-32
\end{tabular}\endgroup%
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-108\\-360\\-48\\-152\\-68\\-144
\end{tabular}\endgroup%
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-18\\-108\\-18\\-68\\-38\\-96
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-144\\-32\\-144\\-96\\-272
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-39024, 1152, 2448; 1152, -34, -72; 2448, -72, -143],[144, 2, 1],[28, 7, -1, -1, 6, 35, 62, 173, 21, 61, 24, 45; 981, 234, -36, -34, 215, 1248, 2205, 6138, 744, 2158, 847, 1584; -18, 0, 1, 0, -6, -32, -54, -144, -17, -48, -18, -32],183,11]
%
% END OF LATTICE L184.11
%
\hbox{}\par\smallskip%
%
% LATTICE L184.12
%
\leavevmode%
\llap{\spacer}%
${L_{184.12}}=\pdualwitharg{2}(\main({L_{184.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{4,1}{\infty}{72}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{6,5}{\infty b}{8}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2166840\\262152\\-21888
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
262152\\-31716\\2648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21888\\2648\\-221
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-126667\\-1127736\\-964296
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15314\\136343\\116584
\end{tabular}\endgroup%
\kern3pt%
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-1271\\-11316\\-9677
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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521\\4644\\4032
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
361\\3222\\2844
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49\\438\\394
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31\\278\\260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-44\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-45\\-44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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72\\-72\\-72\\-216\\-360\\-288
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-72\\72\\0\\-72\\-216\\-216
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-72\\0\\4\\0\\-24\\-32
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-216\\-72\\0\\8\\-8\\-24
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-288\\-216\\-32\\-24\\0\\4
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-1368\\-1224\\-216\\-216\\-72\\0
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\kern3pt%
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-216\\-288\\-68\\-96\\-72\\-32
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-216\\-360\\-96\\-152\\-136\\-72
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-72\\-216\\-72\\-136\\-152\\-96
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-32\\-72\\-96\\-68
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2166840, 262152, -21888; 262152, -31716, 2648; -21888, 2648, -221],[72, 4, 1],[521, 361, 49, 31, -5, -5, 37, 197, 75, 155, 191, 129; 4644, 3222, 438, 278, -44, -45, 324, 1746, 666, 1378, 1700, 1149; 4032, 2844, 394, 260, -32, -44, 216, 1404, 550, 1156, 1448, 988],183,12]
%
% END OF LATTICE L184.12
%
\hbox{}\par\smallskip%
%
% LATTICE L184.13
%
\leavevmode%
\llap{\spacer}%
${L_{184.13}}=\pdualwitharg{2}(\main({L_{184.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{2,1}{\infty a}{72}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{12,11}{\infty}{8}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23688\\1872\\1368
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1872\\-140\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1368\\-108\\-79
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6209\\-1656\\109296
\end{tabular}\endgroup%
\kern3pt%
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-360\\96\\-6337
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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247\\-72\\4356
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
167\\-54\\2952
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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11\\-4\\195
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\HardButStrongLineBreak\kern3pt%
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13\\-6\\232
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\HardButStrongLineBreak\kern3pt%
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-3\\0\\-52
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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72\\-72\\-36\\-216\\-360\\-288
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-72\\72\\0\\-72\\-216\\-216
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-36\\0\\1\\0\\-12\\-16
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-216\\-72\\0\\8\\-8\\-24
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-1368\\-1224\\-108\\-216\\-72\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
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-108\\-144\\-17\\-48\\-36\\-16
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-216\\-360\\-48\\-152\\-136\\-72
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-72\\-216\\-36\\-136\\-152\\-96
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-72\\-16\\-72\\-96\\-68
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-23688, 1872, 1368; 1872, -140, -108; 1368, -108, -79],[72, 4, 1],[247, 167, 11, 13, -3, -2, 23, 103, 19, 77, 93, 62; -72, -54, -4, -6, 0, 1, 0, -18, -4, -18, -24, -17; 4356, 2952, 195, 232, -52, -36, 396, 1800, 333, 1352, 1636, 1092],183,13]
%
% END OF LATTICE L184.13
%
\hbox{}\par\smallskip%
%
% LATTICE L184.14
%
\leavevmode%
\llap{\spacer}%
${L_{184.14}}=\pdualwitharg{2}({L_{184.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{2,1}{\infty}{36}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{6,5}{\infty z}{1}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-16920, -1872, 504; -1872, -200, 56; 504, 56, -15],[72, 8, 1],[10, 7, 0, -1, 0, 3, 8, 29, 16, 13, 6, 13; -9, -9, -1, 1, 1, 2, 0, -9, -7, -7, -4, -10; 297, 198, -4, -30, 3, 104, 261, 918, 500, 402, 183, 392],183,14]
%
% END OF LATTICE L184.14
%
\hbox{}\par\smallskip%
%
% LATTICE L184.15
%
\leavevmode%
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${L_{184.15}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{184.2}}))$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[36, 0, 0; 0, -5922, 684; 0, 684, -79],[36, 18, 1],[9, 12, 17, 4, 3, 4, 3, 0, -1, 0, 1, 4; 59, 103, 168, 45, 41, 66, 67, 23, 0, -3, 1, 18; 504, 882, 1440, 386, 352, 567, 576, 198, 0, -26, 8, 153],183,15]
%
% END OF LATTICE L184.15
%
\hbox{}\par\smallskip%
%
% LATTICE L184.16
%
\leavevmode%
\llap{\spacer}%
${L_{184.16}}=\pdualwitharg{3}({L_{184.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-260280, 4608, 4536; 4608, -72, -81; 4536, -81, -79],[72, 9, 1],[113, 175, 135, 69, 59, 45, 85, 23, -3, -3, 7, 21; 416, 652, 506, 260, 224, 172, 328, 92, -10, -12, 24, 76; 6048, 9360, 7218, 3688, 3152, 2403, 4536, 1224, -162, -160, 376, 1125],183,16]
%
% END OF LATTICE L184.16
%
\hbox{}\par\smallskip%
%
% LATTICE L184.17
%
\leavevmode%
\llap{\spacer}%
${L_{184.17}}=\pdualwitharg{2}({L_{184.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7920, 144, -1728; 144, 24, 32; -1728, 32, -377],[144, 8, 1],[7, -20, -37, -290, -439, -223, -169, -142, -71, -196, -47, 7; 1, -1, -3, -27, -45, -24, -19, -17, -9, -27, -9, 0; -32, 92, 170, 1332, 2016, 1024, 776, 652, 326, 900, 216, -32],183,17]
%
% END OF LATTICE L184.17
%
\hbox{}\par\smallskip%
%
% LATTICE L184.18
%
\leavevmode%
\llap{\spacer}%
${L_{184.18}}=\pdualwitharg{2}({L_{184.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1208880, -25056, 9648; -25056, -504, 200; 9648, 200, -77],[144, 8, 1],[-2, 9, 15, 113, 166, 83, 62, 51, 25, 67, 14, -3; 1, -1, -3, -27, -45, -24, -19, -17, -9, -27, -9, 0; -248, 1124, 1870, 14076, 20664, 10328, 7712, 6340, 3106, 8316, 1728, -376],183,18]
%
% END OF LATTICE L184.18
%
\hbox{}\par\smallskip%
%
% LATTICE L184.19
%
\leavevmode%
\llap{\spacer}%
${L_{184.19}}=\pdualwitharg{2}({L_{184.4}})$%
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{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3863088, 23328, 29088; 23328, -136, -176; 29088, -176, -219],[144, 8, 1],[-2, 9, 15, 113, 166, 83, 62, 51, 25, 67, 14, -3; -17, 83, 136, 1017, 1485, 740, 551, 451, 220, 585, 117, -28; -252, 1128, 1882, 14184, 20844, 10424, 7788, 6408, 3142, 8424, 1764, -376],183,19]
%
% END OF LATTICE L184.19
%
\hbox{}\par\smallskip%
%
% LATTICE L184.20
%
\leavevmode%
\llap{\spacer}%
${L_{184.20}}=\pdualwitharg{3}(\main({L_{184.3}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-198504, 2016, 5328; 2016, -18, -54; 5328, -54, -143],[72, 18, 1],[-9, -12, -23, -7, -15, -55, -33, -9, -5, 0, 1, -1; 14, 26, 55, 18, 42, 164, 106, 34, 25, 2, -2, -4; -342, -459, -882, -269, -578, -2124, -1278, -351, -198, -1, 38, -36],183,20]
%
% END OF LATTICE L184.20
%
\hbox{}\par\smallskip%
%
% LATTICE L184.21
%
\leavevmode%
\llap{\spacer}%
${L_{184.21}}=\pdualwitharg{3}(\main({L_{184.4}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19512, 1224, 576; 1224, -54, -36; 576, -36, -17],[72, 18, 1],[7, 13, 55, 9, 21, 41, 53, 17, 25, 1, -1, -1; 0, -3, -16, -3, -8, -17, -24, -9, -16, -1, 0, 1; 234, 441, 1872, 307, 718, 1404, 1818, 585, 864, 35, -34, -36],183,21]
%
% END OF LATTICE L184.21
%
\hbox{}\par\smallskip%
%
% LATTICE L184.22
%
\leavevmode%
\llap{\spacer}%
${L_{184.22}}=\pdualwitharg{3}({L_{184.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-397008, -112032, 12672; -112032, -31518, 3564; 12672, 3564, -403],[144, 18, 1],[-15, -7, -23, -12, -9, -1, 1, 0, -5, -9, -33, -55; 662, 305, 992, 511, 370, 28, -42, 5, 248, 419, 1490, 2452; 5380, 2476, 8046, 4140, 2988, 216, -340, 44, 2034, 3420, 12132, 19944],183,22]
%
% END OF LATTICE L184.22
%
\hbox{}\par\smallskip%
%
% LATTICE L184.23
%
\leavevmode%
\llap{\spacer}%
${L_{184.23}}=\pdualwitharg{3}({L_{184.3}})$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-277776, -48096, 4896; -48096, -7902, 792; 4896, 792, -79],[144, 18, 1],[57, 25, 39, 38, 23, -3, -3, 2, 15, 43, 139, 219; -1474, -647, -1010, -985, -598, 76, 78, -51, -386, -1109, -3590, -5660; -11252, -4940, -7713, -7524, -4572, 576, 596, -388, -2943, -8460, -27396, -43200],183,23]
%
% END OF LATTICE L184.23
%
\hbox{}\par\smallskip%
%
% LATTICE L184.24
%
\leavevmode%
\llap{\spacer}%
${L_{184.24}}=\pdualwitharg{3}({L_{184.4}})$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-60336, -24480, 2880; -24480, -9666, 1134; 2880, 1134, -133],[144, 18, 1],[9, 21, 41, 53, 17, 25, 1, -1, -1, 7, 13, 55; -241, -566, -1110, -1442, -467, -696, -29, 26, 30, -178, -343, -1464; -1862, -4376, -8586, -11160, -3618, -5400, -226, 200, 234, -1368, -2646, -11304],183,24]
%
% END OF LATTICE L184.24
%
\hbox{}\par\smallskip%
%
% LATTICE L184.25
%
\leavevmode%
\llap{\spacer}%
${L_{184.25}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{184.5}}))$%
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{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-39024, -29952, 3600; -29952, -21762, 2610; 3600, 2610, -313],[144, 18, 1],[12, 29, 29, 77, 26, 41, 2, -1, -1, 7, 16, 71; -421, -1022, -1025, -2730, -927, -1472, -73, 34, 37, -234, -555, -2480; -3374, -8192, -8217, -21888, -7434, -11808, -586, 272, 297, -1872, -4446, -19872],183,25]
%
% END OF LATTICE L184.25
%
\hbox{}\par\smallskip%
%
% LATTICE L184.26
%
\leavevmode%
\llap{\spacer}%
${L_{184.26}}=\pdualwitharg{2.3}(\main({L_{184.3}}))$%
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{$1\above{1pt}{1pt}{1}{7}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8568, 7416, 1872; 7416, -6372, -1620; 1872, -1620, -409],[72, 36, 1],[141, 121, 185, 175, 47, -7, -7, 13, 83, 227, 355, 275; -8, -6, -8, -6, 0, 1, 0, -2, -8, -18, -24, -17; 680, 580, 882, 828, 216, -36, -32, 68, 414, 1116, 1728, 1332],183,26]
%
% END OF LATTICE L184.26
%
\hbox{}\par\smallskip%
%
% LATTICE L184.27
%
\leavevmode%
\llap{\spacer}%
${L_{184.27}}=\pdualwitharg{2.3}(\main({L_{184.4}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{2,1}{\infty a}{8}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{12,7}{\infty}{72}\above{1pt}{1pt}{l}{2}{36}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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1872\\-108\\-108
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1368\\-108\\-79
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-23688, 1872, 1368; 1872, -108, -108; 1368, -108, -79],[72, 36, 1],[61, 53, 41, 79, 23, -2, -3, 5, 17, 95, 151, 118; -8, -6, -4, -6, 0, 1, 0, -2, -4, -18, -24, -17; 1060, 920, 711, 1368, 396, -36, -52, 88, 297, 1656, 2628, 2052],183,27]
%
% END OF LATTICE L184.27
%
\hbox{}\par\smallskip%
%
% LATTICE L184.28
%
\leavevmode%
\llap{\spacer}%
${L_{184.28}}=\pdualwitharg{2.3}({L_{184.1}})$%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-177624, -187776, 5040; -187776, -198504, 5328; 5040, 5328, -143],[72, 72, 1],[7, 13, 55, 9, 21, 82, 53, 17, 25, 1, -1, -2; -9, -12, -46, -7, -15, -55, -33, -9, -10, 0, 1, -1; -90, 9, 216, 55, 178, 828, 630, 261, 504, 35, 2, -108],183,28]
%
% END OF LATTICE L184.28
%
\hbox{}\par\smallskip%
%
% LATTICE L184.29
%
\leavevmode%
\llap{\spacer}%
${L_{184.29}}=\pdualwitharg{2.3}({L_{184.3}})$%
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{$1\above{1pt}{1pt}{1}{7}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[144, -144, 0; -144, -8424, -1872; 0, -1872, -409],[144, 72, 1],[-81, -76, -125, -130, -47, -1, 7, 2, -23, -92, -175, -295; -85, -79, -129, -133, -47, 0, 7, 1, -27, -101, -187, -312; 392, 364, 594, 612, 216, 0, -32, -4, 126, 468, 864, 1440],183,29]
%
% END OF LATTICE L184.29
%
\hbox{}\par\smallskip%
%
% LATTICE L184.30
%
\leavevmode%
\llap{\spacer}%
${L_{184.30}}=\pdualwitharg{2.3}({L_{184.2}})$%
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{$1\above{1pt}{1pt}{-}{3}[8\above{1pt}{1pt}{-}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-65808, -18576, -3888; -18576, -3960, -792; -3888, -792, -157],[144, 72, 1],[38, 33, 51, 49, 14, -3, -2, 3, 21, 59, 94, 147; -1105, -961, -1487, -1431, -411, 86, 59, -85, -605, -1707, -2727, -4270; 4640, 4036, 6246, 6012, 1728, -360, -248, 356, 2538, 7164, 11448, 17928],183,30]
%
% END OF LATTICE L184.30
%
\hbox{}\par\smallskip%
%
% LATTICE L184.31
%
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\llap{\spacer}%
${L_{184.31}}=\pdualwitharg{2.3}({L_{184.4}})$%
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{$1\above{1pt}{1pt}{-}{5}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-432, 15840, -288; 15840, -566856, 10296; -288, 10296, -187],[144, 72, 1],[38, 33, 51, 49, 14, -3, -2, 3, 21, 59, 94, 147; 23, 19, 28, 25, 5, -4, -1, 3, 16, 41, 61, 92; 1204, 992, 1458, 1296, 252, -216, -52, 160, 846, 2160, 3204, 4824],183,31]
%
% END OF LATTICE L184.31
%
\hbox{}\par\smallskip%
%
% LATTICE L184.32
%
\leavevmode%
\llap{\spacer}%
${L_{184.32}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{184.5}}))$%
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{$1\above{1pt}{1pt}{1}{1}[8\above{1pt}{1pt}{1}{}16\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-94608, 141984, -2736; 141984, -213048, 4104; -2736, 4104, -79],[144, 72, 1],[57, 50, 39, 76, 23, -3, -3, 4, 15, 86, 139, 219; 53, 47, 37, 73, 23, -2, -3, 3, 13, 77, 127, 202; 772, 704, 567, 1152, 396, 0, -52, 16, 153, 1008, 1764, 2880],183,32]
%
% END OF LATTICE L184.32
%
\hbox{}\par\smallskip%
%
% LATTICE L184.33
%
\leavevmode%
\llap{\spacer}%
${L_{184.33}}=\pdualwitharg{3}({L_{184.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{8,1}{\infty b}{32}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{*}{2}{288}\above{1pt}{1pt}{48,7}{\infty z}{72}\above{1pt}{1pt}{b}{2}{576}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with {$ \pdualwitharg{3}({L_{184.6}})$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
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-607680\\-156672\\18432
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-156672\\-40392\\4752
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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4591\\-17712\\0
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0\\0\\0
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16\\-49\\108
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-607680, -156672, 18432; -156672, -40392, 4752; 18432, 4752, -559],[576, 72, 1],[2, -1, -1, 7, 16, 71, 12, 29, 29, 77, 26, 41; -11, 2, 6, -10, -49, -240, -43, -110, -114, -314, -113, -192; -28, -16, 18, 144, 108, 288, 28, 16, -18, -144, -108, -288],183,33]
%
% END OF LATTICE L184.33
%
\hbox{}\par\smallskip%
%
% LATTICE L184.34
%
\leavevmode%
\llap{\spacer}%
${L_{184.34}}=\pdualwitharg{3}({L_{184.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{1}{7}64\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{8,1}{\infty a}{32}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{288}\above{1pt}{1pt}{48,31}{\infty z}{72}\above{1pt}{1pt}{l}{2}{576}\above{1pt}{1pt}{r}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with {$ \pdualwitharg{3}({L_{184.5}})$}%
; isometric to own %
2-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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-11952576\\-1032192\\123840
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-1032192\\-88776\\10656
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
123840\\10656\\-1279
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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34671\\3192976\\29956608
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2926\\269457\\2528064
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-352\\-32416\\-304129
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0\\0\\0
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4\\365\\3428
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-1\\-94\\-880
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7\\662\\6192
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22\\2039\\19116
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103\\9520\\89280
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{$\left.\llap{\phantom{%
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0\\0\\0
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-72\\-432\\-324\\-2448\\-1368\\-3456
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-576\\-576\\-5184\\-3456\\-9792
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-11952576, -1032192, 123840; -1032192, -88776, 10656; 123840, 10656, -1279],[576, 72, 1],[4, -1, -1, 7, 22, 103, 18, 45, 23, 125, 44, 73; 365, -94, -91, 662, 2039, 9520, 1661, 4146, 2117, 11494, 4039, 6688; 3428, -880, -855, 6192, 19116, 89280, 15580, 38896, 19863, 107856, 37908, 62784],183,34]
%
% END OF LATTICE L184.34
%

\medskip%
%------------------------------WEYL-GROUP-W185--------------------------------
%
\leavevmode\llap{}%
$W_{185}$%
\qquad\llap{12} lattices, $\chi=30$%
\hfill%
$22242224\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L185.1
%
\leavevmode%
\llap{\spacer}%
${L_{185.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{50}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-221700\\73800\\1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73800\\-24566\\-601
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1800\\-601\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61499\\183000\\70500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20541\\-61123\\-23547
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-328\\-976\\-377
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\3\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
294\\875\\325
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
124\\369\\141
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
287\\854\\330
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-6\\-28\\-27\\-125\\-9\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\50\\0\\-100\\-125\\-675\\-75\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-9\\-75\\-15\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-100\\0\\4\\-2\\-50\\-18\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-6\\-28
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\50\\0\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28\\-100\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-125\\-9\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-125\\-675\\-75\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-75\\-15\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-50\\-18\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-221700, 73800, 1800; 73800, -24566, -601; 1800, -601, -10],[300, 1, 1],[1, 294, 124, 287, 204, 731, -1, -41; 3, 875, 369, 854, 607, 2175, -3, -122; -1, 325, 141, 330, 236, 850, 0, -48],184,1]
%
% END OF LATTICE L185.1
%
\hbox{}\par\smallskip%
%
% LATTICE L185.2
%
\leavevmode%
\llap{\spacer}%
${L_{185.2}}=\pfillwitharg{2}({L_{185.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{s}{2}{50}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-479550\\5025\\6750
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5025\\-46\\-71
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6750\\-71\\-95
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30524\\86025\\2103450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-341\\-962\\-23498
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-429\\-1209\\-29563
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\166\\4066
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
283\\800\\19500
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\66\\1584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\12\\275
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-6\\-14\\-27\\-125\\-9\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\50\\0\\-50\\-125\\-675\\-75\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0\\-9\\-75\\-15\\-9
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-50\\0\\1\\-1\\-25\\-9\\-10
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-6\\-14
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\50\\0\\-50
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\6\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-50\\0\\1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-125\\-9\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-125\\-675\\-75\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-75\\-15\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-25\\-9\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-479550, 5025, 6750; 5025, -46, -71; 6750, -71, -95],[75, 1, 1],[59, 283, 23, 4, -4, -8, 10, 29; 166, 800, 66, 12, -11, -25, 27, 81; 4066, 19500, 1584, 275, -276, -550, 690, 1999],184,2]
%
% END OF LATTICE L185.2
%
\hbox{}\par\smallskip%
%
% LATTICE L185.3
%
\leavevmode%
\llap{\spacer}%
${L_{185.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{185.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{s}{2}{150}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72992850\\-123675\\23313450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-123675\\210\\-39501
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23313450\\-39501\\7446167
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15388124\\561875\\-48176250
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26319\\-962\\82398
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4914861\\179459\\-15387163
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4222\\153\\-13218
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20698\\775\\-64800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
612\\25\\-1916
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
505\\23\\-1581
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-6\\-42\\-81\\-375\\-9\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\150\\0\\-150\\-375\\-2025\\-75\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0\\-9\\-75\\-5\\-9
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-150\\0\\3\\-3\\-75\\-9\\-30
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-6\\-42
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\150\\0\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42\\-150\\0\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81\\-375\\-9\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-375\\-2025\\-75\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-75\\-5\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-75\\-9\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[72992850, -123675, 23313450; -123675, 210, -39501; 23313450, -39501, 7446167],[75, 3, 1],[4222, 20698, 612, 505, 23, 527, 237, 2042; 153, 775, 25, 23, 2, 0, 6, 70; -13218, -64800, -1916, -1581, -72, -1650, -742, -6393],184,3]
%
% END OF LATTICE L185.3
%
\hbox{}\par\smallskip%
%
% LATTICE L185.4
%
\leavevmode%
\llap{\spacer}%
${L_{185.4}}=\pdualwitharg{3}({L_{185.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{b}{2}{150}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-753900\\10200\\5100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10200\\-138\\-69
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5100\\-69\\-34
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1399\\105000\\-4200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-1426\\57
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-675\\26
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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7\\550\\-75
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\HardButStrongLineBreak\kern3pt%
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7\\524\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-753900, 10200, 5100; 10200, -138, -69; 5100, -69, -34],[300, 3, 1],[0, 7, 1, 7, 5, 18, 0, -1; 4, 550, 76, 524, 371, 1325, -1, -74; -9, -75, -5, -18, -6, 0, 2, 0],184,4]
%
% END OF LATTICE L185.4
%
\hbox{}\par\smallskip%
%
% LATTICE L185.5
%
\leavevmode%
\llap{\spacer}%
${L_{185.5}}=\pdualwitharg{2}({L_{185.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69484200\\973500\\-17275500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
973500\\13640\\-242036
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17275500\\-242036\\4295119
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-4941\\6625\\-19500
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-1195\\1605\\-4716
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[69484200, 973500, -17275500; 973500, 13640, -242036; -17275500, -242036, 4295119],[300, 4, 1],[-525, -4941, -1195, -1079, -1335, -4359, 79, -37; 703, 6625, 1605, 1453, 1802, 5900, -102, 50; -2072, -19500, -4716, -4258, -5268, -17200, 312, -146],184,5]
%
% END OF LATTICE L185.5
%
\hbox{}\par\smallskip%
%
% LATTICE L185.6
%
\leavevmode%
\llap{\spacer}%
${L_{185.6}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{185.1}}))$%
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{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
992625\\-1027575\\315975
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1027575\\1063775\\-327100
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
315975\\-327100\\100582
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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366\\23\\-1075
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[992625, -1027575, 315975; -1027575, 1063775, -327100; 315975, -327100, 100582],[75, 25, 1],[3376, 654, 1400, 366, 9, 23, 631, 1665; 194, 38, 84, 23, 1, 1, 33, 94; -9975, -1931, -4125, -1075, -25, -69, -1875, -4925],184,6]
%
% END OF LATTICE L185.6
%
\hbox{}\par\smallskip%
%
% LATTICE L185.7
%
\leavevmode%
\llap{\spacer}%
${L_{185.7}}=\pdualwitharg{2.3}({L_{185.1}})$%
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{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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72600\\21300\\-18300
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\kern3pt%
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21300\\15096\\-5328
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[72600, 21300, -18300; 21300, 15096, -5328; -18300, -5328, 4613],[300, 12, 1],[-1262, -12316, -1038, -3000, -3923, -13609, 1, -111; 23, 225, 19, 55, 72, 250, 0, 2; -4980, -48600, -4096, -11838, -15480, -53700, 4, -438],184,7]
%
% END OF LATTICE L185.7
%
\hbox{}\par\smallskip%
%
% LATTICE L185.8
%
\leavevmode%
\llap{\spacer}%
${L_{185.8}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{185.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${150}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{s}{2}{50}\above{1pt}{1pt}{l}{2}{75}\above{1pt}{1pt}{}{4}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-7125\\-751575\\-245550
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-751575\\-78280950\\-25575675
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\kern3pt%
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-245550\\-25575675\\-8355994
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{$\left.\llap{\phantom{%
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0\\0\\0
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7125, -751575, -245550; -751575, -78280950, -25575675; -245550, -25575675, -8355994],[75, 75, 1],[4066, 780, 528, 275, -276, -22, 230, 1999; 62666, 12016, 8122, 4212, -4261, -335, 3559, 30831; -191925, -36801, -24875, -12900, 13050, 1026, -10900, -94425],184,8]
%
% END OF LATTICE L185.8
%
\hbox{}\par\smallskip%
%
% LATTICE L185.9
%
\leavevmode%
\llap{\spacer}%
${L_{185.9}}=\pdualwitharg{5}({L_{185.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\kern3pt%
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-41400\\-250\\-20925
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5916900, -41400, -3001800; -41400, -250, -20925; -3001800, -20925, -1522738],[300, 25, 1],[-37, -266, -2776, -6391, -4528, -647, 37, 913; -151, -1065, -11109, -25570, -18114, -2588, 150, 3652; 75, 539, 5625, 12950, 9175, 1311, -75, -1850],184,9]
%
% END OF LATTICE L185.9
%
\hbox{}\par\smallskip%
%
% LATTICE L185.10
%
\leavevmode%
\llap{\spacer}%
${L_{185.10}}=\pdualwitharg{3.5}({L_{185.1}})$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-2316300\\28957800\\-1160100
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\kern3pt%
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28957800\\-362015850\\14502975
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2316300, 28957800, -1160100; 28957800, -362015850, 14502975; -1160100, 14502975, -581014],[300, 75, 1],[-48, -103, -337, -2263, -1577, -222, 12, 313; -109, -217, -705, -4718, -3281, -461, 27, 650; -2625, -5211, -16925, -113250, -78750, -11064, 650, 15600],184,10]
%
% END OF LATTICE L185.10
%
\hbox{}\par\smallskip%
%
% LATTICE L185.11
%
\leavevmode%
\llap{\spacer}%
${L_{185.11}}=\pdualwitharg{2.5}({L_{185.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${200}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{600}\above{1pt}{1pt}{*}{2}{100}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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704160600\\-463372500\\525916800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-463372500\\304922600\\-346079700
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
525916800\\-346079700\\392792087
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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142630010\\-427934409\\-568012800
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\kern3pt%
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-93929589\\281817990\\374067200
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106580385\\-319774315\\-424448001
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-31489\\94474\\125400
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-12247\\36744\\48772
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-77215\\231666\\307500
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-74289\\222890\\295850
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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200\\0\\-600\\-1400
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0\\8\\0\\-200
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-600\\0\\600\\0
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-1400\\-200\\0\\100
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-2700\\-500\\-900\\-100
\end{tabular}\endgroup%
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-500\\-108\\-300\\-100
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\-300\\-1500\\-900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-100\\-900\\-1000
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[704160600, -463372500, 525916800; -463372500, 304922600, -346079700; 525916800, -346079700, 392792087],[300, 100, 1],[-31489, -12247, -77215, -74289, -97076, -13466, 76, -2850; 94474, 36744, 231666, 222890, 291261, 40403, -225, 8551; 125400, 48772, 307500, 295850, 386600, 53628, -300, 11350],184,11]
%
% END OF LATTICE L185.11
%
\hbox{}\par\smallskip%
%
% LATTICE L185.12
%
\leavevmode%
\llap{\spacer}%
${L_{185.12}}=\pdualwitharg{2.3.5}({L_{185.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${600}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{200}\above{1pt}{1pt}{*}{2}{300}\above{1pt}{1pt}{*}{4}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29281800\\-646434300\\163951200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-646434300\\14270904600\\-3619442700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
163951200\\-3619442700\\917977229
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-4129699\\18693816\\74444400
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\kern3pt%
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91212725\\-412890701\\-1644255000
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-23133618\\104718456\\417020399
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-7415\\33599\\133800
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-2889\\13087\\52116
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-6083\\27547\\109700
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17599\\79665\\317250
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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600\\0\\-600\\-4200\\-8100\\-1500\\-900\\-300
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0
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600\\0\\-600\\-4200
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0\\24\\0\\-600
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-600\\0\\200\\0
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\kern3pt%
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-4200\\-600\\0\\300
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8100\\-1500\\-900\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1500\\-324\\-300\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\-300\\-500\\-900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\-300\\-900\\-3000
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[29281800, -646434300, 163951200; -646434300, 14270904600, -3619442700; 163951200, -3619442700, 917977229],[300, 300, 1],[-7415, -2889, -6083, -17599, -23040, -3202, -8, -674; 33599, 13087, 27547, 79665, 104261, 14485, 25, 3051; 133800, 52116, 109700, 317250, 415200, 57684, 100, 12150],184,12]
%
% END OF LATTICE L185.12
%

\medskip%
%------------------------------WEYL-GROUP-W186--------------------------------
%
\leavevmode\llap{}%
$W_{186}$%
\qquad\llap{12} lattices, $\chi=10$%
\hfill%
$22226$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L186.1
%
\leavevmode%
\llap{\spacer}%
${L_{186.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{b}{2}{100}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{150}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2716400100\\11466900\\-790200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11466900\\-48386\\3309
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-790200\\3309\\-194
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
161\\40203\\29946
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
343\\85650\\63800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-137\\-34210\\-25482
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2656\\-663225\\-494025
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-168\\-41951\\-31249
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-6\\-75\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\100\\0\\-450\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4\\0\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-450\\0\\150\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-50\\-4\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2716400100, 11466900, -790200; 11466900, -48386, 3309; -790200, 3309, -194],[300, 1, 1],[161, 343, -137, -2656, -168; 40203, 85650, -34210, -663225, -41951; 29946, 63800, -25482, -494025, -31249],185,1]
%
% END OF LATTICE L186.1
%
\hbox{}\par\smallskip%
%
% LATTICE L186.2
%
\leavevmode%
\llap{\spacer}%
${L_{186.2}}=\pfillwitharg{2}({L_{186.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{150}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7829850\\350250\\16875
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
350250\\-15667\\-752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16875\\-752\\-23
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-119\\-2688\\576
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-114\\-2575\\550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
51\\1152\\-247
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1876\\42375\\-9075
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
114\\2575\\-551
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\0\\-3\\-75\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\25\\0\\-225\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\1\\0\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-225\\0\\150\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-25\\-2\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-7829850, 350250, 16875; 350250, -15667, -752; 16875, -752, -23],[75, 1, 1],[-119, -114, 51, 1876, 114; -2688, -2575, 1152, 42375, 2575; 576, 550, -247, -9075, -551],185,2]
%
% END OF LATTICE L186.2
%
\hbox{}\par\smallskip%
%
% LATTICE L186.3
%
\leavevmode%
\llap{\spacer}%
${L_{186.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{186.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{l}{2}{75}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{r}{2}{50}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
315119175\\36324300\\102446250
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36324300\\4187163\\11809146
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
102446250\\11809146\\33305603
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17\\-2\\53
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3072\\425\\-9600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
790\\110\\-2469
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10109\\1425\\-31600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2305\\327\\-7206
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-3\\-25\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\75\\0\\-225\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\3\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-225\\0\\50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-75\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[315119175, 36324300, 102446250; 36324300, 4187163, 11809146; 102446250, 11809146, 33305603],[75, 3, 1],[-17, 3072, 790, 10109, 2305; -2, 425, 110, 1425, 327; 53, -9600, -2469, -31600, -7206],185,3]
%
% END OF LATTICE L186.3
%
\hbox{}\par\smallskip%
%
% LATTICE L186.4
%
\leavevmode%
\llap{\spacer}%
${L_{186.4}}=\pdualwitharg{3}({L_{186.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{b}{2}{300}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{50}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10130700\\22200\\30900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22200\\-42\\-69
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30900\\-69\\-94
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\112\\575
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61\\3400\\17550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\166\\864
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-225\\-1150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-111\\-576
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-6\\-25\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\300\\0\\-450\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-450\\0\\50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-150\\-12\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10130700, 22200, 30900; 22200, -42, -69; 30900, -69, -94],[300, 3, 1],[2, 61, 3, -4, -2; 112, 3400, 166, -225, -111; 575, 17550, 864, -1150, -576],185,4]
%
% END OF LATTICE L186.4
%
\hbox{}\par\smallskip%
%
% LATTICE L186.5
%
\leavevmode%
\llap{\spacer}%
${L_{186.5}}=\pdualwitharg{2}({L_{186.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{*}{2}{100}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{600}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3882015911400\\-71015676300\\-983728425600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71015676300\\1299125608\\17995840572
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-983728425600\\17995840572\\249283268647
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8971\\-144\\35412
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1153\\0\\-4550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5215\\-89\\20586
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
625777\\-10275\\2470200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
64571\\-1055\\254888
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\0\\-12\\-300\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\100\\0\\-900\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\4\\0\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\-900\\0\\600\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-100\\-8\\0\\8
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3882015911400, -71015676300, -983728425600; -71015676300, 1299125608, 17995840572; -983728425600, 17995840572, 249283268647],[300, 4, 1],[8971, -1153, 5215, 625777, 64571; -144, 0, -89, -10275, -1055; 35412, -4550, 20586, 2470200, 254888],185,5]
%
% END OF LATTICE L186.5
%
\hbox{}\par\smallskip%
%
% LATTICE L186.6
%
\leavevmode%
\llap{\spacer}%
${L_{186.6}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{186.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${150}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{25}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{s}{2}{50}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3580350\\2394075\\698625
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2394075\\1600850\\467150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
698625\\467150\\136321
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\3\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\0\\41
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\2\\275
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\9\\456
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-191\\23\\900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
150\\0\\-75\\-75\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0\\-9\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\0\\25\\0\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-9\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-25\\-50\\0\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3580350, 2394075, 698625; 2394075, 1600850, 467150; 698625, 467150, 136321],[75, 25, 1],[-2, -8, -55, -95, -191; 3, 0, 2, 9, 23; 0, 41, 275, 456, 900],185,6]
%
% END OF LATTICE L186.6
%
\hbox{}\par\smallskip%
%
% LATTICE L186.7
%
\leavevmode%
\llap{\spacer}%
${L_{186.7}}=\pdualwitharg{2.3}({L_{186.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{*}{2}{300}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{200}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1637290200\\-12857100\\-414857100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12857100\\100968\\3257736
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-414857100\\3257736\\105116621
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1878\\7\\-7412
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24742\\75\\-97650
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1163\\2\\-4590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1799\\0\\-7100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-301\\2\\-1188
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-12\\-100\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\300\\0\\-900\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\12\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-900\\0\\200\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-300\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1637290200, -12857100, -414857100; -12857100, 100968, 3257736; -414857100, 3257736, 105116621],[300, 12, 1],[-1878, -24742, -1163, -1799, -301; 7, 75, 2, 0, 2; -7412, -97650, -4590, -7100, -1188],185,7]
%
% END OF LATTICE L186.7
%
\hbox{}\par\smallskip%
%
% LATTICE L186.8
%
\leavevmode%
\llap{\spacer}%
${L_{186.8}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{186.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${50}\above{1pt}{1pt}{l}{2}{3}\above{1pt}{1pt}{}{2}{75}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{s}{2}{150}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1725\\232350\\75525
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
232350\\-21515025\\-6994650
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
75525\\-6994650\\-2273998
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
192\\-3696\\11375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\-425\\1308
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-247\\4752\\-14625
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-121\\2331\\-7174
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-551\\10625\\-32700
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\0\\-75\\-25\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\3\\0\\-9\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\0\\75\\0\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-9\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-75\\-150\\0\\150
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1725, 232350, 75525; 232350, -21515025, -6994650; 75525, -6994650, -2273998],[75, 75, 1],[192, 22, -247, -121, -551; -3696, -425, 4752, 2331, 10625; 11375, 1308, -14625, -7174, -32700],185,8]
%
% END OF LATTICE L186.8
%
\hbox{}\par\smallskip%
%
% LATTICE L186.9
%
\leavevmode%
\llap{\spacer}%
${L_{186.9}}=\pdualwitharg{5}({L_{186.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${150}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{100}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{s}{2}{50}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-207300\\529200\\-11700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
529200\\-1209650\\26375
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11700\\26375\\-574
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
161\\597\\24150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\26\\1052
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-137\\-508\\-20550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-76\\-282\\-11409
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-84\\-312\\-12625
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
150\\0\\-150\\-75\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-18\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-150\\0\\100\\0\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-18\\0\\6\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-50\\-100\\0\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-207300, 529200, -11700; 529200, -1209650, 26375; -11700, 26375, -574],[300, 25, 1],[161, 7, -137, -76, -84; 597, 26, -508, -282, -312; 24150, 1052, -20550, -11409, -12625],185,9]
%
% END OF LATTICE L186.9
%
\hbox{}\par\smallskip%
%
% LATTICE L186.10
%
\leavevmode%
\llap{\spacer}%
${L_{186.10}}=\pdualwitharg{3.5}({L_{186.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${50}\above{1pt}{1pt}{b}{2}{12}\above{1pt}{1pt}{*}{2}{300}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{s}{2}{150}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-137100\\12000\\6000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12000\\-1050\\-525
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6000\\-525\\-262
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\20\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\34\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-23\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\0\\-150\\-25\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-18\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-150\\0\\300\\0\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-18\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-75\\-150\\-300\\0\\150
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-137100, 12000, 6000; 12000, -1050, -525; 6000, -525, -262],[300, 75, 1],[0, 1, 3, 0, -2; 12, 20, 34, -1, -23; -25, -18, 0, 2, 0],185,10]
%
% END OF LATTICE L186.10
%
\hbox{}\par\smallskip%
%
% LATTICE L186.11
%
\leavevmode%
\llap{\spacer}%
${L_{186.11}}=\pdualwitharg{2.5}({L_{186.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}3\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${600}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{b}{2}{100}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{s}{2}{200}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
87535411800\\722695500\\21634184100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
722695500\\5966600\\178612600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21634184100\\178612600\\5346840919
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-607\\1677\\2400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-111\\-146
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\631\\950
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1660\\4569\\6564
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4426\\12223\\17500
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
600\\0\\-300\\-300\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-36\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\0\\100\\0\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\-36\\0\\24\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\-100\\-200\\0\\200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[87535411800, 722695500, 21634184100; 722695500, 5966600, 178612600; 21634184100, 178612600, 5346840919],[300, 100, 1],[-607, 37, -240, -1660, -4426; 1677, -111, 631, 4569, 12223; 2400, -146, 950, 6564, 17500],185,11]
%
% END OF LATTICE L186.11
%
\hbox{}\par\smallskip%
%
% LATTICE L186.12
%
\leavevmode%
\llap{\spacer}%
${L_{186.12}}=\pdualwitharg{2.3.5}({L_{186.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${200}\above{1pt}{1pt}{*}{2}{12}\above{1pt}{1pt}{b}{2}{300}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{s}{2}{600}\above{1pt}{1pt}{}{6}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
79800\\46500\\-11700
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
46500\\53400\\-13500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11700\\-13500\\3413
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-1270\\-5000
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-538\\-2118
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-343\\-1350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-381\\-1500
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\0\\-300\\-100\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\12\\0\\-36\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\0\\300\\0\\-600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-36\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\-300\\-600\\0\\600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[79800, 46500, -11700; 46500, 53400, -13500; -11700, -13500, 3413],[300, 300, 1],[7, 3, 2, 0, 2; -1270, -538, -343, 1, -381; -5000, -2118, -1350, 4, -1500],185,12]
%
% END OF LATTICE L186.12
%

\medskip%
%------------------------------WEYL-GROUP-W187--------------------------------
%
\leavevmode\llap{}%
$W_{187}$%
\qquad\llap{4} lattices, $\chi=8$%
\hfill%
$22232$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L187.1
%
\leavevmode%
\llap{\spacer}%
${L_{187.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{-}{3}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57840\\-880\\1040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-880\\-2\\13
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1040\\13\\-18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-256\\-1056
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-15\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\88\\352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\17\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-35\\-141
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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16\\0\\-48\\-32\\0
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0\\10\\0\\-10\\-5
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-48\\0\\16\\0\\-8
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-32\\-10\\0\\2\\-1
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0\\-5\\-8\\-1\\2
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-57840, -880, 1040; -880, -2, 13; 1040, 13, -18],[80, 1, 1],[-15, -1, 5, 1, -2; -256, -15, 88, 17, -35; -1056, -70, 352, 70, -141],186,1]
%
% END OF LATTICE L187.1
%
\hbox{}\par\smallskip%
%
% LATTICE L187.2
%
\leavevmode%
\llap{\spacer}%
${L_{187.2}}=\pdualwitharg{5}({L_{187.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{b}{2}{80}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{+}{3}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2480\\-400\\-720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-10\\-75
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-75\\-178
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\368\\-480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\-6
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-120\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-23\\30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\49\\-65
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-240\\-160\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-10\\-5
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-240\\0\\80\\0\\-40
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-160\\-10\\0\\10\\-5
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-40\\-5\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2480, -400, -720; -400, -10, -75; -720, -75, -178],[80, 5, 1],[81, 1, -27, -5, 11; 368, 5, -120, -23, 49; -480, -6, 160, 30, -65],186,2]
%
% END OF LATTICE L187.2
%
\hbox{}\par\smallskip%
%
% LATTICE L187.3
%
\leavevmode%
\llap{\spacer}%
${L_{187.3}}=\pdualwitharg{2}({L_{187.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{160}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{32}\above{1pt}{1pt}{-}{3}{32}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-400\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-928\\-192
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-192\\-39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\2\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17\\10\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-2\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-6\\-32\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\160\\0\\-160\\-80
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\0\\4\\0\\-16
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-160\\0\\32\\-16
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-16\\-16\\32
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-160, -400, -80; -400, -928, -192; -80, -192, -39],[80, 16, 1],[2, 17, 1, -3, -2; 2, 10, 0, -2, 1; -13, -80, -2, 16, 0],186,3]
%
% END OF LATTICE L187.3
%
\hbox{}\par\smallskip%
%
% LATTICE L187.4
%
\leavevmode%
\llap{\spacer}%
${L_{187.4}}=\pdualwitharg{2.5}({L_{187.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{32}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{*}{2}{160}\above{1pt}{1pt}{+}{3}{160}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18720\\-46800\\1120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46800\\-116960\\2800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1120\\2800\\-67
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-2\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\2\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\-1\\160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-30\\-160\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\32\\0\\-160\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\0\\20\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-160\\0\\160\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-80\\-80\\-80\\160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-18720, -46800, 1120; -46800, -116960, 2800; 1120, 2800, -67],[80, 80, 1],[7, 3, -3, -5, 12; -2, -2, 0, 2, -1; 35, -32, -50, 0, 160],186,4]
%
% END OF LATTICE L187.4
%

\medskip%
%------------------------------WEYL-GROUP-W188--------------------------------
%
\leavevmode\llap{}%
$W_{188}$%
\qquad\llap{4} lattices, $\chi=18$%
\hfill%
$42\slashinfty24|\rtimes D_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L188.1
%
\leavevmode%
\llap{\spacer}%
${L_{188.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{*}{4}{4}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{8,3}{\infty}{20}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16720\\480\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
480\\-11\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
240\\-8\\-3
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\15\\39
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-15\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\110\\260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\62\\151
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-2\\-5\\-10\\-1
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\4\\0\\-40\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\0\\5\\-10\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-40\\-10\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-10\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-16720, 480, 240; 480, -11, -8; 240, -8, -3],[80, 1, 1],[1, -1, -1, 7, 4; 15, -16, -15, 110, 62; 39, -38, -40, 260, 151],187,1]
%
% END OF LATTICE L188.1
%
\hbox{}\par\smallskip%
%
% LATTICE L188.2
%
\leavevmode%
\llap{\spacer}%
${L_{188.2}}=\pdualwitharg{5}({L_{188.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{}{4}{5}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{8,7}{\infty z}{1}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-544400\\3280\\6560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3280\\-15\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6560\\-40\\-79
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\39\\395
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\87\\950
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\20\\238
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-79
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-6\\-80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-5\\-10\\-5\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\5\\0\\-10\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\4\\-2\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-10\\-2\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-90\\-40\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-544400, 3280, 6560; 3280, -15, -40; 6560, -40, -79],[80, 5, 1],[5, 12, 3, -1, -1; 39, 87, 20, -8, -6; 395, 950, 238, -79, -80],187,2]
%
% END OF LATTICE L188.2
%
\hbox{}\par\smallskip%
%
% LATTICE L188.3
%
\leavevmode%
\llap{\spacer}%
${L_{188.3}}=\pdualwitharg{2}({L_{188.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${32}\above{1pt}{1pt}{}{4}{16}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{4,1}{\infty a}{80}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{*}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26480\\31360\\3040
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
31360\\-37104\\-3600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3040\\-3600\\-349
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-1\\176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\1\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\0\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
43\\-10\\480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\-9\\600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32\\-16\\-80\\-80\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\16\\0\\-160\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\80\\-80\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-160\\-80\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-144\\-160\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-26480, 31360, 3040; 31360, -37104, -3600; 3040, -3600, -349],[80, 16, 1],[19, -8, -23, 43, 58; -1, 1, 0, -10, -9; 176, -80, -200, 480, 600],187,3]
%
% END OF LATTICE L188.3
%
\hbox{}\par\smallskip%
%
% LATTICE L188.4
%
\leavevmode%
\llap{\spacer}%
${L_{188.4}}=\pdualwitharg{2.5}({L_{188.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${160}\above{1pt}{1pt}{*}{4}{80}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{8,5}{\infty}{16}\above{1pt}{1pt}{l}{2}{80}\above{1pt}{1pt}{}{4}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105440\\-105520\\2080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-105520\\-105520\\2080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2080\\2080\\-41
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\12\\560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\19\\520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\1\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
160\\-80\\-80\\-80\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\80\\0\\-160\\-720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\16\\-16\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-160\\-16\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-720\\-160\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-105440, -105520, 2080; -105520, -105520, 2080; 2080, 2080, -41],[80, 80, 1],[-1, -9, -2, 0, 1; 12, 19, 1, -3, -1; 560, 520, -48, -152, 0],187,4]
%
% END OF LATTICE L188.4
%

\medskip%
%------------------------------WEYL-GROUP-W189--------------------------------
%
\leavevmode\llap{}%
$W_{189}$%
\qquad\llap{36} lattices, $\chi=36$%
\hfill%
$2\slashtwo2\slashinfty2\slashtwo2\slashinfty\rtimes D_{4}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L189.1
%
\leavevmode%
\llap{\spacer}%
${L_{189.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{5}%
\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{s}{2}{400}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{5,4}{\infty b}{80}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{400}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{40,21}{\infty}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2375600\\-43600\\-45200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43600\\-795\\-825
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45200\\-825\\-856
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-298\\340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\360\\-400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\302\\-344
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\2064\\-2360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\2576\\-2960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\279\\-322
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\2920\\-3400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-21\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-160\\-240\\-30\\-400\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\400\\0\\-400\\-1200\\-200\\-3600\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\4\\0\\-80\\-18\\-400\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-400\\0\\80\\-80\\-40\\-1200\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1200\\-80\\-80\\80\\0\\-400\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-200\\-18\\-40\\0\\1\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-3600\\-400\\-1200\\-400\\0\\400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-200\\-30\\-120\\-80\\-5\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2375600, -43600, -45200; -43600, -795, -825; -45200, -825, -856],[400, 5, 1],[-1, 1, 1, 7, 9, 1, 11, 0; -298, 360, 302, 2064, 2576, 279, 2920, -21; 340, -400, -344, -2360, -2960, -322, -3400, 20],188,1]
%
% END OF LATTICE L189.1
%
\hbox{}\par\smallskip%
%
% LATTICE L189.2
%
\leavevmode%
\llap{\spacer}%
${L_{189.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{5}%
\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{400}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{20,9}{\infty z}{80}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{400}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{40,1}{\infty}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13903600\\108800\\40400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108800\\-795\\-325
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40400\\-325\\-116
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-418\\-2660
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-720\\-4600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\342\\2176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
87\\3304\\21040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
149\\5656\\36040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\759\\4838
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
291\\11040\\70400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\379\\2420
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-160\\-240\\-30\\-400\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\400\\0\\-400\\-1200\\-200\\-3600\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\4\\0\\-80\\-18\\-400\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-400\\0\\80\\-80\\-40\\-1200\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1200\\-80\\-80\\80\\0\\-400\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-200\\-18\\-40\\0\\1\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-3600\\-400\\-1200\\-400\\0\\400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-200\\-30\\-120\\-80\\-5\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-13903600, 108800, 40400; 108800, -795, -325; 40400, -325, -116],[400, 5, 1],[-11, -19, 9, 87, 149, 20, 291, 10; -418, -720, 342, 3304, 5656, 759, 11040, 379; -2660, -4600, 2176, 21040, 36040, 4838, 70400, 2420],188,2]
%
% END OF LATTICE L189.2
%
\hbox{}\par\smallskip%
%
% LATTICE L189.3
%
\leavevmode%
\llap{\spacer}%
${L_{189.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{5}%
\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{100}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{40,39}{\infty}{20}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{40,31}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1072400\\288400\\-71200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
288400\\-75055\\20390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-71200\\20390\\-4111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
957\\2376\\-4790
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1019\\2530\\-5100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-377\\-936\\1887
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1916\\-4757\\9590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6769\\-16806\\33880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3691\\-9164\\18474
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6833\\-16965\\34200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-990\\-2458\\4955
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-10\\-40\\-120\\-60\\-100\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\100\\0\\-50\\-300\\-200\\-450\\-100
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\1\\0\\-20\\-18\\-50\\-15
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-50\\0\\5\\-10\\-20\\-75\\-30
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-300\\-20\\-10\\20\\0\\-50\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-200\\-18\\-20\\0\\4\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-450\\-50\\-75\\-50\\0\\25\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-100\\-15\\-30\\-40\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1072400, 288400, -71200; 288400, -75055, 20390; -71200, 20390, -4111],[400, 5, 1],[957, 1019, -377, -1916, -6769, -3691, -6833, -990; 2376, 2530, -936, -4757, -16806, -9164, -16965, -2458; -4790, -5100, 1887, 9590, 33880, 18474, 34200, 4955],188,3]
%
% END OF LATTICE L189.3
%
\hbox{}\par\smallskip%
%
% LATTICE L189.4
%
\leavevmode%
\llap{\spacer}%
${L_{189.4}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{40,39}{\infty z}{5}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{*}{2}{100}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{40,11}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-475600\\-47600\\-44800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47600\\-4670\\-4585
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44800\\-4585\\-4111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\528\\490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-106\\565\\525
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\-208\\-193
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
503\\-2682\\-2490
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
510\\-2719\\-2525
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
595\\-3172\\-2946
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2373\\-12650\\-11750
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
209\\-1114\\-1035
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-10\\-80\\-60\\-60\\-200\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\25\\0\\-50\\-75\\-100\\-450\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\1\\0\\-10\\-18\\-100\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-50\\0\\20\\-10\\-40\\-300\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-75\\-10\\-10\\5\\0\\-50\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-100\\-18\\-40\\0\\4\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-450\\-100\\-300\\-50\\0\\100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-50\\-15\\-60\\-20\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-475600, -47600, -44800; -47600, -4670, -4585; -44800, -4585, -4111],[400, 5, 1],[-99, -106, 39, 503, 510, 595, 2373, 209; 528, 565, -208, -2682, -2719, -3172, -12650, -1114; 490, 525, -193, -2490, -2525, -2946, -11750, -1035],188,4]
%
% END OF LATTICE L189.4
%
\hbox{}\par\smallskip%
%
% LATTICE L189.5
%
\leavevmode%
\llap{\spacer}%
${L_{189.5}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{5}%
\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{s}{2}{400}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{10,9}{\infty b}{80}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{400}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{20,1}{\infty}$}%
\nopagebreak\par%
shares genus with 2-dual${}\iso{}$5-dual; isometric to own %
2.5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1775600\\164400\\40400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
164400\\-15220\\-3740
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40400\\-3740\\-919
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\28\\-70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-60\\200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-30\\78
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-184\\440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-196\\400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\29
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-120\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11\\-45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-160\\-240\\-30\\-400\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\400\\0\\-400\\-1200\\-200\\-3600\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\4\\0\\-80\\-18\\-400\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-400\\0\\80\\-80\\-40\\-1200\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1200\\-80\\-80\\80\\0\\-400\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-200\\-18\\-40\\0\\1\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-3600\\-400\\-1200\\-400\\0\\400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-200\\-30\\-120\\-80\\-5\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1775600, 164400, 40400; 164400, -15220, -3740; 40400, -3740, -919],[400, 20, 1],[1, -1, -1, -7, -9, -1, -11, 0; 28, -60, -30, -184, -196, -18, -120, 11; -70, 200, 78, 440, 400, 29, 0, -45],188,5]
%
% END OF LATTICE L189.5
%
\hbox{}\par\smallskip%
%
% LATTICE L189.6
%
\leavevmode%
\llap{\spacer}%
${L_{189.6}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{5}%
\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{l}{2}{100}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{40,39}{\infty}{20}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{100}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{20,11}{\infty}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-456400\\-268000\\-44800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-268000\\-152620\\-25140
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-44800\\-25140\\-4111
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-99\\528\\-2150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-106\\565\\-2300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\-208\\847
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
397\\-2117\\8620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
702\\-3743\\15240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
383\\-2042\\8314
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1419\\-7565\\30800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
103\\-549\\2235
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-10\\-80\\-120\\-60\\-200\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\100\\0\\-100\\-300\\-200\\-900\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\1\\0\\-20\\-18\\-100\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-100\\0\\20\\-20\\-40\\-300\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-300\\-20\\-20\\20\\0\\-100\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-200\\-18\\-40\\0\\4\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-900\\-100\\-300\\-100\\0\\100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-100\\-15\\-60\\-40\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-456400, -268000, -44800; -268000, -152620, -25140; -44800, -25140, -4111],[400, 20, 1],[-99, -106, 39, 397, 702, 383, 1419, 103; 528, 565, -208, -2117, -3743, -2042, -7565, -549; -2150, -2300, 847, 8620, 15240, 8314, 30800, 2235],188,6]
%
% END OF LATTICE L189.6
%
\hbox{}\par\smallskip%
%
% LATTICE L189.7
%
\leavevmode%
\llap{\spacer}%
${L_{189.7}}=\pfillwitharg{5}({L_{189.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-5680, 80, 720; 80, -1, -10; 720, -10, -91],[80, 1, 1],[3, 1, 5, 2, 1, -1, -1, -3; 160, 18, 40, 0, -20, 0, 16, 120; 0, 5, 32, 15, 10, -8, -10, -40],188,7]
%
% END OF LATTICE L189.7
%
\hbox{}\par\smallskip%
%
% LATTICE L189.8
%
\leavevmode%
\llap{\spacer}%
${L_{189.8}}=\pfillwitharg{5}({L_{189.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{8,1}{\infty}{20}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{4,1}{\infty z}$}%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-548080, 8000, 7120; 8000, -116, -105; 7120, -105, -91],[80, 1, 1],[77, 8, 15, -2, -11, 1, 9, 63; 3240, 338, 640, -80, -460, 40, 376, 2640; 2280, 235, 432, -65, -330, 32, 270, 1880],188,8]
%
% END OF LATTICE L189.8
%
\hbox{}\par\smallskip%
%
% LATTICE L189.9
%
\leavevmode%
\llap{\spacer}%
${L_{189.9}}=\pfillwitharg{5}({L_{189.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-209680, -82960, 4560; -82960, -32823, 1804; 4560, 1804, -99],[80, 1, 1],[-261, -165, -71, -73, 11, 15, -4, -52; 700, 442, 190, 195, -30, -40, 11, 140; 730, 452, 191, 190, -40, -38, 16, 155],188,9]
%
% END OF LATTICE L189.9
%
\hbox{}\par\smallskip%
%
% LATTICE L189.10
%
\leavevmode%
\llap{\spacer}%
${L_{189.10}}=\pfillwitharg{5}({L_{189.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
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-30\\-80\\-18\\-80\\0\\4\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-144\\-40\\-240\\-80\\0\\16\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-80\\-30\\-240\\-160\\-20\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-5680, 160, 720; 160, -4, -20; 720, -20, -91],[80, 4, 1],[2, 5, 1, 3, -3, -1, -1, 1; 0, 20, 9, 80, 60, 8, 0, -10; 15, 32, 5, 0, -40, -10, -8, 10],188,10]
%
% END OF LATTICE L189.10
%
\hbox{}\par\smallskip%
%
% LATTICE L189.11
%
\leavevmode%
\llap{\spacer}%
${L_{189.11}}=\pfillwitharg{5}({L_{189.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{8,3}{\infty}{20}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{4,3}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3280\\-2960\\1760
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2960\\-2620\\1536
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1760\\1536\\-891
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-83\\220\\215
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-160\\425\\416
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-185\\492\\482
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-291\\775\\760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-114\\305\\300
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-4\\11\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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17\\-45\\-44
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-30\\-30
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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5\\0\\-10\\-40\\-60\\-15\\-20\\-10
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-15\\-20\\-18\\-20\\0\\1\\0\\-10
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-20\\-36\\-40\\-60\\-20\\0\\4\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-40\\-60\\-120\\-80\\-10\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-3280, -2960, 1760; -2960, -2620, 1536; 1760, 1536, -891],[80, 4, 1],[-83, -160, -185, -291, -114, -4, 17, 11; 220, 425, 492, 775, 305, 11, -45, -30; 215, 416, 482, 760, 300, 11, -44, -30],188,11]
%
% END OF LATTICE L189.11
%
\hbox{}\par\smallskip%
%
% LATTICE L189.12
%
\leavevmode%
\llap{\spacer}%
${L_{189.12}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{189.6}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{8,1}{\infty}{20}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{4,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-160\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\20\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-60\\200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\-16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\5\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-25\\80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-56\\184
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\-35\\116
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-120\\400
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-160\\-240\\-240\\-80\\-80
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\kern3pt%
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0\\4\\0\\-20\\-60\\-80\\-36\\-80
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\kern3pt%
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-160\\-20\\0\\20\\-20\\-80\\-60\\-240
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-60\\-80\\-20\\20\\0\\-20\\-160
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-80\\-144\\-80\\0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-36\\-80\\-60\\-20\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-80\\-240\\-240\\-160\\-80\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[720, -160, 0; -160, 20, -4; 0, -4, -1],[80, 4, 1],[-13, -1, 1, 1, -6, -13, -8, -27; -60, -5, 4, 5, -25, -56, -35, -120; 200, 16, -16, -20, 80, 184, 116, 400],188,12]
%
% END OF LATTICE L189.12
%
\hbox{}\par\smallskip%
%
% LATTICE L189.13
%
\leavevmode%
\llap{\spacer}%
${L_{189.13}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{189.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{8,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{2,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-290800\\1280\\6800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1280\\-5\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6800\\-30\\-159
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-24\\-296
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-24\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\24\\-480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-2\\-215
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-16\\-384
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-80\\-32\\-24\\-240\\-40\\-16
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0\\20\\0\\-20\\-30\\-400\\-90\\-80
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-80\\0\\80\\0\\-40\\-720\\-200\\-240
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-240\\-400\\-720\\-80\\0\\80\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-90\\-200\\-30\\-5\\0\\5\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-80\\-240\\-48\\-16\\-80\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-290800, 1280, 6800; 1280, -5, -30; 6800, -30, -159],[80, 5, 1],[-7, -5, -1, 1, 0, -11, -5, -9; -24, -24, -16, 4, 4, 24, -2, -16; -296, -210, -40, 42, -1, -480, -215, -384],188,13]
%
% END OF LATTICE L189.13
%
\hbox{}\par\smallskip%
%
% LATTICE L189.14
%
\leavevmode%
\llap{\spacer}%
${L_{189.14}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{189.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{8,5}{\infty z}{1}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{4,3}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-238640\\-2720\\6160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2720\\-25\\70
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6160\\70\\-159
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\32\\904
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\16\\510
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-8\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-118
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\4\\79
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\112\\2640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\38\\945
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\56\\1456
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-80\\-32\\-24\\-240\\-40\\-16
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20\\-30\\-400\\-90\\-80
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\kern3pt%
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-80\\0\\80\\0\\-40\\-720\\-200\\-240
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-238640, -2720, 6160; -2720, -25, 70; 6160, 70, -159],[80, 5, 1],[23, 13, -3, -3, 2, 67, 24, 37; 32, 16, -8, -4, 4, 112, 38, 56; 904, 510, -120, -118, 79, 2640, 945, 1456],188,14]
%
% END OF LATTICE L189.14
%
\hbox{}\par\smallskip%
%
% LATTICE L189.15
%
\leavevmode%
\llap{\spacer}%
${L_{189.15}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{189.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{8,3}{\infty z}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-40400\\880\\880
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\kern3pt%
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880\\-15\\-20
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880\\-20\\-19
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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911\\6384\\34960
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-6\\-15\\-20\\-2
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-15\\-50\\-90\\-20
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-90\\-200\\-60
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-2\\-20\\-60\\-24
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-40400, 880, 880; 880, -15, -20; 880, -20, -19],[80, 5, 1],[1, 7, 19, 7, 2, 2, -1, -1; 6, 47, 132, 50, 15, 16, -6, -8; 39, 270, 730, 268, 76, 75, -40, -38],188,15]
%
% END OF LATTICE L189.15
%
\hbox{}\par\smallskip%
%
% LATTICE L189.16
%
\leavevmode%
\llap{\spacer}%
${L_{189.16}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{189.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{80}\above{1pt}{1pt}{8,3}{\infty}{80}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{2,1}{\infty z}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5680\\-5440\\720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5440\\-5200\\688
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\688\\-91
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\15
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\10\\112
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2\\9\\82
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3\\40\\320
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\30\\200
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-1\\4\\22
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\HardButStrongLineBreak\kern3pt%
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-1\\0\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-5\\-30
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-10\\-80\\-120\\-30\\-40\\-10
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\kern3pt%
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0\\16\\0\\-80\\-240\\-80\\-144\\-80
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\kern3pt%
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-30\\-80\\-36\\-80\\0\\4\\0\\-20
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-10\\-80\\-60\\-240\\-160\\-20\\0\\20
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5680, -5440, 720; -5440, -5200, 688; 720, 688, -91],[80, 16, 1],[2, 5, 2, 3, -3, -1, -1, 1; 0, 10, 9, 40, 30, 4, 0, -5; 15, 112, 82, 320, 200, 22, -8, -30],188,16]
%
% END OF LATTICE L189.16
%
\hbox{}\par\smallskip%
%
% LATTICE L189.17
%
\leavevmode%
\llap{\spacer}%
${L_{189.17}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{189.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{80}\above{1pt}{1pt}{4,3}{\infty a}{80}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{2,1}{\infty a}$}%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
320\\-80\\80
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\kern3pt%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\48\\9
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-31\\25\\150
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-67\\55\\328
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-41\\34\\202
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-137\\115\\680
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63\\55\\320
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-2\\2\\11
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7\\-5\\-32
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6\\-5\\-30
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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-30\\-40\\-18\\-40\\0\\1\\0\\-10
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-80\\-144\\-80\\-240\\-80\\0\\16\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-80\\-60\\-240\\-160\\-10\\0\\20
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[320, -80, 80; -80, -400, 48; 80, 48, 9],[80, 16, 1],[-31, -67, -41, -137, -63, -2, 7, 6; 25, 55, 34, 115, 55, 2, -5, -5; 150, 328, 202, 680, 320, 11, -32, -30],188,17]
%
% END OF LATTICE L189.17
%
\hbox{}\par\smallskip%
%
% LATTICE L189.18
%
\leavevmode%
\llap{\spacer}%
${L_{189.18}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{189.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1008\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-32\\-1
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-15\\400
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\kern3pt%
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29\\86\\-2320
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1\\3\\-81
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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-1\\0\\-8
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-32
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-320\\-480\\-240\\-160\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-240\\-160\\-144\\-160
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-320\\-80\\0\\80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-480\\-240\\-160\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-160\\-144\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-144\\-160\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-160\\-240\\-480
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[80, 0, 0; 0, -1008, -32; 0, -32, -1],[80, 16, 1],[-4, -1, 0, 1, -1, -2, -3, -6; -5, 0, 1, 0, -10, -9, -10, -15; 120, -8, -32, 0, 280, 248, 272, 400],188,18]
%
% END OF LATTICE L189.18
%
\hbox{}\par\smallskip%
%
% LATTICE L189.19
%
\leavevmode%
\llap{\spacer}%
${L_{189.19}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{189.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{2,1}{\infty b}{16}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{4,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270640\\2480\\6560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2480\\-20\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6560\\-60\\-159
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-2\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\8\\-210
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\16\\-296
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\28\\-384
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\19\\-215
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\56\\-480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-32\\-48\\-30\\-80\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-80\\-240\\-200\\-720\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0\\-80\\-90\\-400\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-80\\0\\16\\-16\\-40\\-240\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-240\\-80\\-16\\16\\0\\-80\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-200\\-90\\-40\\0\\5\\0\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-720\\-400\\-240\\-80\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-40\\-30\\-24\\-16\\-5\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-270640, 2480, 6560; 2480, -20, -60; 6560, -60, -159],[80, 20, 1],[1, -1, -5, -7, -9, -5, -11, 0; -2, -4, 8, 16, 28, 19, 56, 2; 42, -40, -210, -296, -384, -215, -480, -1],188,19]
%
% END OF LATTICE L189.19
%
\hbox{}\par\smallskip%
%
% LATTICE L189.20
%
\leavevmode%
\llap{\spacer}%
${L_{189.20}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{189.6}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{8,7}{\infty}{4}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{4,3}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27280\\1520\\720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1520\\-60\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-40\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-1\\152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-3\\268
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-10\\730
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-9\\540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-10\\-16\\-24\\-60\\-40\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20\\-60\\-200\\-180\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\5\\0\\-20\\-90\\-100\\-15
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-20\\0\\4\\-4\\-40\\-60\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-60\\-20\\-4\\4\\0\\-20\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-200\\-90\\-40\\0\\20\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-180\\-100\\-60\\-20\\0\\20\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-20\\-15\\-12\\-8\\-10\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-27280, 1520, 720; 1520, -60, -40; 720, -40, -19],[80, 20, 1],[-1, -1, 2, 4, 7, 19, 14, 1; 0, 1, 0, -1, -3, -10, -9, -1; -38, -40, 75, 152, 268, 730, 540, 39],188,20]
%
% END OF LATTICE L189.20
%
\hbox{}\par\smallskip%
%
% LATTICE L189.21
%
\leavevmode%
\llap{\spacer}%
${L_{189.21}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{189.6}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{4,1}{\infty}{16}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{8,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4080\\-5360\\-2080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5360\\-2700\\-1000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2080\\-1000\\-369
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\83\\-208
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\80\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-359\\900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-632\\1584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\-1012\\2536
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
48\\-1309\\3280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\-1820\\4560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-107\\268
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-32\\-48\\-60\\-80\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-80\\-240\\-400\\-720\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\20\\0\\-80\\-180\\-400\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32\\-80\\0\\16\\-16\\-80\\-240\\-48
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-240\\-80\\-16\\16\\0\\-80\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-400\\-180\\-80\\0\\20\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-720\\-400\\-240\\-80\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-80\\-60\\-48\\-32\\-20\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4080, -5360, -2080; -5360, -2700, -1000; -2080, -1000, -369],[80, 20, 1],[-3, -3, 13, 23, 37, 48, 67, 4; 83, 80, -359, -632, -1012, -1309, -1820, -107; -208, -200, 900, 1584, 2536, 3280, 4560, 268],188,21]
%
% END OF LATTICE L189.21
%
\hbox{}\par\smallskip%
%
% LATTICE L189.22
%
\leavevmode%
\llap{\spacer}%
${L_{189.22}}=\pdualwitharg{5}({L_{189.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{0}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{*}{2}{100}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{40,9}{\infty z}{5}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{10,1}{\infty}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1092400\\73200\\-10000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
73200\\-4905\\670
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10000\\670\\-91
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-408\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-21\\-320\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-16\\-8
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\HardButStrongLineBreak\kern3pt%
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5\\76\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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2\\32\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-3\\-40\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-190\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\-432\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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80\\0\\-80\\-160\\-120\\-240\\-200\\-80
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-120\\-150\\-40\\-10\\5\\0\\-25\\-80
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-240\\-400\\-144\\-80\\0\\16\\0\\-80
\end{tabular}\endgroup%
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-200\\-450\\-200\\-150\\-25\\0\\25\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-400\\-240\\-240\\-80\\-80\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1092400, 73200, -10000; 73200, -4905, 670; -10000, 670, -91],[400, 5, 1],[-27, -21, -1, 5, 2, -3, -13, -29; -408, -320, -16, 76, 32, -40, -190, -432; -40, -50, -8, 10, 15, 32, 25, 0],188,22]
%
% END OF LATTICE L189.22
%
\hbox{}\par\smallskip%
%
% LATTICE L189.23
%
\leavevmode%
\llap{\spacer}%
${L_{189.23}}=\pdualwitharg{5}({L_{189.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{s}{2}{100}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{40,29}{\infty z}{5}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{25}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{20,11}{\infty z}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12111600\\52800\\33200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52800\\-220\\-145
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33200\\-145\\-91
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\144\\5240
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\120\\4550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\352
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-28\\-1050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-16\\-705
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-368
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\50\\1375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\136\\4520
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-160\\-120\\-240\\-200\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\100\\0\\-100\\-150\\-400\\-450\\-400
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\kern3pt%
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-80\\0\\16\\0\\-40\\-144\\-200\\-240
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-100\\0\\20\\-10\\-80\\-150\\-240
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-150\\-40\\-10\\5\\0\\-25\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-400\\-144\\-80\\0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-450\\-200\\-150\\-25\\0\\25\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-400\\-240\\-240\\-80\\-80\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-12111600, 52800, 33200; 52800, -220, -145; 33200, -145, -91],[400, 5, 1],[15, 13, 1, -3, -2, -1, 4, 13; 144, 120, 8, -28, -16, 0, 50, 136; 5240, 4550, 352, -1050, -705, -368, 1375, 4520],188,23]
%
% END OF LATTICE L189.23
%
\hbox{}\par\smallskip%
%
% LATTICE L189.24
%
\leavevmode%
\llap{\spacer}%
${L_{189.24}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{189.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{b}{2}{80}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{4,3}{\infty b}{16}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{80}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{2,1}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-244880\\-257440\\6240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-257440\\-270640\\6560
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6240\\6560\\-159
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-1\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-5\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\-7\\24
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-9\\176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-10\\330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-11\\640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\39
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-20\\-32\\-48\\-60\\-80\\-2
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\kern3pt%
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0\\80\\0\\-80\\-240\\-400\\-720\\-40
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-20\\0\\20\\0\\-80\\-180\\-400\\-30
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-32\\-80\\0\\16\\-16\\-80\\-240\\-24
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-240\\-80\\-16\\16\\0\\-80\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-400\\-180\\-80\\0\\20\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-720\\-400\\-240\\-80\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-40\\-30\\-24\\-16\\-10\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-244880, -257440, 6240; -257440, -270640, 6560; 6240, 6560, -159],[80, 80, 1],[-1, -2, 4, 8, 14, 19, 28, 1; 1, -1, -5, -7, -9, -10, -11, 0; 2, -120, -50, 24, 176, 330, 640, 39],188,24]
%
% END OF LATTICE L189.24
%
\hbox{}\par\smallskip%
%
% LATTICE L189.25
%
\leavevmode%
\llap{\spacer}%
${L_{189.25}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{189.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{l}{2}{80}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{8,7}{\infty}{16}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{b}{2}{80}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{2,1}{\infty b}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27280\\-27280\\720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27280\\-27200\\720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\720\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-38
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-1\\224
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-3\\296
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14\\-5\\330
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-9\\360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-10\\-32\\-48\\-60\\-80\\-4
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\kern3pt%
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0\\80\\0\\-80\\-240\\-400\\-720\\-80
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-10\\0\\5\\0\\-40\\-90\\-200\\-30
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-32\\-80\\0\\16\\-16\\-80\\-240\\-48
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\-240\\-40\\-16\\16\\0\\-80\\-32
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-400\\-90\\-80\\0\\20\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-720\\-200\\-240\\-80\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-80\\-30\\-48\\-32\\-20\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-27280, -27280, 720; -27280, -27200, 720; 720, 720, -19],[80, 80, 1],[-1, -1, 2, 7, 11, 14, 19, 1; 0, 1, 0, -1, -3, -5, -9, -1; -38, 0, 75, 224, 296, 330, 360, -2],188,25]
%
% END OF LATTICE L189.25
%
\hbox{}\par\smallskip%
%
% LATTICE L189.26
%
\leavevmode%
\llap{\spacer}%
${L_{189.26}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{189.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${16}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{4,1}{\infty b}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4720\\-1920
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1920\\-781
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-99\\240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\890\\-2160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33\\363\\-881
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\13\\-32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-49\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-36\\88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-80\\-64\\-96\\-240\\-160\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-80\\-240\\-800\\-720\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\80\\0\\-160\\-720\\-800\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\-80\\0\\16\\-16\\-160\\-240\\-96
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\0\\-80\\-64
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\80\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-64\\-80\\0\\16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-96\\-240\\-160\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-800\\-720\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-720\\-800\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16\\-160\\-240\\-96
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[80, 0, 0; 0, -4720, -1920; 0, -1920, -781],[80, 80, 1],[0, 1, 0, -1, -3, -10, -9, -2; 13, 0, -49, -36, -46, -99, -50, 3; -32, 0, 120, 88, 112, 240, 120, -8],188,26]
%
% END OF LATTICE L189.26
%
\hbox{}\par\smallskip%
%
% LATTICE L189.27
%
\leavevmode%
\llap{\spacer}%
${L_{189.27}}=\pdualwitharg{2}({L_{189.5}})\iso \pdualwitharg{5}({L_{189.5}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{s}{2}{100}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{20,9}{\infty z}{5}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{25}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{20,11}{\infty}$}%
\nopagebreak\par%
shares genus with 2-dual${}\iso{}$5-dual; isometric to own %
2.5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111600\\21600\\-3200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21600\\-4180\\620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3200\\620\\-91
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\72\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\60\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\4\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-14\\10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-8\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\32
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\25\\25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\68\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-160\\-120\\-240\\-200\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\100\\0\\-100\\-150\\-400\\-450\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\16\\0\\-40\\-144\\-200\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-100\\0\\20\\-10\\-80\\-150\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-150\\-40\\-10\\5\\0\\-25\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-400\\-144\\-80\\0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-450\\-200\\-150\\-25\\0\\25\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-400\\-240\\-240\\-80\\-80\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-111600, 21600, -3200; 21600, -4180, 620; -3200, 620, -91],[400, 20, 1],[15, 13, 1, -3, -2, -1, 4, 13; 72, 60, 4, -14, -8, 0, 25, 68; -40, -50, -8, 10, 15, 32, 25, 0],188,27]
%
% END OF LATTICE L189.27
%
\hbox{}\par\smallskip%
%
% LATTICE L189.28
%
\leavevmode%
\llap{\spacer}%
${L_{189.28}}=\pdualwitharg{5}({L_{189.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{}{2}{25}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{20,19}{\infty z}{5}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{100}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{40,11}{\infty}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-136400\\-52400\\11200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-52400\\-19420\\4160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11200\\4160\\-891
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\317\\1580
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\75\\375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-41\\-204
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-38\\-190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\204\\1015
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\405\\2016
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\2380\\11850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\763\\3800
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-80\\-60\\-60\\-200\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\25\\0\\-50\\-75\\-100\\-450\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\4\\0\\-20\\-36\\-200\\-60
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-50\\0\\20\\-10\\-40\\-300\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-75\\-20\\-10\\5\\0\\-50\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-100\\-36\\-40\\0\\4\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-450\\-200\\-300\\-50\\0\\100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-100\\-60\\-120\\-40\\-20\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-136400, -52400, 11200; -52400, -19420, 4160; 11200, 4160, -891],[400, 20, 1],[8, 2, -1, -1, 5, 10, 59, 19; 317, 75, -41, -38, 204, 405, 2380, 763; 1580, 375, -204, -190, 1015, 2016, 11850, 3800],188,28]
%
% END OF LATTICE L189.28
%
\hbox{}\par\smallskip%
%
% LATTICE L189.29
%
\leavevmode%
\llap{\spacer}%
${L_{189.29}}=\pdualwitharg{2.5}({L_{189.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{400}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{40,1}{\infty}{20}\above{1pt}{1pt}{r}{2}{400}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{20,9}{\infty}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-380\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-8\\120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\16
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\-20\\200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-3\\36
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-12\\160
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-400\\-160\\-240\\-1200\\-80\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20\\-60\\-400\\-36\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\0\\400\\0\\-400\\-3600\\-400\\-1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-20\\0\\20\\-20\\-400\\-60\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-60\\-400\\-20\\20\\0\\-20\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1200\\-400\\-3600\\-400\\0\\400\\0\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-36\\-400\\-60\\-20\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-80\\-1200\\-240\\-160\\-400\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[400, 0, 0; 0, -380, -20; 0, -20, -1],[400, 20, 1],[1, 0, -1, 0, 1, 9, 1, 3; -8, -1, 0, 1, -1, -20, -3, -12; 120, 16, 0, -20, 0, 200, 36, 160],188,29]
%
% END OF LATTICE L189.29
%
\hbox{}\par\smallskip%
%
% LATTICE L189.30
%
\leavevmode%
\llap{\spacer}%
${L_{189.30}}=\pdualwitharg{2}({L_{189.6}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{l}{2}{100}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{20,1}{\infty}{80}\above{1pt}{1pt}{l}{2}{100}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{40,9}{\infty}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3600\\800\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
800\\180\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\5\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-16\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-25\\-100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-8\\-48
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-100\\-160\\-240\\-300\\-80\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\0\\-80\\-240\\-400\\-144\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\0\\100\\0\\-400\\-900\\-400\\-300
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-80\\0\\80\\-80\\-400\\-240\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-240\\-400\\-80\\80\\0\\-80\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\-400\\-900\\-400\\0\\100\\0\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-144\\-400\\-240\\-80\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-80\\-300\\-240\\-160\\-100\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[3600, 800, 0; 800, 180, 0; 0, 0, -1],[400, 20, 1],[-1, -1, -1, 1, 3, 4, 1, 0; 3, 4, 5, -4, -16, -25, -8, -3; -20, -8, 0, 0, -40, -100, -48, -40],188,30]
%
% END OF LATTICE L189.30
%
\hbox{}\par\smallskip%
%
% LATTICE L189.31
%
\leavevmode%
\llap{\spacer}%
${L_{189.31}}=\pdualwitharg{2.5}({L_{189.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{b}{2}{400}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{80}\above{1pt}{1pt}{20,19}{\infty b}{80}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{400}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{10,1}{\infty}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1072400\\-270400\\-316800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270400\\-68080\\-79840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-316800\\-79840\\-93571
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\91\\-230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
118\\235\\-600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-69\\174
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-368\\-743\\1880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-674\\-1357\\3440
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-187\\-376\\954
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1412\\-2835\\7200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-108\\275
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-20\\-160\\-240\\-60\\-400\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\400\\0\\-400\\-1200\\-400\\-3600\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\4\\0\\-80\\-36\\-400\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-400\\0\\80\\-80\\-80\\-1200\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1200\\-80\\-80\\80\\0\\-400\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-400\\-36\\-80\\0\\4\\0\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-3600\\-400\\-1200\\-400\\0\\400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-200\\-30\\-120\\-80\\-10\\0\\5
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1072400, -270400, -316800; -270400, -68080, -79840; -316800, -79840, -93571],[400, 80, 1],[45, 118, -34, -368, -674, -187, -1412, -54; 91, 235, -69, -743, -1357, -376, -2835, -108; -230, -600, 174, 1880, 3440, 954, 7200, 275],188,31]
%
% END OF LATTICE L189.31
%
\hbox{}\par\smallskip%
%
% LATTICE L189.32
%
\leavevmode%
\llap{\spacer}%
${L_{189.32}}=\pdualwitharg{2}({L_{189.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}16\above{1pt}{1pt}{2}{0}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{s}{2}{100}\above{1pt}{1pt}{s}{2}{16}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{10,9}{\infty z}{5}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{100}\above{1pt}{1pt}{l}{2}{80}\above{1pt}{1pt}{40,11}{\infty}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3560400\\-142400\\18000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-142400\\-5680\\720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18000\\720\\-91
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\1160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\-5\\750
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\1\\-190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\5\\432
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\10\\1850
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\3\\1600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-160\\-120\\-240\\-400\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\100\\0\\-100\\-150\\-400\\-900\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\0\\16\\0\\-40\\-144\\-400\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-100\\0\\20\\-10\\-80\\-300\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-150\\-40\\-10\\5\\0\\-50\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-400\\-144\\-80\\0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-900\\-400\\-300\\-50\\0\\100\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-400\\-240\\-240\\-80\\-80\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3560400, -142400, 18000; -142400, -5680, 720; 18000, 720, -91],[400, 80, 1],[6, 4, 0, -1, 0, 2, 9, 8; -3, -5, -1, 1, 2, 5, 10, 3; 1160, 750, -8, -190, 15, 432, 1850, 1600],188,32]
%
% END OF LATTICE L189.32
%
\hbox{}\par\smallskip%
%
% LATTICE L189.33
%
\leavevmode%
\llap{\spacer}%
${L_{189.33}}=\pdualwitharg{2.5}({L_{189.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{l}{2}{400}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{40,39}{\infty}{80}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{400}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{10,1}{\infty b}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\-400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-2880\\-1200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1200\\-439
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\11\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-4\\11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-57\\-58\\160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-69\\-72\\200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-15\\42
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61\\-70\\200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\4\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\0\\-10\\-160\\-240\\-60\\-400\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\400\\0\\-400\\-1200\\-400\\-3600\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\0\\1\\0\\-40\\-18\\-200\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-400\\0\\80\\-80\\-80\\-1200\\-240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-240\\-1200\\-40\\-80\\80\\0\\-400\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-400\\-18\\-80\\0\\4\\0\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\-3600\\-200\\-1200\\-400\\0\\400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-400\\-30\\-240\\-160\\-20\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[400, -400, 0; -400, -2880, -1200; 0, -1200, -439],[400, 80, 1],[11, -1, -4, -57, -69, -14, -61, 5; 11, 0, -4, -58, -72, -15, -70, 4; -30, 0, 11, 160, 200, 42, 200, -10],188,33]
%
% END OF LATTICE L189.33
%
\hbox{}\par\smallskip%
%
% LATTICE L189.34
%
\leavevmode%
\llap{\spacer}%
${L_{189.34}}=\pdualwitharg{2}({L_{189.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}16\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{l}{2}{400}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{40,1}{\infty}{80}\above{1pt}{1pt}{b}{2}{400}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{40,9}{\infty}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
400\\-400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-400\\480\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-4\\-40
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-1\\-8
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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1\\-1\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-10\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-3\\-48
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1200\\-800\\-3600\\-800\\0\\400\\0\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-144\\-800\\-240\\-80\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-160\\-1200\\-480\\-320\\-400\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[400, -400, 0; -400, 480, 0; 0, 0, -1],[400, 80, 1],[-3, -1, -1, 1, 1, -1, -1, -3; -4, -1, 0, 1, -1, -10, -3, -6; -40, -8, 0, 0, -40, -200, -48, -80],188,34]
%
% END OF LATTICE L189.34
%
\hbox{}\par\smallskip%
%
% LATTICE L189.35
%
\leavevmode%
\llap{\spacer}%
${L_{189.35}}=\pdualwitharg{2}({L_{189.4}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}16\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{b}{2}{400}\above{1pt}{1pt}{l}{2}{80}\above{1pt}{1pt}{40,21}{\infty}{80}\above{1pt}{1pt}{l}{2}{400}\above{1pt}{1pt}{}{2}{16}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{20,9}{\infty a}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8935600\\-2072000\\55200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2072000\\-479920\\12800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
55200\\12800\\-341
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
81\\-4\\12960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-1\\2552
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\0\\-3400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\1\\-3200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42\\-1\\6760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
289\\-10\\46400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78\\-3\\12512
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
143\\-6\\22920
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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80\\0\\-400\\-320\\-480\\-1200\\-160\\-80
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\kern3pt%
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0\\16\\0\\-80\\-240\\-800\\-144\\-160
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\kern3pt%
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\kern3pt%
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-480\\-240\\-800\\-80\\80\\0\\-80\\-320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1200\\-800\\-3600\\-800\\0\\400\\0\\-400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-144\\-800\\-240\\-80\\0\\16\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-160\\-1200\\-480\\-320\\-400\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-8935600, -2072000, 55200; -2072000, -479920, 12800; 55200, 12800, -341],[400, 80, 1],[81, 16, -21, -20, 42, 289, 78, 143; -4, -1, 0, 1, -1, -10, -3, -6; 12960, 2552, -3400, -3200, 6760, 46400, 12512, 22920],188,35]
%
% END OF LATTICE L189.35
%
\hbox{}\par\smallskip%
%
% LATTICE L189.36
%
\leavevmode%
\llap{\spacer}%
${L_{189.36}}=\pdualwitharg{2}({L_{189.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}16\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${80}\above{1pt}{1pt}{b}{2}{100}\above{1pt}{1pt}{b}{2}{16}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{10,9}{\infty b}{20}\above{1pt}{1pt}{l}{2}{16}\above{1pt}{1pt}{}{2}{25}\above{1pt}{1pt}{}{2}{80}\above{1pt}{1pt}{40,31}{\infty}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95512400\\-15821600\\405200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15821600\\-2620720\\67120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
405200\\67120\\-1719
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10\\-39\\-3880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-45\\-4350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\31\\3096
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\103\\10150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
53\\212\\20770
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62\\249\\24336
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
62\\250\\24375
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\179\\17360
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
80\\0\\-80\\-160\\-240\\-240\\-200\\-80
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0\\100\\0\\-100\\-300\\-400\\-450\\-400
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\kern3pt%
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-240\\-400\\-144\\-80\\0\\16\\0\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-450\\-200\\-150\\-50\\0\\25\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-80\\-400\\-240\\-240\\-160\\-80\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-95512400, -15821600, 405200; -15821600, -2620720, 67120; 405200, 67120, -1719],[400, 80, 1],[-10, -11, 8, 26, 53, 62, 62, 44; -39, -45, 31, 103, 212, 249, 250, 179; -3880, -4350, 3096, 10150, 20770, 24336, 24375, 17360],188,36]
%
% END OF LATTICE L189.36
%

\medskip%
%------------------------------WEYL-GROUP-W190--------------------------------
%
\leavevmode\llap{}%
$W_{190}$%
\qquad\llap{6} lattices, $\chi=6$%
\hfill%
$6223$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L190.1
%
\leavevmode%
\llap{\spacer}%
${L_{190.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{s}{2}{54}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{-}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-124524\\2052\\-29592
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2052\\-30\\507
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29592\\507\\-6934
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\-410\\81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-285\\56
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
52\\819\\-162
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\427\\-84
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\-27\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\0\\54\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-124524, 2052, -29592; 2052, -30, 507; -29592, 507, -6934],[108, 3, 1],[-26, -18, 52, 27; -410, -285, 819, 427; 81, 56, -162, -84],189,1]
%
% END OF LATTICE L190.1
%
\hbox{}\par\smallskip%
%
% LATTICE L190.2
%
\leavevmode%
\llap{\spacer}%
${L_{190.2}}=\pfillwitharg{2}({L_{190.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${6}\above{1pt}{1pt}{}{6}{2}\above{1pt}{1pt}{s}{2}{54}\above{1pt}{1pt}{s}{2}{6}\above{1pt}{1pt}{+}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10233\\567\\2187
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
567\\-30\\-126
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2187\\-126\\-451
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\94\\27
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\67\\20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-189\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-101\\-30
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6\\-3\\-27\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\2\\0\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\0\\54\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\0\\6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-10233, 567, 2187; 567, -30, -126; 2187, -126, -451],[27, 3, 1],[11, 8, -22, -12; 94, 67, -189, -101; 27, 20, -54, -30],189,2]
%
% END OF LATTICE L190.2
%
\hbox{}\par\smallskip%
%
% LATTICE L190.3
%
\leavevmode%
\llap{\spacer}%
${L_{190.3}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{190.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{}{6}{54}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{s}{2}{18}\above{1pt}{1pt}{-}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-513\\216\\108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\-90\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108\\-45\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\9\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-27\\-9\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\54\\0\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-54\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-513, 216, 108; 216, -90, -45; 108, -45, -22],[27, 9, 1],[0, 4, 0, -2; 4, 9, -1, -5; -9, 0, 2, 0],189,3]
%
% END OF LATTICE L190.3
%
\hbox{}\par\smallskip%
%
% LATTICE L190.4
%
\leavevmode%
\llap{\spacer}%
${L_{190.4}}=\pdualwitharg{3}({L_{190.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{}{6}{54}\above{1pt}{1pt}{s}{2}{2}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{+}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2052\\432\\216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\-90\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\-45\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\9\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\-27\\-9\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\54\\0\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-54\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2052, 432, 216; 432, -90, -45; 216, -45, -22],[108, 9, 1],[0, 2, 0, -1; 4, 9, -1, -5; -9, 0, 2, 0],189,4]
%
% END OF LATTICE L190.4
%
\hbox{}\par\smallskip%
%
% LATTICE L190.5
%
\leavevmode%
\llap{\spacer}%
${L_{190.5}}=\pdualwitharg{2}({L_{190.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{}{6}{8}\above{1pt}{1pt}{s}{2}{216}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{-}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22322520\\5509188\\-5671080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5509188\\1359672\\-1399620
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5671080\\-1399620\\1440749
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-300\\5\\-1176
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-283\\2\\-1112
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-439\\0\\-1728
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-95\\2\\-372
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
24\\-12\\-108\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\8\\0\\-24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\0\\216\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-24\\0\\24
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[22322520, 5509188, -5671080; 5509188, 1359672, -1399620; -5671080, -1399620, 1440749],[108, 12, 1],[-300, -283, -439, -95; 5, 2, 0, 2; -1176, -1112, -1728, -372],189,5]
%
% END OF LATTICE L190.5
%
\hbox{}\par\smallskip%
%
% LATTICE L190.6
%
\leavevmode%
\llap{\spacer}%
${L_{190.6}}=\pdualwitharg{2.3}({L_{190.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{}{6}{216}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{+}{3}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1512\\-324\\648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-324\\504\\-468
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
648\\-468\\527
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\-109\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-81\\-108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\-55\\-72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\-108\\-36\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\216\\0\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-216\\0\\72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1512, -324, 648; -324, 504, -468; 648, -468, 527],[108, 36, 1],[38, 29, -1, 19; -109, -81, 3, -55; -144, -108, 4, -72],189,6]
%
% END OF LATTICE L190.6
%

\medskip%
%------------------------------WEYL-GROUP-W191--------------------------------
%
\leavevmode\llap{}%
$W_{191}$%
\qquad\llap{22} lattices, $\chi=36$%
\hfill%
$2\infty222\infty22\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L191.1
%
\leavevmode%
\llap{\spacer}%
${L_{191.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{3,2}{\infty b}{4}\above{1pt}{1pt}{r}{2}{54}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9739332\\-38988\\-110160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38988\\156\\441
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110160\\441\\1246
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9217\\-230400\\-732672
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\949\\3021
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
104\\2600\\8267
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-18\\-82
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\60\\244
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\522\\2025
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-36\\-216\\-60\\-24\\-12\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-4\\-54\\-24\\-16\\-20\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-4\\4\\0\\-12\\-20\\-52\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-54\\0\\54\\0\\-54\\-216\\-1026
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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12\\0\\-36\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-4\\-54
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-4\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-54\\0\\54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-24\\-12\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-16\\-20\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-20\\-52\\-216
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[9739332, -38988, -110160; -38988, 156, 441; -110160, 441, 1246],[108, 3, 1],[-1, -1, 3, 25, 9, 5, 5, 11; 4, -18, 60, 522, 196, 118, 140, 378; -90, -82, 244, 2025, 726, 400, 392, 837],190,1]
%
% END OF LATTICE L191.1
%
\hbox{}\par\smallskip%
%
% LATTICE L191.2
%
\leavevmode%
\llap{\spacer}%
${L_{191.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{12,5}{\infty a}{8}\above{1pt}{1pt}{s}{2}{108}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4970376\\-835272\\32616
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-835272\\-140367\\5481
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32616\\5481\\-214
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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-76321\\541872\\2243808
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12820\\91021\\376908
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-85\\604\\2512
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-373\\2646\\10908
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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24\\0\\-72\\-432
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0\\2\\-4\\-54
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-72\\-4\\8\\0
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-432\\-54\\0\\108
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-120\\-24\\-24\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-8\\-20\\-54
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0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-4970376, -835272, 32616; -835272, -140367, 5481; 32616, 5481, -214],[216, 3, 1],[-11, -14, -85, -373, -69, -6, 5, 13; 80, 100, 604, 2646, 488, 42, -36, -90; 372, 427, 2512, 10908, 1980, 161, -160, -324],190,2]
%
% END OF LATTICE L191.2
%
\hbox{}\par\smallskip%
%
% LATTICE L191.3
%
\leavevmode%
\llap{\spacer}%
${L_{191.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{m}%
\EasyButWeakLineBreak%
{${24}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{12,5}{\infty b}{8}\above{1pt}{1pt}{l}{2}{27}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-758376\\-250128\\4752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-250128\\-82497\\1566
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-338005\\1030248\\283392
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-111523\\339925\\93504
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2290\\-6980\\-1921
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-979\\2984\\820
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2123\\6471\\1782
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{$\left.\llap{\phantom{%
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-120\\-24\\-24\\0
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-24\\-8\\-20\\-27
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-758376, -250128, 4752; -250128, -82497, 1566; 4752, 1566, -25],[216, 3, 1],[-147, -167, -979, -2123, -769, -62, 63, 62; 448, 509, 2984, 6471, 2344, 189, -192, -189; 120, 139, 820, 1782, 648, 53, -52, -54],190,3]
%
% END OF LATTICE L191.3
%
\hbox{}\par\smallskip%
%
% LATTICE L191.4
%
\leavevmode%
\llap{\spacer}%
${L_{191.4}}=\pfillwitharg{2}({L_{191.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\-27\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\12\\-9
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-9\\-53
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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33\\62\\-13
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271\\504\\-108
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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3\\0\\-9\\-108
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0\\1\\-1\\-27
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-9\\-1\\1\\0
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-108\\-27\\0\\54
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-6\\-3\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-4\\-5\\-27
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-5\\-13\\-108
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-27\\-108\\-1026
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[54, -27, 0; -27, 12, -9; 0, -9, -53],[27, 3, 1],[8, 13, 33, 271, 22, 2, -3, -1; 16, 25, 62, 504, 40, 3, -6, 0; -3, -5, -13, -108, -9, -1, 1, 0],190,4]
%
% END OF LATTICE L191.4
%
\hbox{}\par\smallskip%
%
% LATTICE L191.5
%
\leavevmode%
\llap{\spacer}%
${L_{191.5}}=\pfillwitharg{2}({L_{191.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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737694\\-15174\\-21438
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15174\\312\\441
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-21438\\441\\623
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-3601\\-45000\\-91800
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76\\949\\1938
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104\\1300\\2651
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\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\right.$}%
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-1\\-9\\-28
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\30\\82
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\261\\675
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{$\left.\llap{\phantom{%
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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6\\0\\-18\\-108
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-2\\-27
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-2\\2\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-27\\0\\27
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-12\\-6\\0
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\-8\\-10\\-27
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-10\\-26\\-108
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[737694, -15174, -21438; -15174, 312, 441; -21438, 441, 623],[54, 3, 1],[-1, -1, 3, 25, 9, 5, 5, 11; 2, -9, 30, 261, 98, 59, 70, 189; -36, -28, 82, 675, 240, 130, 122, 243],190,5]
%
% END OF LATTICE L191.5
%
\hbox{}\par\smallskip%
%
% LATTICE L191.6
%
\leavevmode%
\llap{\spacer}%
${L_{191.6}}=\main({L_{191.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{12,5}{\infty}{4}\above{1pt}{1pt}{b}{2}{54}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-210708\\-72252\\4752
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72252\\-24774\\1629
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4752\\1629\\-107
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-22321\\79236\\214272
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7640\\27121\\73344
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-69\\244\\648
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\-18\\-52
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-45\\-108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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12\\0\\-36\\-216
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0\\1\\-2\\-27
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-216\\-27\\0\\54
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\-12\\-12\\0
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-12\\-4\\-10\\-27
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-12\\-10\\-52\\-216
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0\\-27\\-216\\-1026
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-210708, -72252, 4752; -72252, -24774, 1629; 4752, 1629, -107],[108, 3, 1],[-69, -6, 5, 13, -11, -14, -85, -373; 244, 21, -18, -45, 40, 50, 302, 1323; 648, 53, -52, -108, 120, 139, 820, 3564],190,6]
%
% END OF LATTICE L191.6
%
\hbox{}\par\smallskip%
%
% LATTICE L191.7
%
\leavevmode%
\llap{\spacer}%
${L_{191.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{191.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{27}\above{1pt}{1pt}{6,1}{\infty}{27}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1431\\-162\\-729
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-162\\-18\\-81
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-729\\-81\\-364
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
131\\-6270\\1188
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\-761\\144
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70\\-3325\\629
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3\\182\\-36
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-1\\126\\-27
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-9\\0
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0\\-9\\2
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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9\\0\\-81\\-36\\-45\\-54\\-27\\0
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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9\\0\\-81\\-36
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0\\27\\-27\\-27
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-81\\-27\\27\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-27\\0\\2
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-54\\-27\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-108\\-135\\-27
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-135\\-351\\-108
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-27\\-108\\-38
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1431, -162, -729; -162, -18, -81; -729, -81, -364],[27, 9, 1],[-3, -1, 1, 0, -1, -5, -13, -4; 182, 126, -9, -9, 8, 159, 579, 199; -36, -27, 0, 2, 0, -27, -108, -38],190,7]
%
% END OF LATTICE L191.7
%
\hbox{}\par\smallskip%
%
% LATTICE L191.8
%
\leavevmode%
\llap{\spacer}%
${L_{191.8}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{191.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{12,11}{\infty z}{1}\above{1pt}{1pt}{}{2}{54}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
700704\\-21330\\336420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21330\\498\\-10242
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
336420\\-10242\\161521
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1033813\\-16074\\2152224
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61100\\949\\-127200
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-496132\\-7714\\1032863
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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391\\6\\-814
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-25
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[700704, -21330, 336420; -21330, 498, -10242; 336420, -10242, 161521],[54, 6, 1],[-843, 391, 12, -5110, -1601, -2835, -2456, -16886; -13, 6, 0, -81, -25, -44, -38, -261; 1755, -814, -25, 10638, 3333, 5902, 5113, 35154],190,8]
%
% END OF LATTICE L191.8
%
\hbox{}\par\smallskip%
%
% LATTICE L191.9
%
\leavevmode%
\llap{\spacer}%
${L_{191.9}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{191.2}}))$%
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-2957364\\-1235880\\360027
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7076106, -2957364, 861732; -2957364, -1235880, 360027; 861732, 360027, -104807],[54, 9, 1],[14935, 11233, -37, -432, -67, 11069, 44641, 7866; -48538, -36507, 120, 1404, 218, -35973, -145080, -25564; -43938, -33048, 108, 1271, 198, -32562, -131328, -23141],190,9]
%
% END OF LATTICE L191.9
%
\hbox{}\par\smallskip%
%
% LATTICE L191.10
%
\leavevmode%
\llap{\spacer}%
${L_{191.10}}=\pdualwitharg{3}(\main({L_{191.3}}))$%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-540, 108, 432; 108, -18, -81; 432, -81, -337],[108, 9, 1],[-33, -13, -1, 1, 1, -11, -95, -17; 112, 39, -6, -3, 4, 48, 354, 61; -72, -27, 0, 2, 0, -27, -216, -38],190,10]
%
% END OF LATTICE L191.10
%
\hbox{}\par\smallskip%
%
% LATTICE L191.11
%
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\llap{\spacer}%
${L_{191.11}}=\pdualwitharg{3}({L_{191.1}})$%
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-216\\-432\\-540\\-54
\end{tabular}\endgroup%
\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-14152212, -2957364, 1723464; -2957364, -617940, 360027; 1723464, 360027, -209614],[108, 9, 1],[14935, 11233, -37, -432, -67, 11069, 44641, 7866; -97076, -73014, 240, 2808, 436, -71946, -290160, -51128; -43938, -33048, 108, 1271, 198, -32562, -131328, -23141],190,11]
%
% END OF LATTICE L191.11
%
\hbox{}\par\smallskip%
%
% LATTICE L191.12
%
\leavevmode%
\llap{\spacer}%
${L_{191.12}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{191.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{108}\above{1pt}{1pt}{12,7}{\infty z}{27}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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6636086460\\-1454655618\\3244884570
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1454655618\\318866094\\-711291150
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3244884570\\-711291150\\1586669483
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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1266469073\\-161243082\\-2662329384
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\kern3pt%
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-277622560\\35346079\\583608960
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{$\left[\!\llap{\phantom{%
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-19937\\2543\\41913
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}}\right.$}%
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-45\\-108\\-27\\0
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-108\\-432\\-270\\-54
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-27\\-270\\-351\\-108
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[6636086460, -1454655618, 3244884570; -1454655618, 318866094, -711291150; 3244884570, -711291150, 1586669483],[54, 18, 1],[-19937, -25169, 169, -39, -3045, -43777, -69115, -22943; 2543, 3228, -12, 5, 383, 5550, 8790, 2921; 41913, 52920, -351, 82, 6399, 92016, 145287, 48230],190,12]
%
% END OF LATTICE L191.12
%
\hbox{}\par\smallskip%
%
% LATTICE L191.13
%
\leavevmode%
\llap{\spacer}%
${L_{191.13}}=\pdualwitharg{2}(\main({L_{191.3}}))$%
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{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${3}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{12,11}{\infty}{4}\above{1pt}{1pt}{*}{2}{216}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4833000\\-138348\\-1240056
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-138348\\3972\\35496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1240056\\35496\\318175
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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-1941265\\-935136\\-7461504
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56303\\27121\\216408
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498004\\239896\\1914143
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-672\\-323\\-2583
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-282\\-135\\-1084
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-13\\-6\\-50
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-253\\-126\\-972
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-12\\-16\\-20\\-108
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4833000, -138348, -1240056; -138348, 3972, 35496; -1240056, 35496, 318175],[108, 12, 1],[-672, -282, -13, -253, -121, -511, -1573, -14021; -323, -135, -6, -126, -59, -247, -758, -6750; -2583, -1084, -50, -972, -465, -1964, -6046, -53892],190,13]
%
% END OF LATTICE L191.13
%
\hbox{}\par\smallskip%
%
% LATTICE L191.14
%
\leavevmode%
\llap{\spacer}%
${L_{191.14}}=\pdualwitharg{2}({L_{191.1}})$%
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{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{6,5}{\infty z}{1}\above{1pt}{1pt}{r}{2}{216}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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5940432\\-121284\\-1542888
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\kern3pt%
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-121284\\1992\\31500
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\kern3pt%
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-1542888\\31500\\400729
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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14276\\-81\\54972
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5940432, -121284, -1542888; -121284, 1992, 31500; -1542888, 31500, 400729],[108, 12, 1],[2370, -549, -20, 14276, 4486, 3977, 3448, 47428; -13, 3, 0, -81, -25, -22, -19, -261; 9126, -2114, -77, 54972, 17274, 15314, 13277, 182628],190,14]
%
% END OF LATTICE L191.14
%
\hbox{}\par\smallskip%
%
% LATTICE L191.15
%
\leavevmode%
\llap{\spacer}%
${L_{191.15}}=\pdualwitharg{3}({L_{191.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{b}{2}{54}\above{1pt}{1pt}{12,1}{\infty b}{216}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7675992\\8640\\71928
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\kern3pt%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7675992, 8640, 71928; 8640, -9, -81; 71928, -81, -674],[216, 9, 1],[33, 13, 1, -1, -1, 11, 95, 17; 224, 78, -12, -6, 8, 96, 708, 122; 3492, 1377, 108, -106, -108, 1161, 10044, 1798],190,15]
%
% END OF LATTICE L191.15
%
\hbox{}\par\smallskip%
%
% LATTICE L191.16
%
\leavevmode%
\llap{\spacer}%
${L_{191.16}}=\pdualwitharg{3}({L_{191.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{r}{2}{54}\above{1pt}{1pt}{12,1}{\infty a}{216}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-136843128\\45719640\\631584
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45719640\\-15275043\\-211014
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\kern3pt%
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631584\\-211014\\-2915
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-136843128, 45719640, 631584; 45719640, -15275043, -211014; 631584, -211014, -2915],[216, 9, 1],[655, 242, -7, -9, 5, 253, 1987, 174; 1304, 483, -12, -18, 8, 501, 3948, 346; 47520, 17469, -648, -647, 504, 18549, 144720, 12653],190,16]
%
% END OF LATTICE L191.16
%
\hbox{}\par\smallskip%
%
% LATTICE L191.17
%
\leavevmode%
\llap{\spacer}%
${L_{191.17}}=\pdualwitharg{2.3}(\main({L_{191.3}}))$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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3456\\648\\365
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[36396, 7344, 3456; 7344, 1548, 648; 3456, 648, 365],[108, 36, 1],[-157, -212, -13, -1, -20, -319, -1049, -353; 478, 645, 39, 3, 61, 972, 3195, 1075; 639, 864, 54, 4, 81, 1296, 4266, 1436],190,17]
%
% END OF LATTICE L191.17
%
\hbox{}\par\smallskip%
%
% LATTICE L191.18
%
\leavevmode%
\llap{\spacer}%
${L_{191.18}}=\pdualwitharg{2.3}({L_{191.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
49093070112\\7913056428\\6437967588
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\kern3pt%
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7913056428\\1275464376\\1037702484
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6437967588\\1037702484\\844262267
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[49093070112, 7913056428, 6437967588; 7913056428, 1275464376, 1037702484; 6437967588, 1037702484, 844262267],[108, 36, 1],[-15406, -9677, 91, -30, -2382, -17005, -26773, -17758; 48761, 30645, -279, 95, 7529, 53790, 84714, 56195; 57546, 36126, -351, 112, 8910, 63558, 100035, 66344],190,18]
%
% END OF LATTICE L191.18
%
\hbox{}\par\smallskip%
%
% LATTICE L191.19
%
\leavevmode%
\llap{\spacer}%
${L_{191.19}}=\pdualwitharg{2}({L_{191.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${12}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{24,11}{\infty z}{4}\above{1pt}{1pt}{s}{2}{216}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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15180264\\1523880\\-143208
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\kern3pt%
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1523880\\152976\\-14376
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\kern3pt%
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-143208\\-14376\\1351
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{$\left.\llap{\phantom{%
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14849\\-47520\\1069200
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1490\\-4769\\107280
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15180264, 1523880, -143208; 1523880, 152976, -14376; -143208, -14376, 1351],[216, 24, 1],[-4, 0, -1, -25, -6, -10, -9, -65; 19, 3, 0, 36, 13, 29, 32, 252; -222, 32, -106, -2268, -498, -752, -614, -4212],190,19]
%
% END OF LATTICE L191.19
%
\hbox{}\par\smallskip%
%
% LATTICE L191.20
%
\leavevmode%
\llap{\spacer}%
${L_{191.20}}=\pdualwitharg{2}({L_{191.3}})$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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5400\\-50760\\4536
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-50760\\477264\\-42648
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4536\\-42648\\3811
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[5400, -50760, 4536; -50760, 477264, -42648; 4536, -42648, 3811],[216, 24, 1],[-2, 0, -1, -25, -3, -10, -9, -65; 14, 5, -1, 36, 8, 39, 45, 360; 159, 56, -10, 432, 93, 448, 514, 4104],190,20]
%
% END OF LATTICE L191.20
%
\hbox{}\par\smallskip%
%
% LATTICE L191.21
%
\leavevmode%
\llap{\spacer}%
${L_{191.21}}=\pdualwitharg{2.3}({L_{191.2}})$%
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{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\kern3pt%
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673908552\\-743009616\\-162237240
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-611233992, 673908552, 147148920; 673908552, -743009616, -162237240; 147148920, -162237240, -35424739],[216, 72, 1],[-7022, -10574, 13, 407, 36, -10384, -20971, -7393; -1853, -2799, 0, 108, 13, -2721, -5520, -1948; -20682, -31104, 54, 1196, 90, -30672, -61830, -21788],190,21]
%
% END OF LATTICE L191.21
%
\hbox{}\par\smallskip%
%
% LATTICE L191.22
%
\leavevmode%
\llap{\spacer}%
${L_{191.22}}=\pdualwitharg{2.3}({L_{191.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{432}\above{1pt}{1pt}{24,7}{\infty z}{108}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-3628368\\-3627288\\671760
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\kern3pt%
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-3627288\\-3605112\\668016
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\kern3pt%
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671760\\668016\\-123775
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{$\left.\llap{\phantom{%
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0\\0\\0
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1573452\\15372865\\91507032
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3628368, -3627288, 671760; -3627288, -3605112, 668016; 671760, 668016, -123775],[216, 72, 1],[-925, -2797, -1, 108, 7, -2711, -5507, -1944; -9037, -27324, -9, 1055, 68, -26490, -53805, -18993; -53793, -162648, -54, 6280, 405, -157680, -320274, -113056],190,22]
%
% END OF LATTICE L191.22
%

\medskip%
%------------------------------WEYL-GROUP-W192--------------------------------
%
\leavevmode\llap{}%
$W_{192}$%
\qquad\llap{22} lattices, $\chi=36$%
\hfill%
$2\infty222\infty22\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L192.1
%
\leavevmode%
\llap{\spacer}%
${L_{192.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
\EasyButWeakLineBreak%
{${108}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{3,1}{\infty a}{4}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1255068\\-800172\\84348
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-800172\\-510132\\53769
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-31051\\67500\\178200
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19688\\42799\\112992
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2819\\-6192\\-16794
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-11\\24\\64
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-121\\266\\723
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-864\\-100\\-44\\-6
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0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1255068, -800172, 84348; -800172, -510132, 53769; 84348, 53769, -5666],[108, 3, 1],[2819, 243, -11, -121, -1991, -151, 57, 144; -6192, -534, 24, 266, 4392, 334, -124, -316; -16794, -1450, 64, 723, 12042, 922, -328, -855],191,1]
%
% END OF LATTICE L192.1
%
\hbox{}\par\smallskip%
%
% LATTICE L192.2
%
\leavevmode%
\llap{\spacer}%
${L_{192.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-603720\\-301536\\3672
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-301536\\-150603\\1833
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-603720, -301536, 3672; -301536, -150603, 1833; 3672, 1833, -22],[216, 3, 1],[-1453, -99, -125, -51, -347, -1, 25, 1; 3024, 206, 260, 106, 720, 2, -52, -2; 9396, 637, 796, 318, 2052, -1, -160, 0],191,2]
%
% END OF LATTICE L192.2
%
\hbox{}\par\smallskip%
%
% LATTICE L192.3
%
\leavevmode%
\llap{\spacer}%
${L_{192.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1015848\\10368\\5184
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10368\\-105\\-54
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5184\\-54\\-25
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{$\left.\llap{\phantom{%
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\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3167\\226512\\166320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34\\-2432\\-1785
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-1001\\-736
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\4248\\3024
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\359\\259
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\644\\472
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\214\\159
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
216\\0\\-216\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-4\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-4\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-9\\0\\3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15768\\-864\\-648\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-864\\-50\\-44\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-648\\-44\\-56\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-12\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-1015848, 10368, 5184; 10368, -105, -54; 5184, -54, -25],[216, 3, 1],[59, 5, 9, 3, 85, 3, -1, -1; 4248, 359, 644, 214, 6048, 213, -72, -71; 3024, 259, 472, 159, 4536, 161, -52, -54],191,3]
%
% END OF LATTICE L192.3
%
\hbox{}\par\smallskip%
%
% LATTICE L192.4
%
\leavevmode%
\llap{\spacer}%
${L_{192.4}}=\pfillwitharg{2}({L_{192.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{3}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${27}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{6,1}{\infty}{1}\above{1pt}{1pt}{r}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-27\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-12\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-27\\-53
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-55\\18\\54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\7\\24
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-48\\16\\47
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70\\0\\-81
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-1\\-11
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-2\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\-4\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\0\\-27\\-108\\-1971\\-216\\-81\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-1\\-9\\-216\\-25\\-11\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-1\\1\\0\\-81\\-11\\-7\\-6
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-9\\0\\6\\0\\-3\\-6\\-12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\0\\-27\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-1\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-1\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-9\\0\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1971\\-216\\-81\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-25\\-11\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81\\-11\\-7\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-6\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-54, -27, -54; -27, -12, -27; -54, -27, -53],[27, 3, 1],[70, 10, 7, 7, 38, 2, -1, -1; 0, -1, -2, -4, -36, -3, 0, 2; -81, -11, -7, -6, -27, -1, 1, 0],191,4]
%
% END OF LATTICE L192.4
%
\hbox{}\par\smallskip%
%
% LATTICE L192.5
%
\leavevmode%
\llap{\spacer}%
${L_{192.5}}=\pfillwitharg{2}({L_{192.2}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${54}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{6,1}{\infty a}{2}\above{1pt}{1pt}{}{2}{3}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-627534\\-46116\\42174
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46116\\-3381\\3096
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
42174\\3096\\-2833
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2699\\33750\\76950
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
190\\2374\\5415
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-178\\-2225\\-5074
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
205\\2196\\5454
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16\\167\\421
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-62\\-142
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-158\\-381
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\0\\-54\\-108\\-3942\\-432\\-162\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-2\\-9\\-432\\-50\\-22\\-3
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54\\0\\-54\\-108
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-2\\-9
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-2\\2\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-9\\0\\3
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3942\\-432\\-162\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-432\\-50\\-22\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-162\\-22\\-14\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-6\\-6
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-627534, -46116, 42174; -46116, -3381, 3096; 42174, 3096, -2833],[54, 3, 1],[205, 16, -5, -14, -277, -24, 1, 12; 2196, 167, -62, -158, -3096, -267, 12, 133; 5454, 421, -142, -381, -7506, -649, 28, 324],191,5]
%
% END OF LATTICE L192.5
%
\hbox{}\par\smallskip%
%
% LATTICE L192.6
%
\leavevmode%
\llap{\spacer}%
${L_{192.6}}=\main({L_{192.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${108}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{12,1}{\infty}{4}\above{1pt}{1pt}{b}{2}{6}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33588\\0\\648
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\6\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
648\\-3\\-11
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
359\\9360\\18360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-53\\-102
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-156\\-307
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\360\\648
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-1
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-26\\-52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\0
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108\\0\\-108\\-216\\-7884\\-432\\-324\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2\\-9\\-432\\-25\\-22\\-3
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-9\\0\\6\\0\\-3\\-12\\-12
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108\\0\\-108\\-216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-2\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-2\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-9\\0\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7884\\-432\\-324\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-432\\-25\\-22\\-3
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-324\\-22\\-28\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-3\\-12\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-33588, 0, 648; 0, 6, -3; 648, -3, -11],[108, 3, 1],[13, 0, -1, 0, 59, 4, 5, 2; 360, 1, -26, -1, 1512, 103, 130, 53; 648, -1, -52, 0, 3024, 205, 256, 102],191,6]
%
% END OF LATTICE L192.6
%
\hbox{}\par\smallskip%
%
% LATTICE L192.7
%
\leavevmode%
\llap{\spacer}%
${L_{192.7}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{192.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-3}{3}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{27}\above{1pt}{1pt}{6,5}{\infty}{27}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1431\\-729\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-729\\-360\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-27\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47\\48\\-1728
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26\\25\\-936
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\2\\-73
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\-3\\297
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-6\\243
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-4\\99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-27\\-36\\-73\\-216\\-81\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\27\\-27\\-81\\-216\\-675\\-297\\-27
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-27\\27\\0\\-81\\-297\\-189\\-54
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-27\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\27\\-27\\-81
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-27\\27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-81\\0\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73\\-216\\-81\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-675\\-297\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81\\-297\\-189\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-27\\-54\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1431, -729, -54; -729, -360, -27; -54, -27, -2],[27, 9, 1],[-3, -11, -7, -2, -1, -1, 1, 0; 0, -3, -6, -4, -4, -9, 0, 2; 70, 297, 243, 99, 74, 135, -27, -27],191,7]
%
% END OF LATTICE L192.7
%
\hbox{}\par\smallskip%
%
% LATTICE L192.8
%
\leavevmode%
\llap{\spacer}%
${L_{192.8}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{192.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${27}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{12,7}{\infty z}{1}\above{1pt}{1pt}{}{2}{6}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38529054\\-1899072\\18493974
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1899072\\93606\\-911556
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18493974\\-911556\\8877121
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
802790\\-45315\\-1677132
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-42075\\2374\\87900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
385407\\-21755\\-805165
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-21157\\1188\\44199
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5545\\311\\11584
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1695\\95\\3541
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1459\\82\\3048
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\0\\-27\\-108\\-1971\\-432\\-81\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-2\\-18\\-432\\-100\\-22\\-6
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\kern3pt%
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-27\\-2\\1\\0\\-81\\-22\\-7\\-6
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-108\\-18\\0\\6\\0\\-6\\-6\\-12
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\0\\-27\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-2\\-18
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-2\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-18\\0\\6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1971\\-432\\-81\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-432\\-100\\-22\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81\\-22\\-7\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-6\\-6\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[38529054, -1899072, 18493974; -1899072, 93606, -911556; 18493974, -911556, 8877121],[54, 6, 1],[-21157, -5545, -1695, -1459, -9137, -1187, 12, -224; 1188, 311, 95, 82, 522, 69, 0, 13; 44199, 11584, 3541, 3048, 19089, 2480, -25, 468],191,8]
%
% END OF LATTICE L192.8
%
\hbox{}\par\smallskip%
%
% LATTICE L192.9
%
\leavevmode%
\llap{\spacer}%
${L_{192.9}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{192.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{54}\above{1pt}{1pt}{6,5}{\infty a}{54}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24570\\-7722\\-3996
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7722\\-1773\\-1107
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3996\\-1107\\-616
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4661\\11592\\-51030
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2035\\5059\\-22275
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
888\\2208\\-9721
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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177\\442\\-1942
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
401\\1002\\-4401
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-12\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50\\-125\\549
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2\\0\\-54\\-36\\-146\\-432\\-162\\0
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\kern3pt%
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-36\\-81\\0\\9\\0\\-27\\-54\\-18
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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2\\0\\-54\\-36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\54\\-54\\-81
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-54\\54\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-81\\0\\9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-146\\-432\\-162\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-432\\-1350\\-594\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-162\\-594\\-378\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-27\\-54\\-18
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-24570, -7722, -3996; -7722, -1773, -1107; -3996, -1107, -616],[54, 9, 1],[177, 401, -5, -50, -29, 43, 227, 87; 442, 1002, -12, -125, -74, 102, 564, 217; -1942, -4401, 54, 549, 322, -459, -2484, -954],191,9]
%
% END OF LATTICE L192.9
%
\hbox{}\par\smallskip%
%
% LATTICE L192.10
%
\leavevmode%
\llap{\spacer}%
${L_{192.10}}=\pdualwitharg{3}(\main({L_{192.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{}{2}{27}\above{1pt}{1pt}{12,5}{\infty}{108}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5724\\1512\\108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1512\\-333\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108\\-27\\-2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-120\\1080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\19\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-19
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\0\\-184
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-3\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-12\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-4\\99
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-108\\-72\\-292\\-432\\-324\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\27\\-54\\-81\\-432\\-675\\-594\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-54\\108\\0\\-324\\-594\\-756\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-81\\0\\18\\0\\-27\\-108\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-108\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\27\\-54\\-81
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-54\\108\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-81\\0\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-292\\-432\\-324\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-432\\-675\\-594\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-324\\-594\\-756\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-27\\-108\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5724, 1512, 108; 1512, -333, -27; 108, -27, -2],[108, 9, 1],[-3, -4, -1, 1, 3, 4, 1, -1; 0, -3, -12, -4, -8, -9, 0, 2; -184, -216, 54, 99, 256, 324, 54, -81],191,10]
%
% END OF LATTICE L192.10
%
\hbox{}\par\smallskip%
%
% LATTICE L192.11
%
\leavevmode%
\llap{\spacer}%
${L_{192.11}}=\pdualwitharg{3}({L_{192.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}4\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${4}\above{1pt}{1pt}{*}{2}{108}\above{1pt}{1pt}{3,2}{\infty b}{108}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\333
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\333\\-308
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\-96\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-209\\-234
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\192\\215
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-22\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-6\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\96\\108
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\41\\45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-108\\-72\\-292\\-864\\-324\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\108\\-108\\-162\\-864\\-2700\\-1188\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-108\\108\\0\\-324\\-1188\\-756\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-162\\0\\18\\0\\-54\\-108\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4\\0\\-108\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\108\\-108\\-162
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-108\\108\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-162\\0\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-292\\-864\\-324\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-864\\-2700\\-1188\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-324\\-1188\\-756\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-54\\-108\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[108, 0, 0; 0, -360, 333; 0, 333, -308],[108, 9, 1],[3, 11, 7, 1, 1, 1, -1, 0; -22, -6, 96, 41, 86, 198, 0, -25; -22, 0, 108, 45, 94, 216, 0, -27],191,11]
%
% END OF LATTICE L192.11
%
\hbox{}\par\smallskip%
%
% LATTICE L192.12
%
\leavevmode%
\llap{\spacer}%
${L_{192.12}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{192.2}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{3}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{108}\above{1pt}{1pt}{12,11}{\infty z}{27}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67359546\\4324482\\32624586
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4324482\\277632\\2094498
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
32624586\\2094498\\15801229
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1094995\\79035\\2250342
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-70104\\5059\\144072
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-530352\\38280\\1089935
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
586\\-37\\-1205
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2336\\-129\\-4806
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
65\\6\\-135
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35\\-2\\-72
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-27\\-36\\-73\\-432\\-81\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\108\\-54\\-162\\-432\\-2700\\-594\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-54\\27\\0\\-81\\-594\\-189\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-162\\0\\18\\0\\-54\\-54\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-27\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\108\\-54\\-162
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-54\\27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-162\\0\\18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-73\\-432\\-81\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-432\\-2700\\-594\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-81\\-594\\-189\\-54
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-54\\-54\\-36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[67359546, 4324482, 32624586; 4324482, 277632, 2094498; 32624586, 2094498, 15801229],[54, 18, 1],[586, 2336, 65, 35, 938, 6808, 2221, 727; -37, -129, 6, -2, -73, -531, -171, -53; -1205, -4806, -135, -72, -1927, -13986, -4563, -1494],191,12]
%
% END OF LATTICE L192.12
%
\hbox{}\par\smallskip%
%
% LATTICE L192.13
%
\leavevmode%
\llap{\spacer}%
${L_{192.13}}=\pdualwitharg{2}(\main({L_{192.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${27}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{12,7}{\infty}{4}\above{1pt}{1pt}{*}{2}{24}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1544616\\-92988\\-401652
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-92988\\5604\\24180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-401652\\24180\\104443
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-110827\\846\\-426384
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6812\\-53\\26208
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28820\\-220\\110879
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1031\\18\\-3969
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\3\\-616
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\0\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-2\\12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-432\\-100\\-44\\-12
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1544616, -92988, -401652; -92988, 5604, 24180; -401652, 24180, 104443],[108, 12, 1],[-1031, -160, -13, 3, -1327, -364, -249, -265; 18, 3, 0, -2, 0, 1, 2, 4; -3969, -616, -50, 12, -5103, -1400, -958, -1020],191,13]
%
% END OF LATTICE L192.13
%
\hbox{}\par\smallskip%
%
% LATTICE L192.14
%
\leavevmode%
\llap{\spacer}%
${L_{192.14}}=\pdualwitharg{2}({L_{192.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${108}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{6,1}{\infty z}{1}\above{1pt}{1pt}{r}{2}{24}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
624286224\\-15288804\\-159533496
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15288804\\374424\\3906984
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-159533496\\3906984\\40768057
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1578529\\-1725696\\-6011712
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39150\\42799\\149100
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
403245\\440840\\1535729
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8494\\9315\\32346
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
551\\607\\2098
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-5\\-23
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
208\\229\\792
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-432\\-36\\0\\24
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7884\\-864\\-162\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-864\\-100\\-22\\-12
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-162\\-22\\-7\\-12
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[624286224, -15288804, -159533496; -15288804, 374424, 3906984; -159533496, 3906984, 40768057],[108, 12, 1],[8494, 551, -6, 208, 19694, 2581, 789, 1358; 9315, 607, -5, 229, 21501, 2817, 861, 1483; 32346, 2098, -23, 792, 75006, 9830, 3005, 5172],191,14]
%
% END OF LATTICE L192.14
%
\hbox{}\par\smallskip%
%
% LATTICE L192.15
%
\leavevmode%
\llap{\spacer}%
${L_{192.15}}=\pdualwitharg{3}({L_{192.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{b}{2}{54}\above{1pt}{1pt}{12,5}{\infty a}{216}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-179928\\11016\\432
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11016\\-666\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432\\-27\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-360\\6480
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0\\19\\-360
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-19
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-3\\-36\\-368
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-4\\-51\\-432
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-24\\108
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\8\\198
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{$\left.\llap{\phantom{%
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0\\0\\0
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\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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0\\54\\-108\\-162\\-864\\-1350\\-1188\\-54
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-216\\-144
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\54\\-108\\-162
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-144\\-162\\0\\36
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-584\\-864\\-648\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-864\\-1350\\-1188\\-54
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-648\\-1188\\-1512\\-216
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-179928, 11016, 432; 11016, -666, -27; 432, -27, -1],[216, 9, 1],[-3, -4, -1, 1, 3, 4, 1, -1; -36, -51, -24, 8, 28, 39, 12, -10; -368, -432, 108, 198, 512, 648, 108, -162],191,15]
%
% END OF LATTICE L192.15
%
\hbox{}\par\smallskip%
%
% LATTICE L192.16
%
\leavevmode%
\llap{\spacer}%
${L_{192.16}}=\pdualwitharg{3}({L_{192.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{54}\above{1pt}{1pt}{12,5}{\infty b}{216}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-58611384\\-412776\\-32615568
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-412776\\-2907\\-229698
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32615568\\-229698\\-18149635
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-30049\\-1074216\\67608
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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181\\3376\\-368
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251\\4893\\-513
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101\\2724\\-216
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-23\\-290\\45
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-584\\-864\\-648\\0
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-864\\-1350\\-1188\\-27
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-58611384, -412776, -32615568; -412776, -2907, -229698; -32615568, -229698, -18149635],[216, 9, 1],[181, 251, 101, -23, -149, -203, -53, 27; 3376, 4893, 2724, -290, -2232, -3177, -1008, 433; -368, -513, -216, 45, 296, 405, 108, -54],191,16]
%
% END OF LATTICE L192.16
%
\hbox{}\par\smallskip%
%
% LATTICE L192.17
%
\leavevmode%
\llap{\spacer}%
${L_{192.17}}=\pdualwitharg{2.3}(\main({L_{192.3}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{}{2}{108}\above{1pt}{1pt}{12,11}{\infty}{108}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
977292\\-8100\\260172
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8100\\72\\-2160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
260172\\-2160\\69265
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-1834\\5362\\7055
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1207\\-3534\\-4644
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800\\-2343\\-3078
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262\\-767\\-1008
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[977292, -8100, 260172; -8100, 72, -2160; 260172, -2160, 69265],[108, 36, 1],[164, 1207, 800, 262, 98, 365, -14, 0; -480, -3534, -2343, -767, -286, -1062, 45, 1; -631, -4644, -3078, -1008, -377, -1404, 54, 0],191,17]
%
% END OF LATTICE L192.17
%
\hbox{}\par\smallskip%
%
% LATTICE L192.18
%
\leavevmode%
\llap{\spacer}%
${L_{192.18}}=\pdualwitharg{2.3}({L_{192.1}})$%
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{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
432648\\69228\\58320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
69228\\11088\\9324
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58320\\9324\\7867
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-8461\\24210\\34020
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{$\left[\!\llap{\phantom{%
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233\\-666\\-938
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845\\-2415\\-3402
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275\\-786\\-1107
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179\\-512\\-720
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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-864\\-2700\\-594\\-108
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[432648, 69228, 58320; 69228, 11088, 9324; 58320, 9324, 7867],[108, 36, 1],[233, 845, 275, 179, 143, 283, 7, 9; -666, -2415, -786, -512, -410, -813, -21, -26; -938, -3402, -1107, -720, -574, -1134, -27, -36],191,18]
%
% END OF LATTICE L192.18
%
\hbox{}\par\smallskip%
%
% LATTICE L192.19
%
\leavevmode%
\llap{\spacer}%
${L_{192.19}}=\pdualwitharg{2}({L_{192.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${108}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{24,19}{\infty z}{4}\above{1pt}{1pt}{s}{2}{24}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72889200\\-7019784\\702000
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7019784\\-676056\\67608
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
702000\\67608\\-6761
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-72889200, -7019784, 702000; -7019784, -676056, 67608; 702000, 67608, -6761],[216, 24, 1],[-125, -23, 0, 12, 143, 27, 1, -11; 288, 50, -1, -25, -216, -34, 5, 29; -10098, -1888, -10, 996, 12690, 2464, 154, -852],191,19]
%
% END OF LATTICE L192.19
%
\hbox{}\par\smallskip%
%
% LATTICE L192.20
%
\leavevmode%
\llap{\spacer}%
${L_{192.20}}=\pdualwitharg{2}({L_{192.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{-2}{2}{\cdot}1\above{1pt}{1pt}{1}{}3\above{1pt}{1pt}{-}{}27\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${27}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{24,7}{\infty z}{4}\above{1pt}{1pt}{l}{2}{24}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-731625264\\4448520\\6860160
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4448520\\-27048\\-41712
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6860160\\-41712\\-64325
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{$\left.\llap{\phantom{%
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}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-731625264, 4448520, 6860160; 4448520, -27048, -41712; 6860160, -41712, -64325],[216, 24, 1],[-49, -21, -1, 12, 112, 49, 8, -5; 144, 50, -1, -25, -108, -34, 5, 29; -5319, -2272, -106, 1296, 12015, 5248, 850, -552],191,20]
%
% END OF LATTICE L192.20
%
\hbox{}\par\smallskip%
%
% LATTICE L192.21
%
\leavevmode%
\llap{\spacer}%
${L_{192.21}}=\pdualwitharg{2.3}({L_{192.2}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3592080, 526824, -601344; 526824, -77256, 88200; -601344, 88200, -100667],[216, 72, 1],[183, 809, -13, -99, -5, 259, 280, 188; 410, 1818, -27, -223, -18, 534, 615, 419; -734, -3240, 54, 396, 14, -1080, -1134, -756],191,21]
%
% END OF LATTICE L192.21
%
\hbox{}\par\smallskip%
%
% LATTICE L192.22
%
\leavevmode%
\llap{\spacer}%
${L_{192.22}}=\pdualwitharg{2.3}({L_{192.3}})$%
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{$\left[\!\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-6740495568, 74694744, 40224816; 74694744, -827640, -445752; 40224816, -445752, -240047],[216, 72, 1],[-278, -2563, -1, 324, 73, 103, -614, -529; -534, -4926, -3, 623, 142, 222, -1173, -1015; -45593, -420336, -162, 53136, 11969, 16848, -100710, -86760],191,22]
%
% END OF LATTICE L192.22
%

\medskip%
%------------------------------WEYL-GROUP-W193--------------------------------
%
\leavevmode\llap{}%
$W_{193}$%
\qquad\llap{6} lattices, $\chi=12$%
\hfill%
$222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L193.1
%
\leavevmode%
\llap{\spacer}%
${L_{193.1}}$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}4\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}3\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{1}{}$}\spacer%
\instructions{2}%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[756, 324, -108; 324, 138, -51; -108, -51, -10],[108, 3, 1],[-1, -115, -10, -9, 25, 5; 2, 252, 22, 20, -54, -11; 0, -54, -5, -6, 0, 2],192,1]
%
% END OF LATTICE L193.1
%
\hbox{}\par\smallskip%
%
% LATTICE L193.2
%
\leavevmode%
\llap{\spacer}%
${L_{193.2}}=\pfillwitharg{2}({L_{193.1}})$%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-54, -54, -27; -54, -51, -27; -27, -27, -13],[27, 3, 1],[-1, 11, 3, 3, 7, -1; 1, 0, -1, -2, -9, 0; 0, -27, -5, -3, 0, 2],192,2]
%
% END OF LATTICE L193.2
%
\hbox{}\par\smallskip%
%
% LATTICE L193.3
%
\leavevmode%
\llap{\spacer}%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[54, 162, -27; 162, 495, -81; -27, -81, 13],[27, 9, 1],[-3, -2, -2, 2, 1, -1; 1, 0, -3, -2, -1, 0; 0, -5, -27, -9, -4, 0],192,3]
%
% END OF LATTICE L193.3
%
\hbox{}\par\smallskip%
%
% LATTICE L193.4
%
\leavevmode%
\llap{\spacer}%
${L_{193.4}}=\pdualwitharg{3}({L_{193.1}})$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[540, -108, 0; -108, -18, 9; 0, 9, -2],[108, 9, 1],[1, 1, 1, -1, -1, -1; 2, 4, 6, -4, -6, -9; 0, 14, 27, -18, -32, -54],192,4]
%
% END OF LATTICE L193.4
%
\hbox{}\par\smallskip%
%
% LATTICE L193.5
%
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${L_{193.5}}=\pdualwitharg{2}({L_{193.1}})$%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8\\0\\-108\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
12\\0\\-12
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\108\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12\\0\\8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24\\-108\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-1080\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-108\\-20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[462456, -31428, -115128; -31428, 2136, 7824; -115128, 7824, 28661],[108, 12, 1],[-11, 124, 36, 48, 209, 1; -8, 45, 17, 25, 108, 0; -42, 486, 140, 186, 810, 4],192,5]
%
% END OF LATTICE L193.5
%
\hbox{}\par\smallskip%
%
% LATTICE L193.6
%
\leavevmode%
\llap{\spacer}%
${L_{193.6}}=\pdualwitharg{2.3}({L_{193.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}4\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{1}{}27\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{*}{2}{216}\above{1pt}{1pt}{*}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
70200\\5076\\14364
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5076\\360\\1044
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14364\\1044\\2935
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1111\\3120\\4320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37\\103\\144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-259\\728\\1007
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\145\\198
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\51\\70
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28\\-81\\-108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-108\\-72\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-36\\-40\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\0\\216\\0\\-108\\-540
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\0\\216
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-36\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-40\\-108
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-108\\-540
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[70200, 5076, 14364; 5076, 360, 1044; 14364, 1044, 2935],[108, 36, 1],[-51, -18, 28, 14, -19, -139; 145, 51, -81, -41, 53, 393; 198, 70, -108, -54, 74, 540],192,6]
%
% END OF LATTICE L193.6
%

\medskip%
%------------------------------WEYL-GROUP-W194--------------------------------
%
\leavevmode\llap{}%
$W_{194}$%
\qquad\llap{32} lattices, $\chi=18$%
\hfill%
$2\infty2222$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L194.1
%
\leavevmode%
\llap{\spacer}%
${L_{194.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{25,5,2*}%
\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{60,49}{\infty a}{40}\above{1pt}{1pt}{b}{2}{450}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{50}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8854200\\1479600\\145800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1479600\\-239810\\-20665
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
145800\\-20665\\-562
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12367\\-92304\\185688
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1020\\-7613\\15315
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1007\\7516\\-15120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2068\\15435\\-31050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-731\\-5456\\10976
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2689\\-20070\\40375
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\0\\-360\\-1800\\-216\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-20\\-225\\-40\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-20\\40\\0\\-40\\-100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-225\\0\\450\\0\\-225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-40\\-40\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-25\\-100\\-225\\0\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-8854200, 1479600, 145800; 1479600, -239810, -20665; 145800, -20665, -562],[1800, 5, 1],[-12367, -1020, 1007, 2068, -731, -2689; -92304, -7613, 7516, 15435, -5456, -20070; 185688, 15315, -15120, -31050, 10976, 40375],193,1]
%
% END OF LATTICE L194.1
%
\hbox{}\par\smallskip%
%
% LATTICE L194.2
%
\leavevmode%
\llap{\spacer}%
${L_{194.2}}=\pfillwitharg{2.5}({L_{194.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{6,1}{\infty a}{10}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51767190\\1483380\\519390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1483380\\-42506\\-14883
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
519390\\-14883\\-5210
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-967\\-34074\\936
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-174\\-6130\\165
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\2080\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58\\2043\\-54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-1798\\52
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78\\-2749\\77
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-90\\-180\\-54\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-10\\-45\\-20\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-10\\10\\0\\-10\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-45\\0\\18\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-20\\-10\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-10\\-9\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-51767190, 1483380, 519390; 1483380, -42506, -14883; 519390, -14883, -5210],[90, 1, 1],[-967, -174, 59, 58, -51, -78; -34074, -6130, 2080, 2043, -1798, -2749; 936, 165, -60, -54, 52, 77],193,2]
%
% END OF LATTICE L194.2
%
\hbox{}\par\smallskip%
%
% LATTICE L194.3
%
\leavevmode%
\llap{\spacer}%
${L_{194.3}}=\pfillwitharg{5}({L_{194.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{12,1}{\infty a}{40}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1041541560\\-7672680\\319680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7672680\\-56522\\2355
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
319680\\2355\\-98
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
137\\-18432\\3960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-270\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\3900\\-880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\135\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\-2824\\640
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\-1345\\299
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
72\\0\\-360\\-360\\-216\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-20\\-45\\-40\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-20\\40\\0\\-40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-45\\0\\18\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-40\\-40\\0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-20\\-9\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-1041541560, -7672680, 319680; -7672680, -56522, 2355; 319680, 2355, -98],[360, 1, 1],[137, 2, -29, -1, 21, 10; -18432, -270, 3900, 135, -2824, -1345; 3960, 35, -880, -18, 640, 299],193,3]
%
% END OF LATTICE L194.3
%
\hbox{}\par\smallskip%
%
% LATTICE L194.4
%
\leavevmode%
\llap{\spacer}%
${L_{194.4}}=\pfillwitharg{2}({L_{194.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{30,19}{\infty b}{10}\above{1pt}{1pt}{r}{2}{450}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{r}{2}{50}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-890550\\339750\\-38250
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
339750\\-129610\\14595
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-38250\\14595\\-1642
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-277\\-558\\1494
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-73\\190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\58\\-160
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\90\\-225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-25\\-50\\138
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-142\\-285\\775
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
18\\0\\-90\\-900\\-54\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-10\\-225\\-20\\-25
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-10\\10\\0\\-10\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\-225\\0\\450\\0\\-225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-20\\-10\\0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-25\\-50\\-225\\0\\50
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-890550, 339750, -38250; 339750, -129610, 14595; -38250, 14595, -1642],[450, 5, 1],[-277, -36, 29, 44, -25, -142; -558, -73, 58, 90, -50, -285; 1494, 190, -160, -225, 138, 775],193,4]
%
% END OF LATTICE L194.4
%
\hbox{}\par\smallskip%
%
% LATTICE L194.5
%
\leavevmode%
\llap{\spacer}%
${L_{194.5}}=\pdualwitharg{2}(\pfillwitharg{2.5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{12,7}{\infty z}{5}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19042920\\4992570\\9451530
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4992570\\1309160\\2477958
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9451530\\2477958\\4691057
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-79396\\-4752\\162477
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33996\\-2035\\69570
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-667\\-40\\1365
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-607\\-36\\1242
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4246\\-254\\8689
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10645\\-637\\21784
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-45\\-180\\-27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\-10\\-90\\-20\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-10\\5\\0\\-5\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-90\\0\\36\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-20\\-5\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-10\\-18\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[19042920, 4992570, 9451530; 4992570, 1309160, 2477958; 9451530, 2477958, 4691057],[90, 2, 1],[-79396, -33996, -667, -607, -4246, -10645; -4752, -2035, -40, -36, -254, -637; 162477, 69570, 1365, 1242, 8689, 21784],193,5]
%
% END OF LATTICE L194.5
%
\hbox{}\par\smallskip%
%
% LATTICE L194.6
%
\leavevmode%
\llap{\spacer}%
${L_{194.6}}=\pdualwitharg{5}(\pfillwitharg{2.5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${90}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{6,1}{\infty a}{2}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-178110\\60030\\3960
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
60030\\-20230\\-1335
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3960\\-1335\\-88
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
109\\198\\1890
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\3\\44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-6\\-44
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\26\\190
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\25\\205
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\0\\-90\\-900\\-270\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-2\\-45\\-20\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-2\\2\\0\\-10\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\-45\\0\\90\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\-20\\-10\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-10\\-45\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-178110, 60030, 3960; 60030, -20230, -1335; 3960, -1335, -88],[90, 5, 1],[109, 2, -3, -1, 13, 13; 198, 3, -6, 0, 26, 25; 1890, 44, -44, -45, 190, 205],193,6]
%
% END OF LATTICE L194.6
%
\hbox{}\par\smallskip%
%
% LATTICE L194.7
%
\leavevmode%
\llap{\spacer}%
${L_{194.7}}=\pdualwitharg{3}(\pfillwitharg{2.5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{6,5}{\infty a}{90}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
47304090\\16693560\\5204070
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
16693560\\5891220\\1836513
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5204070\\1836513\\572516
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3313\\-1056\\33502
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6381\\-2035\\64530
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-249\\-80\\2520
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13\\-4\\131
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1597\\-508\\16146
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2000\\-637\\20223
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-90\\-20\\-54\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\90\\-90\\-45\\-180\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-90\\90\\0\\-90\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-45\\0\\2\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-180\\-90\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-90\\-9\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[47304090, 16693560, 5204070; 16693560, 5891220, 1836513; 5204070, 1836513, 572516],[90, 9, 1],[-3313, -6381, -249, -13, -1597, -2000; -1056, -2035, -80, -4, -508, -637; 33502, 64530, 2520, 131, 16146, 20223],193,7]
%
% END OF LATTICE L194.7
%
\hbox{}\par\smallskip%
%
% LATTICE L194.8
%
\leavevmode%
\llap{\spacer}%
${L_{194.8}}=\pdualwitharg{2.5}(\pfillwitharg{2.5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{12,7}{\infty z}{1}\above{1pt}{1pt}{r}{2}{180}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2938860\\-50130\\1431720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-50130\\880\\-24420
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1431720\\-24420\\697489
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10175\\1458\\-20835
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
881\\126\\-1804
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\3\\-43
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
44\\9\\-90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
525\\76\\-1075
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1343\\193\\-2750
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-45\\-900\\-135\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\-2\\-90\\-20\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-2\\1\\0\\-5\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\-90\\0\\180\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-20\\-5\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-10\\-90\\0\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2938860, -50130, 1431720; -50130, 880, -24420; 1431720, -24420, 697489],[90, 10, 1],[10175, 881, 21, 44, 525, 1343; 1458, 126, 3, 9, 76, 193; -20835, -1804, -43, -90, -1075, -2750],193,8]
%
% END OF LATTICE L194.8
%
\hbox{}\par\smallskip%
%
% LATTICE L194.9
%
\leavevmode%
\llap{\spacer}%
${L_{194.9}}=\pdualwitharg{2.3}(\pfillwitharg{2.5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{180}\above{1pt}{1pt}{12,11}{\infty z}{45}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27658470420\\-145470870\\-13572666630
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-145470870\\-765108\\-71386002
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-13572666630\\-71386002\\-6660429035
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28865\\-1893\\-58801
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
93309\\-6130\\-190080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15927\\1040\\32445
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3450\\227\\7028
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13771\\-899\\-28053
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41989\\-2749\\-85536
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-45\\-20\\-27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\180\\-90\\-90\\-180\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-90\\45\\0\\-45\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-90\\0\\4\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-180\\-45\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-90\\-18\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-27658470420, -145470870, -13572666630; -145470870, -765108, -71386002; -13572666630, -71386002, -6660429035],[90, 18, 1],[28865, 93309, -15927, -3450, 13771, 41989; -1893, -6130, 1040, 227, -899, -2749; -58801, -190080, 32445, 7028, -28053, -85536],193,9]
%
% END OF LATTICE L194.9
%
\hbox{}\par\smallskip%
%
% LATTICE L194.10
%
\leavevmode%
\llap{\spacer}%
${L_{194.10}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${360}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{12,1}{\infty a}{8}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26280\\7560\\2160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7560\\2090\\795
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2160\\795\\-178
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3635\\11088\\5400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-60\\183\\89
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
59\\-180\\-88
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
121\\-369\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-215\\656\\320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-158\\482\\235
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\-360\\-1800\\-1080\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\-4\\-45\\-40\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-4\\8\\0\\-40\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-45\\0\\90\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\-40\\-40\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-20\\-45\\0\\10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[26280, 7560, 2160; 7560, 2090, 795; 2160, 795, -178],[360, 5, 1],[-3635, -60, 59, 121, -215, -158; 11088, 183, -180, -369, 656, 482; 5400, 89, -88, -180, 320, 235],193,10]
%
% END OF LATTICE L194.10
%
\hbox{}\par\smallskip%
%
% LATTICE L194.11
%
\leavevmode%
\llap{\spacer}%
${L_{194.11}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${450}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{30,1}{\infty a}{10}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}{50}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2110950\\30600\\708750
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30600\\-440\\-10275
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
708750\\-10275\\-237962
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4133\\3690\\12150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
148\\134\\435
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51\\-44\\-150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49\\-45\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
221\\190\\650
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
67\\59\\197
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
450\\0\\-450\\-900\\-1350\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-10\\-45\\-100\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-450\\-10\\10\\0\\-50\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\-45\\0\\18\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1350\\-100\\-50\\0\\50\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-10\\-9\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2110950, 30600, 708750; 30600, -440, -10275; 708750, -10275, -237962],[450, 5, 1],[4133, 148, -51, -49, 221, 67; 3690, 134, -44, -45, 190, 59; 12150, 435, -150, -144, 650, 197],193,11]
%
% END OF LATTICE L194.11
%
\hbox{}\par\smallskip%
%
% LATTICE L194.12
%
\leavevmode%
\llap{\spacer}%
${L_{194.12}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{24,7}{\infty z}{20}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216583920\\22333320\\-1261800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22333320\\-2302928\\130112
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1261800\\130112\\-7351
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
284\\2835\\1431
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
181\\1805\\880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\-430\\-230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-47\\-468\\-216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\200\\107
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
102\\1019\\528
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-90\\-360\\-27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\-40\\-360\\-40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-40\\20\\0\\-10\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-360\\0\\144\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-40\\-10\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-40\\-72\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-216583920, 22333320, -1261800; 22333320, -2302928, 130112; -1261800, 130112, -7351],[360, 8, 1],[284, 181, -43, -47, 20, 102; 2835, 1805, -430, -468, 200, 1019; 1431, 880, -230, -216, 107, 528],193,12]
%
% END OF LATTICE L194.12
%
\hbox{}\par\smallskip%
%
% LATTICE L194.13
%
\leavevmode%
\llap{\spacer}%
${L_{194.13}}=\pdualwitharg{3}(\pfillwitharg{5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{12,5}{\infty b}{360}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2872440\\198360\\-51480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
198360\\-13698\\3555
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-51480\\3555\\-922
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-33\\-488\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22\\-330\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\420\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\15\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-21\\-304\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-205\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-360\\-40\\-216\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\90\\-180\\-45\\-360\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-180\\360\\0\\-360\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-45\\0\\2\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-360\\-360\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-180\\-9\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2872440, 198360, -51480; 198360, -13698, 3555; -51480, 3555, -922],[360, 9, 1],[-33, -22, 29, 1, -21, -14; -488, -330, 420, 15, -304, -205; -40, -45, 0, 2, 0, -9],193,13]
%
% END OF LATTICE L194.13
%
\hbox{}\par\smallskip%
%
% LATTICE L194.14
%
\leavevmode%
\llap{\spacer}%
${L_{194.14}}=\pdualwitharg{3.5}(\pfillwitharg{2.5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${10}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{6,5}{\infty a}{18}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{l}{2}{90}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
245250\\30330\\-15390
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
30330\\3960\\-1845
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15390\\-1845\\982
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-375\\1074\\-3860
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-146\\418\\-1503
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\20\\-72
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\3\\-10
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-175\\502\\-1800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-223\\639\\-2295
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-90\\-100\\-270\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\-18\\-45\\-180\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-18\\18\\0\\-90\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-45\\0\\10\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\-180\\-90\\0\\90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-90\\-45\\0\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[245250, 30330, -15390; 30330, 3960, -1845; -15390, -1845, 982],[90, 45, 1],[-375, -146, -7, -1, -175, -223; 1074, 418, 20, 3, 502, 639; -3860, -1503, -72, -10, -1800, -2295],193,14]
%
% END OF LATTICE L194.14
%
\hbox{}\par\smallskip%
%
% LATTICE L194.15
%
\leavevmode%
\llap{\spacer}%
${L_{194.15}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{60,19}{\infty z}{5}\above{1pt}{1pt}{r}{2}{900}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{100}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
491832900\\2918250\\245115000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2918250\\17320\\1454370
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
245115000\\1454370\\122158081
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11818\\1656\\-23733
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5119\\716\\-10280
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
122\\17\\-245
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
224\\45\\-450
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
608\\86\\-1221
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7793\\1095\\-15650
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-45\\-900\\-27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\-10\\-450\\-20\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-10\\5\\0\\-5\\-50
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\-450\\0\\900\\0\\-450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-20\\-5\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-50\\-50\\-450\\0\\100
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[491832900, 2918250, 245115000; 2918250, 17320, 1454370; 245115000, 1454370, 122158081],[450, 10, 1],[11818, 5119, 122, 224, 608, 7793; 1656, 716, 17, 45, 86, 1095; -23733, -10280, -245, -450, -1221, -15650],193,15]
%
% END OF LATTICE L194.15
%
\hbox{}\par\smallskip%
%
% LATTICE L194.16
%
\leavevmode%
\llap{\spacer}%
${L_{194.16}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${225}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{60,31}{\infty z}{5}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
34712100\\-723150\\17114850
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-723150\\15080\\-356550
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
17114850\\-356550\\8438501
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
26746\\495\\-54225
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2264\\43\\-4590
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\1\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\0\\-144
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1492\\25\\-3025
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
729\\13\\-1478
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
225\\0\\-225\\-900\\-675\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\-10\\-90\\-100\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-225\\-10\\5\\0\\-25\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\-90\\0\\36\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-675\\-100\\-25\\0\\25\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-10\\-10\\-18\\0\\4
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[34712100, -723150, 17114850; -723150, 15080, -356550; 17114850, -356550, 8438501],[450, 10, 1],[26746, 2264, 37, 71, 1492, 729; 495, 43, 1, 0, 25, 13; -54225, -4590, -75, -144, -3025, -1478],193,16]
%
% END OF LATTICE L194.16
%
\hbox{}\par\smallskip%
%
% LATTICE L194.17
%
\leavevmode%
\llap{\spacer}%
${L_{194.17}}=\pdualwitharg{2.3.5}(\pfillwitharg{2.5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{12,11}{\infty z}{9}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{r}{2}{180}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3983400\\-11513430\\-5597910
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11513430\\-33277680\\-16179840
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5597910\\-16179840\\-7866751
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\-2306\\4735
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\-1060\\2178
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\443\\-909
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\141\\-290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-1908\\3915
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\-4297\\8820
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-45\\-100\\-135\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\36\\-18\\-90\\-180\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-18\\9\\0\\-45\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-90\\0\\20\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-180\\-45\\0\\45\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-90\\-90\\0\\180
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3983400, -11513430, -5597910; -11513430, -33277680, -16179840; -5597910, -16179840, -7866751],[90, 90, 1],[11, 3, -3, 0, 13, 25; -2306, -1060, 443, 141, -1908, -4297; 4735, 2178, -909, -290, 3915, 8820],193,17]
%
% END OF LATTICE L194.17
%
\hbox{}\par\smallskip%
%
% LATTICE L194.18
%
\leavevmode%
\llap{\spacer}%
${L_{194.18}}=\pdualwitharg{5}({L_{194.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1800}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{60,1}{\infty a}{40}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{l}{2}{200}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1476919800\\-18318600\\383400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18318600\\-227210\\4755
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
383400\\4755\\-98
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-937\\75960\\19800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\162\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
41\\-3324\\-880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-81\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-149\\12080\\3200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\1135\\299
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1800\\0\\-1800\\-1800\\-5400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-20\\-45\\-200\\-5
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-20\\40\\0\\-200\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-45\\0\\18\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5400\\-200\\-200\\0\\200\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-5\\-20\\-9\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1476919800, -18318600, 383400; -18318600, -227210, 4755; 383400, 4755, -98],[1800, 5, 1],[-937, -2, 41, 1, -149, -14; 75960, 162, -3324, -81, 12080, 1135; 19800, 35, -880, -18, 3200, 299],193,18]
%
% END OF LATTICE L194.18
%
\hbox{}\par\smallskip%
%
% LATTICE L194.19
%
\leavevmode%
\llap{\spacer}%
${L_{194.19}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{24,7}{\infty z}{4}\above{1pt}{1pt}{*}{2}{720}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-893664720\\-61043400\\-11062440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61043400\\-4169680\\-755640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11062440\\-755640\\-136939
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\2169\\-11565
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\235\\-1216
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-83\\458
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-225\\1080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\193\\-1065
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\870\\-4720
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-90\\-1800\\-135\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\16\\-8\\-360\\-40\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-8\\4\\0\\-10\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-360\\0\\720\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-40\\-10\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-40\\-360\\0\\80
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-893664720, -61043400, -11062440; -61043400, -4169680, -755640; -11062440, -755640, -136939],[360, 40, 1],[-5, -1, 0, 2, 0, -1; 2169, 235, -83, -225, 193, 870; -11565, -1216, 458, 1080, -1065, -4720],193,19]
%
% END OF LATTICE L194.19
%
\hbox{}\par\smallskip%
%
% LATTICE L194.20
%
\leavevmode%
\llap{\spacer}%
${L_{194.20}}=\pdualwitharg{3.5}(\pfillwitharg{5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{12,5}{\infty b}{72}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{l}{2}{360}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-158040\\-135000\\5400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135000\\-113490\\4545
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5400\\4545\\-182
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\120\\3200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\99
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-44\\-1188
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-25
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\160\\4320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\77\\2070
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\0\\-360\\-200\\-1080\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\-36\\-45\\-360\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-36\\72\\0\\-360\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-45\\0\\10\\0\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\-360\\-360\\0\\360\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-180\\-45\\0\\90
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-158040, -135000, 5400; -135000, -113490, 4545; 5400, 4545, -182],[360, 45, 1],[7, 0, -3, 0, 11, 5; 120, 4, -44, -1, 160, 77; 3200, 99, -1188, -25, 4320, 2070],193,20]
%
% END OF LATTICE L194.20
%
\hbox{}\par\smallskip%
%
% LATTICE L194.21
%
\leavevmode%
\llap{\spacer}%
${L_{194.21}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{30,29}{\infty a}{90}\above{1pt}{1pt}{r}{2}{50}\above{1pt}{1pt}{l}{2}{18}\above{1pt}{1pt}{r}{2}{450}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
408613050\\5643000\\90458550
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5643000\\77940\\1249245
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90458550\\1249245\\20025668
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1491\\368\\-6758
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2909\\716\\-13185
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
139\\34\\-630
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\5\\-50
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
687\\172\\-3114
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4418\\1095\\-20025
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-90\\-100\\-54\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\90\\-90\\-225\\-180\\-225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-90\\90\\0\\-90\\-450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-225\\0\\50\\0\\-225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-180\\-90\\0\\18\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-225\\-450\\-225\\0\\450
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[408613050, 5643000, 90458550; 5643000, 77940, 1249245; 90458550, 1249245, 20025668],[450, 45, 1],[1491, 2909, 139, 11, 687, 4418; 368, 716, 34, 5, 172, 1095; -6758, -13185, -630, -50, -3114, -20025],193,21]
%
% END OF LATTICE L194.21
%
\hbox{}\par\smallskip%
%
% LATTICE L194.22
%
\leavevmode%
\llap{\spacer}%
${L_{194.22}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${50}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{30,11}{\infty b}{90}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{450}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
320319450\\4660200\\71527500
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4660200\\67860\\1040625
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71527500\\1040625\\15972128
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16537\\110\\74050
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6291\\43\\28170
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-201\\2\\900
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\0\\103
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8341\\50\\37350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2032\\13\\9099
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50\\0\\-450\\-100\\-1350\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\90\\-90\\-45\\-900\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-450\\-90\\90\\0\\-450\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-45\\0\\2\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1350\\-900\\-450\\0\\450\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-90\\-9\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[320319450, 4660200, 71527500; 4660200, 67860, 1040625; 71527500, 1040625, 15972128],[450, 45, 1],[-16537, -6291, -201, -23, -8341, -2032; 110, 43, 2, 0, 50, 13; 74050, 28170, 900, 103, 37350, 9099],193,22]
%
% END OF LATTICE L194.22
%
\hbox{}\par\smallskip%
%
% LATTICE L194.23
%
\leavevmode%
\llap{\spacer}%
${L_{194.23}}=\pdualwitharg{2.3}(\pfillwitharg{5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{720}\above{1pt}{1pt}{24,23}{\infty z}{180}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61213680\\-14397480\\-3423240
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14397480\\-3386160\\-805104
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3423240\\-805104\\-191423
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
131\\-2048\\6271
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1076\\-16815\\51480
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-43\\675\\-2070
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\703\\-2152
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
20\\-314\\963
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
281\\-4396\\13464
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-90\\-40\\-27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\720\\-360\\-360\\-360\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-360\\180\\0\\-90\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40\\-360\\0\\16\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-360\\-90\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-360\\-72\\0\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-61213680, -14397480, -3423240; -14397480, -3386160, -805104; -3423240, -805104, -191423],[360, 72, 1],[131, 1076, -43, -45, 20, 281; -2048, -16815, 675, 703, -314, -4396; 6271, 51480, -2070, -2152, 963, 13464],193,23]
%
% END OF LATTICE L194.23
%
\hbox{}\par\smallskip%
%
% LATTICE L194.24
%
\leavevmode%
\llap{\spacer}%
${L_{194.24}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{180}\above{1pt}{1pt}{60,59}{\infty z}{45}\above{1pt}{1pt}{r}{2}{100}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{900}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40007502900\\-305538750\\-19942064550
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-305538750\\-2332980\\-152298270
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19942064550\\-152298270\\-9940283939
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71683\\-31\\-143809
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
166346\\-73\\-333720
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-68122\\29\\136665
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-22331\\10\\44800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
58737\\-25\\-117837
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
664397\\-285\\-1332900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-45\\-100\\-27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\180\\-90\\-450\\-180\\-450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-45\\-90\\45\\0\\-45\\-450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-450\\0\\100\\0\\-450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-180\\-45\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-450\\-450\\-450\\0\\900
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-40007502900, -305538750, -19942064550; -305538750, -2332980, -152298270; -19942064550, -152298270, -9940283939],[450, 90, 1],[71683, 166346, -68122, -22331, 58737, 664397; -31, -73, 29, 10, -25, -285; -143809, -333720, 136665, 44800, -117837, -1332900],193,24]
%
% END OF LATTICE L194.24
%
\hbox{}\par\smallskip%
%
% LATTICE L194.25
%
\leavevmode%
\llap{\spacer}%
${L_{194.25}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${25}\above{1pt}{1pt}{r}{2}{180}\above{1pt}{1pt}{60,11}{\infty z}{45}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{225}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-203143994100\\40279950\\-101259942750
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40279950\\-7920\\20078100
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-101259942750\\20078100\\-50474423579
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-451521\\205\\905825
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-291063\\134\\583920
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
50133\\-22\\-100575
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10708\\-5\\-21482
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-217243\\95\\435825
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-131741\\59\\264294
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\-225\\-100\\-675\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\180\\-90\\-90\\-900\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-225\\-90\\45\\0\\-225\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-90\\0\\4\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-675\\-900\\-225\\0\\225\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-90\\-90\\-18\\0\\36
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-203143994100, 40279950, -101259942750; 40279950, -7920, 20078100; -101259942750, 20078100, -50474423579],[450, 90, 1],[-451521, -291063, 50133, 10708, -217243, -131741; 205, 134, -22, -5, 95, 59; 905825, 583920, -100575, -21482, 435825, 264294],193,25]
%
% END OF LATTICE L194.25
%
\hbox{}\par\smallskip%
%
% LATTICE L194.26
%
\leavevmode%
\llap{\spacer}%
${L_{194.26}}=\pdualwitharg{2}({L_{194.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{120,79}{\infty z}{20}\above{1pt}{1pt}{*}{2}{3600}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{400}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7653243600\\-40120200\\7500600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-40120200\\-210320\\39320
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7500600\\39320\\-7351
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\189\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\86\\-560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-43\\-230
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\-45\\1800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\107
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\415\\1200
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-90\\-1800\\-27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\-40\\-1800\\-40\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-40\\20\\0\\-10\\-200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-1800\\0\\3600\\0\\-1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-40\\-10\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-200\\-200\\-1800\\0\\400
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7653243600, -40120200, 7500600; -40120200, -210320, 39320; 7500600, 39320, -7351],[1800, 40, 1],[-1, -1, 0, 2, 0, -1; 189, 86, -43, -45, 20, 415; -9, -560, -230, 1800, 107, 1200],193,26]
%
% END OF LATTICE L194.26
%
\hbox{}\par\smallskip%
%
% LATTICE L194.27
%
\leavevmode%
\llap{\spacer}%
${L_{194.27}}=\pdualwitharg{2.5}({L_{194.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${225}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{120,31}{\infty z}{20}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{l}{2}{25}\above{1pt}{1pt}{r}{2}{16}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-78663600\\14596200\\-366908400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14596200\\-2696720\\68441400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-366908400\\68441400\\-1700175911
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5036843\\14990670\\-483525
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
635849\\1892416\\-61040
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-155108\\-461633\\14890
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-162754\\-484389\\15624
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360686\\1073475\\-34625
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
364509\\1084853\\-34992
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
225\\0\\-450\\-1800\\-675\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\80\\-40\\-360\\-200\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-450\\-40\\20\\0\\-50\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-360\\0\\144\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-675\\-200\\-50\\0\\25\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-40\\-72\\0\\16
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-78663600, 14596200, -366908400; 14596200, -2696720, 68441400; -366908400, 68441400, -1700175911],[1800, 40, 1],[5036843, 635849, -155108, -162754, 360686, 364509; 14990670, 1892416, -461633, -484389, 1073475, 1084853; -483525, -61040, 14890, 15624, -34625, -34992],193,27]
%
% END OF LATTICE L194.27
%
\hbox{}\par\smallskip%
%
% LATTICE L194.28
%
\leavevmode%
\llap{\spacer}%
${L_{194.28}}=\pdualwitharg{3}({L_{194.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{60,29}{\infty b}{360}\above{1pt}{1pt}{b}{2}{50}\above{1pt}{1pt}{l}{2}{72}\above{1pt}{1pt}{r}{2}{450}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-111003406200\\-7383319200\\-49100400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7383319200\\-491096610\\-3265875
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-49100400\\-3265875\\-21718
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
657\\-10488\\91792
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
103\\-1644\\14355
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1217\\19428\\-170100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\415\\-3625
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
885\\-14128\\123696
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2122\\-33875\\296550
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-360\\-200\\-216\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\90\\-180\\-225\\-360\\-225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-180\\360\\0\\-360\\-900
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-225\\0\\50\\0\\-225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-216\\-360\\-360\\0\\72\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-225\\-900\\-225\\0\\450
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-111003406200, -7383319200, -49100400; -7383319200, -491096610, -3265875; -49100400, -3265875, -21718],[1800, 45, 1],[657, 103, -1217, -26, 885, 2122; -10488, -1644, 19428, 415, -14128, -33875; 91792, 14355, -170100, -3625, 123696, 296550],193,28]
%
% END OF LATTICE L194.28
%
\hbox{}\par\smallskip%
%
% LATTICE L194.29
%
\leavevmode%
\llap{\spacer}%
${L_{194.29}}=\pdualwitharg{3.5}({L_{194.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${200}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{60,41}{\infty a}{360}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{l}{2}{1800}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-790200\\54000\\27000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
54000\\-3690\\-1845
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27000\\-1845\\-922
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\200\\-200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\22\\-45
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-44\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11\\160\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\19\\-9
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
200\\0\\-1800\\-200\\-5400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\90\\-180\\-45\\-1800\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-180\\360\\0\\-1800\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-45\\0\\2\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5400\\-1800\\-1800\\0\\1800\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-180\\-9\\0\\18
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-790200, 54000, 27000; 54000, -3690, -1845; 27000, -1845, -922],[1800, 45, 1],[7, 0, -3, 0, 11, 1; 200, 22, -44, -1, 160, 19; -200, -45, 0, 2, 0, -9],193,29]
%
% END OF LATTICE L194.29
%
\hbox{}\par\smallskip%
%
% LATTICE L194.30
%
\leavevmode%
\llap{\spacer}%
${L_{194.30}}=\pdualwitharg{2.3.5}(\pfillwitharg{5}({L_{194.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{24,23}{\infty z}{36}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{r}{2}{720}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7094160\\-23413320\\-53277480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23413320\\-77259600\\-175806720
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-53277480\\-175806720\\-400053811
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4791\\43524\\-19765
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
6231\\56602\\-25704
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-999\\-9077\\4122
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1057\\-9601\\4360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2323\\21107\\-9585
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13874\\126047\\-57240
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5\\0\\-90\\-200\\-135\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\144\\-72\\-360\\-360\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-72\\36\\0\\-90\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-360\\0\\80\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-360\\-90\\0\\45\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-360\\-360\\0\\720
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-7094160, -23413320, -53277480; -23413320, -77259600, -175806720; -53277480, -175806720, -400053811],[360, 360, 1],[4791, 6231, -999, -1057, 2323, 13874; 43524, 56602, -9077, -9601, 21107, 126047; -19765, -25704, 4122, 4360, -9585, -57240],193,30]
%
% END OF LATTICE L194.30
%
\hbox{}\par\smallskip%
%
% LATTICE L194.31
%
\leavevmode%
\llap{\spacer}%
${L_{194.31}}=\pdualwitharg{2.3}({L_{194.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{720}\above{1pt}{1pt}{120,119}{\infty z}{180}\above{1pt}{1pt}{*}{2}{400}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{r}{2}{3600}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-154184400\\211138200\\7653600
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
211138200\\-289064880\\-10480680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7653600\\-10480680\\-379919
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1995\\41\\-41321
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12949\\266\\-268200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2090\\-43\\43290
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2192\\-45\\45400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-972\\20\\-20133
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-28939\\595\\-599400
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\-90\\-200\\-27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\720\\-360\\-1800\\-360\\-1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-360\\180\\0\\-90\\-1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-1800\\0\\400\\0\\-1800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-360\\-90\\0\\9\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1800\\-1800\\-1800\\0\\3600
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-154184400, 211138200, 7653600; 211138200, -289064880, -10480680; 7653600, -10480680, -379919],[1800, 360, 1],[-1995, -12949, 2090, 2192, -972, -28939; 41, 266, -43, -45, 20, 595; -41321, -268200, 43290, 45400, -20133, -599400],193,31]
%
% END OF LATTICE L194.31
%
\hbox{}\par\smallskip%
%
% LATTICE L194.32
%
\leavevmode%
\llap{\spacer}%
${L_{194.32}}=\pdualwitharg{2.3.5}({L_{194.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${25}\above{1pt}{1pt}{r}{2}{720}\above{1pt}{1pt}{120,71}{\infty z}{180}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{l}{2}{225}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{l}{2}$}%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-645303600\\-198801000\\-160754400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-198801000\\-61213680\\-49524120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160754400\\-49524120\\-40046231
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
78126\\655\\-314425
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
128361\\1076\\-516600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5121\\-43\\20610
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5369\\-45\\21608
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
11908\\100\\-47925
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
33508\\281\\-134856
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
25\\0\\-450\\-200\\-675\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\720\\-360\\-360\\-1800\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-450\\-360\\180\\0\\-450\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-360\\0\\16\\0\\-72
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-675\\-1800\\-450\\0\\225\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-360\\-360\\-72\\0\\144
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-645303600, -198801000, -160754400; -198801000, -61213680, -49524120; -160754400, -49524120, -40046231],[1800, 360, 1],[78126, 128361, -5121, -5369, 11908, 33508; 655, 1076, -43, -45, 100, 281; -314425, -516600, 20610, 21608, -47925, -134856],193,32]
%
% END OF LATTICE L194.32
%

\medskip%
%------------------------------WEYL-GROUP-W195--------------------------------
%
\leavevmode\llap{}%
$W_{195}$%
\qquad\llap{32} lattices, $\chi=72$%
\hfill%
$2\infty2|2\infty2|2\infty2|2\infty2|\rtimes D_{4}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L195.1
%
\leavevmode%
\llap{\spacer}%
${L_{195.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\spacer%
\instructions{25,5,2*}%
\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{60,29}{\infty b}{40}\above{1pt}{1pt}{b}{2}{50}\above{1pt}{1pt}{b}{2}{360}\above{1pt}{1pt}{10,1}{\infty z}{90}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12431315950200\\-85733217000\\-591012000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-85733217000\\-591263590\\-4075945
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-591012000\\-4075945\\-28098
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-726032161\\104802166080\\68540893200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5007114\\722772931\\472695405
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34524\\4983512\\3259229
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1839\\265456\\173928
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1134\\163691\\107200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1639\\236588\\154780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-631\\91085\\59475
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2147\\309924\\201780
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-632\\91233\\59040
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-120\\-200\\-2520\\-2160\\-1288\\-720\\-840\\-200\\-360\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-20\\-75\\-1260\\-1170\\-720\\-415\\-520\\-150\\-360\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-20\\40\\0\\-1080\\-1260\\-840\\-520\\-760\\-300\\-1080\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-75\\0\\50\\0\\-225\\-200\\-150\\-300\\-175\\-900\\-450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-1260\\-1080\\0\\360\\-180\\-360\\-360\\-1080\\-900\\-6840\\-4680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2160\\-1170\\-1260\\-225\\-180\\90\\0\\-45\\-360\\-450\\-4680\\-3735
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8\\0\\-120\\-200\\-2520\\-2160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-20\\-75\\-1260\\-1170
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-120\\-20\\40\\0\\-1080\\-1260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-75\\0\\50\\0\\-225
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-1260\\-1080\\0\\360\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2160\\-1170\\-1260\\-225\\-180\\90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1288\\-720\\-840\\-200\\-360\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-415\\-520\\-150\\-360\\-45
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-840\\-520\\-760\\-300\\-1080\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-200\\-150\\-300\\-175\\-900\\-450
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-360\\-1080\\-900\\-6840\\-4680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-360\\-450\\-4680\\-3735
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-12431315950200, -85733217000, -591012000; -85733217000, -591263590, -4075945; -591012000, -4075945, -28098],[1800, 5, 1],[-1839, -1134, -1639, -631, -2147, -632, -177, 0, 127, 1, -2389, -2770; 265456, 163691, 236588, 91085, 309924, 91233, 25552, 1, -18332, -145, 344844, 399843; 173928, 107200, 154780, 59475, 201780, 59040, 16392, -145, -12040, 0, 226440, 262125],194,1]
%
% END OF LATTICE L195.1
%
\hbox{}\par\smallskip%
%
% LATTICE L195.2
%
\leavevmode%
\llap{\spacer}%
${L_{195.2}}=\pfillwitharg{2.5}({L_{195.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{6,5}{\infty a}{10}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{l}{2}{90}\above{1pt}{1pt}{1,0}{\infty}{90}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-32514390\\932400\\-61200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
932400\\-26738\\1755
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-61200\\1755\\-112
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2081501\\72588420\\52830
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-59691\\-2081611\\-1515
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
4334\\151140\\109
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-485\\-16914\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-319\\-11125\\-15
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
39\\1360\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
40\\1395\\2
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-893\\-31140\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2977\\-103815\\-45
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\0\\-30\\-20\\-630\\-1080\\-322\\-360\\-210\\-20\\-90\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\10\\-10\\-15\\-630\\-1170\\-360\\-415\\-260\\-30\\-180\\-45
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-15\\0\\2\\0\\-45\\-20\\-30\\-30\\-7\\-90\\-90
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-630\\-270\\0\\90\\-90\\-90\\-180\\-270\\-90\\-1710\\-2340
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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0\\10\\-10\\-15\\-630\\-1170
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-30\\-10\\10\\0\\-270\\-630
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-20\\-15\\0\\2\\0\\-45
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-630\\-630\\-270\\0\\90\\-90
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-1080\\-1170\\-630\\-45\\-90\\90
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-322\\-360\\-210\\-20\\-90\\0
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\kern3pt%
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-360\\-415\\-260\\-30\\-180\\-45
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\kern3pt%
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-210\\-260\\-190\\-30\\-270\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\-30\\-30\\-7\\-90\\-90
\end{tabular}\endgroup%
\kern3pt%
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-90\\-180\\-270\\-90\\-1710\\-2340
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-180\\-90\\-2340\\-3735
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-32514390, 932400, -61200; 932400, -26738, 1755; -61200, 1755, -112],[90, 1, 1],[-485, -319, 39, 40, -893, -2977, -1091, -1454, -1221, -237, -2653, -2342; -16914, -11125, 1360, 1395, -31140, -103815, -38046, -50705, -42580, -8265, -92520, -81675; -20, -15, 0, 2, 0, -45, -20, -30, -30, -7, -90, -90],194,2]
%
% END OF LATTICE L195.2
%
\hbox{}\par\smallskip%
%
% LATTICE L195.3
%
\leavevmode%
\llap{\spacer}%
${L_{195.3}}=\pfillwitharg{5}({L_{195.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${8}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{12,5}{\infty b}{40}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{360}\above{1pt}{1pt}{2,1}{\infty z}{90}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-522723960\\-3601800\\231480
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3601800\\-24818\\1595
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
231480\\1595\\-102
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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-1517329\\220197600\\-185040
\end{tabular}\endgroup%
\kern3pt%
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-10455\\1517249\\-1275
\end{tabular}\endgroup%
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656\\-95200\\79
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-583\\84608\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-362\\52535\\-30
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-531\\77060\\-60
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-42\\6095\\-7
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-743\\107820\\-180
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-236\\34245\\-90
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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8\\0\\-120\\-40\\-2520\\-2160\\-1288\\-720\\-840\\-40\\-360\\0
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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8\\0\\-120\\-40\\-2520\\-2160
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0\\10\\-20\\-15\\-1260\\-1170
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-120\\-20\\40\\0\\-1080\\-1260
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-40\\-15\\0\\2\\0\\-45
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-2520\\-1260\\-1080\\0\\360\\-180
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-2160\\-1170\\-1260\\-45\\-180\\90
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-1288\\-720\\-840\\-40\\-360\\0
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-720\\-415\\-520\\-30\\-360\\-45
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-840\\-520\\-760\\-60\\-1080\\-360
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\kern3pt%
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-40\\-30\\-60\\-7\\-180\\-90
\end{tabular}\endgroup%
\kern3pt%
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-360\\-360\\-1080\\-180\\-6840\\-4680
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-360\\-90\\-4680\\-3735
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-522723960, -3601800, 231480; -3601800, -24818, 1595; 231480, 1595, -102],[360, 1, 1],[-583, -362, -531, -42, -743, -236, -73, -7, 39, 1, -733, -871; 84608, 52535, 77060, 6095, 107820, 34245, 10592, 1015, -5660, -145, 106380, 126405; -40, -30, -60, -7, -180, -90, -40, -15, 0, 2, 0, -45],194,3]
%
% END OF LATTICE L195.3
%
\hbox{}\par\smallskip%
%
% LATTICE L195.4
%
\leavevmode%
\llap{\spacer}%
${L_{195.4}}=\pfillwitharg{2}({L_{195.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${2}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{10,9}{\infty a}{90}\above{1pt}{1pt}{r}{2}{50}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{15,11}{\infty}{10}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2794950\\35550\\1121400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
35550\\-440\\-14265
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1121400\\-14265\\-449932
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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1363949\\415800\\3386250
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\kern3pt%
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-16887\\-5149\\-41925
\end{tabular}\endgroup%
\kern3pt%
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-547312\\-166848\\-1358801
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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2863\\870\\7108
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\HardButStrongLineBreak\kern3pt%
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8392\\2547\\20835
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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3335\\1008\\8280
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\HardButStrongLineBreak\kern3pt%
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-141\\-45\\-350
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-145\\-44\\-360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
284\\89\\705
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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-100\\-225\\0\\50\\0\\-75
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-210\\-630\\-270\\0\\10\\-10
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-360\\-1170\\-630\\-75\\-10\\10
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-322\\-1080\\-630\\-100\\-30\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\-3735\\-2340\\-450\\-180\\-45
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-630\\-2340\\-1710\\-450\\-270\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-100\\-450\\-450\\-175\\-150\\-150
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-30\\-180\\-270\\-150\\-190\\-260
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-180\\-150\\-260\\-415
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-2794950, 35550, 1121400; 35550, -440, -14265; 1121400, -14265, -449932],[450, 5, 1],[2863, 8392, 3335, -141, -145, 284, 601, 3299, 4459, 2306, 2743, 3613; 870, 2547, 1008, -45, -44, 89, 186, 1017, 1368, 705, 836, 1099; 7108, 20835, 8280, -350, -360, 705, 1492, 8190, 11070, 5725, 6810, 8970],194,4]
%
% END OF LATTICE L195.4
%
\hbox{}\par\smallskip%
%
% LATTICE L195.5
%
\leavevmode%
\llap{\spacer}%
${L_{195.5}}=\pdualwitharg{2}(\pfillwitharg{2.5}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{12,11}{\infty z}{5}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{1,0}{\infty}{180}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
14726520\\5828850\\7366770
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5828850\\2307296\\2915820
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14726520, 5828850, 7366770; 5828850, 2307296, 2915820; 7366770, 2915820, 3685141],[90, 2, 1],[-21975, -28661, -682, -1, -101716, -484663, -81777, -204781, -77132, -25937, -131726, -215663; -1284, -1675, -40, 0, -5940, -28305, -4776, -11960, -4505, -1515, -7695, -12600; 44945, 58620, 1395, 2, 208035, 991260, 167255, 418830, 157755, 53048, 269415, 441090],194,5]
%
% END OF LATTICE L195.5
%
\hbox{}\par\smallskip%
%
% LATTICE L195.6
%
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${L_{195.6}}=\pdualwitharg{5}(\pfillwitharg{2.5}({L_{195.1}}))$%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-170190, 1980, 3870; 1980, -20, -45; 3870, -45, -88],[90, 5, 1],[99, 25, 19, 16, 31, 23, 21, 2, -1, -1, 23, 58; -20, -6, -6, -7, -18, -18, -20, -3, 0, 2, 0, -9; 4350, 1099, 836, 705, 1368, 1017, 930, 89, -44, -45, 1008, 2547],194,6]
%
% END OF LATTICE L195.6
%
\hbox{}\par\smallskip%
%
% LATTICE L195.7
%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[68502690, 26669250, 7614990; 26669250, 10382832, 2964645; 7614990, 2964645, 846508],[90, 9, 1],[-9125, -5946, -279, -1, -4703, -11202, -34019, -42591, -32079, -5392, -6083, -4977; -2568, -1675, -80, 0, -1320, -3145, -9552, -11960, -9010, -1515, -1710, -1400; 91080, 59355, 2790, 9, 46930, 111785, 339480, 425025, 320130, 53811, 60710, 49675],194,7]
%
% END OF LATTICE L195.7
%
\hbox{}\par\smallskip%
%
% LATTICE L195.8
%
\leavevmode%
\llap{\spacer}%
${L_{195.8}}=\pdualwitharg{2.5}(\pfillwitharg{2.5}({L_{195.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[4237740, 3870, 2069550; 3870, 20, 1890; 2069550, 1890, 1010689],[90, 10, 1],[12563, 6292, 2370, 3985, 4048, 6628, 3377, 881, 21, 0, 3125, 14891; 67, 33, 12, 19, 18, 27, 13, 3, 0, 1, 18, 81; -25725, -12884, -4853, -8160, -8289, -13572, -6915, -1804, -43, 0, -6399, -30492],194,8]
%
% END OF LATTICE L195.8
%
\hbox{}\par\smallskip%
%
% LATTICE L195.9
%
\leavevmode%
\llap{\spacer}%
${L_{195.9}}=\pdualwitharg{2.3}(\pfillwitharg{2.5}({L_{195.1}}))$%
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{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-21802479120, -102436650, -10699426980; -102436650, -481284, -50270130; -10699426980, -50270130, -5250675259],[90, 18, 1],[117325, 154239, -9477, -19334, 24111, 160493, 264583, 705054, 295908, 114811, 71366, 125938; -8457, -11125, 680, 1395, -1730, -11535, -19023, -50705, -21290, -8265, -5140, -9075; -238995, -314190, 19305, 39384, -49115, -326930, -538965, -1436220, -602775, -233874, -145375, -256540],194,9]
%
% END OF LATTICE L195.9
%
\hbox{}\par\smallskip%
%
% LATTICE L195.10
%
\leavevmode%
\llap{\spacer}%
${L_{195.10}}=\pdualwitharg{5}(\pfillwitharg{5}({L_{195.1}}))$%
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{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${40}\above{1pt}{1pt}{r}{2}{2}\above{1pt}{1pt}{12,5}{\infty b}{8}\above{1pt}{1pt}{b}{2}{10}\above{1pt}{1pt}{b}{2}{72}\above{1pt}{1pt}{2,1}{\infty z}{18}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\1\\-1
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0\\-1\\0
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19\\828\\1656
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-158040, 0, 1800; 0, 10, -5; 1800, -5, -18],[360, 5, 1],[7, 0, -1, 0, 19, 22, 73, 9, 13, 5, 17, 5; 320, 1, -44, -1, 828, 963, 3200, 395, 572, 221, 756, 225; 600, -1, -88, 0, 1656, 1917, 6360, 784, 1132, 435, 1476, 432],194,10]
%
% END OF LATTICE L195.10
%
\hbox{}\par\smallskip%
%
% LATTICE L195.11
%
\leavevmode%
\llap{\spacer}%
${L_{195.11}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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67\\2988\\-90
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6\\267\\-7
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31\\1376\\-30
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37\\1639\\-30
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-850950, 19350, 9900; 19350, -440, -225; 9900, -225, -112],[450, 5, 1],[61, 59, 67, 6, 31, 37, 139, 76, 23, -1, -1, 8; 2730, 2637, 2988, 267, 1376, 1639, 6150, 3357, 1008, -45, -44, 359; -100, -90, -90, -7, -30, -30, -100, -45, 0, 2, 0, -15],194,11]
%
% END OF LATTICE L195.11
%
\hbox{}\par\smallskip%
%
% LATTICE L195.12
%
\leavevmode%
\llap{\spacer}%
${L_{195.12}}=\pdualwitharg{2}(\pfillwitharg{5}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{720}\above{1pt}{1pt}{8,7}{\infty z}{180}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{3,2}{\infty b}{80}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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69415560\\-7322160\\-858704
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8140680\\-858704\\-100703
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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23812676\\208021905\\151157160
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-7819\\-68310\\-49590
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-1489\\-13008\\-9448
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{$\left.\llap{\phantom{%
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\end{tabular}\endgroup%
\kern3pt%
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-40\\-720\\-360\\-56\\-120\\-240
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0\\-360\\-720\\-240\\-1040\\-3320
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-658073520, 69415560, 8140680; 69415560, -7322160, -858704; 8140680, -858704, -100703],[360, 8, 1],[-639, -12892, -7819, -1489, -4113, -10324, -1995, -22667, -4034, 172, 162, -1529; -5583, -112635, -68310, -13008, -35930, -90185, -17427, -198000, -35235, 1503, 1415, -13360; -4049, -81720, -49590, -9448, -26110, -65560, -12671, -144000, -25650, 1088, 1030, -9680],194,12]
%
% END OF LATTICE L195.12
%
\hbox{}\par\smallskip%
%
% LATTICE L195.13
%
\leavevmode%
\llap{\spacer}%
${L_{195.13}}=\pdualwitharg{3}(\pfillwitharg{5}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{4,1}{\infty a}{40}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{b}{2}{360}\above{1pt}{1pt}{6,1}{\infty z}{90}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-30704760\\1981080\\-55080
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1981080\\-127818\\3555
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-55080\\3555\\-98
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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337951\\5026320\\-7603920
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-21830\\-324676\\491175
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-2\\-30\\35
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\HardButStrongLineBreak\kern3pt%
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39\\580\\-880
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\HardButStrongLineBreak\kern3pt%
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1\\15\\-18
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\HardButStrongLineBreak\kern3pt%
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-733\\-10900\\16560
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-856\\-12730\\19305
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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-360\\-45\\0\\18\\0\\-135
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-7560\\-1260\\-1080\\0\\360\\-180
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-360\\-90\\-180\\-63\\-540\\-270
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-360\\-270\\-4680\\-3735
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-30704760, 1981080, -55080; 1981080, -127818, 3555; -55080, 3555, -98],[360, 9, 1],[-179, -2, 39, 1, -733, -856, -1709, -352, -511, -119, -683, -206; -2664, -30, 580, 15, -10900, -12730, -25416, -5235, -7600, -1770, -10160, -3065; 3960, 35, -880, -18, 16560, 19305, 38520, 7930, 11500, 2673, 15300, 4590],194,13]
%
% END OF LATTICE L195.13
%
\hbox{}\par\smallskip%
%
% LATTICE L195.14
%
\leavevmode%
\llap{\spacer}%
${L_{195.14}}=\pdualwitharg{3.5}(\pfillwitharg{2.5}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${90}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{6,1}{\infty a}{18}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{1,0}{\infty}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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642330\\-3060\\-63000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3060\\90\\315
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-63000\\315\\6182
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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137279\\-278720\\1413360
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-660\\1339\\-6795
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-13464\\27336\\-138619
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-4161\\8456\\-42840
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-1042\\2117\\-10728
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-785\\1594\\-8082
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-660\\1339\\-6795
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\HardButStrongLineBreak\kern3pt%
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-149\\302\\-1534
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-122\\247\\-1256
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-90\\-36\\-54\\-90\\-38\\-52
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0\\-9\\-36\\-90\\-52\\-83
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[642330, -3060, -63000; -3060, 90, 315; -63000, 315, 6182],[90, 45, 1],[-4161, -1042, -785, -660, -149, -122, -1119, -146, -7, 0, -115, -274; 8456, 2117, 1594, 1339, 302, 247, 2264, 295, 14, 1, 234, 557; -42840, -10728, -8082, -6795, -1534, -1256, -11520, -1503, -72, 0, -1184, -2821],194,14]
%
% END OF LATTICE L195.14
%
\hbox{}\par\smallskip%
%
% LATTICE L195.15
%
\leavevmode%
\llap{\spacer}%
${L_{195.15}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{180}\above{1pt}{1pt}{20,19}{\infty z}{45}\above{1pt}{1pt}{r}{2}{100}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{15,11}{\infty}{20}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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57873600\\-814950\\28275300
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-814950\\11480\\-398160
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28275300\\-398160\\13814461
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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14295599\\366795\\-29249550
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6984400\\179205\\-14290451
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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4927\\123\\-10081
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29471\\729\\-60300
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[57873600, -814950, 28275300; -814950, 11480, -398160; 28275300, -398160, 13814461],[450, 10, 1],[4927, 29471, 6356, 171, 22, 1437, 1153, 11569, 7324, 7429, 4538, 12243; 123, 729, 153, 0, 1, 43, 33, 324, 198, 195, 116, 308; -10081, -60300, -13005, -350, -45, -2940, -2359, -23670, -14985, -15200, -9285, -25050],194,15]
%
% END OF LATTICE L195.15
%
\hbox{}\par\smallskip%
%
% LATTICE L195.16
%
\leavevmode%
\llap{\spacer}%
${L_{195.16}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{195.1}}))$%
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{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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shares genus with 5-dual\nopagebreak\par%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[849600, 221850, 423900; 221850, 60080, 110700; 423900, 110700, 211501],[450, 10, 1],[17607, 34537, 21076, 4147, 12326, 32715, 65313, 77405, 16238, -1, 112, 4599; 165, 324, 198, 39, 116, 308, 615, 729, 153, 0, 1, 43; -35375, -69390, -42345, -8332, -24765, -65730, -131225, -155520, -32625, 2, -225, -9240],194,16]
%
% END OF LATTICE L195.16
%
\hbox{}\par\smallskip%
%
% LATTICE L195.17
%
\leavevmode%
\llap{\spacer}%
${L_{195.17}}=\pdualwitharg{2.3.5}(\pfillwitharg{2.5}({L_{195.1}}))$%
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{$1\above{1pt}{1pt}{1}{1}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1884060, -7697790, -3742740; -7697790, -31450320, -15291450; -3742740, -15291450, -7434851],[90, 90, 1],[-10, -6, -3, -7, -1, -2, -10, -3, 0, 2, 0, -1; -60538, -30551, -11591, -19471, -2089, -3085, -12622, -2371, 617, 1181, -1569, -7889; 124515, 62838, 23841, 40050, 4297, 6346, 25965, 4878, -1269, -2430, 3227, 16226],194,17]
%
% END OF LATTICE L195.17
%
\hbox{}\par\smallskip%
%
% LATTICE L195.18
%
\leavevmode%
\llap{\spacer}%
${L_{195.18}}=\pdualwitharg{5}({L_{195.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${200}\above{1pt}{1pt}{r}{2}{90}\above{1pt}{1pt}{20,1}{\infty a}{360}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{30,29}{\infty z}{10}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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-790200\\18000\\9000
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-790200, 18000, 9000; 18000, -410, -205; 9000, -205, -102],[1800, 5, 1],[73, 22, 19, 0, -1, 0, 7, 5, 17, 1, 13, 9; 3280, 981, 828, -1, -44, 7, 400, 261, 828, 47, 596, 407; -200, -45, 0, 2, 0, -15, -200, -90, -180, -7, -60, -30],194,18]
%
% END OF LATTICE L195.18
%
\hbox{}\par\smallskip%
%
% LATTICE L195.19
%
\leavevmode%
\llap{\spacer}%
${L_{195.19}}=\pdualwitharg{2.5}(\pfillwitharg{5}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{8,7}{\infty z}{36}\above{1pt}{1pt}{*}{2}{80}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{3,2}{\infty b}{16}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-528034320\\-64853640\\-128781360
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-64853640\\-7965360\\-15817040
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-128781360\\-15817040\\-31408259
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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21364\\-5\\-87595
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10435\\0\\-42786
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-528034320, -64853640, -128781360; -64853640, -7965360, -15817040; -128781360, -15817040, -31408259],[360, 40, 1],[21364, 50644, 10435, -449, -419, 1292, 3826, 17369, 12236, 13044, 7976, 21379; -5, -9, 0, 2, 0, -3, -5, -18, -9, -7, -3, -6; -87595, -207648, -42786, 1840, 1718, -5296, -15685, -71208, -50166, -53480, -32702, -87656],194,19]
%
% END OF LATTICE L195.19
%
\hbox{}\par\smallskip%
%
% LATTICE L195.20
%
\leavevmode%
\llap{\spacer}%
${L_{195.20}}=\pdualwitharg{3.5}(\pfillwitharg{5}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${360}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{12,1}{\infty a}{72}\above{1pt}{1pt}{b}{2}{90}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{2,1}{\infty z}{2}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
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{$\left.\llap{\phantom{%
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11\\483\\964
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-158040, -7920, 5760; -7920, -270, 225; 5760, 225, -178],[360, 45, 1],[61, 11, 29, 22, 17, 11, 259, 25, 19, -2, -1, 1; 2720, 489, 1284, 971, 748, 483, 11360, 1095, 828, -91, -44, 45; 5400, 972, 2556, 1935, 1492, 964, 22680, 2187, 1656, -180, -88, 89],194,20]
%
% END OF LATTICE L195.20
%
\hbox{}\par\smallskip%
%
% LATTICE L195.21
%
\leavevmode%
\llap{\spacer}%
${L_{195.21}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{10,9}{\infty a}{10}\above{1pt}{1pt}{r}{2}{450}\above{1pt}{1pt}{l}{2}{90}\above{1pt}{1pt}{15,1}{\infty}{90}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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11003850\\51660\\2454615
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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258918499\\-1100580\\-1160688600
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1211100\\-5149\\-5429160
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-1054\\0\\4725
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-8643\\43\\38745
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{$\left.\llap{\phantom{%
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0\\0\\0
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[2344707450, 11003850, 523030950; 11003850, 51660, 2454615; 523030950, 2454615, 116671858],[450, 45, 1],[-59515, -19781, -8537, -1054, -261, -8643, -13885, -7744, -9813, -44821, -54789, -73932; 246, 81, 34, 0, 2, 43, 66, 36, 44, 195, 232, 308; 266796, 88675, 38270, 4725, 1170, 38745, 62244, 34715, 43990, 200925, 245610, 331425],194,21]
%
% END OF LATTICE L195.21
%
\hbox{}\par\smallskip%
%
% LATTICE L195.22
%
\leavevmode%
\llap{\spacer}%
${L_{195.22}}=\pdualwitharg{3.5}(\pfillwitharg{2}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}2\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${450}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{10,1}{\infty a}{10}\above{1pt}{1pt}{r}{2}{18}\above{1pt}{1pt}{l}{2}{90}\above{1pt}{1pt}{15,4}{\infty}{90}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1539450\\643950\\341550
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643950\\270360\\142875
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341550\\142875\\75778
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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14239\\330\\-64800
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1548\\36\\-7045
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1883\\44\\-8570
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1663\\39\\-7569
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9867\\232\\-44910
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13080\\308\\-59535
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[1539450, 643950, 341550; 643950, 270360, 142875; 341550, 142875, 75778],[450, 45, 1],[14239, 1548, 1883, 1663, 9867, 13080, 52201, 3435, 1439, -2, 99, 1869; 330, 36, 44, 39, 232, 308, 1230, 81, 34, 0, 2, 43; -64800, -7045, -8570, -7569, -44910, -59535, -237600, -15635, -6550, 9, -450, -8505],194,22]
%
% END OF LATTICE L195.22
%
\hbox{}\par\smallskip%
%
% LATTICE L195.23
%
\leavevmode%
\llap{\spacer}%
${L_{195.23}}=\pdualwitharg{2.3}(\pfillwitharg{5}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-37959120, 6524280, 16694640; 6524280, -1121328, -2869416; 16694640, -2869416, -7342399],[360, 72, 1],[-14397, -24191, -9162, -9216, -14846, -22013, -3021, -1936, 453, 507, -11281, -56368; -5, -10, -5, -7, -15, -30, -5, -5, 0, 2, 0, -15; -32733, -55000, -20830, -20952, -33750, -50040, -6867, -4400, 1030, 1152, -25650, -128160],194,23]
%
% END OF LATTICE L195.23
%
\hbox{}\par\smallskip%
%
% LATTICE L195.24
%
\leavevmode%
\llap{\spacer}%
${L_{195.24}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{20,19}{\infty z}{5}\above{1pt}{1pt}{r}{2}{900}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{15,1}{\infty}{180}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-448300128600, 60413850, -223461183600; 60413850, -7920, 30114090; -223461183600, 30114090, -111387210019],[450, 90, 1],[-3026312, -1971263, -391690, 298106, 153270, -600429, -635300, -774946, -523713, -4875121, -2899479, -7638198; 435, 283, 56, -45, -22, 89, 93, 113, 76, 705, 418, 1099; 6071283, 3954680, 785795, -598050, -307485, 1204560, 1274517, 1554670, 1050655, 9780300, 5816835, 15323490],194,24]
%
% END OF LATTICE L195.24
%
\hbox{}\par\smallskip%
%
% LATTICE L195.25
%
\leavevmode%
\llap{\spacer}%
${L_{195.25}}=\pdualwitharg{2.3.5}(\pfillwitharg{2}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}2\above{1pt}{1pt}{2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${225}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{20,11}{\infty z}{5}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{15,4}{\infty}{180}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-15894021600, -11222550, -7922474100; -11222550, -7920, -5593950; -7922474100, -5593950, -3949006579],[450, 90, 1],[-105088, -22346, -12469, -19757, -50043, -117582, -218812, -26239, -3726, 3562, 1458, -28173; 1365, 293, 166, 267, 688, 1639, 3075, 373, 56, -45, -22, 359; 210825, 44830, 25015, 39636, 100395, 235890, 438975, 52640, 7475, -7146, -2925, 56520],194,25]
%
% END OF LATTICE L195.25
%
\hbox{}\par\smallskip%
%
% LATTICE L195.26
%
\leavevmode%
\llap{\spacer}%
${L_{195.26}}=\pdualwitharg{2}({L_{195.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${1}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{120,119}{\infty z}{20}\above{1pt}{1pt}{*}{2}{400}\above{1pt}{1pt}{*}{2}{180}\above{1pt}{1pt}{5,1}{\infty a}{720}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-49143600\\-9048600\\-17190000
\end{tabular}\endgroup%
\kern3pt%
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-9048600\\-1654320\\-3165080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-17190000\\-3165080\\-6012911
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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125976059\\1253880\\-360806400
\end{tabular}\endgroup%
\kern3pt%
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23302979\\231941\\-66741760
\end{tabular}\endgroup%
\kern3pt%
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44065700\\438600\\-126208001
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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43999\\438\\-126017
\end{tabular}\endgroup%
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220105\\2191\\-630400
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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82082\\817\\-235090
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\HardButStrongLineBreak\kern3pt%
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134144\\1335\\-384200
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
125726\\1251\\-360090
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
178235\\1773\\-510480
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-49143600, -9048600, -17190000; -9048600, -1654320, -3165080; -17190000, -3165080, -6012911],[1800, 40, 1],[43999, 220105, 82082, 134144, 125726, 178235, 7843, 13184, -4319, -4539, 107563, 521632; 438, 2191, 817, 1335, 1251, 1773, 78, 131, -43, -45, 1071, 5193; -126017, -630400, -235090, -384200, -360090, -510480, -22463, -37760, 12370, 13000, -308070, -1494000],194,26]
%
% END OF LATTICE L195.26
%
\hbox{}\par\smallskip%
%
% LATTICE L195.27
%
\leavevmode%
\llap{\spacer}%
${L_{195.27}}=\pdualwitharg{2.5}({L_{195.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${25}\above{1pt}{1pt}{r}{2}{720}\above{1pt}{1pt}{40,31}{\infty z}{180}\above{1pt}{1pt}{*}{2}{16}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{15,14}{\infty b}{80}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-141483600\\-16749000\\-35281800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16749000\\-1978320\\-4176680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-35281800\\-4176680\\-8798231
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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147957749\\2975400\\-594738000
\end{tabular}\endgroup%
\kern3pt%
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17114645\\344171\\-68794840
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36894260\\741936\\-148301921
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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131249\\2640\\-527575
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298414\\6003\\-1199520
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\HardButStrongLineBreak\kern3pt%
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53221\\1071\\-213930
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-2241\\-45\\9008
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-2137\\-43\\8590
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19962\\401\\-80240
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-141483600, -16749000, -35281800; -16749000, -1978320, -4176680; -35281800, -4176680, -8798231],[1800, 40, 1],[131249, 298414, 53221, -2241, -2137, 19962, 41801, 168821, 102524, 19546, 54052, 135783; 2640, 6003, 1071, -45, -43, 401, 840, 3393, 2061, 393, 1087, 2731; -527575, -1199520, -213930, 9008, 8590, -80240, -168025, -678600, -412110, -78568, -217270, -545800],194,27]
%
% END OF LATTICE L195.27
%
\hbox{}\par\smallskip%
%
% LATTICE L195.28
%
\leavevmode%
\llap{\spacer}%
${L_{195.28}}=\pdualwitharg{3}({L_{195.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${72}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{20,9}{\infty a}{40}\above{1pt}{1pt}{b}{2}{450}\above{1pt}{1pt}{b}{2}{360}\above{1pt}{1pt}{30,1}{\infty z}{90}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
361800\\-1731600\\88200
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1731600\\-5420610\\340965
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
88200\\340965\\-20978
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1264319\\3107840\\55828800
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
5021721\\12343951\\221745015
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-308178\\-757536\\-13608271
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-12159\\-29888\\-536904
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1002\\-2463\\-44245
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
991\\2436\\43760
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2028\\4985\\89550
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-18637\\-45812\\-822960
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-24616\\-60509\\-1086975
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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0\\10\\-20\\-225\\-1260\\-1170
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-360\\-20\\40\\0\\-1080\\-1260
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-1800\\-225\\0\\450\\0\\-675
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-7560\\-1260\\-1080\\0\\360\\-180
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-6480\\-1170\\-1260\\-675\\-180\\90
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\kern3pt%
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-2160\\-415\\-520\\-450\\-360\\-45
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2520\\-520\\-760\\-900\\-1080\\-360
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\kern3pt%
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-1080\\-360\\-1080\\-2700\\-6840\\-4680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-360\\-1350\\-4680\\-3735
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[361800, -1731600, 88200; -1731600, -5420610, 340965; 88200, 340965, -20978],[1800, 45, 1],[-12159, -1002, 991, 2028, -18637, -24616, -51057, -10851, -16795, -21783, -28775, -10943; -29888, -2463, 2436, 4985, -45812, -60509, -125504, -26673, -41284, -53545, -70732, -26899; -536904, -44245, 43760, 89550, -822960, -1086975, -2254536, -479150, -741620, -961875, -1270620, -483210],194,28]
%
% END OF LATTICE L195.28
%
\hbox{}\par\smallskip%
%
% LATTICE L195.29
%
\leavevmode%
\llap{\spacer}%
${L_{195.29}}=\pdualwitharg{3.5}({L_{195.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{{\rm II}}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-}{}25\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${1800}\above{1pt}{1pt}{r}{2}{10}\above{1pt}{1pt}{20,1}{\infty a}{40}\above{1pt}{1pt}{b}{2}{18}\above{1pt}{1pt}{b}{2}{360}\above{1pt}{1pt}{30,19}{\infty z}{90}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-790200\\-162000\\9000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-162000\\-31410\\1755
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9000\\1755\\-98
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1439\\63840\\1274400
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
267\\11836\\236295
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-15\\-665\\-13276
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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21\\1000\\19800
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\35
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-44\\-880
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
19\\828\\16560
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
22\\966\\19305
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1800\\0\\-1800\\-1800\\-37800\\-32400\\-289800\\-10800\\-12600\\-1800\\-5400\\0
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\kern3pt%
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0\\10\\-20\\-45\\-1260\\-1170\\-10800\\-415\\-520\\-90\\-360\\-45
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1800\\0\\-1800\\-1800\\-37800\\-32400
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0\\10\\-20\\-45\\-1260\\-1170
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-1800\\-20\\40\\0\\-1080\\-1260
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-1800\\-45\\0\\18\\0\\-135
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-37800\\-1260\\-1080\\0\\360\\-180
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\kern3pt%
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-32400\\-1170\\-1260\\-135\\-180\\90
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-289800\\-10800\\-12600\\-1800\\-5400\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-10800\\-415\\-520\\-90\\-360\\-45
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12600\\-520\\-760\\-180\\-1080\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-90\\-180\\-63\\-540\\-270
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5400\\-360\\-1080\\-540\\-6840\\-4680
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-45\\-360\\-270\\-4680\\-3735
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-790200, -162000, 9000; -162000, -31410, 1755; 9000, 1755, -98],[1800, 45, 1],[21, 0, -1, 0, 19, 22, 219, 9, 13, 3, 17, 5; 1000, 2, -44, -1, 828, 966, 9640, 397, 576, 134, 768, 231; 19800, 35, -880, -18, 16560, 19305, 192600, 7930, 11500, 2673, 15300, 4590],194,29]
%
% END OF LATTICE L195.29
%
\hbox{}\par\smallskip%
%
% LATTICE L195.30
%
\leavevmode%
\llap{\spacer}%
${L_{195.30}}=\pdualwitharg{2.3.5}(\pfillwitharg{5}({L_{195.1}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{2}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{r}{2}{144}\above{1pt}{1pt}{24,7}{\infty z}{36}\above{1pt}{1pt}{*}{2}{720}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{1,0}{\infty b}{16}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-639738000\\-581208840\\-141750000
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-581208840\\-528034320\\-128781360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-141750000\\-128781360\\-31408259
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13189\\-94282120\\386519760
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\kern3pt%
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11985\\-85668781\\351208440
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2923\\-20893604\\85655591
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\9433\\-38655
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\14915\\-61128
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\11009\\-45126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7\\36269\\-148680
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\7567\\-31022
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\20561\\-84296
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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-1800\\-1080\\0\\720\\0\\-360
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-639738000, -581208840, -141750000; -581208840, -528034320, -128781360; -141750000, -128781360, -31408259],[360, 360, 1],[-5, -6, -3, -7, -1, -2, -5, -3, 0, 2, 0, -1; 9433, 14915, 11009, 36269, 7567, 20561, 62047, 49417, 10435, -529, -419, 883; -38655, -61128, -45126, -148680, -31022, -84296, -254385, -202608, -42786, 2160, 1718, -3616],194,30]
%
% END OF LATTICE L195.30
%
\hbox{}\par\smallskip%
%
% LATTICE L195.31
%
\leavevmode%
\llap{\spacer}%
${L_{195.31}}=\pdualwitharg{2.3}({L_{195.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{80}\above{1pt}{1pt}{40,39}{\infty z}{20}\above{1pt}{1pt}{*}{2}{3600}\above{1pt}{1pt}{*}{2}{180}\above{1pt}{1pt}{15,1}{\infty a}{720}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-4468323600\\-411841800\\-181130400
\end{tabular}\endgroup%
\kern3pt%
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-411841800\\-37959120\\-16694640
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-181130400\\-16694640\\-7342399
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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15589\\-128523960\\291844800
\end{tabular}\endgroup%
\kern3pt%
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1437\\-11846629\\26900640
\end{tabular}\endgroup%
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632\\-5210208\\11831039
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1\\2556\\-5787
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\HardButStrongLineBreak\kern3pt%
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-1\\1471\\-3320
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-453\\1030
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\HardButStrongLineBreak\kern3pt%
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2\\-1605\\3600
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\11281\\-25650
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\54973\\-124920
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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9\\0\\-90\\-1800\\-1890\\-6480
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-1800\\-1800\\0\\3600\\0\\-5400
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-1890\\-2520\\-540\\0\\180\\-360
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0\\-360\\-720\\-10800\\-9360\\-29880
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4468323600, -411841800, -181130400; -411841800, -37959120, -16694640; -181130400, -16694640, -7342399],[1800, 360, 1],[-1, -1, 0, 2, 0, -3, -1, -2, -1, -7, -3, -6; 2556, 1471, -453, -1605, 11281, 54973, 13932, 23261, 8697, 42825, 13451, 19223; -5787, -3320, 1030, 3600, -25650, -124920, -31653, -52840, -19750, -97200, -30510, -43560],194,31]
%
% END OF LATTICE L195.31
%
\hbox{}\par\smallskip%
%
% LATTICE L195.32
%
\leavevmode%
\llap{\spacer}%
${L_{195.32}}=\pdualwitharg{2.3.5}({L_{195.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{{\rm II}}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{-2}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{1}{}25\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${225}\above{1pt}{1pt}{r}{2}{720}\above{1pt}{1pt}{120,31}{\infty z}{180}\above{1pt}{1pt}{*}{2}{144}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{5,4}{\infty b}{80}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
shares genus with 5-dual\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-34743600\\-67545000\\831879000
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-67545000\\-131235120\\1616271480
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
831879000\\1616271480\\-19905748759
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
56031249\\-2802674875\\-225225000
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
108884000\\-5446361649\\-437673600
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\HardButStrongLineBreak\kern3pt%
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264919\\-13251248\\-1064880
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\HardButStrongLineBreak\kern3pt%
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53221\\-2662121\\-213930
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-2257\\112891\\9072
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-2137\\106893\\8590
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8797\\-440016\\-35360
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-34743600, -67545000, 831879000; -67545000, -131235120, 1616271480; 831879000, 1616271480, -19905748759],[1800, 360, 1],[337922, 264919, 53221, -2257, -2137, 8797, 69578, 101831, 69029, 43007, 42887, 113453; -16902845, -13251248, -2662121, 112891, 106893, -440016, -3480245, -5093533, -3452806, -2151200, -2145202, -5674911; -1358325, -1064880, -213930, 9072, 8590, -35360, -279675, -409320, -277470, -172872, -172390, -456040],194,32]
%
% END OF LATTICE L195.32
%

\medskip%
%------------------------------WEYL-GROUP-W196--------------------------------
%
\leavevmode\llap{}%
$W_{196}$%
\qquad\llap{92} lattices, $\chi=36$%
\hfill%
$2222222222\rtimes C_{2}$%
\nopagebreak\smallskip\hrule\nopagebreak\medskip%
%
% LATTICE L196.1
%
\leavevmode%
\llap{\spacer}%
${L_{196.1}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-239400\\-1651\\-720
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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85\\-12276\\-108
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17\\-2460\\-10
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-1\\145\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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\kern3pt%
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-34713720, -239400, -104400; -239400, -1651, -720; -104400, -720, -313],[360, 1, 1],[107, 25, 85, 17, -1, -79, -5, -31, -1, 7; -15435, -3608, -12276, -2460, 140, 11340, 719, 4464, 145, -1008; -180, -38, -108, -10, 12, 270, 14, 72, 0, -16],195,1]
%
% END OF LATTICE L196.1
%
\hbox{}\par\smallskip%
%
% LATTICE L196.2
%
\leavevmode%
\llap{\spacer}%
${L_{196.2}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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\kern3pt%
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-2682360\\-18499\\360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-388942920, -2682360, 52200; -2682360, -18499, 360; 52200, 360, -7],[360, 1, 1],[17, 3, 31, 6, 5, 28, 2, 5, -1, -1; -2475, -436, -4500, -870, -724, -4050, -289, -720, 145, 144; -540, -55, -288, -5, 48, 495, 50, 252, 0, -52],195,2]
%
% END OF LATTICE L196.2
%
\hbox{}\par\smallskip%
%
% LATTICE L196.3
%
\leavevmode%
\llap{\spacer}%
${L_{196.3}}$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2359080, 1049040, 8280; 1049040, -466489, -3682; 8280, -3682, -29],[360, 1, 1],[-403, -35, -49, 31, 41, 763, 75, 193, 9, -25; -900, -78, -108, 70, 92, 1710, 168, 432, 20, -56; -810, -92, -288, -40, 24, 720, 82, 252, 30, -28],195,3]
%
% END OF LATTICE L196.3
%
\hbox{}\par\smallskip%
%
% LATTICE L196.4
%
\leavevmode%
\llap{\spacer}%
${L_{196.4}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}16\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
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0\\0\\0
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[229680, 56880, 21600; 56880, 14086, 5346; 21600, 5346, 1993],[720, 2, 1],[259, -1, -635, -1291, -657, -30623, -836, -2693, -1027, -6; -1080, 4, 2646, 5380, 2738, 127620, 3484, 11223, 4280, 25; 90, 0, -216, -440, -224, -10440, -285, -918, -350, -2],195,4]
%
% END OF LATTICE L196.4
%
\hbox{}\par\smallskip%
%
% LATTICE L196.5
%
\leavevmode%
\llap{\spacer}%
${L_{196.5}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}16\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-452880\\152640\\720
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-452880, 152640, 720; 152640, -51434, -244; 720, -244, -1],[720, 2, 1],[16, -7, -19, 7, 19, 1331, 81, 149, 37, 6; 45, -20, -54, 20, 54, 3780, 230, 423, 105, 17; 495, -168, -504, 160, 496, 35280, 2154, 3978, 995, 166],195,5]
%
% END OF LATTICE L196.5
%
\hbox{}\par\smallskip%
%
% LATTICE L196.6
%
\leavevmode%
\llap{\spacer}%
${L_{196.6}}$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}16\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\spacer%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2159280, 8640, 8640; 8640, -34, -36; 8640, -36, -31],[720, 2, 1],[17, 7, 10, 17, 2, 101, 2, 7, -1, -1; 3060, 1256, 1791, 3040, 357, 18000, 356, 1242, -180, -178; 1170, 488, 702, 1200, 142, 7200, 143, 504, -70, -72],195,6]
%
% END OF LATTICE L196.6
%
\hbox{}\par\smallskip%
%
% LATTICE L196.7
%
\leavevmode%
\llap{\spacer}%
${L_{196.7}}$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{4}16\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%EXPLICIT%[[-2720880, 7200, 7200; 7200, -2, -20; 7200, -20, -19],[720, 2, 1],[101, 47, 41, 37, 1, 1, -1, -1, 2, 7; 1980, 920, 801, 720, 19, 0, -20, -18, 40, 138; 36135, 16816, 14670, 13240, 358, 360, -358, -360, 715, 2504],195,7]
%
% END OF LATTICE L196.7
%
\hbox{}\par\smallskip%
%
% LATTICE L196.8
%
\leavevmode%
\llap{\spacer}%
${L_{196.8}}=\pfillwitharg{2}({L_{196.1}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-7247970, 207090, -23940; 207090, -5917, 684; -23940, 684, -79],[90, 1, 1],[7, 4, 31, 16, 9, 128, 11, 23, -1, -3; 225, 140, 1098, 570, 322, 4590, 395, 828, -35, -108; -180, -1, 108, 85, 60, 945, 86, 198, 0, -26],195,8]
%
% END OF LATTICE L196.8
%
\hbox{}\par\smallskip%
%
% LATTICE L196.9
%
\leavevmode%
\llap{\spacer}%
${L_{196.9}}=\main({L_{196.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[-244260, 97560, 3060; 97560, -38966, -1223; 3060, -1223, -37],[180, 1, 1],[1264, 116, 271, 2, -35, -274, -6, 125, 108, 79; 3105, 285, 666, 5, -86, -675, -15, 306, 265, 194; 1890, 172, 396, 0, -52, -360, -2, 216, 170, 120],195,9]
%
% END OF LATTICE L196.9
%
\hbox{}\par\smallskip%
%
% LATTICE L196.10
%
\leavevmode%
\llap{\spacer}%
${L_{196.10}}=\pfillwitharg{2}({L_{196.4}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7\\315\\306
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\940\\700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\581\\394
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
673\\30060\\19620
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\1697\\1095
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
128\\5715\\3636
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
27\\1205\\745
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
2\\89\\44
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
45\\0\\-90\\-180\\-90\\-4140\\-225\\-720\\-135\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\4\\0\\-20\\-16\\-900\\-52\\-180\\-40\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\18\\0\\-18\\-1440\\-90\\-342\\-90\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-20\\0\\20\\0\\-540\\-40\\-180\\-60\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-16\\-18\\0\\2\\0\\-2\\-18\\-10\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4140\\-900\\-1440\\-540\\0\\180\\0\\-180\\-180\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-225\\-52\\-90\\-40\\-2\\0\\1\\0\\-5\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-180\\-342\\-180\\-18\\-180\\0\\18\\0\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-135\\-40\\-90\\-60\\-10\\-180\\-5\\0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-18\\-20\\-6\\-180\\-8\\-18\\0\\2
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[160831620, -3655260, 83160; -3655260, 83074, -1890; 83160, -1890, 43],[180, 2, 1],[1, -1, 7, 21, 13, 673, 38, 128, 27, 2; 45, -44, 315, 940, 581, 30060, 1697, 5715, 1205, 89; 45, 0, 306, 700, 394, 19620, 1095, 3636, 745, 44],195,10]
%
% END OF LATTICE L196.10
%
\hbox{}\par\smallskip%
%
% LATTICE L196.11
%
\leavevmode%
\llap{\spacer}%
${L_{196.11}}=\main({L_{196.5}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${90}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{360}\above{1pt}{1pt}{l}{2}{2}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
114840\\28800\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
28800\\7222\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-90\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-11\\45\\90
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\4\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
13\\-54\\-162
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
29\\-120\\-340
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
15\\-62\\-174
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
697\\-2880\\-8100
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
38\\-157\\-442
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
61\\-252\\-711
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
23\\-95\\-270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\0\\-180\\-360\\-180\\-8280\\-450\\-720\\-270\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-40\\-32\\-1800\\-104\\-180\\-80\\-4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\36\\0\\-36\\-2880\\-180\\-342\\-180\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-40\\0\\40\\0\\-1080\\-80\\-180\\-120\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-32\\-36\\0\\4\\0\\-4\\-18\\-20\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8280\\-1800\\-2880\\-1080\\0\\360\\0\\-180\\-360\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-450\\-104\\-180\\-80\\-4\\0\\2\\0\\-10\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-180\\-342\\-180\\-18\\-180\\0\\9\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\-80\\-180\\-120\\-20\\-360\\-10\\0\\10\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-18\\-20\\-6\\-180\\-8\\-9\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[114840, 28800, -360; 28800, 7222, -90; -360, -90, 1],[360, 2, 1],[-11, -1, 13, 29, 15, 697, 38, 61, 23, 0; 45, 4, -54, -120, -62, -2880, -157, -252, -95, 0; 90, 4, -162, -340, -174, -8100, -442, -711, -270, -1],195,11]
%
% END OF LATTICE L196.11
%
\hbox{}\par\smallskip%
%
% LATTICE L196.12
%
\leavevmode%
\llap{\spacer}%
${L_{196.12}}=\main({L_{196.6}})$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{-}{}]\above{1pt}{1pt}{}{4}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${360}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}{90}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\0\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\0\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-5\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-22
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-23\\-225\\-540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-52\\-120
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-45\\-99
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-40\\-80
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\-360\\-360\\-360\\-8280\\-1800\\-1440\\-1080\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-10\\-16\\-450\\-104\\-90\\-80\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\36\\0\\-36\\-1440\\-360\\-342\\-360\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-10\\0\\10\\0\\-270\\-80\\-90\\-120\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-16\\-36\\0\\4\\0\\-8\\-18\\-40\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8280\\-450\\-1440\\-270\\0\\90\\0\\-90\\-360\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-104\\-360\\-80\\-8\\0\\8\\0\\-40\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1440\\-90\\-342\\-90\\-18\\-90\\0\\9\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\-80\\-360\\-120\\-40\\-360\\-40\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-2\\-18\\-10\\-6\\-90\\-16\\-9\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%IMPLICIT%[[360, 0, 0; 0, 2, 0; 0, 0, -1],[360, 2, 1],[1, 0, -1, -1, -1, -23, -5, -4, -3, 0; 0, 1, 0, -5, -8, -225, -52, -45, -40, -1; 0, 0, -18, -20, -22, -540, -120, -99, -80, -1],195,12]
%
% END OF LATTICE L196.12
%
\hbox{}\par\smallskip%
%
% LATTICE L196.13
%
\leavevmode%
\llap{\spacer}%
${L_{196.13}}=\pdualwitharg{2}(\pfillwitharg{2}({L_{196.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${90}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
791190\\-93420\\384210
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-93420\\11866\\-45342
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
384210\\-45342\\186577
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1885456\\-109395\\3856050
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
407563\\23646\\-833530
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-910351\\-52819\\1861809
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-82996\\-4815\\169740
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-12135\\-704\\24818
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27614\\-1602\\56475
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19133\\-1110\\39130
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3654\\-212\\7473
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
90\\0\\-90\\-180\\-90\\-4140\\-450\\-720\\-270\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\2\\0\\-10\\-8\\-450\\-52\\-90\\-40\\-2
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\9\\0\\-9\\-720\\-90\\-171\\-90\\-9
\end{tabular}\endgroup%
\kern3pt%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[791190, -93420, 384210; -93420, 11866, -45342; 384210, -45342, 186577],[90, 2, 1],[-82996, -12135, -27614, -19133, -3654, -88409, -6910, -6667, 88, -155; -4815, -704, -1602, -1110, -212, -5130, -401, -387, 5, -9; 169740, 24818, 56475, 39130, 7473, 180810, 14132, 13635, -180, 317],195,13]
%
% END OF LATTICE L196.13
%
\hbox{}\par\smallskip%
%
% LATTICE L196.14
%
\leavevmode%
\llap{\spacer}%
${L_{196.14}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{196.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{90}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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90\\0\\0
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0\\5\\0
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{$\left.\llap{\phantom{%
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-1\\-2\\-10
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[90, 0, 0; 0, 5, 0; 0, 0, -1],[90, 5, 1],[-8, -5, -19, -1, -1, -1, 0, 1, 0, -1; -45, -26, -90, -4, -2, 0, 1, 0, -1, -8; -126, -75, -270, -13, -10, -9, 0, 0, -2, -20],195,14]
%
% END OF LATTICE L196.14
%
\hbox{}\par\smallskip%
%
% LATTICE L196.15
%
\leavevmode%
\llap{\spacer}%
${L_{196.15}}=\pdualwitharg{2}(\main({L_{196.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}4\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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15429240\\334260\\3907620
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\kern3pt%
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334260\\7244\\84656
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{$\left[\!\llap{\phantom{%
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751\\1294\\-3076
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901\\1548\\-3690
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15429240, 334260, 3907620; 334260, 7244, 84656; 3907620, 84656, 989647],[180, 4, 1],[8657, 751, 901, 171, 21, 6283, 909, 2087, 1489, 311; 14940, 1294, 1548, 290, 36, 10890, 1576, 3618, 2580, 538; -35460, -3076, -3690, -700, -86, -25740, -3724, -8550, -6100, -1274],195,15]
%
% END OF LATTICE L196.15
%
\hbox{}\par\smallskip%
%
% LATTICE L196.16
%
\leavevmode%
\llap{\spacer}%
${L_{196.16}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{196.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-2}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0
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0\\9\\0\\-45\\-72
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-90\\-45\\0\\45\\0
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-90\\-72\\-18\\0\\18
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-225\\-234\\-90\\-180\\-18
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-135\\-180\\-90\\-270\\-90
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[15030, 10530, 1800; 10530, 53397, 2538; 1800, 2538, 251],[90, 9, 1],[-1934, -2545, -1287, -4013, -1533, -2061, -1450, -311, 18, -65; -535, -704, -356, -1110, -424, -570, -401, -86, 5, -18; 19280, 25371, 12830, 40005, 15282, 20545, 14454, 3100, -180, 648],195,16]
%
% END OF LATTICE L196.16
%
\hbox{}\par\smallskip%
%
% LATTICE L196.17
%
\leavevmode%
\llap{\spacer}%
${L_{196.17}}=\pdualwitharg{2.5}(\pfillwitharg{2}({L_{196.1}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-3870\\0\\-1890
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0\\10\\0
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-1890\\0\\-923
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-1653\\285\\3419
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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226\\-45\\-468
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145\\-26\\-300
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283\\-45\\-585
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\HardButStrongLineBreak\kern3pt%
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31\\-4\\-64
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17\\-1\\-35
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-828\\-450\\-720\\-54\\0
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-450\\-260\\-450\\-40\\-10
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-54\\-40\\-90\\-12\\-10
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-3870, 0, -1890; 0, 10, 0; -1890, 0, -923],[90, 10, 1],[226, 145, 283, 31, 17, 35, 0, -22, -2, 12; -45, -26, -45, -4, -1, 0, 1, 0, -1, -4; -468, -300, -585, -64, -35, -72, 0, 45, 4, -25],195,17]
%
% END OF LATTICE L196.17
%
\hbox{}\par\smallskip%
%
% LATTICE L196.18
%
\leavevmode%
\llap{\spacer}%
${L_{196.18}}=\pdualwitharg{5}(\main({L_{196.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{s}{2}{10}\above{1pt}{1pt}{b}{2}{180}\above{1pt}{1pt}{b}{2}{2}\above{1pt}{1pt}{b}{2}{20}\above{1pt}{1pt}{b}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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20\\79\\1950
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61\\234\\5940
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2\\7\\194
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left[\!\llap{\phantom{%
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-54\\-40\\-180\\-12\\-20
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-121860, -540, 1260; -540, 10, 5; 1260, 5, -13],[180, 5, 1],[38, 20, 61, 2, -1, -2, 0, 11, 2, 9; 153, 79, 234, 7, -6, -9, 1, 54, 9, 38; 3708, 1950, 5940, 194, -100, -198, 0, 1080, 196, 880],195,18]
%
% END OF LATTICE L196.18
%
\hbox{}\par\smallskip%
%
% LATTICE L196.19
%
\leavevmode%
\llap{\spacer}%
${L_{196.19}}=\pdualwitharg{2}(\main({L_{196.6}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{5}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{0}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${45}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{l}{2}{20}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{180}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-5553720\\156960\\-11880
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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-73\\-2565\\225
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-13\\-457\\36
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\HardButStrongLineBreak\kern3pt%
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7\\245\\-40
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11\\386\\-44
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\HardButStrongLineBreak\kern3pt%
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191\\6705\\-720
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\HardButStrongLineBreak\kern3pt%
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9\\316\\-33
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\HardButStrongLineBreak\kern3pt%
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41\\1440\\-144
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-35\\5
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-9\\-316\\32
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5553720, 156960, -11880; 156960, -4436, 336; -11880, 336, -23],[360, 4, 1],[-73, -13, -23, 7, 11, 191, 9, 41, -1, -9; -2565, -457, -810, 245, 386, 6705, 316, 1440, -35, -316; 225, 36, 36, -40, -44, -720, -33, -144, 5, 32],195,19]
%
% END OF LATTICE L196.19
%
\hbox{}\par\smallskip%
%
% LATTICE L196.20
%
\leavevmode%
\llap{\spacer}%
${L_{196.20}}=\pdualwitharg{2}(\main({L_{196.5}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}[4\above{1pt}{1pt}{1}{}8\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{2}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${180}\above{1pt}{1pt}{l}{2}{4}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{180}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{s}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-299160\\-8548\\-1872
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\kern3pt%
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-65520\\-1872\\-409
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{$\left.\llap{\phantom{%
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0\\0\\0
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{$\left[\!\llap{\phantom{%
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197\\-7110\\990
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19\\-685\\92
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47\\-1692\\216
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\HardButStrongLineBreak\kern3pt%
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1\\-35\\0
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-7\\252\\-32
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-9\\320\\-22
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-11\\378\\36
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9\\-330\\70
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11\\-398\\60
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-10469880, -299160, -65520; -299160, -8548, -1872; -65520, -1872, -409],[360, 4, 1],[197, 19, 47, 1, -7, -107, -9, -11, 9, 11; -7110, -685, -1692, -35, 252, 3825, 320, 378, -330, -398; 990, 92, 216, 0, -32, -360, -22, 36, 70, 60],195,20]
%
% END OF LATTICE L196.20
%
\hbox{}\par\smallskip%
%
% LATTICE L196.21
%
\leavevmode%
\llap{\spacer}%
${L_{196.21}}=\pdualwitharg{2.3}(\pfillwitharg{2}({L_{196.1}}))$%
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{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{2}{}]\above{1pt}{1pt}{}{1}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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{$\left.\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-4985393310, -23042880, -2446560270; -23042880, -106506, -11308194; -2446560270, -11308194, -1200638903],[90, 18, 1],[-1438, -1997, -519, -3271, -597, -1347, -788, -38, 486, 40; 25, 140, 61, 570, 161, 510, 395, 46, -35, -54; 2930, 4068, 1057, 6660, 1215, 2740, 1602, 77, -990, -81],195,21]
%
% END OF LATTICE L196.21
%
\hbox{}\par\smallskip%
%
% LATTICE L196.22
%
\leavevmode%
\llap{\spacer}%
${L_{196.22}}=\pdualwitharg{3}(\main({L_{196.2}}))$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{2}4\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{1}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
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{$\left[\!\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10\\0\\-20\\-180\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\18\\0\\-90\\-144
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-20\\0\\4\\0\\-36
\end{tabular}\endgroup%
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-180\\-90\\0\\90\\0
\end{tabular}\endgroup%
\kern3pt%
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-180\\-144\\-36\\0\\36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-460\\-450\\-160\\-270\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-450\\-468\\-180\\-360\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-160\\-180\\-76\\-180\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-270\\-360\\-180\\-540\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-36\\-36\\-180\\-108
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[14940, -3060, 360; -3060, 558, -63; 360, -63, 7],[180, 9, 1],[-3, -3, -1, -1, 1, 3, 3, 1, 1, -1; -65, -65, -22, -25, 18, 55, 55, 18, 15, -22; -430, -432, -148, -180, 108, 340, 342, 112, 90, -144],195,22]
%
% END OF LATTICE L196.22
%
\hbox{}\par\smallskip%
%
% LATTICE L196.23
%
\leavevmode%
\llap{\spacer}%
${L_{196.23}}=\pdualwitharg{5}(\pfillwitharg{2}({L_{196.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{90}\above{1pt}{1pt}{}{2}{4}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}{90}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{10}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
7380\\900\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
900\\110\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-9\\-9
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\-8\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\9\\0
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\8\\-4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\23\\-20
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-29\\216\\-252
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-8\\59\\-75
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-26\\189\\-270
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\7\\-13
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-1\\-10
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-90\\-36\\-90\\-828\\-225\\-720\\-27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20\\-80\\-900\\-260\\-900\\-40\\-20
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\0\\90\\0\\-90\\-1440\\-450\\-1710\\-90\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-20\\0\\4\\0\\-108\\-40\\-180\\-12\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-90\\-80\\-90\\0\\10\\0\\-10\\-90\\-10\\-30
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-828\\-900\\-1440\\-108\\0\\36\\0\\-180\\-36\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-225\\-260\\-450\\-40\\-10\\0\\5\\0\\-5\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-900\\-1710\\-180\\-90\\-180\\0\\90\\0\\-90
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-40\\-90\\-12\\-10\\-36\\-5\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-90\\-20\\-30\\-180\\-40\\-90\\0\\10
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{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[7380, 900, 0; 900, 110, 0; 0, 0, -1],[180, 10, 1],[1, 1, -1, -1, -3, -29, -8, -26, -1, 0; -9, -8, 9, 8, 23, 216, 59, 189, 7, -1; -9, 0, 0, -4, -20, -252, -75, -270, -13, -10],195,23]
%
% END OF LATTICE L196.23
%
\hbox{}\par\smallskip%
%
% LATTICE L196.24
%
\leavevmode%
\llap{\spacer}%
${L_{196.24}}=\pdualwitharg{5}({L_{196.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}8\above{1pt}{1pt}{-}{5}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{r}{2}{20}\above{1pt}{1pt}{*}{2}{360}\above{1pt}{1pt}{*}{2}{4}\above{1pt}{1pt}{*}{2}{40}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{l}{2}{5}\above{1pt}{1pt}{}{2}{360}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-45\\-126
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-52\\-150
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19\\-180\\-540
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-8\\-26
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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-1\\-4\\-20
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\HardButStrongLineBreak\kern3pt%
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-1\\0\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
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0\\1\\0
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1\\0\\0
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\HardButStrongLineBreak\kern3pt%
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0\\-1\\-2
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-16\\-40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-180\\-36\\-180\\-828\\-225\\-1440\\-27\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20\\-160\\-900\\-260\\-1800\\-40\\-40
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\0\\360\\0\\-360\\-2880\\-900\\-6840\\-180\\-360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-36\\-20\\0\\4\\0\\-108\\-40\\-360\\-12\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-180\\-160\\-360\\0\\40\\0\\-20\\-360\\-20\\-120
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-828\\-900\\-2880\\-108\\0\\36\\0\\-360\\-36\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-225\\-260\\-900\\-40\\-20\\0\\5\\0\\-5\\-80
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1440\\-1800\\-6840\\-360\\-360\\-360\\0\\360\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-27\\-40\\-180\\-12\\-20\\-36\\-5\\0\\1\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-360\\-40\\-120\\-360\\-80\\-360\\0\\40
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[360, 0, 0; 0, 5, 0; 0, 0, -1],[360, 5, 1],[-4, -5, -19, -1, -1, -1, 0, 1, 0, -1; -45, -52, -180, -8, -4, 0, 1, 0, -1, -16; -126, -150, -540, -26, -20, -18, 0, 0, -2, -40],195,24]
%
% END OF LATTICE L196.24
%
\hbox{}\par\smallskip%
%
% LATTICE L196.25
%
\leavevmode%
\llap{\spacer}%
${L_{196.25}}=\pdualwitharg{5}({L_{196.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${9}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{r}{2}{360}\above{1pt}{1pt}{l}{2}{1}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2408040\\0\\8640
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\5\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8640\\0\\-31
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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8351\\-20880\\2317680
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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10\\-26\\2775
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61\\-180\\16920
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\HardButStrongLineBreak\kern3pt%
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1\\-4\\277
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\HardButStrongLineBreak\kern3pt%
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-1\\-4\\-280
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
9\\0\\-180\\-18\\-180\\-414\\-225\\-1440\\-27\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-180\\-80\\-360\\0\\40\\0\\-20\\-360\\-20\\-120
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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9\\0\\-180\\-18\\-180
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0\\5\\0\\-5\\-80
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-180\\0\\360\\0\\-360
\end{tabular}\endgroup%
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-18\\-5\\0\\1\\0
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-180\\-80\\-360\\0\\40
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-414\\-225\\-1440\\-27\\0
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
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-27\\-20\\-180\\-6\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-360\\-20\\-120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-2408040, 0, 8640; 0, 5, 0; 8640, 0, -31],[360, 5, 1],[19, 10, 61, 1, -1, -1, 0, 11, 1, 9; -45, -26, -180, -4, -4, 0, 1, 0, -1, -16; 5274, 2775, 16920, 277, -280, -279, 0, 3060, 278, 2500],195,25]
%
% END OF LATTICE L196.25
%
\hbox{}\par\smallskip%
%
% LATTICE L196.26
%
\leavevmode%
\llap{\spacer}%
${L_{196.26}}=\pdualwitharg{5}({L_{196.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{1}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${36}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{s}{2}{360}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-19080\\360\\360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-5\\-10
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-10\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
575\\19296\\10080
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-14\\-470\\-245
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-6\\-201\\-106
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
37\\1242\\648
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
21\\704\\370
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
71\\2376\\1260
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\100\\54
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\32\\20
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-360\\-72\\-360\\-1656\\-900\\-2880\\-108\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20\\-160\\-900\\-520\\-1800\\-80\\-40
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\360\\0\\-360\\-2880\\-1800\\-6840\\-360\\-360
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-20\\0\\4\\0\\-108\\-80\\-360\\-24\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-160\\-360\\0\\40\\0\\-40\\-360\\-40\\-120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
36\\0\\-360\\-72\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\20\\0\\-20\\-160
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\360\\0\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-72\\-20\\0\\4\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-160\\-360\\0\\40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1656\\-900\\-2880\\-108\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-900\\-520\\-1800\\-80\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2880\\-1800\\-6840\\-360\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-108\\-80\\-360\\-24\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-40\\-360\\-40\\-120
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-19080, 360, 360; 360, -5, -10; 360, -10, -1],[360, 5, 1],[37, 21, 71, 3, 1, -1, -1, 1, 1, 7; 1242, 704, 2376, 100, 32, -36, -34, 36, 34, 236; 648, 370, 1260, 54, 20, -18, -20, 0, 16, 120],195,26]
%
% END OF LATTICE L196.26
%
\hbox{}\par\smallskip%
%
% LATTICE L196.27
%
\leavevmode%
\llap{\spacer}%
${L_{196.27}}=\pdualwitharg{2}({L_{196.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{1}8\above{1pt}{1pt}{-2}{4}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${360}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{b}{2}{36}\above{1pt}{1pt}{b}{2}{40}\above{1pt}{1pt}{b}{2}{4}\above{1pt}{1pt}{b}{2}{360}\above{1pt}{1pt}{l}{2}{8}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{40}\above{1pt}{1pt}{}{2}{1}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
113760\\360\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\-8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
1\\45\\360
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\4
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-36\\-342
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-75\\-700
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-38\\-354
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\-1755\\-16380
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-191\\-1784
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-4\\-153\\-1431
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-3\\-115\\-1080
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\-360\\-720\\-360\\-16560\\-1800\\-1440\\-1080\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-40\\-32\\-1800\\-208\\-180\\-160\\-4
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\36\\0\\-36\\-2880\\-360\\-342\\-360\\-18
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-40\\0\\40\\0\\-1080\\-160\\-180\\-240\\-20
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-32\\-36\\0\\4\\0\\-8\\-18\\-40\\-6
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16560\\-1800\\-2880\\-1080\\0\\360\\0\\-180\\-720\\-180
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-208\\-360\\-160\\-8\\0\\8\\0\\-40\\-16
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1440\\-180\\-342\\-180\\-18\\-180\\0\\9\\0\\-9
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\-160\\-360\\-240\\-40\\-720\\-40\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-4\\-18\\-20\\-6\\-180\\-16\\-9\\0\\1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[113760, 360, -360; 360, -8, 0; -360, 0, 1],[360, 8, 1],[1, 0, -1, -2, -1, -46, -5, -4, -3, 0; 45, 1, -36, -75, -38, -1755, -191, -153, -115, 0; 360, 4, -342, -700, -354, -16380, -1784, -1431, -1080, -1],195,27]
%
% END OF LATTICE L196.27
%
\hbox{}\par\smallskip%
%
% LATTICE L196.28
%
\leavevmode%
\llap{\spacer}%
${L_{196.28}}=\pdualwitharg{2}({L_{196.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{1}{7}8\above{1pt}{1pt}{-2}{6}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${360}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{l}{2}{40}\above{1pt}{1pt}{r}{2}{4}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\-1
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\hfil\penalty500%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-46\\-225\\-1080
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-5\\-26\\-120
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
3\\15\\71
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{$\left.\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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1\\0\\0
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\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\0
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\HardButStrongLineBreak\kern3pt%
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-1\\0\\-18
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2\\-5\\-40
\end{tabular}\endgroup%
\HardButStrongLineBreak\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1\\-4\\-22
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
\discretionary{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\-360\\-720\\-360\\-16560\\-1800\\-2880\\-1080\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-40\\-32\\-1800\\-208\\-360\\-160\\-8
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\36\\0\\-36\\-2880\\-360\\-684\\-360\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-40\\0\\40\\0\\-1080\\-160\\-360\\-240\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-32\\-36\\0\\4\\0\\-8\\-36\\-40\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0\\0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
360\\0\\-360\\-720\\-360
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\8\\0\\-40\\-32
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\0\\36\\0\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-720\\-40\\0\\40\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-360\\-32\\-36\\0\\4
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-16560\\-1800\\-2880\\-1080\\0
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1800\\-208\\-360\\-160\\-8
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-2880\\-360\\-684\\-360\\-36
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1080\\-160\\-360\\-240\\-40
\end{tabular}\endgroup%
\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-36\\-40\\-12
\end{tabular}\endgroup%
{$\left.\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0\\0\\0
\end{tabular}\endgroup%
}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[360, 0, 0; 0, 8, 0; 0, 0, -1],[360, 8, 1],[1, 0, -1, -2, -1, -46, -5, -8, -3, 0; 0, 1, 0, -5, -4, -225, -26, -45, -20, -1; 0, 0, -18, -40, -22, -1080, -120, -198, -80, -2],195,28]
%
% END OF LATTICE L196.28
%
\hbox{}\par\smallskip%
%
% LATTICE L196.29
%
\leavevmode%
\llap{\spacer}%
${L_{196.29}}=\pdualwitharg{2}({L_{196.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-}{3}8\above{1pt}{1pt}{2}{6}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{1}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{-}{}$}\EasyButWeakLineBreak%
{${360}\above{1pt}{1pt}{b}{2}{8}\above{1pt}{1pt}{s}{2}{36}\above{1pt}{1pt}{s}{2}{40}\above{1pt}{1pt}{s}{2}{4}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-204840\\10080\\-2520
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
10080\\-488\\120
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-2520\\120\\-29
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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37\\1368\\2368
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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11\\450\\900
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0\\1\\4
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-2\\-81\\-162
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-2\\-85\\-180
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0\\-3\\-14
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
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{$\left[\!\llap{\phantom{%
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}}\right.$}%
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{$\left.\llap{\phantom{%
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}}{}{\hbox{%
{$\left[\!\llap{\phantom{%
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0\\8\\0\\-40\\-32
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-360\\0\\36\\0\\-36
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-720\\-40\\0\\40\\0
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-360\\-32\\-36\\0\\4
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-16560\\-1800\\-2880\\-1080\\0
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-1800\\-208\\-360\\-160\\-8
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-2880\\-360\\-684\\-360\\-36
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-1080\\-160\\-360\\-240\\-40
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0\\-8\\-36\\-40\\-12
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-204840, 10080, -2520; 10080, -488, 120; -2520, 120, -29],[360, 8, 1],[11, 0, -2, -2, 0, 34, 5, 11, 7, 1; 450, 1, -81, -85, -3, 1215, 184, 414, 270, 40; 900, 4, -162, -180, -14, 1980, 316, 738, 500, 78],195,29]
%
% END OF LATTICE L196.29
%
\hbox{}\par\smallskip%
%
% LATTICE L196.30
%
\leavevmode%
\llap{\spacer}%
${L_{196.30}}=\pdualwitharg{3}(\pfillwitharg{2}({L_{196.4}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{-}{}2\above{1pt}{1pt}{1}{}4\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{5}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{36}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{180}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}{20}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{}{2}{18}\above{1pt}{1pt}{}{2}$}%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
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0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
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-1328220\\332100\\-7560
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
332100\\-83034\\1890
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-7560\\1890\\-43
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\EasyButWeakLineBreak%
{$\left[\!\llap{\phantom{%
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0\\0\\0
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4\\20\\175
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-1\\-4\\0
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-1\\-5\\-44
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\HardButStrongLineBreak\kern3pt%
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1\\0\\-180
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9\\41\\216
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77\\360\\2260
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43\\202\\1305
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18\\85\\566
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42\\200\\1395
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8\\39\\306
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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0\\-36\\-18\\-180\\-54\\-180\\-72\\-18\\0\\18
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-1328220, 332100, -7560; 332100, -83034, 1890; -7560, 1890, -43],[180, 18, 1],[4, -1, -1, 1, 9, 77, 43, 18, 42, 8; 20, -4, -5, 0, 41, 360, 202, 85, 200, 39; 175, 0, -44, -180, 216, 2260, 1305, 566, 1395, 306],195,30]
%
% END OF LATTICE L196.30
%
\hbox{}\par\smallskip%
%
% LATTICE L196.31
%
\leavevmode%
\llap{\spacer}%
${L_{196.31}}=\pdualwitharg{3}({L_{196.1}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{4}8\above{1pt}{1pt}{1}{1}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{r}{2}{36}\above{1pt}{1pt}{*}{2}{8}\above{1pt}{1pt}{*}{2}{180}\above{1pt}{1pt}{*}{2}{72}\above{1pt}{1pt}{*}{2}{20}\above{1pt}{1pt}{l}{2}{9}\above{1pt}{1pt}{}{2}{8}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{}{2}{72}\above{1pt}{1pt}{}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-5026680\\-335160\\-162000
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\kern3pt%
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-335160\\-22347\\-10800
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-162000\\-10800\\-5209
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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25\\-401\\54
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27\\-432\\56
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1\\-15\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\right.$}%
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0\\-72\\-72\\-360\\-216\\-360\\-144\\-72\\0\\72
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-5026680, -335160, -162000; -335160, -22347, -10800; -162000, -10800, -5209],[360, 9, 1],[-283, -625, -257, -577, -91, -9, 25, 27, 1, -139; 4525, 9992, 4108, 9220, 1452, 140, -401, -432, -15, 2224; -580, -1278, -524, -1170, -180, -10, 54, 56, 0, -288],195,31]
%
% END OF LATTICE L196.31
%
\hbox{}\par\smallskip%
%
% LATTICE L196.32
%
\leavevmode%
\llap{\spacer}%
${L_{196.32}}=\pdualwitharg{3}({L_{196.2}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{-2}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${5}\above{1pt}{1pt}{}{2}{9}\above{1pt}{1pt}{r}{2}{8}\above{1pt}{1pt}{l}{2}{45}\above{1pt}{1pt}{r}{2}{72}\above{1pt}{1pt}{l}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
\nopagebreak\par\leavevmode%
{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-652680\\43560\\-2160
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\kern3pt%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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{$\left.\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-652680, 43560, -2160; 43560, -2907, 144; -2160, 144, -7],[360, 9, 1],[-7, -7, -5, -4, 1, 2, 2, 1, -1, -5; -115, -116, -84, -70, 12, 30, 31, 16, -15, -80; -220, -243, -200, -225, -72, -5, 18, 20, 0, -108],195,32]
%
% END OF LATTICE L196.32
%
\hbox{}\par\smallskip%
%
% LATTICE L196.33
%
\leavevmode%
\llap{\spacer}%
${L_{196.33}}=\pdualwitharg{3}({L_{196.3}})$%
\nopagebreak\par%
{$1\above{1pt}{1pt}{2}{6}8\above{1pt}{1pt}{-}{3}{\cdot}1\above{1pt}{1pt}{-}{}9\above{1pt}{1pt}{2}{}{\cdot}1\above{1pt}{1pt}{-2}{}5\above{1pt}{1pt}{1}{}$}\EasyButWeakLineBreak%
{${20}\above{1pt}{1pt}{*}{2}{36}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{s}{2}{180}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{s}{2}$}\relax$\,(\times2)$%
\nopagebreak\par%
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{$\left[\!\llap{\phantom{%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\0\\0
\end{tabular}\endgroup%
}}\right.$}%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
8280\\-1440\\720
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\kern3pt%
\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-1440\\171\\-81
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
720\\-81\\38
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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-1\\-320\\-720
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0\\19\\45
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-8\\-19
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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}}\right.$}%
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-3\\-190\\-350
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-3\\-194\\-360
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-1\\-68\\-128
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-1\\-90\\-180
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1\\44\\72
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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{$\left[\!\llap{\phantom{%
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}}\!\right]$}%
}}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[8280, -1440, 720; -1440, 171, -81; 720, -81, 38],[360, 9, 1],[-3, -3, -1, -1, 1, 3, 3, 1, 1, -1; -190, -194, -68, -90, 44, 150, 154, 52, 50, -60; -350, -360, -128, -180, 72, 260, 270, 92, 90, -108],195,33]
%
% END OF LATTICE L196.33
%
\hbox{}\par\smallskip%
%
% LATTICE L196.34
%
\leavevmode%
\llap{\spacer}%
${L_{196.34}}=\pdualwitharg{5}(\main({L_{196.5}}))$%
\nopagebreak\par%
{$[1\above{1pt}{1pt}{1}{}2\above{1pt}{1pt}{1}{}]\above{1pt}{1pt}{}{6}8\above{1pt}{1pt}{1}{7}{\cdot}1\above{1pt}{1pt}{2}{}9\above{1pt}{1pt}{-}{}{\cdot}1\above{1pt}{1pt}{-}{}5\above{1pt}{1pt}{-2}{}$}\EasyButWeakLineBreak%
{${18}\above{1pt}{1pt}{r}{2}{40}\above{1pt}{1pt}{s}{2}{180}\above{1pt}{1pt}{s}{2}{8}\above{1pt}{1pt}{s}{2}{20}\above{1pt}{1pt}{s}{2}{72}\above{1pt}{1pt}{l}{2}{10}\above{1pt}{1pt}{}{2}{45}\above{1pt}{1pt}{}{2}{2}\above{1pt}{1pt}{}{2}{5}\above{1pt}{1pt}{}{2}$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
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{$\left[\!\llap{\phantom{%
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0\\0\\0
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1\\63\\-234
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1\\60\\-220
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\HardButStrongLineBreak\kern3pt%
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-1\\-64\\236
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-3\\-188\\690
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\HardButStrongLineBreak\kern3pt%
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-29\\-1800\\6588
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\HardButStrongLineBreak\kern3pt%
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-8\\-495\\1810
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-13\\-801\\2925
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-1\\-61\\222
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\1\\-5
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{$\left.\llap{\phantom{%
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0\\0\\0
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}}\!\right]$}%
\GramMatrix{%
\par\leavevmode\begin{center}%
{$\left[\!\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
-54\\-80\\-180\\-24\\-20\\-72\\-10\\0\\2\\0
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\begingroup \smaller\smaller\smaller\begin{tabular}{@{}c@{}}%
0\\-20\\-90\\-20\\-30\\-180\\-40\\-45\\0\\5
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{$\left.\llap{\phantom{%
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0\\0\\0\\0\\0\\0\\0\\0\\0\\0
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}}\!\right]$}%
\end{center}%
}% end GramMatrix
% PARI-readable data for this lattice:
% [IP matrix, Elementary Divisors, Simple Roots, Weyl Group #, Lattice # with this Weyl group]
%DUALITY%[[-43560, 4680, 1080; 4680, -370, -80; 1080, -80, -17],[360, 10, 1],[1, 1, -1, -1, -3, -29, -8, -13, -1, 0; 63, 60, -72, -64, -188, -1800, -495, -801, -61, 1; -234, -220, 270, 236, 690, 6588, 1810, 2925, 222, -5],195,34]
%
% END OF LATTICE L196.34
%
\hbox{}\par\smallskip%
%
% LATTICE L196.35
%
\leavev